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from sympy import S, Sum, I, lambdify, re, im, log, simplify, zeta, pi
from sympy.abc import x
from sympy.core.relational import Eq, Ne
from sympy.functions.elementary.exponential import exp
from sympy.logic.boolalg import Or
from sympy.sets.fancysets import Range
from sympy.stats import (P, E, variance, density, characteristic_function,
where, moment_generating_function)
from sympy.stats.drv_types import (PoissonDistribution, GeometricDistribution,
Poisson, Geometric, Logarithmic, NegativeBinomial, YuleSimon, Zeta)
from sympy.stats.rv import sample
from sympy.utilities.pytest import slow
def test_PoissonDistribution():
l = 3
p = PoissonDistribution(l)
assert abs(p.cdf(10).evalf() - 1) < .001
assert p.expectation(x, x) == l
assert p.expectation(x**2, x) - p.expectation(x, x)**2 == l
def test_Poisson():
l = 3
x = Poisson('x', l)
assert E(x) == l
assert variance(x) == l
assert density(x) == PoissonDistribution(l)
assert isinstance(E(x, evaluate=False), Sum)
assert isinstance(E(2*x, evaluate=False), Sum)
def test_GeometricDistribution():
p = S.One / 5
d = GeometricDistribution(p)
assert d.expectation(x, x) == 1/p
assert d.expectation(x**2, x) - d.expectation(x, x)**2 == (1-p)/p**2
assert abs(d.cdf(20000).evalf() - 1) < .001
def test_Logarithmic():
p = S.One / 2
x = Logarithmic('x', p)
assert E(x) == -p / ((1 - p) * log(1 - p))
assert variance(x) == -1/log(2)**2 + 2/log(2)
assert E(2*x**2 + 3*x + 4) == 4 + 7 / log(2)
assert isinstance(E(x, evaluate=False), Sum)
def test_negative_binomial():
r = 5
p = S(1) / 3
x = NegativeBinomial('x', r, p)
assert E(x) == p*r / (1-p)
assert variance(x) == p*r / (1-p)**2
assert E(x**5 + 2*x + 3) == S(9207)/4
assert isinstance(E(x, evaluate=False), Sum)
def test_yule_simon():
rho = S(3)
x = YuleSimon('x', rho)
assert simplify(E(x)) == rho / (rho - 1)
assert simplify(variance(x)) == rho**2 / ((rho - 1)**2 * (rho - 2))
assert isinstance(E(x, evaluate=False), Sum)
def test_zeta():
s = S(5)
x = Zeta('x', s)
assert E(x) == zeta(s-1) / zeta(s)
assert simplify(variance(x)) == (zeta(s) * zeta(s-2) - zeta(s-1)**2) / zeta(s)**2
@slow
def test_sample_discrete():
X, Y, Z = Geometric('X', S(1)/2), Poisson('Y', 4), Poisson('Z', 1000)
W = Poisson('W', S(1)/100)
assert sample(X) in X.pspace.domain.set
assert sample(Y) in Y.pspace.domain.set
assert sample(Z) in Z.pspace.domain.set
assert sample(W) in W.pspace.domain.set
def test_discrete_probability():
X = Geometric('X', S(1)/5)
Y = Poisson('Y', 4)
G = Geometric('e', x)
assert P(Eq(X, 3)) == S(16)/125
assert P(X < 3) == S(9)/25
assert P(X > 3) == S(64)/125
assert P(X >= 3) == S(16)/25
assert P(X <= 3) == S(61)/125
assert P(Ne(X, 3)) == S(109)/125
assert P(Eq(Y, 3)) == 32*exp(-4)/3
assert P(Y < 3) == 13*exp(-4)
assert P(Y > 3).equals(32*(-S(71)/32 + 3*exp(4)/32)*exp(-4)/3)
assert P(Y >= 3).equals(32*(-S(39)/32 + 3*exp(4)/32)*exp(-4)/3)
assert P(Y <= 3) == 71*exp(-4)/3
assert P(Ne(Y, 3)).equals(
13*exp(-4) + 32*(-S(71)/32 + 3*exp(4)/32)*exp(-4)/3)
assert P(X < S.Infinity) is S.One
assert P(X > S.Infinity) is S.Zero
assert P(G < 3) == x*(-x + 1) + x
assert P(Eq(G, 3)) == x*(-x + 1)**2
def test_precomputed_characteristic_functions():
import mpmath
def test_cf(dist, support_lower_limit, support_upper_limit):
pdf = density(dist)
t = S('t')
x = S('x')
# first function is the hardcoded CF of the distribution
cf1 = lambdify([t], characteristic_function(dist)(t), 'mpmath')
# second function is the Fourier transform of the density function
f = lambdify([x, t], pdf(x)*exp(I*x*t), 'mpmath')
cf2 = lambda t: mpmath.nsum(lambda x: f(x, t), [support_lower_limit, support_upper_limit], maxdegree=10)
# compare the two functions at various points
for test_point in [2, 5, 8, 11]:
n1 = cf1(test_point)
n2 = cf2(test_point)
assert abs(re(n1) - re(n2)) < 1e-12
assert abs(im(n1) - im(n2)) < 1e-12
test_cf(Geometric('g', S(1)/3), 1, mpmath.inf)
test_cf(Logarithmic('l', S(1)/5), 1, mpmath.inf)
test_cf(NegativeBinomial('n', 5, S(1)/7), 0, mpmath.inf)
test_cf(Poisson('p', 5), 0, mpmath.inf)
test_cf(YuleSimon('y', 5), 1, mpmath.inf)
test_cf(Zeta('z', 5), 1, mpmath.inf)
def test_moment_generating_functions():
t = S('t')
geometric_mgf = moment_generating_function(Geometric('g', S(1)/2))(t)
assert geometric_mgf.diff(t).subs(t, 0) == 2
logarithmic_mgf = moment_generating_function(Logarithmic('l', S(1)/2))(t)
assert logarithmic_mgf.diff(t).subs(t, 0) == 1/log(2)
negative_binomial_mgf = moment_generating_function(NegativeBinomial('n', 5, S(1)/3))(t)
assert negative_binomial_mgf.diff(t).subs(t, 0) == S(5)/2
poisson_mgf = moment_generating_function(Poisson('p', 5))(t)
assert poisson_mgf.diff(t).subs(t, 0) == 5
yule_simon_mgf = moment_generating_function(YuleSimon('y', 3))(t)
assert simplify(yule_simon_mgf.diff(t).subs(t, 0)) == S(3)/2
zeta_mgf = moment_generating_function(Zeta('z', 5))(t)
assert zeta_mgf.diff(t).subs(t, 0) == pi**4/(90*zeta(5))
def test_Or():
X = Geometric('X', S(1)/2)
P(Or(X < 3, X > 4)) == S(13)/16
P(Or(X > 2, X > 1)) == P(X > 1)
P(Or(X >= 3, X < 3)) == 1
def test_where():
X = Geometric('X', S(1)/5)
Y = Poisson('Y', 4)
assert where(X**2 > 4).set == Range(3, S.Infinity, 1)
assert where(X**2 >= 4).set == Range(2, S.Infinity, 1)
assert where(Y**2 < 9).set == Range(0, 3, 1)
assert where(Y**2 <= 9).set == Range(0, 4, 1)
def test_conditional():
X = Geometric('X', S(2)/3)
Y = Poisson('Y', 3)
assert P(X > 2, X > 3) == 1
assert P(X > 3, X > 2) == S(1)/3
assert P(Y > 2, Y < 2) == 0
assert P(Eq(Y, 3), Y >= 0) == 9*exp(-3)/2
assert P(Eq(Y, 3), Eq(Y, 2)) == 0
assert P(X < 2, Eq(X, 2)) == 0
assert P(X > 2, Eq(X, 3)) == 1
def test_product_spaces():
X1 = Geometric('X1', S(1)/2)
X2 = Geometric('X2', S(1)/3)
assert str(P(X1 + X2 < 3, evaluate=False)) == """Sum(Piecewise((2**(X2 - n - 2)*(2/3)**(X2 - 1)/6, """\
+ """(-X2 + n + 3 >= 1) & (-X2 + n + 3 < oo)), (0, True)), (X2, 1, oo), (n, -oo, -1))"""
assert str(P(X1 + X2 > 3)) == """Sum(Piecewise((2**(X2 - n - 2)*(2/3)**(X2 - 1)/6, """ +\
"""(-X2 + n + 3 >= 1) & (-X2 + n + 3 < oo)), (0, True)), (X2, 1, oo), (n, 1, oo))"""
assert str(P(Eq(X1 + X2, 3))) == """Sum(Piecewise((2**(X2 - 2)*(2/3)**(X2 - 1)/6, """ +\
"""X2 <= 2), (0, True)), (X2, 1, oo))"""
|
bdc2e4fdea3f6ad7566f4a227072dc7512fadebfde0ffadf0327840ec9f04b09
|
from sympy import (Symbol, Abs, exp, S, N, pi, simplify, Interval, erf, erfc, Ne,
Eq, log, lowergamma, uppergamma, Sum, symbols, sqrt, And, gamma, beta,
Piecewise, Integral, sin, cos, besseli, factorial, binomial,
floor, expand_func, Rational, I, re, im, lambdify, hyper, diff, Or, Mul)
from sympy.core.compatibility import range
from sympy.external import import_module
from sympy.stats import (P, E, where, density, variance, covariance, skewness,
given, pspace, cdf, characteristic_function, ContinuousRV, sample,
Arcsin, Benini, Beta, BetaPrime, Cauchy,
Chi, ChiSquared,
ChiNoncentral, Dagum, Erlang, Exponential,
FDistribution, FisherZ, Frechet, Gamma, GammaInverse,
Gompertz, Gumbel, Kumaraswamy, Laplace, Logistic,
LogNormal, Maxwell, Nakagami, Normal, Pareto,
QuadraticU, RaisedCosine, Rayleigh, ShiftedGompertz,
StudentT, Trapezoidal, Triangular, Uniform, UniformSum,
VonMises, Weibull, WignerSemicircle, correlation,
moment, cmoment, smoment)
from sympy.stats.crv_types import NormalDistribution
from sympy.stats.joint_rv import JointPSpace
from sympy.utilities.pytest import raises, XFAIL, slow, skip
from sympy.utilities.randtest import verify_numerically as tn
oo = S.Infinity
x, y, z = map(Symbol, 'xyz')
def test_single_normal():
mu = Symbol('mu', real=True, finite=True)
sigma = Symbol('sigma', real=True, positive=True, finite=True)
X = Normal('x', 0, 1)
Y = X*sigma + mu
assert simplify(E(Y)) == mu
assert simplify(variance(Y)) == sigma**2
pdf = density(Y)
x = Symbol('x')
assert (pdf(x) ==
2**S.Half*exp(-(mu - x)**2/(2*sigma**2))/(2*pi**S.Half*sigma))
assert P(X**2 < 1) == erf(2**S.Half/2)
assert E(X, Eq(X, mu)) == mu
@XFAIL
def test_conditional_1d():
X = Normal('x', 0, 1)
Y = given(X, X >= 0)
assert density(Y) == 2 * density(X)
assert Y.pspace.domain.set == Interval(0, oo)
assert E(Y) == sqrt(2) / sqrt(pi)
assert E(X**2) == E(Y**2)
def test_ContinuousDomain():
X = Normal('x', 0, 1)
assert where(X**2 <= 1).set == Interval(-1, 1)
assert where(X**2 <= 1).symbol == X.symbol
where(And(X**2 <= 1, X >= 0)).set == Interval(0, 1)
raises(ValueError, lambda: where(sin(X) > 1))
Y = given(X, X >= 0)
assert Y.pspace.domain.set == Interval(0, oo)
@slow
def test_multiple_normal():
X, Y = Normal('x', 0, 1), Normal('y', 0, 1)
assert E(X + Y) == 0
assert variance(X + Y) == 2
assert variance(X + X) == 4
assert covariance(X, Y) == 0
assert covariance(2*X + Y, -X) == -2*variance(X)
assert skewness(X) == 0
assert skewness(X + Y) == 0
assert correlation(X, Y) == 0
assert correlation(X, X + Y) == correlation(X, X - Y)
assert moment(X, 2) == 1
assert cmoment(X, 3) == 0
assert moment(X + Y, 4) == 12
assert cmoment(X, 2) == variance(X)
assert smoment(X*X, 2) == 1
assert smoment(X + Y, 3) == skewness(X + Y)
assert E(X, Eq(X + Y, 0)) == 0
assert variance(X, Eq(X + Y, 0)) == S.Half
def test_symbolic():
mu1, mu2 = symbols('mu1 mu2', real=True, finite=True)
s1, s2 = symbols('sigma1 sigma2', real=True, finite=True, positive=True)
rate = Symbol('lambda', real=True, positive=True, finite=True)
X = Normal('x', mu1, s1)
Y = Normal('y', mu2, s2)
Z = Exponential('z', rate)
a, b, c = symbols('a b c', real=True, finite=True)
assert E(X) == mu1
assert E(X + Y) == mu1 + mu2
assert E(a*X + b) == a*E(X) + b
assert variance(X) == s1**2
assert simplify(variance(X + a*Y + b)) == variance(X) + a**2*variance(Y)
assert E(Z) == 1/rate
assert E(a*Z + b) == a*E(Z) + b
assert E(X + a*Z + b) == mu1 + a/rate + b
def test_cdf():
X = Normal('x', 0, 1)
d = cdf(X)
assert P(X < 1) == d(1).rewrite(erfc)
assert d(0) == S.Half
d = cdf(X, X > 0) # given X>0
assert d(0) == 0
Y = Exponential('y', 10)
d = cdf(Y)
assert d(-5) == 0
assert P(Y > 3) == 1 - d(3)
raises(ValueError, lambda: cdf(X + Y))
Z = Exponential('z', 1)
f = cdf(Z)
z = Symbol('z')
assert f(z) == Piecewise((1 - exp(-z), z >= 0), (0, True))
def test_characteristic_function():
X = Uniform('x', 0, 1)
cf = characteristic_function(X)
assert cf(1) == -I*(-1 + exp(I))
Y = Normal('y', 1, 1)
cf = characteristic_function(Y)
assert cf(0) == 1
assert simplify(cf(1)) == exp(I - S(1)/2)
Z = Exponential('z', 5)
cf = characteristic_function(Z)
assert cf(0) == 1
assert simplify(cf(1)) == S(25)/26 + 5*I/26
def test_sample_continuous():
z = Symbol('z')
Z = ContinuousRV(z, exp(-z), set=Interval(0, oo))
assert sample(Z) in Z.pspace.domain.set
sym, val = list(Z.pspace.sample().items())[0]
assert sym == Z and val in Interval(0, oo)
assert density(Z)(-1) == 0
def test_ContinuousRV():
x = Symbol('x')
pdf = sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)) # Normal distribution
# X and Y should be equivalent
X = ContinuousRV(x, pdf)
Y = Normal('y', 0, 1)
assert variance(X) == variance(Y)
assert P(X > 0) == P(Y > 0)
def test_arcsin():
from sympy import asin
a = Symbol("a", real=True)
b = Symbol("b", real=True)
X = Arcsin('x', a, b)
assert density(X)(x) == 1/(pi*sqrt((-x + b)*(x - a)))
assert cdf(X)(x) == Piecewise((0, a > x),
(2*asin(sqrt((-a + x)/(-a + b)))/pi, b >= x),
(1, True))
def test_benini():
alpha = Symbol("alpha", positive=True)
beta = Symbol("beta", positive=True)
sigma = Symbol("sigma", positive=True)
X = Benini('x', alpha, beta, sigma)
assert density(X)(x) == ((alpha/x + 2*beta*log(x/sigma)/x)
*exp(-alpha*log(x/sigma) - beta*log(x/sigma)**2))
alpha = Symbol("alpha", positive=False)
raises(ValueError, lambda: Benini('x', alpha, beta, sigma))
beta = Symbol("beta", positive=False)
raises(ValueError, lambda: Benini('x', alpha, beta, sigma))
alpha = Symbol("alpha", positive=True)
raises(ValueError, lambda: Benini('x', alpha, beta, sigma))
beta = Symbol("beta", positive=True)
sigma = Symbol("sigma", positive=False)
raises(ValueError, lambda: Benini('x', alpha, beta, sigma))
def test_beta():
a, b = symbols('alpha beta', positive=True)
B = Beta('x', a, b)
assert pspace(B).domain.set == Interval(0, 1)
dens = density(B)
x = Symbol('x')
assert dens(x) == x**(a - 1)*(1 - x)**(b - 1) / beta(a, b)
assert simplify(E(B)) == a / (a + b)
assert simplify(variance(B)) == a*b / (a**3 + 3*a**2*b + a**2 + 3*a*b**2 + 2*a*b + b**3 + b**2)
# Full symbolic solution is too much, test with numeric version
a, b = 1, 2
B = Beta('x', a, b)
assert expand_func(E(B)) == a / S(a + b)
assert expand_func(variance(B)) == (a*b) / S((a + b)**2 * (a + b + 1))
def test_betaprime():
alpha = Symbol("alpha", positive=True)
betap = Symbol("beta", positive=True)
X = BetaPrime('x', alpha, betap)
assert density(X)(x) == x**(alpha - 1)*(x + 1)**(-alpha - betap)/beta(alpha, betap)
alpha = Symbol("alpha", positive=False)
raises(ValueError, lambda: BetaPrime('x', alpha, betap))
alpha = Symbol("alpha", positive=True)
betap = Symbol("beta", positive=False)
raises(ValueError, lambda: BetaPrime('x', alpha, betap))
def test_cauchy():
x0 = Symbol("x0")
gamma = Symbol("gamma", positive=True)
X = Cauchy('x', x0, gamma)
assert density(X)(x) == 1/(pi*gamma*(1 + (x - x0)**2/gamma**2))
gamma = Symbol("gamma", positive=False)
raises(ValueError, lambda: Cauchy('x', x0, gamma))
def test_chi():
k = Symbol("k", integer=True)
X = Chi('x', k)
assert density(X)(x) == 2**(-k/2 + 1)*x**(k - 1)*exp(-x**2/2)/gamma(k/2)
k = Symbol("k", integer=True, positive=False)
raises(ValueError, lambda: Chi('x', k))
k = Symbol("k", integer=False, positive=True)
raises(ValueError, lambda: Chi('x', k))
def test_chi_noncentral():
k = Symbol("k", integer=True)
l = Symbol("l")
X = ChiNoncentral("x", k, l)
assert density(X)(x) == (x**k*l*(x*l)**(-k/2)*
exp(-x**2/2 - l**2/2)*besseli(k/2 - 1, x*l))
k = Symbol("k", integer=True, positive=False)
raises(ValueError, lambda: ChiNoncentral('x', k, l))
k = Symbol("k", integer=True, positive=True)
l = Symbol("l", positive=False)
raises(ValueError, lambda: ChiNoncentral('x', k, l))
k = Symbol("k", integer=False)
l = Symbol("l", positive=True)
raises(ValueError, lambda: ChiNoncentral('x', k, l))
def test_chi_squared():
k = Symbol("k", integer=True)
X = ChiSquared('x', k)
assert density(X)(x) == 2**(-k/2)*x**(k/2 - 1)*exp(-x/2)/gamma(k/2)
assert cdf(X)(x) == Piecewise((lowergamma(k/2, x/2)/gamma(k/2), x >= 0), (0, True))
assert E(X) == k
assert variance(X) == 2*k
X = ChiSquared('x', 15)
assert cdf(X)(3) == -14873*sqrt(6)*exp(-S(3)/2)/(5005*sqrt(pi)) + erf(sqrt(6)/2)
k = Symbol("k", integer=True, positive=False)
raises(ValueError, lambda: ChiSquared('x', k))
k = Symbol("k", integer=False, positive=True)
raises(ValueError, lambda: ChiSquared('x', k))
def test_dagum():
p = Symbol("p", positive=True)
b = Symbol("b", positive=True)
a = Symbol("a", positive=True)
X = Dagum('x', p, a, b)
assert density(X)(x) == a*p*(x/b)**(a*p)*((x/b)**a + 1)**(-p - 1)/x
assert cdf(X)(x) == Piecewise(((1 + (x/b)**(-a))**(-p), x >= 0),
(0, True))
p = Symbol("p", positive=False)
raises(ValueError, lambda: Dagum('x', p, a, b))
p = Symbol("p", positive=True)
b = Symbol("b", positive=False)
raises(ValueError, lambda: Dagum('x', p, a, b))
b = Symbol("b", positive=True)
a = Symbol("a", positive=False)
raises(ValueError, lambda: Dagum('x', p, a, b))
def test_erlang():
k = Symbol("k", integer=True, positive=True)
l = Symbol("l", positive=True)
X = Erlang("x", k, l)
assert density(X)(x) == x**(k - 1)*l**k*exp(-x*l)/gamma(k)
assert cdf(X)(x) == Piecewise((lowergamma(k, l*x)/gamma(k), x > 0),
(0, True))
def test_exponential():
rate = Symbol('lambda', positive=True, real=True, finite=True)
X = Exponential('x', rate)
assert E(X) == 1/rate
assert variance(X) == 1/rate**2
assert skewness(X) == 2
assert skewness(X) == smoment(X, 3)
assert smoment(2*X, 4) == smoment(X, 4)
assert moment(X, 3) == 3*2*1/rate**3
assert P(X > 0) == S(1)
assert P(X > 1) == exp(-rate)
assert P(X > 10) == exp(-10*rate)
assert where(X <= 1).set == Interval(0, 1)
def test_f_distribution():
d1 = Symbol("d1", positive=True)
d2 = Symbol("d2", positive=True)
X = FDistribution("x", d1, d2)
assert density(X)(x) == (d2**(d2/2)*sqrt((d1*x)**d1*(d1*x + d2)**(-d1 - d2))
/(x*beta(d1/2, d2/2)))
d1 = Symbol("d1", positive=False)
raises(ValueError, lambda: FDistribution('x', d1, d1))
d1 = Symbol("d1", positive=True, integer=False)
raises(ValueError, lambda: FDistribution('x', d1, d1))
d1 = Symbol("d1", positive=True)
d2 = Symbol("d2", positive=False)
raises(ValueError, lambda: FDistribution('x', d1, d2))
d2 = Symbol("d2", positive=True, integer=False)
raises(ValueError, lambda: FDistribution('x', d1, d2))
def test_fisher_z():
d1 = Symbol("d1", positive=True)
d2 = Symbol("d2", positive=True)
X = FisherZ("x", d1, d2)
assert density(X)(x) == (2*d1**(d1/2)*d2**(d2/2)*(d1*exp(2*x) + d2)
**(-d1/2 - d2/2)*exp(d1*x)/beta(d1/2, d2/2))
def test_frechet():
a = Symbol("a", positive=True)
s = Symbol("s", positive=True)
m = Symbol("m", real=True)
X = Frechet("x", a, s=s, m=m)
assert density(X)(x) == a*((x - m)/s)**(-a - 1)*exp(-((x - m)/s)**(-a))/s
assert cdf(X)(x) == Piecewise((exp(-((-m + x)/s)**(-a)), m <= x), (0, True))
def test_gamma():
k = Symbol("k", positive=True)
theta = Symbol("theta", positive=True)
X = Gamma('x', k, theta)
assert density(X)(x) == x**(k - 1)*theta**(-k)*exp(-x/theta)/gamma(k)
assert cdf(X, meijerg=True)(z) == Piecewise(
(-k*lowergamma(k, 0)/gamma(k + 1) +
k*lowergamma(k, z/theta)/gamma(k + 1), z >= 0),
(0, True))
# assert simplify(variance(X)) == k*theta**2 # handled numerically below
assert E(X) == moment(X, 1)
k, theta = symbols('k theta', real=True, finite=True, positive=True)
X = Gamma('x', k, theta)
assert E(X) == k*theta
assert variance(X) == k*theta**2
assert simplify(skewness(X)) == 2/sqrt(k)
def test_gamma_inverse():
a = Symbol("a", positive=True)
b = Symbol("b", positive=True)
X = GammaInverse("x", a, b)
assert density(X)(x) == x**(-a - 1)*b**a*exp(-b/x)/gamma(a)
assert cdf(X)(x) == Piecewise((uppergamma(a, b/x)/gamma(a), x > 0), (0, True))
def test_sampling_gamma_inverse():
scipy = import_module('scipy')
if not scipy:
skip('Scipy not installed. Abort tests for sampling of gamma inverse.')
X = GammaInverse("x", 1, 1)
assert sample(X) in X.pspace.domain.set
def test_gompertz():
b = Symbol("b", positive=True)
eta = Symbol("eta", positive=True)
X = Gompertz("x", b, eta)
assert density(X)(x) == b*eta*exp(eta)*exp(b*x)*exp(-eta*exp(b*x))
def test_gumbel():
beta = Symbol("beta", positive=True)
mu = Symbol("mu")
x = Symbol("x")
X = Gumbel("x", beta, mu)
assert simplify(density(X)(x)) == exp((beta*exp((mu - x)/beta) + mu - x)/beta)/beta
def test_kumaraswamy():
a = Symbol("a", positive=True)
b = Symbol("b", positive=True)
X = Kumaraswamy("x", a, b)
assert density(X)(x) == x**(a - 1)*a*b*(-x**a + 1)**(b - 1)
assert cdf(X)(x) == Piecewise((0, x < 0),
(-(-x**a + 1)**b + 1, x <= 1),
(1, True))
def test_laplace():
mu = Symbol("mu")
b = Symbol("b", positive=True)
X = Laplace('x', mu, b)
assert density(X)(x) == exp(-Abs(x - mu)/b)/(2*b)
assert cdf(X)(x) == Piecewise((exp((-mu + x)/b)/2, mu > x),
(-exp((mu - x)/b)/2 + 1, True))
def test_logistic():
mu = Symbol("mu", real=True)
s = Symbol("s", positive=True)
X = Logistic('x', mu, s)
assert density(X)(x) == exp((-x + mu)/s)/(s*(exp((-x + mu)/s) + 1)**2)
assert cdf(X)(x) == 1/(exp((mu - x)/s) + 1)
def test_lognormal():
mean = Symbol('mu', real=True, finite=True)
std = Symbol('sigma', positive=True, real=True, finite=True)
X = LogNormal('x', mean, std)
# The sympy integrator can't do this too well
#assert E(X) == exp(mean+std**2/2)
#assert variance(X) == (exp(std**2)-1) * exp(2*mean + std**2)
# Right now, only density function and sampling works
# Test sampling: Only e^mean in sample std of 0
for i in range(3):
X = LogNormal('x', i, 0)
assert S(sample(X)) == N(exp(i))
# The sympy integrator can't do this too well
#assert E(X) ==
mu = Symbol("mu", real=True)
sigma = Symbol("sigma", positive=True)
X = LogNormal('x', mu, sigma)
assert density(X)(x) == (sqrt(2)*exp(-(-mu + log(x))**2
/(2*sigma**2))/(2*x*sqrt(pi)*sigma))
X = LogNormal('x', 0, 1) # Mean 0, standard deviation 1
assert density(X)(x) == sqrt(2)*exp(-log(x)**2/2)/(2*x*sqrt(pi))
def test_maxwell():
a = Symbol("a", positive=True)
X = Maxwell('x', a)
assert density(X)(x) == (sqrt(2)*x**2*exp(-x**2/(2*a**2))/
(sqrt(pi)*a**3))
assert E(X) == 2*sqrt(2)*a/sqrt(pi)
assert simplify(variance(X)) == a**2*(-8 + 3*pi)/pi
def test_nakagami():
mu = Symbol("mu", positive=True)
omega = Symbol("omega", positive=True)
X = Nakagami('x', mu, omega)
assert density(X)(x) == (2*x**(2*mu - 1)*mu**mu*omega**(-mu)
*exp(-x**2*mu/omega)/gamma(mu))
assert simplify(E(X)) == (sqrt(mu)*sqrt(omega)
*gamma(mu + S.Half)/gamma(mu + 1))
assert simplify(variance(X)) == (
omega - omega*gamma(mu + S(1)/2)**2/(gamma(mu)*gamma(mu + 1)))
assert cdf(X)(x) == Piecewise(
(lowergamma(mu, mu*x**2/omega)/gamma(mu), x > 0),
(0, True))
def test_pareto():
xm, beta = symbols('xm beta', positive=True, finite=True)
alpha = beta + 5
X = Pareto('x', xm, alpha)
dens = density(X)
x = Symbol('x')
assert dens(x) == x**(-(alpha + 1))*xm**(alpha)*(alpha)
assert simplify(E(X)) == alpha*xm/(alpha-1)
# computation of taylor series for MGF still too slow
#assert simplify(variance(X)) == xm**2*alpha / ((alpha-1)**2*(alpha-2))
def test_pareto_numeric():
xm, beta = 3, 2
alpha = beta + 5
X = Pareto('x', xm, alpha)
assert E(X) == alpha*xm/S(alpha - 1)
assert variance(X) == xm**2*alpha / S(((alpha - 1)**2*(alpha - 2)))
# Skewness tests too slow. Try shortcutting function?
def test_raised_cosine():
mu = Symbol("mu", real=True)
s = Symbol("s", positive=True)
X = RaisedCosine("x", mu, s)
assert density(X)(x) == (Piecewise(((cos(pi*(x - mu)/s) + 1)/(2*s),
And(x <= mu + s, mu - s <= x)), (0, True)))
def test_rayleigh():
sigma = Symbol("sigma", positive=True)
X = Rayleigh('x', sigma)
assert density(X)(x) == x*exp(-x**2/(2*sigma**2))/sigma**2
assert E(X) == sqrt(2)*sqrt(pi)*sigma/2
assert variance(X) == -pi*sigma**2/2 + 2*sigma**2
def test_shiftedgompertz():
b = Symbol("b", positive=True)
eta = Symbol("eta", positive=True)
X = ShiftedGompertz("x", b, eta)
assert density(X)(x) == b*(eta*(1 - exp(-b*x)) + 1)*exp(-b*x)*exp(-eta*exp(-b*x))
def test_studentt():
nu = Symbol("nu", positive=True)
X = StudentT('x', nu)
assert density(X)(x) == (1 + x**2/nu)**(-nu/2 - S(1)/2)/(sqrt(nu)*beta(S(1)/2, nu/2))
assert cdf(X)(x) == S(1)/2 + x*gamma(nu/2 + S(1)/2)*hyper((S(1)/2, nu/2 + S(1)/2),
(S(3)/2,), -x**2/nu)/(sqrt(pi)*sqrt(nu)*gamma(nu/2))
def test_trapezoidal():
a = Symbol("a", real=True)
b = Symbol("b", real=True)
c = Symbol("c", real=True)
d = Symbol("d", real=True)
X = Trapezoidal('x', a, b, c, d)
assert density(X)(x) == Piecewise(((-2*a + 2*x)/((-a + b)*(-a - b + c + d)), (a <= x) & (x < b)),
(2/(-a - b + c + d), (b <= x) & (x < c)),
((2*d - 2*x)/((-c + d)*(-a - b + c + d)), (c <= x) & (x <= d)),
(0, True))
X = Trapezoidal('x', 0, 1, 2, 3)
assert E(X) == S(3)/2
assert variance(X) == S(5)/12
assert P(X < 2) == S(3)/4
@XFAIL
def test_triangular():
a = Symbol("a")
b = Symbol("b")
c = Symbol("c")
X = Triangular('x', a, b, c)
assert density(X)(x) == Piecewise(
((2*x - 2*a)/((-a + b)*(-a + c)), And(a <= x, x < c)),
(2/(-a + b), x == c),
((-2*x + 2*b)/((-a + b)*(b - c)), And(x <= b, c < x)),
(0, True))
def test_quadratic_u():
a = Symbol("a", real=True)
b = Symbol("b", real=True)
X = QuadraticU("x", a, b)
assert density(X)(x) == (Piecewise((12*(x - a/2 - b/2)**2/(-a + b)**3,
And(x <= b, a <= x)), (0, True)))
def test_uniform():
l = Symbol('l', real=True, finite=True)
w = Symbol('w', positive=True, finite=True)
X = Uniform('x', l, l + w)
assert simplify(E(X)) == l + w/2
assert simplify(variance(X)) == w**2/12
# With numbers all is well
X = Uniform('x', 3, 5)
assert P(X < 3) == 0 and P(X > 5) == 0
assert P(X < 4) == P(X > 4) == S.Half
z = Symbol('z')
p = density(X)(z)
assert p.subs(z, 3.7) == S(1)/2
assert p.subs(z, -1) == 0
assert p.subs(z, 6) == 0
c = cdf(X)
assert c(2) == 0 and c(3) == 0
assert c(S(7)/2) == S(1)/4
assert c(5) == 1 and c(6) == 1
def test_uniform_P():
""" This stopped working because SingleContinuousPSpace.compute_density no
longer calls integrate on a DiracDelta but rather just solves directly.
integrate used to call UniformDistribution.expectation which special-cased
subsed out the Min and Max terms that Uniform produces
I decided to regress on this class for general cleanliness (and I suspect
speed) of the algorithm.
"""
l = Symbol('l', real=True, finite=True)
w = Symbol('w', positive=True, finite=True)
X = Uniform('x', l, l + w)
assert P(X < l) == 0 and P(X > l + w) == 0
@XFAIL
def test_uniformsum():
n = Symbol("n", integer=True)
_k = Symbol("k")
X = UniformSum('x', n)
assert density(X)(x) == (Sum((-1)**_k*(-_k + x)**(n - 1)
*binomial(n, _k), (_k, 0, floor(x)))/factorial(n - 1))
def test_von_mises():
mu = Symbol("mu")
k = Symbol("k", positive=True)
X = VonMises("x", mu, k)
assert density(X)(x) == exp(k*cos(x - mu))/(2*pi*besseli(0, k))
def test_weibull():
a, b = symbols('a b', positive=True)
X = Weibull('x', a, b)
assert simplify(E(X)) == simplify(a * gamma(1 + 1/b))
assert simplify(variance(X)) == simplify(a**2 * gamma(1 + 2/b) - E(X)**2)
assert simplify(skewness(X)) == (2*gamma(1 + 1/b)**3 - 3*gamma(1 + 1/b)*gamma(1 + 2/b) + gamma(1 + 3/b))/(-gamma(1 + 1/b)**2 + gamma(1 + 2/b))**(S(3)/2)
def test_weibull_numeric():
# Test for integers and rationals
a = 1
bvals = [S.Half, 1, S(3)/2, 5]
for b in bvals:
X = Weibull('x', a, b)
assert simplify(E(X)) == expand_func(a * gamma(1 + 1/S(b)))
assert simplify(variance(X)) == simplify(
a**2 * gamma(1 + 2/S(b)) - E(X)**2)
# Not testing Skew... it's slow with int/frac values > 3/2
def test_wignersemicircle():
R = Symbol("R", positive=True)
X = WignerSemicircle('x', R)
assert density(X)(x) == 2*sqrt(-x**2 + R**2)/(pi*R**2)
assert E(X) == 0
def test_prefab_sampling():
N = Normal('X', 0, 1)
L = LogNormal('L', 0, 1)
E = Exponential('Ex', 1)
P = Pareto('P', 1, 3)
W = Weibull('W', 1, 1)
U = Uniform('U', 0, 1)
B = Beta('B', 2, 5)
G = Gamma('G', 1, 3)
variables = [N, L, E, P, W, U, B, G]
niter = 10
for var in variables:
for i in range(niter):
assert sample(var) in var.pspace.domain.set
def test_input_value_assertions():
a, b = symbols('a b')
p, q = symbols('p q', positive=True)
m, n = symbols('m n', positive=False, real=True)
raises(ValueError, lambda: Normal('x', 3, 0))
raises(ValueError, lambda: Normal('x', m, n))
Normal('X', a, p) # No error raised
raises(ValueError, lambda: Exponential('x', m))
Exponential('Ex', p) # No error raised
for fn in [Pareto, Weibull, Beta, Gamma]:
raises(ValueError, lambda: fn('x', m, p))
raises(ValueError, lambda: fn('x', p, n))
fn('x', p, q) # No error raised
@XFAIL
def test_unevaluated():
X = Normal('x', 0, 1)
assert E(X, evaluate=False) == (
Integral(sqrt(2)*x*exp(-x**2/2)/(2*sqrt(pi)), (x, -oo, oo)))
assert E(X + 1, evaluate=False) == (
Integral(sqrt(2)*x*exp(-x**2/2)/(2*sqrt(pi)), (x, -oo, oo)) + 1)
assert P(X > 0, evaluate=False) == (
Integral(sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)), (x, 0, oo)))
assert P(X > 0, X**2 < 1, evaluate=False) == (
Integral(sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)*
Integral(sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)),
(x, -1, 1))), (x, 0, 1)))
def test_probability_unevaluated():
T = Normal('T', 30, 3)
assert type(P(T > 33, evaluate=False)) == Integral
def test_density_unevaluated():
X = Normal('X', 0, 1)
Y = Normal('Y', 0, 2)
assert isinstance(density(X+Y, evaluate=False)(z), Integral)
def test_NormalDistribution():
nd = NormalDistribution(0, 1)
x = Symbol('x')
assert nd.cdf(x) == erf(sqrt(2)*x/2)/2 + S.One/2
assert isinstance(nd.sample(), float) or nd.sample().is_Number
assert nd.expectation(1, x) == 1
assert nd.expectation(x, x) == 0
assert nd.expectation(x**2, x) == 1
def test_random_parameters():
mu = Normal('mu', 2, 3)
meas = Normal('T', mu, 1)
assert density(meas, evaluate=False)(z)
assert isinstance(pspace(meas), JointPSpace)
#assert density(meas, evaluate=False)(z) == Integral(mu.pspace.pdf *
# meas.pspace.pdf, (mu.symbol, -oo, oo)).subs(meas.symbol, z)
def test_random_parameters_given():
mu = Normal('mu', 2, 3)
meas = Normal('T', mu, 1)
assert given(meas, Eq(mu, 5)) == Normal('T', 5, 1)
def test_conjugate_priors():
mu = Normal('mu', 2, 3)
x = Normal('x', mu, 1)
assert isinstance(simplify(density(mu, Eq(x, y), evaluate=False)(z)),
Mul)
def test_difficult_univariate():
""" Since using solve in place of deltaintegrate we're able to perform
substantially more complex density computations on single continuous random
variables """
x = Normal('x', 0, 1)
assert density(x**3)
assert density(exp(x**2))
assert density(log(x))
def test_issue_10003():
X = Exponential('x', 3)
G = Gamma('g', 1, 2)
assert P(X < -1) == S.Zero
assert P(G < -1) == S.Zero
@slow
def test_precomputed_cdf():
x = symbols("x", real=True, finite=True)
mu = symbols("mu", real=True, finite=True)
sigma, xm, alpha = symbols("sigma xm alpha", positive=True, finite=True)
n = symbols("n", integer=True, positive=True, finite=True)
distribs = [
Normal("X", mu, sigma),
Pareto("P", xm, alpha),
ChiSquared("C", n),
Exponential("E", sigma),
# LogNormal("L", mu, sigma),
]
for X in distribs:
compdiff = cdf(X)(x) - simplify(X.pspace.density.compute_cdf()(x))
compdiff = simplify(compdiff.rewrite(erfc))
assert compdiff == 0
@slow
def test_precomputed_characteristic_functions():
import mpmath
def test_cf(dist, support_lower_limit, support_upper_limit):
pdf = density(dist)
t = Symbol('t')
x = Symbol('x')
# first function is the hardcoded CF of the distribution
cf1 = lambdify([t], characteristic_function(dist)(t), 'mpmath')
# second function is the Fourier transform of the density function
f = lambdify([x, t], pdf(x)*exp(I*x*t), 'mpmath')
cf2 = lambda t: mpmath.quad(lambda x: f(x, t), [support_lower_limit, support_upper_limit], maxdegree=10)
# compare the two functions at various points
for test_point in [2, 5, 8, 11]:
n1 = cf1(test_point)
n2 = cf2(test_point)
assert abs(re(n1) - re(n2)) < 1e-12
assert abs(im(n1) - im(n2)) < 1e-12
test_cf(Beta('b', 1, 2), 0, 1)
test_cf(Chi('c', 3), 0, mpmath.inf)
test_cf(ChiSquared('c', 2), 0, mpmath.inf)
test_cf(Exponential('e', 6), 0, mpmath.inf)
test_cf(Logistic('l', 1, 2), -mpmath.inf, mpmath.inf)
test_cf(Normal('n', -1, 5), -mpmath.inf, mpmath.inf)
test_cf(RaisedCosine('r', 3, 1), 2, 4)
test_cf(Rayleigh('r', 0.5), 0, mpmath.inf)
test_cf(Uniform('u', -1, 1), -1, 1)
test_cf(WignerSemicircle('w', 3), -3, 3)
def test_long_precomputed_cdf():
x = symbols("x", real=True, finite=True)
distribs = [
Arcsin("A", -5, 9),
Dagum("D", 4, 10, 3),
Erlang("E", 14, 5),
Frechet("F", 2, 6, -3),
Gamma("G", 2, 7),
GammaInverse("GI", 3, 5),
Kumaraswamy("K", 6, 8),
Laplace("LA", -5, 4),
Logistic("L", -6, 7),
Nakagami("N", 2, 7),
StudentT("S", 4)
]
for distr in distribs:
for _ in range(5):
assert tn(diff(cdf(distr)(x), x), density(distr)(x), x, a=0, b=0, c=1, d=0)
US = UniformSum("US", 5)
pdf01 = density(US)(x).subs(floor(x), 0).doit() # pdf on (0, 1)
cdf01 = cdf(US, evaluate=False)(x).subs(floor(x), 0).doit() # cdf on (0, 1)
assert tn(diff(cdf01, x), pdf01, x, a=0, b=0, c=1, d=0)
def test_issue_13324():
X = Uniform('X', 0, 1)
assert E(X, X > Rational(1, 2)) == Rational(3, 4)
assert E(X, X > 0) == Rational(1, 2)
def test_FiniteSet_prob():
x = symbols('x')
E = Exponential('E', 3)
N = Normal('N', 5, 7)
assert P(Eq(E, 1)) is S.Zero
assert P(Eq(N, 2)) is S.Zero
assert P(Eq(N, x)) is S.Zero
def test_prob_neq():
E = Exponential('E', 4)
X = ChiSquared('X', 4)
x = symbols('x')
assert P(Ne(E, 2)) == 1
assert P(Ne(X, 4)) == 1
assert P(Ne(X, 4)) == 1
assert P(Ne(X, 5)) == 1
assert P(Ne(E, x)) == 1
def test_union():
N = Normal('N', 3, 2)
assert simplify(P(N**2 - N > 2)) == \
-erf(sqrt(2))/2 - erfc(sqrt(2)/4)/2 + S(3)/2
assert simplify(P(N**2 - 4 > 0)) == \
-erf(5*sqrt(2)/4)/2 - erfc(sqrt(2)/4)/2 + S(3)/2
def test_Or():
N = Normal('N', 0, 1)
assert simplify(P(Or(N > 2, N < 1))) == \
-erf(sqrt(2))/2 - erfc(sqrt(2)/2)/2 + S(3)/2
assert P(Or(N < 0, N < 1)) == P(N < 1)
assert P(Or(N > 0, N < 0)) == 1
def test_conditional_eq():
E = Exponential('E', 1)
assert P(Eq(E, 1), Eq(E, 1)) == 1
assert P(Eq(E, 1), Eq(E, 2)) == 0
assert P(E > 1, Eq(E, 2)) == 1
assert P(E < 1, Eq(E, 2)) == 0
|
dcdcbd062377e43b83f7d870ebafbb8c2abe9c059f43e68397b1bef86bc4f148
|
from sympy.core.compatibility import range
from sympy.ntheory.generate import Sieve, sieve
from sympy.ntheory.primetest import (mr, is_lucas_prp, is_square,
is_strong_lucas_prp, is_extra_strong_lucas_prp, isprime, is_euler_pseudoprime)
from sympy.utilities.pytest import slow
def test_euler_pseudoprimes():
assert is_euler_pseudoprime(9, 1) == True
assert is_euler_pseudoprime(341, 2) == False
assert is_euler_pseudoprime(121, 3) == True
assert is_euler_pseudoprime(341, 4) == True
assert is_euler_pseudoprime(217, 5) == False
assert is_euler_pseudoprime(185, 6) == False
assert is_euler_pseudoprime(55, 111) == True
assert is_euler_pseudoprime(115, 114) == True
assert is_euler_pseudoprime(49, 117) == True
assert is_euler_pseudoprime(85, 84) == True
assert is_euler_pseudoprime(87, 88) == True
assert is_euler_pseudoprime(49, 128) == True
assert is_euler_pseudoprime(39, 77) == True
assert is_euler_pseudoprime(9881, 30) == True
assert is_euler_pseudoprime(8841, 29) == False
assert is_euler_pseudoprime(8421, 29) == False
assert is_euler_pseudoprime(9997, 19) == True
@slow
def test_prps():
oddcomposites = [n for n in range(1, 10**5) if
n % 2 and not isprime(n)]
# A checksum would be better.
assert sum(oddcomposites) == 2045603465
assert [n for n in oddcomposites if mr(n, [2])] == [
2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141,
52633, 65281, 74665, 80581, 85489, 88357, 90751]
assert [n for n in oddcomposites if mr(n, [3])] == [
121, 703, 1891, 3281, 8401, 8911, 10585, 12403, 16531,
18721, 19345, 23521, 31621, 44287, 47197, 55969, 63139,
74593, 79003, 82513, 87913, 88573, 97567]
assert [n for n in oddcomposites if mr(n, [325])] == [
9, 25, 27, 49, 65, 81, 325, 341, 343, 697, 1141, 2059,
2149, 3097, 3537, 4033, 4681, 4941, 5833, 6517, 7987, 8911,
12403, 12913, 15043, 16021, 20017, 22261, 23221, 24649,
24929, 31841, 35371, 38503, 43213, 44173, 47197, 50041,
55909, 56033, 58969, 59089, 61337, 65441, 68823, 72641,
76793, 78409, 85879]
assert not any(mr(n, [9345883071009581737]) for n in oddcomposites)
assert [n for n in oddcomposites if is_lucas_prp(n)] == [
323, 377, 1159, 1829, 3827, 5459, 5777, 9071, 9179, 10877,
11419, 11663, 13919, 14839, 16109, 16211, 18407, 18971,
19043, 22499, 23407, 24569, 25199, 25877, 26069, 27323,
32759, 34943, 35207, 39059, 39203, 39689, 40309, 44099,
46979, 47879, 50183, 51983, 53663, 56279, 58519, 60377,
63881, 69509, 72389, 73919, 75077, 77219, 79547, 79799,
82983, 84419, 86063, 90287, 94667, 97019, 97439]
assert [n for n in oddcomposites if is_strong_lucas_prp(n)] == [
5459, 5777, 10877, 16109, 18971, 22499, 24569, 25199, 40309,
58519, 75077, 97439]
assert [n for n in oddcomposites if is_extra_strong_lucas_prp(n)
] == [
989, 3239, 5777, 10877, 27971, 29681, 30739, 31631, 39059,
72389, 73919, 75077]
def test_isprime():
s = Sieve()
s.extend(100000)
ps = set(s.primerange(2, 100001))
for n in range(100001):
# if (n in ps) != isprime(n): print n
assert (n in ps) == isprime(n)
assert isprime(179424673)
assert isprime(20678048681)
assert isprime(1968188556461)
assert isprime(2614941710599)
assert isprime(65635624165761929287)
assert isprime(1162566711635022452267983)
assert isprime(77123077103005189615466924501)
assert isprime(3991617775553178702574451996736229)
assert isprime(273952953553395851092382714516720001799)
assert isprime(int('''
531137992816767098689588206552468627329593117727031923199444138200403\
559860852242739162502265229285668889329486246501015346579337652707239\
409519978766587351943831270835393219031728127'''))
# Some Mersenne primes
assert isprime(2**61 - 1)
assert isprime(2**89 - 1)
assert isprime(2**607 - 1)
# (but not all Mersenne's are primes
assert not isprime(2**601 - 1)
# pseudoprimes
#-------------
# to some small bases
assert not isprime(2152302898747)
assert not isprime(3474749660383)
assert not isprime(341550071728321)
assert not isprime(3825123056546413051)
# passes the base set [2, 3, 7, 61, 24251]
assert not isprime(9188353522314541)
# large examples
assert not isprime(877777777777777777777777)
# conjectured psi_12 given at http://mathworld.wolfram.com/StrongPseudoprime.html
assert not isprime(318665857834031151167461)
# conjectured psi_17 given at http://mathworld.wolfram.com/StrongPseudoprime.html
assert not isprime(564132928021909221014087501701)
# Arnault's 1993 number; a factor of it is
# 400958216639499605418306452084546853005188166041132508774506\
# 204738003217070119624271622319159721973358216316508535816696\
# 9145233813917169287527980445796800452592031836601
assert not isprime(int('''
803837457453639491257079614341942108138837688287558145837488917522297\
427376533365218650233616396004545791504202360320876656996676098728404\
396540823292873879185086916685732826776177102938969773947016708230428\
687109997439976544144845341155872450633409279022275296229414984230688\
1685404326457534018329786111298960644845216191652872597534901'''))
# Arnault's 1995 number; can be factored as
# p1*(313*(p1 - 1) + 1)*(353*(p1 - 1) + 1) where p1 is
# 296744956686855105501541746429053327307719917998530433509950\
# 755312768387531717701995942385964281211880336647542183455624\
# 93168782883
assert not isprime(int('''
288714823805077121267142959713039399197760945927972270092651602419743\
230379915273311632898314463922594197780311092934965557841894944174093\
380561511397999942154241693397290542371100275104208013496673175515285\
922696291677532547504444585610194940420003990443211677661994962953925\
045269871932907037356403227370127845389912612030924484149472897688540\
6024976768122077071687938121709811322297802059565867'''))
sieve.extend(3000)
assert isprime(2819)
assert not isprime(2931)
assert not isprime(2.0)
def test_is_square():
assert [i for i in range(25) if is_square(i)] == [0, 1, 4, 9, 16]
|
4189c5cc2b1d0923aa2d1f1e81801426d5e5c2488367946d0e68bce498ca5c78
|
from sympy import (Sieve, binomial_coefficients, binomial_coefficients_list,
Mul, S, Pow, sieve, Symbol, summation, Dummy,
factorial as fac)
from sympy.core.evalf import bitcount
from sympy.core.numbers import Integer, Rational
from sympy.core.compatibility import long, range
from sympy.ntheory import (isprime, n_order, is_primitive_root,
is_quad_residue, legendre_symbol, jacobi_symbol, npartitions, totient,
factorint, primefactors, divisors, randprime, nextprime, prevprime,
primerange, primepi, prime, pollard_rho, perfect_power, multiplicity,
trailing, divisor_count, primorial, pollard_pm1, divisor_sigma,
factorrat, reduced_totient)
from sympy.ntheory.factor_ import (smoothness, smoothness_p,
antidivisors, antidivisor_count, core, digits, udivisors, udivisor_sigma,
udivisor_count, primenu, primeomega, small_trailing, mersenne_prime_exponent,
is_perfect, is_mersenne_prime, is_abundant, is_deficient, is_amicable)
from sympy.ntheory.generate import cycle_length
from sympy.ntheory.multinomial import (
multinomial_coefficients, multinomial_coefficients_iterator)
from sympy.ntheory.bbp_pi import pi_hex_digits
from sympy.ntheory.modular import crt, crt1, crt2, solve_congruence
from sympy.utilities.pytest import raises, slow
from sympy.utilities.iterables import capture
def fac_multiplicity(n, p):
"""Return the power of the prime number p in the
factorization of n!"""
if p > n:
return 0
if p > n//2:
return 1
q, m = n, 0
while q >= p:
q //= p
m += q
return m
def multiproduct(seq=(), start=1):
"""
Return the product of a sequence of factors with multiplicities,
times the value of the parameter ``start``. The input may be a
sequence of (factor, exponent) pairs or a dict of such pairs.
>>> multiproduct({3:7, 2:5}, 4) # = 3**7 * 2**5 * 4
279936
"""
if not seq:
return start
if isinstance(seq, dict):
seq = iter(seq.items())
units = start
multi = []
for base, exp in seq:
if not exp:
continue
elif exp == 1:
units *= base
else:
if exp % 2:
units *= base
multi.append((base, exp//2))
return units * multiproduct(multi)**2
def test_trailing_bitcount():
assert trailing(0) == 0
assert trailing(1) == 0
assert trailing(-1) == 0
assert trailing(2) == 1
assert trailing(7) == 0
assert trailing(-7) == 0
for i in range(100):
assert trailing((1 << i)) == i
assert trailing((1 << i) * 31337) == i
assert trailing((1 << 1000001)) == 1000001
assert trailing((1 << 273956)*7**37) == 273956
# issue 12709
big = small_trailing[-1]*2
assert trailing(-big) == trailing(big)
assert bitcount(-big) == bitcount(big)
def test_multiplicity():
for b in range(2, 20):
for i in range(100):
assert multiplicity(b, b**i) == i
assert multiplicity(b, (b**i) * 23) == i
assert multiplicity(b, (b**i) * 1000249) == i
# Should be fast
assert multiplicity(10, 10**10023) == 10023
# Should exit quickly
assert multiplicity(10**10, 10**10) == 1
# Should raise errors for bad input
raises(ValueError, lambda: multiplicity(1, 1))
raises(ValueError, lambda: multiplicity(1, 2))
raises(ValueError, lambda: multiplicity(1.3, 2))
raises(ValueError, lambda: multiplicity(2, 0))
raises(ValueError, lambda: multiplicity(1.3, 0))
# handles Rationals
assert multiplicity(10, Rational(30, 7)) == 1
assert multiplicity(Rational(2, 7), Rational(4, 7)) == 1
assert multiplicity(Rational(1, 7), Rational(3, 49)) == 2
assert multiplicity(Rational(2, 7), Rational(7, 2)) == -1
assert multiplicity(3, Rational(1, 9)) == -2
def test_perfect_power():
assert perfect_power(0) is False
assert perfect_power(1) is False
assert perfect_power(2) is False
assert perfect_power(3) is False
assert perfect_power(4) == (2, 2)
assert perfect_power(14) is False
assert perfect_power(25) == (5, 2)
assert perfect_power(22) is False
assert perfect_power(22, [2]) is False
assert perfect_power(137**(3*5*13)) == (137, 3*5*13)
assert perfect_power(137**(3*5*13) + 1) is False
assert perfect_power(137**(3*5*13) - 1) is False
assert perfect_power(103005006004**7) == (103005006004, 7)
assert perfect_power(103005006004**7 + 1) is False
assert perfect_power(103005006004**7 - 1) is False
assert perfect_power(103005006004**12) == (103005006004, 12)
assert perfect_power(103005006004**12 + 1) is False
assert perfect_power(103005006004**12 - 1) is False
assert perfect_power(2**10007) == (2, 10007)
assert perfect_power(2**10007 + 1) is False
assert perfect_power(2**10007 - 1) is False
assert perfect_power((9**99 + 1)**60) == (9**99 + 1, 60)
assert perfect_power((9**99 + 1)**60 + 1) is False
assert perfect_power((9**99 + 1)**60 - 1) is False
assert perfect_power((10**40000)**2, big=False) == (10**40000, 2)
assert perfect_power(10**100000) == (10, 100000)
assert perfect_power(10**100001) == (10, 100001)
assert perfect_power(13**4, [3, 5]) is False
assert perfect_power(3**4, [3, 10], factor=0) is False
assert perfect_power(3**3*5**3) == (15, 3)
assert perfect_power(2**3*5**5) is False
assert perfect_power(2*13**4) is False
assert perfect_power(2**5*3**3) is False
def test_factorint():
assert primefactors(123456) == [2, 3, 643]
assert factorint(0) == {0: 1}
assert factorint(1) == {}
assert factorint(-1) == {-1: 1}
assert factorint(-2) == {-1: 1, 2: 1}
assert factorint(-16) == {-1: 1, 2: 4}
assert factorint(2) == {2: 1}
assert factorint(126) == {2: 1, 3: 2, 7: 1}
assert factorint(123456) == {2: 6, 3: 1, 643: 1}
assert factorint(5951757) == {3: 1, 7: 1, 29: 2, 337: 1}
assert factorint(64015937) == {7993: 1, 8009: 1}
assert factorint(2**(2**6) + 1) == {274177: 1, 67280421310721: 1}
assert factorint(0, multiple=True) == [0]
assert factorint(1, multiple=True) == []
assert factorint(-1, multiple=True) == [-1]
assert factorint(-2, multiple=True) == [-1, 2]
assert factorint(-16, multiple=True) == [-1, 2, 2, 2, 2]
assert factorint(2, multiple=True) == [2]
assert factorint(24, multiple=True) == [2, 2, 2, 3]
assert factorint(126, multiple=True) == [2, 3, 3, 7]
assert factorint(123456, multiple=True) == [2, 2, 2, 2, 2, 2, 3, 643]
assert factorint(5951757, multiple=True) == [3, 7, 29, 29, 337]
assert factorint(64015937, multiple=True) == [7993, 8009]
assert factorint(2**(2**6) + 1, multiple=True) == [274177, 67280421310721]
assert factorint(fac(1, evaluate=False)) == {}
assert factorint(fac(7, evaluate=False)) == {2: 4, 3: 2, 5: 1, 7: 1}
assert factorint(fac(15, evaluate=False)) == \
{2: 11, 3: 6, 5: 3, 7: 2, 11: 1, 13: 1}
assert factorint(fac(20, evaluate=False)) == \
{2: 18, 3: 8, 5: 4, 7: 2, 11: 1, 13: 1, 17: 1, 19: 1}
assert factorint(fac(23, evaluate=False)) == \
{2: 19, 3: 9, 5: 4, 7: 3, 11: 2, 13: 1, 17: 1, 19: 1, 23: 1}
assert multiproduct(factorint(fac(200))) == fac(200)
assert multiproduct(factorint(fac(200, evaluate=False))) == fac(200)
for b, e in factorint(fac(150)).items():
assert e == fac_multiplicity(150, b)
for b, e in factorint(fac(150, evaluate=False)).items():
assert e == fac_multiplicity(150, b)
assert factorint(103005006059**7) == {103005006059: 7}
assert factorint(31337**191) == {31337: 191}
assert factorint(2**1000 * 3**500 * 257**127 * 383**60) == \
{2: 1000, 3: 500, 257: 127, 383: 60}
assert len(factorint(fac(10000))) == 1229
assert len(factorint(fac(10000, evaluate=False))) == 1229
assert factorint(12932983746293756928584532764589230) == \
{2: 1, 5: 1, 73: 1, 727719592270351: 1, 63564265087747: 1, 383: 1}
assert factorint(727719592270351) == {727719592270351: 1}
assert factorint(2**64 + 1, use_trial=False) == factorint(2**64 + 1)
for n in range(60000):
assert multiproduct(factorint(n)) == n
assert pollard_rho(2**64 + 1, seed=1) == 274177
assert pollard_rho(19, seed=1) is None
assert factorint(3, limit=2) == {3: 1}
assert factorint(12345) == {3: 1, 5: 1, 823: 1}
assert factorint(
12345, limit=3) == {4115: 1, 3: 1} # the 5 is greater than the limit
assert factorint(1, limit=1) == {}
assert factorint(0, 3) == {0: 1}
assert factorint(12, limit=1) == {12: 1}
assert factorint(30, limit=2) == {2: 1, 15: 1}
assert factorint(16, limit=2) == {2: 4}
assert factorint(124, limit=3) == {2: 2, 31: 1}
assert factorint(4*31**2, limit=3) == {2: 2, 31: 2}
p1 = nextprime(2**32)
p2 = nextprime(2**16)
p3 = nextprime(p2)
assert factorint(p1*p2*p3) == {p1: 1, p2: 1, p3: 1}
assert factorint(13*17*19, limit=15) == {13: 1, 17*19: 1}
assert factorint(1951*15013*15053, limit=2000) == {225990689: 1, 1951: 1}
assert factorint(primorial(17) + 1, use_pm1=0) == \
{long(19026377261): 1, 3467: 1, 277: 1, 105229: 1}
# when prime b is closer than approx sqrt(8*p) to prime p then they are
# "close" and have a trivial factorization
a = nextprime(2**2**8) # 78 digits
b = nextprime(a + 2**2**4)
assert 'Fermat' in capture(lambda: factorint(a*b, verbose=1))
raises(ValueError, lambda: pollard_rho(4))
raises(ValueError, lambda: pollard_pm1(3))
raises(ValueError, lambda: pollard_pm1(10, B=2))
# verbose coverage
n = nextprime(2**16)*nextprime(2**17)*nextprime(1901)
assert 'with primes' in capture(lambda: factorint(n, verbose=1))
capture(lambda: factorint(nextprime(2**16)*1012, verbose=1))
n = nextprime(2**17)
capture(lambda: factorint(n**3, verbose=1)) # perfect power termination
capture(lambda: factorint(2*n, verbose=1)) # factoring complete msg
# exceed 1st
n = nextprime(2**17)
n *= nextprime(n)
assert '1000' in capture(lambda: factorint(n, limit=1000, verbose=1))
n *= nextprime(n)
assert len(factorint(n)) == 3
assert len(factorint(n, limit=p1)) == 3
n *= nextprime(2*n)
# exceed 2nd
assert '2001' in capture(lambda: factorint(n, limit=2000, verbose=1))
assert capture(
lambda: factorint(n, limit=4000, verbose=1)).count('Pollard') == 2
# non-prime pm1 result
n = nextprime(8069)
n *= nextprime(2*n)*nextprime(2*n, 2)
capture(lambda: factorint(n, verbose=1)) # non-prime pm1 result
# factor fermat composite
p1 = nextprime(2**17)
p2 = nextprime(2*p1)
assert factorint((p1*p2**2)**3) == {p1: 3, p2: 6}
# Test for non integer input
raises(ValueError, lambda: factorint(4.5))
def test_divisors_and_divisor_count():
assert divisors(-1) == [1]
assert divisors(0) == []
assert divisors(1) == [1]
assert divisors(2) == [1, 2]
assert divisors(3) == [1, 3]
assert divisors(17) == [1, 17]
assert divisors(10) == [1, 2, 5, 10]
assert divisors(100) == [1, 2, 4, 5, 10, 20, 25, 50, 100]
assert divisors(101) == [1, 101]
assert divisor_count(0) == 0
assert divisor_count(-1) == 1
assert divisor_count(1) == 1
assert divisor_count(6) == 4
assert divisor_count(12) == 6
assert divisor_count(180, 3) == divisor_count(180//3)
assert divisor_count(2*3*5, 7) == 0
def test_udivisors_and_udivisor_count():
assert udivisors(-1) == [1]
assert udivisors(0) == []
assert udivisors(1) == [1]
assert udivisors(2) == [1, 2]
assert udivisors(3) == [1, 3]
assert udivisors(17) == [1, 17]
assert udivisors(10) == [1, 2, 5, 10]
assert udivisors(100) == [1, 4, 25, 100]
assert udivisors(101) == [1, 101]
assert udivisors(1000) == [1, 8, 125, 1000]
assert udivisor_count(0) == 0
assert udivisor_count(-1) == 1
assert udivisor_count(1) == 1
assert udivisor_count(6) == 4
assert udivisor_count(12) == 4
assert udivisor_count(180) == 8
assert udivisor_count(2*3*5*7) == 16
def test_issue_6981():
S = set(divisors(4)).union(set(divisors(Integer(2))))
assert S == {1,2,4}
def test_totient():
assert [totient(k) for k in range(1, 12)] == \
[1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10]
assert totient(5005) == 2880
assert totient(5006) == 2502
assert totient(5009) == 5008
assert totient(2**100) == 2**99
raises(ValueError, lambda: totient(30.1))
raises(ValueError, lambda: totient(20.001))
m = Symbol("m", integer=True)
assert totient(m)
assert totient(m).subs(m, 3**10) == 3**10 - 3**9
assert summation(totient(m), (m, 1, 11)) == 42
n = Symbol("n", integer=True, positive=True)
assert totient(n).is_integer
x=Symbol("x", integer=False)
raises(ValueError, lambda: totient(x))
y=Symbol("y", positive=False)
raises(ValueError, lambda: totient(y))
z=Symbol("z", positive=True, integer=True)
raises(ValueError, lambda: totient(2**(-z)))
def test_reduced_totient():
assert [reduced_totient(k) for k in range(1, 16)] == \
[1, 1, 2, 2, 4, 2, 6, 2, 6, 4, 10, 2, 12, 6, 4]
assert reduced_totient(5005) == 60
assert reduced_totient(5006) == 2502
assert reduced_totient(5009) == 5008
assert reduced_totient(2**100) == 2**98
m = Symbol("m", integer=True)
assert reduced_totient(m)
assert reduced_totient(m).subs(m, 2**3*3**10) == 3**10 - 3**9
assert summation(reduced_totient(m), (m, 1, 16)) == 68
n = Symbol("n", integer=True, positive=True)
assert reduced_totient(n).is_integer
def test_divisor_sigma():
assert [divisor_sigma(k) for k in range(1, 12)] == \
[1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12]
assert [divisor_sigma(k, 2) for k in range(1, 12)] == \
[1, 5, 10, 21, 26, 50, 50, 85, 91, 130, 122]
assert divisor_sigma(23450) == 50592
assert divisor_sigma(23450, 0) == 24
assert divisor_sigma(23450, 1) == 50592
assert divisor_sigma(23450, 2) == 730747500
assert divisor_sigma(23450, 3) == 14666785333344
m = Symbol("m", integer=True)
k = Symbol("k", integer=True)
assert divisor_sigma(m)
assert divisor_sigma(m, k)
assert divisor_sigma(m).subs(m, 3**10) == 88573
assert divisor_sigma(m, k).subs([(m, 3**10), (k, 3)]) == 213810021790597
assert summation(divisor_sigma(m), (m, 1, 11)) == 99
def test_udivisor_sigma():
assert [udivisor_sigma(k) for k in range(1, 12)] == \
[1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12]
assert [udivisor_sigma(k, 3) for k in range(1, 12)] == \
[1, 9, 28, 65, 126, 252, 344, 513, 730, 1134, 1332]
assert udivisor_sigma(23450) == 42432
assert udivisor_sigma(23450, 0) == 16
assert udivisor_sigma(23450, 1) == 42432
assert udivisor_sigma(23450, 2) == 702685000
assert udivisor_sigma(23450, 4) == 321426961814978248
m = Symbol("m", integer=True)
k = Symbol("k", integer=True)
assert udivisor_sigma(m)
assert udivisor_sigma(m, k)
assert udivisor_sigma(m).subs(m, 4**9) == 262145
assert udivisor_sigma(m, k).subs([(m, 4**9), (k, 2)]) == 68719476737
assert summation(udivisor_sigma(m), (m, 2, 15)) == 169
def test_issue_4356():
assert factorint(1030903) == {53: 2, 367: 1}
def test_divisors():
assert divisors(28) == [1, 2, 4, 7, 14, 28]
assert [x for x in divisors(3*5*7, 1)] == [1, 3, 5, 15, 7, 21, 35, 105]
assert divisors(0) == []
def test_divisor_count():
assert divisor_count(0) == 0
assert divisor_count(6) == 4
def test_antidivisors():
assert antidivisors(-1) == []
assert antidivisors(-3) == [2]
assert antidivisors(14) == [3, 4, 9]
assert antidivisors(237) == [2, 5, 6, 11, 19, 25, 43, 95, 158]
assert antidivisors(12345) == [2, 6, 7, 10, 30, 1646, 3527, 4938, 8230]
assert antidivisors(393216) == [262144]
assert sorted(x for x in antidivisors(3*5*7, 1)) == \
[2, 6, 10, 11, 14, 19, 30, 42, 70]
assert antidivisors(1) == []
def test_antidivisor_count():
assert antidivisor_count(0) == 0
assert antidivisor_count(-1) == 0
assert antidivisor_count(-4) == 1
assert antidivisor_count(20) == 3
assert antidivisor_count(25) == 5
assert antidivisor_count(38) == 7
assert antidivisor_count(180) == 6
assert antidivisor_count(2*3*5) == 3
def test_smoothness_and_smoothness_p():
assert smoothness(1) == (1, 1)
assert smoothness(2**4*3**2) == (3, 16)
assert smoothness_p(10431, m=1) == \
(1, [(3, (2, 2, 4)), (19, (1, 5, 5)), (61, (1, 31, 31))])
assert smoothness_p(10431) == \
(-1, [(3, (2, 2, 2)), (19, (1, 3, 9)), (61, (1, 5, 5))])
assert smoothness_p(10431, power=1) == \
(-1, [(3, (2, 2, 2)), (61, (1, 5, 5)), (19, (1, 3, 9))])
assert smoothness_p(21477639576571, visual=1) == \
'p**i=4410317**1 has p-1 B=1787, B-pow=1787\n' + \
'p**i=4869863**1 has p-1 B=2434931, B-pow=2434931'
def test_visual_factorint():
assert factorint(1, visual=1) == 1
forty2 = factorint(42, visual=True)
assert type(forty2) == Mul
assert str(forty2) == '2**1*3**1*7**1'
assert factorint(1, visual=True) is S.One
no = dict(evaluate=False)
assert factorint(42**2, visual=True) == Mul(Pow(2, 2, **no),
Pow(3, 2, **no),
Pow(7, 2, **no), **no)
assert -1 in factorint(-42, visual=True).args
def test_factorrat():
assert str(factorrat(S(12)/1, visual=True)) == '2**2*3**1'
assert str(factorrat(S(1)/1, visual=True)) == '1'
assert str(factorrat(S(25)/14, visual=True)) == '5**2/(2*7)'
assert str(factorrat(S(-25)/14/9, visual=True)) == '-5**2/(2*3**2*7)'
assert factorrat(S(12)/1, multiple=True) == [2, 2, 3]
assert factorrat(S(1)/1, multiple=True) == []
assert factorrat(S(25)/14, multiple=True) == [S(1)/7, S(1)/2, 5, 5]
assert factorrat(S(12)/1, multiple=True) == [2, 2, 3]
assert factorrat(S(-25)/14/9, multiple=True) == \
[-1, S(1)/7, S(1)/3, S(1)/3, S(1)/2, 5, 5]
def test_visual_io():
sm = smoothness_p
fi = factorint
# with smoothness_p
n = 124
d = fi(n)
m = fi(d, visual=True)
t = sm(n)
s = sm(t)
for th in [d, s, t, n, m]:
assert sm(th, visual=True) == s
assert sm(th, visual=1) == s
for th in [d, s, t, n, m]:
assert sm(th, visual=False) == t
assert [sm(th, visual=None) for th in [d, s, t, n, m]] == [s, d, s, t, t]
assert [sm(th, visual=2) for th in [d, s, t, n, m]] == [s, d, s, t, t]
# with factorint
for th in [d, m, n]:
assert fi(th, visual=True) == m
assert fi(th, visual=1) == m
for th in [d, m, n]:
assert fi(th, visual=False) == d
assert [fi(th, visual=None) for th in [d, m, n]] == [m, d, d]
assert [fi(th, visual=0) for th in [d, m, n]] == [m, d, d]
# test reevaluation
no = dict(evaluate=False)
assert sm({4: 2}, visual=False) == sm(16)
assert sm(Mul(*[Pow(k, v, **no) for k, v in {4: 2, 2: 6}.items()], **no),
visual=False) == sm(2**10)
assert fi({4: 2}, visual=False) == fi(16)
assert fi(Mul(*[Pow(k, v, **no) for k, v in {4: 2, 2: 6}.items()], **no),
visual=False) == fi(2**10)
def test_core():
assert core(35**13, 10) == 42875
assert core(210**2) == 1
assert core(7776, 3) == 36
assert core(10**27, 22) == 10**5
assert core(537824) == 14
assert core(1, 6) == 1
def test_digits():
assert all([digits(n, 2)[1:] == [int(d) for d in format(n, 'b')]
for n in range(20)])
assert all([digits(n, 8)[1:] == [int(d) for d in format(n, 'o')]
for n in range(20)])
assert all([digits(n, 16)[1:] == [int(d, 16) for d in format(n, 'x')]
for n in range(20)])
assert digits(2345, 34) == [34, 2, 0, 33]
assert digits(384753, 71) == [71, 1, 5, 23, 4]
assert digits(93409) == [10, 9, 3, 4, 0, 9]
assert digits(-92838, 11) == [-11, 6, 3, 8, 2, 9]
def test_primenu():
assert primenu(2) == 1
assert primenu(2 * 3) == 2
assert primenu(2 * 3 * 5) == 3
assert primenu(3 * 25) == primenu(3) + primenu(25)
assert [primenu(p) for p in primerange(1, 10)] == [1, 1, 1, 1]
assert primenu(fac(50)) == 15
assert primenu(2 ** 9941 - 1) == 1
n = Symbol('n', integer=True)
assert primenu(n)
assert primenu(n).subs(n, 2 ** 31 - 1) == 1
assert summation(primenu(n), (n, 2, 30)) == 43
def test_primeomega():
assert primeomega(2) == 1
assert primeomega(2 * 2) == 2
assert primeomega(2 * 2 * 3) == 3
assert primeomega(3 * 25) == primeomega(3) + primeomega(25)
assert [primeomega(p) for p in primerange(1, 10)] == [1, 1, 1, 1]
assert primeomega(fac(50)) == 108
assert primeomega(2 ** 9941 - 1) == 1
n = Symbol('n', integer=True)
assert primeomega(n)
assert primeomega(n).subs(n, 2 ** 31 - 1) == 1
assert summation(primeomega(n), (n, 2, 30)) == 59
def test_mersenne_prime_exponent():
assert mersenne_prime_exponent(1) == 2
assert mersenne_prime_exponent(4) == 7
assert mersenne_prime_exponent(10) == 89
assert mersenne_prime_exponent(25) == 21701
raises(ValueError, lambda: mersenne_prime_exponent(52))
raises(ValueError, lambda: mersenne_prime_exponent(0))
def test_is_perfect():
assert is_perfect(6) is True
assert is_perfect(15) is False
assert is_perfect(28) is True
assert is_perfect(400) is False
assert is_perfect(496) is True
assert is_perfect(8128) is True
assert is_perfect(10000) is False
def test_is_mersenne_prime():
assert is_mersenne_prime(10) is False
assert is_mersenne_prime(127) is True
assert is_mersenne_prime(511) is False
assert is_mersenne_prime(131071) is True
assert is_mersenne_prime(2147483647) is True
def test_is_abundant():
assert is_abundant(10) is False
assert is_abundant(12) is True
assert is_abundant(18) is True
assert is_abundant(21) is False
assert is_abundant(945) is True
def test_is_deficient():
assert is_deficient(10) is True
assert is_deficient(22) is True
assert is_deficient(56) is False
assert is_deficient(20) is False
assert is_deficient(36) is False
def test_is_amicable():
assert is_amicable(173, 129) is False
assert is_amicable(220, 284) is True
assert is_amicable(8756, 8756) is False
|
db93909d185d39c336ea3604515b35ba42ec3c32393eb1bc1f5c4293fc17db05
|
from collections import defaultdict
from sympy import S, Symbol, Tuple
from sympy.core.compatibility import range
from sympy.ntheory import n_order, is_primitive_root, is_quad_residue, \
legendre_symbol, jacobi_symbol, totient, primerange, sqrt_mod, \
primitive_root, quadratic_residues, is_nthpow_residue, nthroot_mod, \
sqrt_mod_iter, mobius, discrete_log
from sympy.ntheory.residue_ntheory import _primitive_root_prime_iter, \
_discrete_log_trial_mul, _discrete_log_shanks_steps, \
_discrete_log_pollard_rho, _discrete_log_pohlig_hellman
from sympy.polys.domains import ZZ
from sympy.utilities.pytest import raises
def test_residue():
assert n_order(2, 13) == 12
assert [n_order(a, 7) for a in range(1, 7)] == \
[1, 3, 6, 3, 6, 2]
assert n_order(5, 17) == 16
assert n_order(17, 11) == n_order(6, 11)
assert n_order(101, 119) == 6
assert n_order(11, (10**50 + 151)**2) == 10000000000000000000000000000000000000000000000030100000000000000000000000000000000000000000000022650
raises(ValueError, lambda: n_order(6, 9))
assert is_primitive_root(2, 7) is False
assert is_primitive_root(3, 8) is False
assert is_primitive_root(11, 14) is False
assert is_primitive_root(12, 17) == is_primitive_root(29, 17)
raises(ValueError, lambda: is_primitive_root(3, 6))
assert [primitive_root(i) for i in range(2, 31)] == [1, 2, 3, 2, 5, 3, \
None, 2, 3, 2, None, 2, 3, None, None, 3, 5, 2, None, None, 7, 5, \
None, 2, 7, 2, None, 2, None]
for p in primerange(3, 100):
it = _primitive_root_prime_iter(p)
assert len(list(it)) == totient(totient(p))
assert primitive_root(97) == 5
assert primitive_root(97**2) == 5
assert primitive_root(40487) == 5
# note that primitive_root(40487) + 40487 = 40492 is a primitive root
# of 40487**2, but it is not the smallest
assert primitive_root(40487**2) == 10
assert primitive_root(82) == 7
p = 10**50 + 151
assert primitive_root(p) == 11
assert primitive_root(2*p) == 11
assert primitive_root(p**2) == 11
raises(ValueError, lambda: primitive_root(-3))
assert is_quad_residue(3, 7) is False
assert is_quad_residue(10, 13) is True
assert is_quad_residue(12364, 139) == is_quad_residue(12364 % 139, 139)
assert is_quad_residue(207, 251) is True
assert is_quad_residue(0, 1) is True
assert is_quad_residue(1, 1) is True
assert is_quad_residue(0, 2) == is_quad_residue(1, 2) is True
assert is_quad_residue(1, 4) is True
assert is_quad_residue(2, 27) is False
assert is_quad_residue(13122380800, 13604889600) is True
assert [j for j in range(14) if is_quad_residue(j, 14)] == \
[0, 1, 2, 4, 7, 8, 9, 11]
raises(ValueError, lambda: is_quad_residue(1.1, 2))
raises(ValueError, lambda: is_quad_residue(2, 0))
assert quadratic_residues(S.One) == [0]
assert quadratic_residues(1) == [0]
assert quadratic_residues(12) == [0, 1, 4, 9]
assert quadratic_residues(12) == [0, 1, 4, 9]
assert quadratic_residues(13) == [0, 1, 3, 4, 9, 10, 12]
assert [len(quadratic_residues(i)) for i in range(1, 20)] == \
[1, 2, 2, 2, 3, 4, 4, 3, 4, 6, 6, 4, 7, 8, 6, 4, 9, 8, 10]
assert list(sqrt_mod_iter(6, 2)) == [0]
assert sqrt_mod(3, 13) == 4
assert sqrt_mod(3, -13) == 4
assert sqrt_mod(6, 23) == 11
assert sqrt_mod(345, 690) == 345
for p in range(3, 100):
d = defaultdict(list)
for i in range(p):
d[pow(i, 2, p)].append(i)
for i in range(1, p):
it = sqrt_mod_iter(i, p)
v = sqrt_mod(i, p, True)
if v:
v = sorted(v)
assert d[i] == v
else:
assert not d[i]
assert sqrt_mod(9, 27, True) == [3, 6, 12, 15, 21, 24]
assert sqrt_mod(9, 81, True) == [3, 24, 30, 51, 57, 78]
assert sqrt_mod(9, 3**5, True) == [3, 78, 84, 159, 165, 240]
assert sqrt_mod(81, 3**4, True) == [0, 9, 18, 27, 36, 45, 54, 63, 72]
assert sqrt_mod(81, 3**5, True) == [9, 18, 36, 45, 63, 72, 90, 99, 117,\
126, 144, 153, 171, 180, 198, 207, 225, 234]
assert sqrt_mod(81, 3**6, True) == [9, 72, 90, 153, 171, 234, 252, 315,\
333, 396, 414, 477, 495, 558, 576, 639, 657, 720]
assert sqrt_mod(81, 3**7, True) == [9, 234, 252, 477, 495, 720, 738, 963,\
981, 1206, 1224, 1449, 1467, 1692, 1710, 1935, 1953, 2178]
for a, p in [(26214400, 32768000000), (26214400, 16384000000),
(262144, 1048576), (87169610025, 163443018796875),
(22315420166400, 167365651248000000)]:
assert pow(sqrt_mod(a, p), 2, p) == a
n = 70
a, p = 5**2*3**n*2**n, 5**6*3**(n+1)*2**(n+2)
it = sqrt_mod_iter(a, p)
for i in range(10):
assert pow(next(it), 2, p) == a
a, p = 5**2*3**n*2**n, 5**6*3**(n+1)*2**(n+3)
it = sqrt_mod_iter(a, p)
for i in range(2):
assert pow(next(it), 2, p) == a
n = 100
a, p = 5**2*3**n*2**n, 5**6*3**(n+1)*2**(n+1)
it = sqrt_mod_iter(a, p)
for i in range(2):
assert pow(next(it), 2, p) == a
assert type(next(sqrt_mod_iter(9, 27))) is int
assert type(next(sqrt_mod_iter(9, 27, ZZ))) is type(ZZ(1))
assert type(next(sqrt_mod_iter(1, 7, ZZ))) is type(ZZ(1))
assert is_nthpow_residue(2, 1, 5)
#issue 10816
assert is_nthpow_residue(1, 0, 1) is False
assert is_nthpow_residue(1, 0, 2) is True
assert is_nthpow_residue(3, 0, 2) is False
assert is_nthpow_residue(0, 1, 8) is True
assert is_nthpow_residue(2, 3, 2) is False
assert is_nthpow_residue(2, 3, 9) is False
assert is_nthpow_residue(3, 5, 30) is True
assert is_nthpow_residue(21, 11, 20) is True
assert is_nthpow_residue(7, 10, 20) is False
assert is_nthpow_residue(5, 10, 20) is True
assert is_nthpow_residue(3, 10, 48) is False
assert is_nthpow_residue(1, 10, 40) is True
assert is_nthpow_residue(3, 10, 24) is False
assert is_nthpow_residue(1, 10, 24) is True
assert is_nthpow_residue(3, 10, 24) is False
assert is_nthpow_residue(2, 10, 48) is False
assert is_nthpow_residue(81, 3, 972) is False
assert is_nthpow_residue(243, 5, 5103) is True
assert is_nthpow_residue(243, 3, 1240029) is False
x = set([pow(i, 56, 1024) for i in range(1024)])
assert set([a for a in range(1024) if is_nthpow_residue(a, 56, 1024)]) == x
x = set([ pow(i, 256, 2048) for i in range(2048)])
assert set([a for a in range(2048) if is_nthpow_residue(a, 256, 2048)]) == x
x = set([ pow(i, 11, 324000) for i in range(1000)])
assert [ is_nthpow_residue(a, 11, 324000) for a in x]
x = set([ pow(i, 17, 22217575536) for i in range(1000)])
assert [ is_nthpow_residue(a, 17, 22217575536) for a in x]
assert is_nthpow_residue(676, 3, 5364)
assert is_nthpow_residue(9, 12, 36)
assert is_nthpow_residue(32, 10, 41)
assert is_nthpow_residue(4, 2, 64)
assert is_nthpow_residue(31, 4, 41)
assert not is_nthpow_residue(2, 2, 5)
assert is_nthpow_residue(8547, 12, 10007)
assert nthroot_mod(29, 31, 74) == 31
assert nthroot_mod(*Tuple(29, 31, 74)) == 31
assert nthroot_mod(1801, 11, 2663) == 44
for a, q, p in [(51922, 2, 203017), (43, 3, 109), (1801, 11, 2663),
(26118163, 1303, 33333347), (1499, 7, 2663), (595, 6, 2663),
(1714, 12, 2663), (28477, 9, 33343)]:
r = nthroot_mod(a, q, p)
assert pow(r, q, p) == a
assert nthroot_mod(11, 3, 109) is None
raises(NotImplementedError, lambda: nthroot_mod(16, 5, 36))
raises(NotImplementedError, lambda: nthroot_mod(9, 16, 36))
for p in primerange(5, 100):
qv = range(3, p, 4)
for q in qv:
d = defaultdict(list)
for i in range(p):
d[pow(i, q, p)].append(i)
for a in range(1, p - 1):
res = nthroot_mod(a, q, p, True)
if d[a]:
assert d[a] == res
else:
assert res is None
assert legendre_symbol(5, 11) == 1
assert legendre_symbol(25, 41) == 1
assert legendre_symbol(67, 101) == -1
assert legendre_symbol(0, 13) == 0
assert legendre_symbol(9, 3) == 0
raises(ValueError, lambda: legendre_symbol(2, 4))
assert jacobi_symbol(25, 41) == 1
assert jacobi_symbol(-23, 83) == -1
assert jacobi_symbol(3, 9) == 0
assert jacobi_symbol(42, 97) == -1
assert jacobi_symbol(3, 5) == -1
assert jacobi_symbol(7, 9) == 1
assert jacobi_symbol(0, 3) == 0
assert jacobi_symbol(0, 1) == 1
assert jacobi_symbol(2, 1) == 1
assert jacobi_symbol(1, 3) == 1
raises(ValueError, lambda: jacobi_symbol(3, 8))
assert mobius(13*7) == 1
assert mobius(1) == 1
assert mobius(13*7*5) == -1
assert mobius(13**2) == 0
raises(ValueError, lambda: mobius(-3))
p = Symbol('p', integer=True, positive=True, prime=True)
x = Symbol('x', positive=True)
i = Symbol('i', integer=True)
assert mobius(p) == -1
raises(TypeError, lambda: mobius(x))
raises(ValueError, lambda: mobius(i))
assert _discrete_log_trial_mul(587, 2**7, 2) == 7
assert _discrete_log_trial_mul(941, 7**18, 7) == 18
assert _discrete_log_trial_mul(389, 3**81, 3) == 81
assert _discrete_log_trial_mul(191, 19**123, 19) == 123
assert _discrete_log_shanks_steps(442879, 7**2, 7) == 2
assert _discrete_log_shanks_steps(874323, 5**19, 5) == 19
assert _discrete_log_shanks_steps(6876342, 7**71, 7) == 71
assert _discrete_log_shanks_steps(2456747, 3**321, 3) == 321
assert _discrete_log_pollard_rho(6013199, 2**6, 2, rseed=0) == 6
assert _discrete_log_pollard_rho(6138719, 2**19, 2, rseed=0) == 19
assert _discrete_log_pollard_rho(36721943, 2**40, 2, rseed=0) == 40
assert _discrete_log_pollard_rho(24567899, 3**333, 3, rseed=0) == 333
raises(ValueError, lambda: _discrete_log_pollard_rho(11, 7, 31, rseed=0))
raises(ValueError, lambda: _discrete_log_pollard_rho(227, 3**7, 5, rseed=0))
assert _discrete_log_pohlig_hellman(98376431, 11**9, 11) == 9
assert _discrete_log_pohlig_hellman(78723213, 11**31, 11) == 31
assert _discrete_log_pohlig_hellman(32942478, 11**98, 11) == 98
assert _discrete_log_pohlig_hellman(14789363, 11**444, 11) == 444
assert discrete_log(587, 2**9, 2) == 9
assert discrete_log(2456747, 3**51, 3) == 51
assert discrete_log(32942478, 11**127, 11) == 127
assert discrete_log(432751500361, 7**324, 7) == 324
args = 5779, 3528, 6215
assert discrete_log(*args) == 687
assert discrete_log(*Tuple(*args)) == 687
|
f950592a9968249c0a7301b335e53b975652f58145d704a2cc277a13f51be0a8
|
from sympy import S
from sympy.combinatorics.fp_groups import (FpGroup, low_index_subgroups,
reidemeister_presentation, FpSubgroup,
simplify_presentation)
from sympy.combinatorics.free_groups import (free_group, FreeGroup)
from sympy.utilities.pytest import slow
"""
References
==========
[1] Holt, D., Eick, B., O'Brien, E.
"Handbook of Computational Group Theory"
[2] John J. Cannon; Lucien A. Dimino; George Havas; Jane M. Watson
Mathematics of Computation, Vol. 27, No. 123. (Jul., 1973), pp. 463-490.
"Implementation and Analysis of the Todd-Coxeter Algorithm"
[3] PROC. SECOND INTERNAT. CONF. THEORY OF GROUPS, CANBERRA 1973,
pp. 347-356. "A Reidemeister-Schreier program" by George Havas.
http://staff.itee.uq.edu.au/havas/1973cdhw.pdf
"""
def test_low_index_subgroups():
F, x, y = free_group("x, y")
# Example 5.10 from [1] Pg. 194
f = FpGroup(F, [x**2, y**3, (x*y)**4])
L = low_index_subgroups(f, 4)
t1 = [[[0, 0, 0, 0]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 3, 3]],
[[0, 0, 1, 2], [2, 2, 2, 0], [1, 1, 0, 1]],
[[1, 1, 0, 0], [0, 0, 1, 1]]]
for i in range(len(t1)):
assert L[i].table == t1[i]
f = FpGroup(F, [x**2, y**3, (x*y)**7])
L = low_index_subgroups(f, 15)
t2 = [[[0, 0, 0, 0]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[4, 4, 5, 3], [6, 6, 3, 4], [5, 5, 6, 6]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[6, 6, 5, 3], [5, 5, 3, 4], [4, 4, 6, 6]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[6, 6, 5, 3], [7, 7, 3, 4], [4, 4, 8, 9], [5, 5, 10, 11],
[11, 11, 9, 6], [9, 9, 6, 8], [12, 12, 11, 7], [8, 8, 7, 10],
[10, 10, 13, 14], [14, 14, 14, 12], [13, 13, 12, 13]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[6, 6, 5, 3], [7, 7, 3, 4], [4, 4, 8, 9], [5, 5, 10, 11],
[11, 11, 9, 6], [12, 12, 6, 8], [10, 10, 11, 7], [8, 8, 7, 10],
[9, 9, 13, 14], [14, 14, 14, 12], [13, 13, 12, 13]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[6, 6, 5, 3], [7, 7, 3, 4], [4, 4, 8, 9], [5, 5, 10, 11],
[11, 11, 9, 6], [12, 12, 6, 8], [13, 13, 11, 7], [8, 8, 7, 10],
[9, 9, 12, 12], [10, 10, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 3, 3], [2, 2, 5, 6]
, [7, 7, 6, 4], [8, 8, 4, 5], [5, 5, 8, 9], [6, 6, 9, 7],
[10, 10, 7, 8], [9, 9, 11, 12], [11, 11, 12, 10], [13, 13, 10, 11],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 3, 3], [2, 2, 5, 6]
, [7, 7, 6, 4], [8, 8, 4, 5], [5, 5, 8, 9], [6, 6, 9, 7],
[10, 10, 7, 8], [9, 9, 11, 12], [13, 13, 12, 10], [12, 12, 10, 11],
[11, 11, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 4, 4]
, [7, 7, 6, 3], [8, 8, 3, 5], [5, 5, 8, 9], [6, 6, 9, 7],
[10, 10, 7, 8], [9, 9, 11, 12], [13, 13, 12, 10], [12, 12, 10, 11],
[11, 11, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [5, 5, 6, 3], [9, 9, 3, 5], [10, 10, 8, 4], [8, 8, 4, 7],
[6, 6, 10, 11], [7, 7, 11, 9], [12, 12, 9, 10], [11, 11, 13, 14],
[14, 14, 14, 12], [13, 13, 12, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [6, 6, 6, 3], [5, 5, 3, 5], [8, 8, 8, 4], [7, 7, 4, 7]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [6, 6, 3, 5], [10, 10, 8, 4], [11, 11, 4, 7],
[5, 5, 10, 12], [7, 7, 12, 9], [8, 8, 11, 11], [13, 13, 9, 10],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [6, 6, 3, 5], [10, 10, 8, 4], [11, 11, 4, 7],
[5, 5, 12, 11], [7, 7, 10, 10], [8, 8, 9, 12], [13, 13, 11, 9],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [10, 10, 3, 5], [7, 7, 8, 4], [11, 11, 4, 7],
[5, 5, 9, 9], [6, 6, 11, 12], [8, 8, 12, 10], [13, 13, 10, 11],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [10, 10, 3, 5], [7, 7, 8, 4], [11, 11, 4, 7],
[5, 5, 12, 11], [6, 6, 10, 10], [8, 8, 9, 12], [13, 13, 11, 9],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [10, 10, 3, 5], [11, 11, 8, 4], [12, 12, 4, 7],
[5, 5, 9, 9], [6, 6, 12, 13], [7, 7, 11, 11], [8, 8, 13, 10],
[13, 13, 10, 12]],
[[1, 1, 0, 0], [0, 0, 2, 3], [4, 4, 3, 1], [5, 5, 1, 2], [2, 2, 4, 4]
, [3, 3, 6, 7], [7, 7, 7, 5], [6, 6, 5, 6]]]
for i in range(len(t2)):
assert L[i].table == t2[i]
f = FpGroup(F, [x**2, y**3, (x*y)**7])
L = low_index_subgroups(f, 10, [x])
t3 = [[[0, 0, 0, 0]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5], [4, 4, 5, 3],
[6, 6, 3, 4], [5, 5, 6, 6]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5], [6, 6, 5, 3],
[5, 5, 3, 4], [4, 4, 6, 6]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8],
[6, 6, 6, 3], [5, 5, 3, 5], [8, 8, 8, 4], [7, 7, 4, 7]]]
for i in range(len(t3)):
assert L[i].table == t3[i]
def test_subgroup_presentations():
F, x, y = free_group("x, y")
f = FpGroup(F, [x**3, y**5, (x*y)**2])
H = [x*y, x**-1*y**-1*x*y*x]
p1 = reidemeister_presentation(f, H)
assert str(p1) == "((y_1, y_2), (y_1**2, y_2**3, y_2*y_1*y_2*y_1*y_2*y_1))"
H = f.subgroup(H)
assert (H.generators, H.relators) == p1
f = FpGroup(F, [x**3, y**3, (x*y)**3])
H = [x*y, x*y**-1]
p2 = reidemeister_presentation(f, H)
assert str(p2) == "((x_0, y_0), (x_0**3, y_0**3, x_0*y_0*x_0*y_0*x_0*y_0))"
f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3])
H = [x]
p3 = reidemeister_presentation(f, H)
assert str(p3) == "((x_0,), (x_0**4,))"
f = FpGroup(F, [x**3*y**-3, (x*y)**3, (x*y**-1)**2])
H = [x]
p4 = reidemeister_presentation(f, H)
assert str(p4) == "((x_0,), (x_0**6,))"
# this presentation can be improved, the most simplified form
# of presentation is <a, b | a^11, b^2, (a*b)^3, (a^4*b*a^-5*b)^2>
# See [2] Pg 474 group PSL_2(11)
# This is the group PSL_2(11)
F, a, b, c = free_group("a, b, c")
f = FpGroup(F, [a**11, b**5, c**4, (b*c**2)**2, (a*b*c)**3, (a**4*c**2)**3, b**2*c**-1*b**-1*c, a**4*b**-1*a**-1*b])
H = [a, b, c**2]
gens, rels = reidemeister_presentation(f, H)
assert str(gens) == "(b_1, c_3)"
assert len(rels) == 18
@slow
def test_order():
F, x, y = free_group("x, y")
f = FpGroup(F, [x**4, y**2, x*y*x**-1*y])
assert f.order() == 8
f = FpGroup(F, [x*y*x**-1*y**-1, y**2])
assert f.order() == S.Infinity
F, a, b, c = free_group("a, b, c")
f = FpGroup(F, [a**250, b**2, c*b*c**-1*b, c**4, c**-1*a**-1*c*a, a**-1*b**-1*a*b])
assert f.order() == 2000
F, x = free_group("x")
f = FpGroup(F, [])
assert f.order() == S.Infinity
f = FpGroup(free_group('')[0], [])
assert f.order() == 1
def test_fp_subgroup():
def _test_subgroup(K, T, S):
_gens = T(K.generators)
assert all(elem in S for elem in _gens)
assert T.is_injective()
assert T.image().order() == S.order()
F, x, y = free_group("x, y")
f = FpGroup(F, [x**4, y**2, x*y*x**-1*y])
S = FpSubgroup(f, [x*y])
assert (x*y)**-3 in S
K, T = f.subgroup([x*y], homomorphism=True)
assert T(K.generators) == [y*x**-1]
_test_subgroup(K, T, S)
S = FpSubgroup(f, [x**-1*y*x])
assert x**-1*y**4*x in S
assert x**-1*y**4*x**2 not in S
K, T = f.subgroup([x**-1*y*x], homomorphism=True)
assert T(K.generators[0]**3) == y**3
_test_subgroup(K, T, S)
f = FpGroup(F, [x**3, y**5, (x*y)**2])
H = [x*y, x**-1*y**-1*x*y*x]
K, T = f.subgroup(H, homomorphism=True)
S = FpSubgroup(f, H)
_test_subgroup(K, T, S)
def test_permutation_methods():
from sympy.combinatorics.fp_groups import FpSubgroup
F, x, y = free_group("x, y")
# DihedralGroup(8)
G = FpGroup(F, [x**2, y**8, x*y*x**-1*y])
T = G._to_perm_group()[1]
assert T.is_isomorphism()
assert G.center() == [y**4]
# DiheadralGroup(4)
G = FpGroup(F, [x**2, y**4, x*y*x**-1*y])
S = FpSubgroup(G, G.normal_closure([x]))
assert x in S
assert y**-1*x*y in S
# Z_5xZ_4
G = FpGroup(F, [x*y*x**-1*y**-1, y**5, x**4])
assert G.is_abelian
assert G.is_solvable
# AlternatingGroup(5)
G = FpGroup(F, [x**3, y**2, (x*y)**5])
assert not G.is_solvable
# AlternatingGroup(4)
G = FpGroup(F, [x**3, y**2, (x*y)**3])
assert len(G.derived_series()) == 3
S = FpSubgroup(G, G.derived_subgroup())
assert S.order() == 4
def test_simplify_presentation():
# ref #16083
G = simplify_presentation(FpGroup(FreeGroup([]), []))
assert not G.generators
assert not G.relators
|
c34f44a1d69da26c4e831a2fd1917116eeed854efd732cee15c661eb71db0b2c
|
from sympy.combinatorics.generators import symmetric, cyclic, alternating, \
dihedral, rubik
from sympy.combinatorics.permutations import Permutation
from sympy.utilities.pytest import raises
def test_generators():
assert list(cyclic(6)) == [
Permutation([0, 1, 2, 3, 4, 5]),
Permutation([1, 2, 3, 4, 5, 0]),
Permutation([2, 3, 4, 5, 0, 1]),
Permutation([3, 4, 5, 0, 1, 2]),
Permutation([4, 5, 0, 1, 2, 3]),
Permutation([5, 0, 1, 2, 3, 4])]
assert list(cyclic(10)) == [
Permutation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]),
Permutation([1, 2, 3, 4, 5, 6, 7, 8, 9, 0]),
Permutation([2, 3, 4, 5, 6, 7, 8, 9, 0, 1]),
Permutation([3, 4, 5, 6, 7, 8, 9, 0, 1, 2]),
Permutation([4, 5, 6, 7, 8, 9, 0, 1, 2, 3]),
Permutation([5, 6, 7, 8, 9, 0, 1, 2, 3, 4]),
Permutation([6, 7, 8, 9, 0, 1, 2, 3, 4, 5]),
Permutation([7, 8, 9, 0, 1, 2, 3, 4, 5, 6]),
Permutation([8, 9, 0, 1, 2, 3, 4, 5, 6, 7]),
Permutation([9, 0, 1, 2, 3, 4, 5, 6, 7, 8])]
assert list(alternating(4)) == [
Permutation([0, 1, 2, 3]),
Permutation([0, 2, 3, 1]),
Permutation([0, 3, 1, 2]),
Permutation([1, 0, 3, 2]),
Permutation([1, 2, 0, 3]),
Permutation([1, 3, 2, 0]),
Permutation([2, 0, 1, 3]),
Permutation([2, 1, 3, 0]),
Permutation([2, 3, 0, 1]),
Permutation([3, 0, 2, 1]),
Permutation([3, 1, 0, 2]),
Permutation([3, 2, 1, 0])]
assert list(symmetric(3)) == [
Permutation([0, 1, 2]),
Permutation([0, 2, 1]),
Permutation([1, 0, 2]),
Permutation([1, 2, 0]),
Permutation([2, 0, 1]),
Permutation([2, 1, 0])]
assert list(symmetric(4)) == [
Permutation([0, 1, 2, 3]),
Permutation([0, 1, 3, 2]),
Permutation([0, 2, 1, 3]),
Permutation([0, 2, 3, 1]),
Permutation([0, 3, 1, 2]),
Permutation([0, 3, 2, 1]),
Permutation([1, 0, 2, 3]),
Permutation([1, 0, 3, 2]),
Permutation([1, 2, 0, 3]),
Permutation([1, 2, 3, 0]),
Permutation([1, 3, 0, 2]),
Permutation([1, 3, 2, 0]),
Permutation([2, 0, 1, 3]),
Permutation([2, 0, 3, 1]),
Permutation([2, 1, 0, 3]),
Permutation([2, 1, 3, 0]),
Permutation([2, 3, 0, 1]),
Permutation([2, 3, 1, 0]),
Permutation([3, 0, 1, 2]),
Permutation([3, 0, 2, 1]),
Permutation([3, 1, 0, 2]),
Permutation([3, 1, 2, 0]),
Permutation([3, 2, 0, 1]),
Permutation([3, 2, 1, 0])]
assert list(dihedral(1)) == [
Permutation([0, 1]), Permutation([1, 0])]
assert list(dihedral(2)) == [
Permutation([0, 1, 2, 3]),
Permutation([1, 0, 3, 2]),
Permutation([2, 3, 0, 1]),
Permutation([3, 2, 1, 0])]
assert list(dihedral(3)) == [
Permutation([0, 1, 2]),
Permutation([2, 1, 0]),
Permutation([1, 2, 0]),
Permutation([0, 2, 1]),
Permutation([2, 0, 1]),
Permutation([1, 0, 2])]
assert list(dihedral(5)) == [
Permutation([0, 1, 2, 3, 4]),
Permutation([4, 3, 2, 1, 0]),
Permutation([1, 2, 3, 4, 0]),
Permutation([0, 4, 3, 2, 1]),
Permutation([2, 3, 4, 0, 1]),
Permutation([1, 0, 4, 3, 2]),
Permutation([3, 4, 0, 1, 2]),
Permutation([2, 1, 0, 4, 3]),
Permutation([4, 0, 1, 2, 3]),
Permutation([3, 2, 1, 0, 4])]
raises(ValueError, lambda: rubik(1))
|
23a5de07666048a58a235ad8199d8c380eba8c3f6fd6d4bce9fe62384bf990ff
|
from sympy.combinatorics.graycode import (GrayCode, bin_to_gray,
random_bitstring, get_subset_from_bitstring, graycode_subsets,
gray_to_bin)
from sympy.utilities.pytest import raises
def test_graycode():
g = GrayCode(2)
got = []
for i in g.generate_gray():
if i.startswith('0'):
g.skip()
got.append(i)
assert got == '00 11 10'.split()
a = GrayCode(6)
assert a.current == '0'*6
assert a.rank == 0
assert len(list(a.generate_gray())) == 64
codes = ['011001', '011011', '011010',
'011110', '011111', '011101', '011100', '010100', '010101', '010111',
'010110', '010010', '010011', '010001', '010000', '110000', '110001',
'110011', '110010', '110110', '110111', '110101', '110100', '111100',
'111101', '111111', '111110', '111010', '111011', '111001', '111000',
'101000', '101001', '101011', '101010', '101110', '101111', '101101',
'101100', '100100', '100101', '100111', '100110', '100010', '100011',
'100001', '100000']
assert list(a.generate_gray(start='011001')) == codes
assert list(
a.generate_gray(rank=GrayCode(6, start='011001').rank)) == codes
assert a.next().current == '000001'
assert a.next(2).current == '000011'
assert a.next(-1).current == '100000'
a = GrayCode(5, start='10010')
assert a.rank == 28
a = GrayCode(6, start='101000')
assert a.rank == 48
assert GrayCode(6, rank=4).current == '000110'
assert GrayCode(6, rank=4).rank == 4
assert [GrayCode(4, start=s).rank for s in
GrayCode(4).generate_gray()] == [0, 1, 2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15]
a = GrayCode(15, rank=15)
assert a.current == '000000000001000'
assert bin_to_gray('111') == '100'
a = random_bitstring(5)
assert type(a) is str
assert len(a) == 5
assert all(i in ['0', '1'] for i in a)
assert get_subset_from_bitstring(
['a', 'b', 'c', 'd'], '0011') == ['c', 'd']
assert get_subset_from_bitstring('abcd', '1001') == ['a', 'd']
assert list(graycode_subsets(['a', 'b', 'c'])) == \
[[], ['c'], ['b', 'c'], ['b'], ['a', 'b'], ['a', 'b', 'c'],
['a', 'c'], ['a']]
raises(ValueError, lambda: GrayCode(0))
raises(ValueError, lambda: GrayCode(2.2))
raises(ValueError, lambda: GrayCode(2, start=[1, 1, 0]))
raises(ValueError, lambda: GrayCode(2, rank=2.5))
raises(ValueError, lambda: get_subset_from_bitstring(['c', 'a', 'c'], '1100'))
raises(ValueError, lambda: list(GrayCode(3).generate_gray(start="1111")))
def test_live_issue_117():
assert bin_to_gray('0100') == '0110'
assert bin_to_gray('0101') == '0111'
for bits in ('0100', '0101'):
assert gray_to_bin(bin_to_gray(bits)) == bits
|
3931ebee36708a95b611f52c1dd8148e0567fdc6d47f1dcc9e4117569e11e378
|
from sympy.combinatorics.named_groups import (SymmetricGroup, CyclicGroup,
DihedralGroup, AlternatingGroup,
AbelianGroup, RubikGroup)
from sympy.utilities.pytest import raises
def test_SymmetricGroup():
G = SymmetricGroup(5)
elements = list(G.generate())
assert (G.generators[0]).size == 5
assert len(elements) == 120
assert G.is_solvable is False
assert G.is_abelian is False
assert G.is_nilpotent is False
assert G.is_transitive() is True
H = SymmetricGroup(1)
assert H.order() == 1
L = SymmetricGroup(2)
assert L.order() == 2
def test_CyclicGroup():
G = CyclicGroup(10)
elements = list(G.generate())
assert len(elements) == 10
assert (G.derived_subgroup()).order() == 1
assert G.is_abelian is True
assert G.is_solvable is True
assert G.is_nilpotent is True
H = CyclicGroup(1)
assert H.order() == 1
L = CyclicGroup(2)
assert L.order() == 2
def test_DihedralGroup():
G = DihedralGroup(6)
elements = list(G.generate())
assert len(elements) == 12
assert G.is_transitive() is True
assert G.is_abelian is False
assert G.is_solvable is True
assert G.is_nilpotent is False
H = DihedralGroup(1)
assert H.order() == 2
L = DihedralGroup(2)
assert L.order() == 4
assert L.is_abelian is True
assert L.is_nilpotent is True
def test_AlternatingGroup():
G = AlternatingGroup(5)
elements = list(G.generate())
assert len(elements) == 60
assert [perm.is_even for perm in elements] == [True]*60
H = AlternatingGroup(1)
assert H.order() == 1
L = AlternatingGroup(2)
assert L.order() == 1
def test_AbelianGroup():
A = AbelianGroup(3, 3, 3)
assert A.order() == 27
assert A.is_abelian is True
def test_RubikGroup():
raises(ValueError, lambda: RubikGroup(1))
|
cf30f5f0901bc7ae3b8d9246c7b33b1902901ac2bd74cb96e0508c5f8c138e0b
|
from sympy.combinatorics.prufer import Prufer
from sympy.utilities.pytest import raises
def test_prufer():
# number of nodes is optional
assert Prufer([[0, 1], [0, 2], [0, 3], [0, 4]], 5).nodes == 5
assert Prufer([[0, 1], [0, 2], [0, 3], [0, 4]]).nodes == 5
a = Prufer([[0, 1], [0, 2], [0, 3], [0, 4]])
assert a.rank == 0
assert a.nodes == 5
assert a.prufer_repr == [0, 0, 0]
a = Prufer([[2, 4], [1, 4], [1, 3], [0, 5], [0, 4]])
assert a.rank == 924
assert a.nodes == 6
assert a.tree_repr == [[2, 4], [1, 4], [1, 3], [0, 5], [0, 4]]
assert a.prufer_repr == [4, 1, 4, 0]
assert Prufer.edges([0, 1, 2, 3], [1, 4, 5], [1, 4, 6]) == \
([[0, 1], [1, 2], [1, 4], [2, 3], [4, 5], [4, 6]], 7)
assert Prufer([0]*4).size == Prufer([6]*4).size == 1296
# accept iterables but convert to list of lists
tree = [(0, 1), (1, 5), (0, 3), (0, 2), (2, 6), (4, 7), (2, 4)]
tree_lists = [list(t) for t in tree]
assert Prufer(tree).tree_repr == tree_lists
assert sorted(Prufer(set(tree)).tree_repr) == sorted(tree_lists)
raises(ValueError, lambda: Prufer([[1, 2], [3, 4]])) # 0 is missing
raises(ValueError, lambda: Prufer([[2, 3], [3, 4]])) # 0, 1 are missing
assert Prufer(*Prufer.edges([1, 2], [3, 4])).prufer_repr == [1, 3]
raises(ValueError, lambda: Prufer.edges(
[1, 3], [3, 4])) # a broken tree but edges doesn't care
raises(ValueError, lambda: Prufer.edges([1, 2], [5, 6]))
raises(ValueError, lambda: Prufer([[]]))
a = Prufer([[0, 1], [0, 2], [0, 3]])
b = a.next()
assert b.tree_repr == [[0, 2], [0, 1], [1, 3]]
assert b.rank == 1
def test_round_trip():
def doit(t, b):
e, n = Prufer.edges(*t)
t = Prufer(e, n)
a = sorted(t.tree_repr)
b = [i - 1 for i in b]
assert t.prufer_repr == b
assert sorted(Prufer(b).tree_repr) == a
assert Prufer.unrank(t.rank, n).prufer_repr == b
doit([[1, 2]], [])
doit([[2, 1, 3]], [1])
doit([[1, 3, 2]], [3])
doit([[1, 2, 3]], [2])
doit([[2, 1, 4], [1, 3]], [1, 1])
doit([[3, 2, 1, 4]], [2, 1])
doit([[3, 2, 1], [2, 4]], [2, 2])
doit([[1, 3, 2, 4]], [3, 2])
doit([[1, 4, 2, 3]], [4, 2])
doit([[3, 1, 4, 2]], [4, 1])
doit([[4, 2, 1, 3]], [1, 2])
doit([[1, 2, 4, 3]], [2, 4])
doit([[1, 3, 4, 2]], [3, 4])
doit([[2, 4, 1], [4, 3]], [4, 4])
doit([[1, 2, 3, 4]], [2, 3])
doit([[2, 3, 1], [3, 4]], [3, 3])
doit([[1, 4, 3, 2]], [4, 3])
doit([[2, 1, 4, 3]], [1, 4])
doit([[2, 1, 3, 4]], [1, 3])
doit([[6, 2, 1, 4], [1, 3, 5, 8], [3, 7]], [1, 2, 1, 3, 3, 5])
|
50e59a2da5b52a67377043f0208c0c420a825fed110e3120dd5efa3860357908
|
from sympy.combinatorics.fp_groups import FpGroup
from sympy.combinatorics.free_groups import free_group
from sympy.utilities.pytest import raises
def test_rewriting():
F, a, b = free_group("a, b")
G = FpGroup(F, [a*b*a**-1*b**-1])
a, b = G.generators
R = G._rewriting_system
assert R.is_confluent
assert G.reduce(b**-1*a) == a*b**-1
assert G.reduce(b**3*a**4*b**-2*a) == a**5*b
assert G.equals(b**2*a**-1*b, b**4*a**-1*b**-1)
assert R.reduce_using_automaton(b*a*a**2*b**-1) == a**3
assert R.reduce_using_automaton(b**3*a**4*b**-2*a) == a**5*b
assert R.reduce_using_automaton(b**-1*a) == a*b**-1
G = FpGroup(F, [a**3, b**3, (a*b)**2])
R = G._rewriting_system
R.make_confluent()
# R._is_confluent should be set to True after
# a successful run of make_confluent
assert R.is_confluent
# but also the system should actually be confluent
assert R._check_confluence()
assert G.reduce(b*a**-1*b**-1*a**3*b**4*a**-1*b**-15) == a**-1*b**-1
# check for automaton reduction
assert R.reduce_using_automaton(b*a**-1*b**-1*a**3*b**4*a**-1*b**-15) == a**-1*b**-1
G = FpGroup(F, [a**2, b**3, (a*b)**4])
R = G._rewriting_system
assert G.reduce(a**2*b**-2*a**2*b) == b**-1
assert R.reduce_using_automaton(a**2*b**-2*a**2*b) == b**-1
assert G.reduce(a**3*b**-2*a**2*b) == a**-1*b**-1
assert R.reduce_using_automaton(a**3*b**-2*a**2*b) == a**-1*b**-1
# Check after adding a rule
R.add_rule(a**2, b)
assert R.reduce_using_automaton(a**2*b**-2*a**2*b) == b**-1
assert R.reduce_using_automaton(a**4*b**-2*a**2*b**3) == b
R.set_max(15)
raises(RuntimeError, lambda: R.add_rule(a**-3, b))
R.set_max(20)
R.add_rule(a**-3, b)
assert R.add_rule(a, a) == set()
|
1d488ce218fd4840d9c70d33858b2362d135341e3391dbc061d3a97d407bd9a8
|
from sympy.core.compatibility import range
from sympy.combinatorics.perm_groups import (PermutationGroup,
_orbit_transversal)
from sympy.combinatorics.named_groups import SymmetricGroup, CyclicGroup,\
DihedralGroup, AlternatingGroup, AbelianGroup, RubikGroup
from sympy.combinatorics.permutations import Permutation
from sympy.utilities.pytest import skip, XFAIL
from sympy.combinatorics.generators import rubik_cube_generators
from sympy.combinatorics.polyhedron import tetrahedron as Tetra, cube
from sympy.combinatorics.testutil import _verify_bsgs, _verify_centralizer,\
_verify_normal_closure
from sympy.utilities.pytest import raises, slow
rmul = Permutation.rmul
def test_has():
a = Permutation([1, 0])
G = PermutationGroup([a])
assert G.is_abelian
a = Permutation([2, 0, 1])
b = Permutation([2, 1, 0])
G = PermutationGroup([a, b])
assert not G.is_abelian
G = PermutationGroup([a])
assert G.has(a)
assert not G.has(b)
a = Permutation([2, 0, 1, 3, 4, 5])
b = Permutation([0, 2, 1, 3, 4])
assert PermutationGroup(a, b).degree == \
PermutationGroup(a, b).degree == 6
def test_generate():
a = Permutation([1, 0])
g = list(PermutationGroup([a]).generate())
assert g == [Permutation([0, 1]), Permutation([1, 0])]
assert len(list(PermutationGroup(Permutation((0, 1))).generate())) == 1
g = PermutationGroup([a]).generate(method='dimino')
assert list(g) == [Permutation([0, 1]), Permutation([1, 0])]
a = Permutation([2, 0, 1])
b = Permutation([2, 1, 0])
G = PermutationGroup([a, b])
g = G.generate()
v1 = [p.array_form for p in list(g)]
v1.sort()
assert v1 == [[0, 1, 2], [0, 2, 1], [1, 0, 2], [1, 2, 0], [2, 0,
1], [2, 1, 0]]
v2 = list(G.generate(method='dimino', af=True))
assert v1 == sorted(v2)
a = Permutation([2, 0, 1, 3, 4, 5])
b = Permutation([2, 1, 3, 4, 5, 0])
g = PermutationGroup([a, b]).generate(af=True)
assert len(list(g)) == 360
def test_order():
a = Permutation([2, 0, 1, 3, 4, 5, 6, 7, 8, 9])
b = Permutation([2, 1, 3, 4, 5, 6, 7, 8, 9, 0])
g = PermutationGroup([a, b])
assert g.order() == 1814400
assert PermutationGroup().order() == 1
def test_equality():
p_1 = Permutation(0, 1, 3)
p_2 = Permutation(0, 2, 3)
p_3 = Permutation(0, 1, 2)
p_4 = Permutation(0, 1, 3)
g_1 = PermutationGroup(p_1, p_2)
g_2 = PermutationGroup(p_3, p_4)
g_3 = PermutationGroup(p_2, p_1)
assert g_1 == g_2
assert g_1.generators != g_2.generators
assert g_1 == g_3
def test_stabilizer():
S = SymmetricGroup(2)
H = S.stabilizer(0)
assert H.generators == [Permutation(1)]
a = Permutation([2, 0, 1, 3, 4, 5])
b = Permutation([2, 1, 3, 4, 5, 0])
G = PermutationGroup([a, b])
G0 = G.stabilizer(0)
assert G0.order() == 60
gens_cube = [[1, 3, 5, 7, 0, 2, 4, 6], [1, 3, 0, 2, 5, 7, 4, 6]]
gens = [Permutation(p) for p in gens_cube]
G = PermutationGroup(gens)
G2 = G.stabilizer(2)
assert G2.order() == 6
G2_1 = G2.stabilizer(1)
v = list(G2_1.generate(af=True))
assert v == [[0, 1, 2, 3, 4, 5, 6, 7], [3, 1, 2, 0, 7, 5, 6, 4]]
gens = (
(1, 2, 0, 4, 5, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19),
(0, 1, 2, 3, 4, 5, 19, 6, 8, 9, 10, 11, 12, 13, 14,
15, 16, 7, 17, 18),
(0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 16, 11, 12, 13, 14, 15, 8, 17, 10, 19))
gens = [Permutation(p) for p in gens]
G = PermutationGroup(gens)
G2 = G.stabilizer(2)
assert G2.order() == 181440
S = SymmetricGroup(3)
assert [G.order() for G in S.basic_stabilizers] == [6, 2]
def test_center():
# the center of the dihedral group D_n is of order 2 for even n
for i in (4, 6, 10):
D = DihedralGroup(i)
assert (D.center()).order() == 2
# the center of the dihedral group D_n is of order 1 for odd n>2
for i in (3, 5, 7):
D = DihedralGroup(i)
assert (D.center()).order() == 1
# the center of an abelian group is the group itself
for i in (2, 3, 5):
for j in (1, 5, 7):
for k in (1, 1, 11):
G = AbelianGroup(i, j, k)
assert G.center().is_subgroup(G)
# the center of a nonabelian simple group is trivial
for i in(1, 5, 9):
A = AlternatingGroup(i)
assert (A.center()).order() == 1
# brute-force verifications
D = DihedralGroup(5)
A = AlternatingGroup(3)
C = CyclicGroup(4)
G.is_subgroup(D*A*C)
assert _verify_centralizer(G, G)
def test_centralizer():
# the centralizer of the trivial group is the entire group
S = SymmetricGroup(2)
assert S.centralizer(Permutation(list(range(2)))).is_subgroup(S)
A = AlternatingGroup(5)
assert A.centralizer(Permutation(list(range(5)))).is_subgroup(A)
# a centralizer in the trivial group is the trivial group itself
triv = PermutationGroup([Permutation([0, 1, 2, 3])])
D = DihedralGroup(4)
assert triv.centralizer(D).is_subgroup(triv)
# brute-force verifications for centralizers of groups
for i in (4, 5, 6):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
C = CyclicGroup(i)
D = DihedralGroup(i)
for gp in (S, A, C, D):
for gp2 in (S, A, C, D):
if not gp2.is_subgroup(gp):
assert _verify_centralizer(gp, gp2)
# verify the centralizer for all elements of several groups
S = SymmetricGroup(5)
elements = list(S.generate_dimino())
for element in elements:
assert _verify_centralizer(S, element)
A = AlternatingGroup(5)
elements = list(A.generate_dimino())
for element in elements:
assert _verify_centralizer(A, element)
D = DihedralGroup(7)
elements = list(D.generate_dimino())
for element in elements:
assert _verify_centralizer(D, element)
# verify centralizers of small groups within small groups
small = []
for i in (1, 2, 3):
small.append(SymmetricGroup(i))
small.append(AlternatingGroup(i))
small.append(DihedralGroup(i))
small.append(CyclicGroup(i))
for gp in small:
for gp2 in small:
if gp.degree == gp2.degree:
assert _verify_centralizer(gp, gp2)
def test_coset_rank():
gens_cube = [[1, 3, 5, 7, 0, 2, 4, 6], [1, 3, 0, 2, 5, 7, 4, 6]]
gens = [Permutation(p) for p in gens_cube]
G = PermutationGroup(gens)
i = 0
for h in G.generate(af=True):
rk = G.coset_rank(h)
assert rk == i
h1 = G.coset_unrank(rk, af=True)
assert h == h1
i += 1
assert G.coset_unrank(48) == None
assert G.coset_unrank(G.coset_rank(gens[0])) == gens[0]
def test_coset_factor():
a = Permutation([0, 2, 1])
G = PermutationGroup([a])
c = Permutation([2, 1, 0])
assert not G.coset_factor(c)
assert G.coset_rank(c) is None
a = Permutation([2, 0, 1, 3, 4, 5])
b = Permutation([2, 1, 3, 4, 5, 0])
g = PermutationGroup([a, b])
assert g.order() == 360
d = Permutation([1, 0, 2, 3, 4, 5])
assert not g.coset_factor(d.array_form)
assert not g.contains(d)
assert Permutation(2) in G
c = Permutation([1, 0, 2, 3, 5, 4])
v = g.coset_factor(c, True)
tr = g.basic_transversals
p = Permutation.rmul(*[tr[i][v[i]] for i in range(len(g.base))])
assert p == c
v = g.coset_factor(c)
p = Permutation.rmul(*v)
assert p == c
assert g.contains(c)
G = PermutationGroup([Permutation([2, 1, 0])])
p = Permutation([1, 0, 2])
assert G.coset_factor(p) == []
def test_orbits():
a = Permutation([2, 0, 1])
b = Permutation([2, 1, 0])
g = PermutationGroup([a, b])
assert g.orbit(0) == {0, 1, 2}
assert g.orbits() == [{0, 1, 2}]
assert g.is_transitive() and g.is_transitive(strict=False)
assert g.orbit_transversal(0) == \
[Permutation(
[0, 1, 2]), Permutation([2, 0, 1]), Permutation([1, 2, 0])]
assert g.orbit_transversal(0, True) == \
[(0, Permutation([0, 1, 2])), (2, Permutation([2, 0, 1])),
(1, Permutation([1, 2, 0]))]
G = DihedralGroup(6)
transversal, slps = _orbit_transversal(G.degree, G.generators, 0, True, slp=True)
for i, t in transversal:
slp = slps[i]
w = G.identity
for s in slp:
w = G.generators[s]*w
assert w == t
a = Permutation(list(range(1, 100)) + [0])
G = PermutationGroup([a])
assert [min(o) for o in G.orbits()] == [0]
G = PermutationGroup(rubik_cube_generators())
assert [min(o) for o in G.orbits()] == [0, 1]
assert not G.is_transitive() and not G.is_transitive(strict=False)
G = PermutationGroup([Permutation(0, 1, 3), Permutation(3)(0, 1)])
assert not G.is_transitive() and G.is_transitive(strict=False)
assert PermutationGroup(
Permutation(3)).is_transitive(strict=False) is False
def test_is_normal():
gens_s5 = [Permutation(p) for p in [[1, 2, 3, 4, 0], [2, 1, 4, 0, 3]]]
G1 = PermutationGroup(gens_s5)
assert G1.order() == 120
gens_a5 = [Permutation(p) for p in [[1, 0, 3, 2, 4], [2, 1, 4, 3, 0]]]
G2 = PermutationGroup(gens_a5)
assert G2.order() == 60
assert G2.is_normal(G1)
gens3 = [Permutation(p) for p in [[2, 1, 3, 0, 4], [1, 2, 0, 3, 4]]]
G3 = PermutationGroup(gens3)
assert not G3.is_normal(G1)
assert G3.order() == 12
G4 = G1.normal_closure(G3.generators)
assert G4.order() == 60
gens5 = [Permutation(p) for p in [[1, 2, 3, 0, 4], [1, 2, 0, 3, 4]]]
G5 = PermutationGroup(gens5)
assert G5.order() == 24
G6 = G1.normal_closure(G5.generators)
assert G6.order() == 120
assert G1.is_subgroup(G6)
assert not G1.is_subgroup(G4)
assert G2.is_subgroup(G4)
I5 = PermutationGroup(Permutation(4))
assert I5.is_normal(G5)
assert I5.is_normal(G6, strict=False)
p1 = Permutation([1, 0, 2, 3, 4])
p2 = Permutation([0, 1, 2, 4, 3])
p3 = Permutation([3, 4, 2, 1, 0])
id_ = Permutation([0, 1, 2, 3, 4])
H = PermutationGroup([p1, p3])
H_n1 = PermutationGroup([p1, p2])
H_n2_1 = PermutationGroup(p1)
H_n2_2 = PermutationGroup(p2)
H_id = PermutationGroup(id_)
assert H_n1.is_normal(H)
assert H_n2_1.is_normal(H_n1)
assert H_n2_2.is_normal(H_n1)
assert H_id.is_normal(H_n2_1)
assert H_id.is_normal(H_n1)
assert H_id.is_normal(H)
assert not H_n2_1.is_normal(H)
assert not H_n2_2.is_normal(H)
def test_eq():
a = [[1, 2, 0, 3, 4, 5], [1, 0, 2, 3, 4, 5], [2, 1, 0, 3, 4, 5], [
1, 2, 0, 3, 4, 5]]
a = [Permutation(p) for p in a + [[1, 2, 3, 4, 5, 0]]]
g = Permutation([1, 2, 3, 4, 5, 0])
G1, G2, G3 = [PermutationGroup(x) for x in [a[:2], a[2:4], [g, g**2]]]
assert G1.order() == G2.order() == G3.order() == 6
assert G1.is_subgroup(G2)
assert not G1.is_subgroup(G3)
G4 = PermutationGroup([Permutation([0, 1])])
assert not G1.is_subgroup(G4)
assert G4.is_subgroup(G1, 0)
assert PermutationGroup(g, g).is_subgroup(PermutationGroup(g))
assert SymmetricGroup(3).is_subgroup(SymmetricGroup(4), 0)
assert SymmetricGroup(3).is_subgroup(SymmetricGroup(3)*CyclicGroup(5), 0)
assert not CyclicGroup(5).is_subgroup(SymmetricGroup(3)*CyclicGroup(5), 0)
assert CyclicGroup(3).is_subgroup(SymmetricGroup(3)*CyclicGroup(5), 0)
def test_derived_subgroup():
a = Permutation([1, 0, 2, 4, 3])
b = Permutation([0, 1, 3, 2, 4])
G = PermutationGroup([a, b])
C = G.derived_subgroup()
assert C.order() == 3
assert C.is_normal(G)
assert C.is_subgroup(G, 0)
assert not G.is_subgroup(C, 0)
gens_cube = [[1, 3, 5, 7, 0, 2, 4, 6], [1, 3, 0, 2, 5, 7, 4, 6]]
gens = [Permutation(p) for p in gens_cube]
G = PermutationGroup(gens)
C = G.derived_subgroup()
assert C.order() == 12
def test_is_solvable():
a = Permutation([1, 2, 0])
b = Permutation([1, 0, 2])
G = PermutationGroup([a, b])
assert G.is_solvable
G = PermutationGroup([a])
assert G.is_solvable
a = Permutation([1, 2, 3, 4, 0])
b = Permutation([1, 0, 2, 3, 4])
G = PermutationGroup([a, b])
assert not G.is_solvable
P = SymmetricGroup(10)
S = P.sylow_subgroup(3)
assert S.is_solvable
def test_rubik1():
gens = rubik_cube_generators()
gens1 = [gens[-1]] + [p**2 for p in gens[1:]]
G1 = PermutationGroup(gens1)
assert G1.order() == 19508428800
gens2 = [p**2 for p in gens]
G2 = PermutationGroup(gens2)
assert G2.order() == 663552
assert G2.is_subgroup(G1, 0)
C1 = G1.derived_subgroup()
assert C1.order() == 4877107200
assert C1.is_subgroup(G1, 0)
assert not G2.is_subgroup(C1, 0)
G = RubikGroup(2)
assert G.order() == 3674160
@XFAIL
def test_rubik():
skip('takes too much time')
G = PermutationGroup(rubik_cube_generators())
assert G.order() == 43252003274489856000
G1 = PermutationGroup(G[:3])
assert G1.order() == 170659735142400
assert not G1.is_normal(G)
G2 = G.normal_closure(G1.generators)
assert G2.is_subgroup(G)
def test_direct_product():
C = CyclicGroup(4)
D = DihedralGroup(4)
G = C*C*C
assert G.order() == 64
assert G.degree == 12
assert len(G.orbits()) == 3
assert G.is_abelian is True
H = D*C
assert H.order() == 32
assert H.is_abelian is False
def test_orbit_rep():
G = DihedralGroup(6)
assert G.orbit_rep(1, 3) in [Permutation([2, 3, 4, 5, 0, 1]),
Permutation([4, 3, 2, 1, 0, 5])]
H = CyclicGroup(4)*G
assert H.orbit_rep(1, 5) is False
def test_schreier_vector():
G = CyclicGroup(50)
v = [0]*50
v[23] = -1
assert G.schreier_vector(23) == v
H = DihedralGroup(8)
assert H.schreier_vector(2) == [0, 1, -1, 0, 0, 1, 0, 0]
L = SymmetricGroup(4)
assert L.schreier_vector(1) == [1, -1, 0, 0]
def test_random_pr():
D = DihedralGroup(6)
r = 11
n = 3
_random_prec_n = {}
_random_prec_n[0] = {'s': 7, 't': 3, 'x': 2, 'e': -1}
_random_prec_n[1] = {'s': 5, 't': 5, 'x': 1, 'e': -1}
_random_prec_n[2] = {'s': 3, 't': 4, 'x': 2, 'e': 1}
D._random_pr_init(r, n, _random_prec_n=_random_prec_n)
assert D._random_gens[11] == [0, 1, 2, 3, 4, 5]
_random_prec = {'s': 2, 't': 9, 'x': 1, 'e': -1}
assert D.random_pr(_random_prec=_random_prec) == \
Permutation([0, 5, 4, 3, 2, 1])
def test_is_alt_sym():
G = DihedralGroup(10)
assert G.is_alt_sym() is False
S = SymmetricGroup(10)
N_eps = 10
_random_prec = {'N_eps': N_eps,
0: Permutation([[2], [1, 4], [0, 6, 7, 8, 9, 3, 5]]),
1: Permutation([[1, 8, 7, 6, 3, 5, 2, 9], [0, 4]]),
2: Permutation([[5, 8], [4, 7], [0, 1, 2, 3, 6, 9]]),
3: Permutation([[3], [0, 8, 2, 7, 4, 1, 6, 9, 5]]),
4: Permutation([[8], [4, 7, 9], [3, 6], [0, 5, 1, 2]]),
5: Permutation([[6], [0, 2, 4, 5, 1, 8, 3, 9, 7]]),
6: Permutation([[6, 9, 8], [4, 5], [1, 3, 7], [0, 2]]),
7: Permutation([[4], [0, 2, 9, 1, 3, 8, 6, 5, 7]]),
8: Permutation([[1, 5, 6, 3], [0, 2, 7, 8, 4, 9]]),
9: Permutation([[8], [6, 7], [2, 3, 4, 5], [0, 1, 9]])}
assert S.is_alt_sym(_random_prec=_random_prec) is True
A = AlternatingGroup(10)
_random_prec = {'N_eps': N_eps,
0: Permutation([[1, 6, 4, 2, 7, 8, 5, 9, 3], [0]]),
1: Permutation([[1], [0, 5, 8, 4, 9, 2, 3, 6, 7]]),
2: Permutation([[1, 9, 8, 3, 2, 5], [0, 6, 7, 4]]),
3: Permutation([[6, 8, 9], [4, 5], [1, 3, 7, 2], [0]]),
4: Permutation([[8], [5], [4], [2, 6, 9, 3], [1], [0, 7]]),
5: Permutation([[3, 6], [0, 8, 1, 7, 5, 9, 4, 2]]),
6: Permutation([[5], [2, 9], [1, 8, 3], [0, 4, 7, 6]]),
7: Permutation([[1, 8, 4, 7, 2, 3], [0, 6, 9, 5]]),
8: Permutation([[5, 8, 7], [3], [1, 4, 2, 6], [0, 9]]),
9: Permutation([[4, 9, 6], [3, 8], [1, 2], [0, 5, 7]])}
assert A.is_alt_sym(_random_prec=_random_prec) is False
def test_minimal_block():
D = DihedralGroup(6)
block_system = D.minimal_block([0, 3])
for i in range(3):
assert block_system[i] == block_system[i + 3]
S = SymmetricGroup(6)
assert S.minimal_block([0, 1]) == [0, 0, 0, 0, 0, 0]
assert Tetra.pgroup.minimal_block([0, 1]) == [0, 0, 0, 0]
P1 = PermutationGroup(Permutation(1, 5)(2, 4), Permutation(0, 1, 2, 3, 4, 5))
P2 = PermutationGroup(Permutation(0, 1, 2, 3, 4, 5), Permutation(1, 5)(2, 4))
assert P1.minimal_block([0, 2]) == [0, 1, 0, 1, 0, 1]
assert P2.minimal_block([0, 2]) == [0, 1, 0, 1, 0, 1]
def test_minimal_blocks():
P = PermutationGroup(Permutation(1, 5)(2, 4), Permutation(0, 1, 2, 3, 4, 5))
assert P.minimal_blocks() == [[0, 1, 0, 1, 0, 1], [0, 1, 2, 0, 1, 2]]
P = SymmetricGroup(5)
assert P.minimal_blocks() == [[0]*5]
P = PermutationGroup(Permutation(0, 3))
assert P.minimal_blocks() == False
def test_max_div():
S = SymmetricGroup(10)
assert S.max_div == 5
def test_is_primitive():
S = SymmetricGroup(5)
assert S.is_primitive() is True
C = CyclicGroup(7)
assert C.is_primitive() is True
def test_random_stab():
S = SymmetricGroup(5)
_random_el = Permutation([1, 3, 2, 0, 4])
_random_prec = {'rand': _random_el}
g = S.random_stab(2, _random_prec=_random_prec)
assert g == Permutation([1, 3, 2, 0, 4])
h = S.random_stab(1)
assert h(1) == 1
def test_transitivity_degree():
perm = Permutation([1, 2, 0])
C = PermutationGroup([perm])
assert C.transitivity_degree == 1
gen1 = Permutation([1, 2, 0, 3, 4])
gen2 = Permutation([1, 2, 3, 4, 0])
# alternating group of degree 5
Alt = PermutationGroup([gen1, gen2])
assert Alt.transitivity_degree == 3
def test_schreier_sims_random():
assert sorted(Tetra.pgroup.base) == [0, 1]
S = SymmetricGroup(3)
base = [0, 1]
strong_gens = [Permutation([1, 2, 0]), Permutation([1, 0, 2]),
Permutation([0, 2, 1])]
assert S.schreier_sims_random(base, strong_gens, 5) == (base, strong_gens)
D = DihedralGroup(3)
_random_prec = {'g': [Permutation([2, 0, 1]), Permutation([1, 2, 0]),
Permutation([1, 0, 2])]}
base = [0, 1]
strong_gens = [Permutation([1, 2, 0]), Permutation([2, 1, 0]),
Permutation([0, 2, 1])]
assert D.schreier_sims_random([], D.generators, 2,
_random_prec=_random_prec) == (base, strong_gens)
def test_baseswap():
S = SymmetricGroup(4)
S.schreier_sims()
base = S.base
strong_gens = S.strong_gens
assert base == [0, 1, 2]
deterministic = S.baseswap(base, strong_gens, 1, randomized=False)
randomized = S.baseswap(base, strong_gens, 1)
assert deterministic[0] == [0, 2, 1]
assert _verify_bsgs(S, deterministic[0], deterministic[1]) is True
assert randomized[0] == [0, 2, 1]
assert _verify_bsgs(S, randomized[0], randomized[1]) is True
def test_schreier_sims_incremental():
identity = Permutation([0, 1, 2, 3, 4])
TrivialGroup = PermutationGroup([identity])
base, strong_gens = TrivialGroup.schreier_sims_incremental(base=[0, 1, 2])
assert _verify_bsgs(TrivialGroup, base, strong_gens) is True
S = SymmetricGroup(5)
base, strong_gens = S.schreier_sims_incremental(base=[0, 1, 2])
assert _verify_bsgs(S, base, strong_gens) is True
D = DihedralGroup(2)
base, strong_gens = D.schreier_sims_incremental(base=[1])
assert _verify_bsgs(D, base, strong_gens) is True
A = AlternatingGroup(7)
gens = A.generators[:]
gen0 = gens[0]
gen1 = gens[1]
gen1 = rmul(gen1, ~gen0)
gen0 = rmul(gen0, gen1)
gen1 = rmul(gen0, gen1)
base, strong_gens = A.schreier_sims_incremental(base=[0, 1], gens=gens)
assert _verify_bsgs(A, base, strong_gens) is True
C = CyclicGroup(11)
gen = C.generators[0]
base, strong_gens = C.schreier_sims_incremental(gens=[gen**3])
assert _verify_bsgs(C, base, strong_gens) is True
def _subgroup_search(i, j, k):
prop_true = lambda x: True
prop_fix_points = lambda x: [x(point) for point in points] == points
prop_comm_g = lambda x: rmul(x, g) == rmul(g, x)
prop_even = lambda x: x.is_even
for i in range(i, j, k):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
C = CyclicGroup(i)
Sym = S.subgroup_search(prop_true)
assert Sym.is_subgroup(S)
Alt = S.subgroup_search(prop_even)
assert Alt.is_subgroup(A)
Sym = S.subgroup_search(prop_true, init_subgroup=C)
assert Sym.is_subgroup(S)
points = [7]
assert S.stabilizer(7).is_subgroup(S.subgroup_search(prop_fix_points))
points = [3, 4]
assert S.stabilizer(3).stabilizer(4).is_subgroup(
S.subgroup_search(prop_fix_points))
points = [3, 5]
fix35 = A.subgroup_search(prop_fix_points)
points = [5]
fix5 = A.subgroup_search(prop_fix_points)
assert A.subgroup_search(prop_fix_points, init_subgroup=fix35
).is_subgroup(fix5)
base, strong_gens = A.schreier_sims_incremental()
g = A.generators[0]
comm_g = \
A.subgroup_search(prop_comm_g, base=base, strong_gens=strong_gens)
assert _verify_bsgs(comm_g, base, comm_g.generators) is True
assert [prop_comm_g(gen) is True for gen in comm_g.generators]
def test_subgroup_search():
_subgroup_search(10, 15, 2)
@XFAIL
def test_subgroup_search2():
skip('takes too much time')
_subgroup_search(16, 17, 1)
def test_normal_closure():
# the normal closure of the trivial group is trivial
S = SymmetricGroup(3)
identity = Permutation([0, 1, 2])
closure = S.normal_closure(identity)
assert closure.is_trivial
# the normal closure of the entire group is the entire group
A = AlternatingGroup(4)
assert A.normal_closure(A).is_subgroup(A)
# brute-force verifications for subgroups
for i in (3, 4, 5):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
D = DihedralGroup(i)
C = CyclicGroup(i)
for gp in (A, D, C):
assert _verify_normal_closure(S, gp)
# brute-force verifications for all elements of a group
S = SymmetricGroup(5)
elements = list(S.generate_dimino())
for element in elements:
assert _verify_normal_closure(S, element)
# small groups
small = []
for i in (1, 2, 3):
small.append(SymmetricGroup(i))
small.append(AlternatingGroup(i))
small.append(DihedralGroup(i))
small.append(CyclicGroup(i))
for gp in small:
for gp2 in small:
if gp2.is_subgroup(gp, 0) and gp2.degree == gp.degree:
assert _verify_normal_closure(gp, gp2)
def test_derived_series():
# the derived series of the trivial group consists only of the trivial group
triv = PermutationGroup([Permutation([0, 1, 2])])
assert triv.derived_series()[0].is_subgroup(triv)
# the derived series for a simple group consists only of the group itself
for i in (5, 6, 7):
A = AlternatingGroup(i)
assert A.derived_series()[0].is_subgroup(A)
# the derived series for S_4 is S_4 > A_4 > K_4 > triv
S = SymmetricGroup(4)
series = S.derived_series()
assert series[1].is_subgroup(AlternatingGroup(4))
assert series[2].is_subgroup(DihedralGroup(2))
assert series[3].is_trivial
def test_lower_central_series():
# the lower central series of the trivial group consists of the trivial
# group
triv = PermutationGroup([Permutation([0, 1, 2])])
assert triv.lower_central_series()[0].is_subgroup(triv)
# the lower central series of a simple group consists of the group itself
for i in (5, 6, 7):
A = AlternatingGroup(i)
assert A.lower_central_series()[0].is_subgroup(A)
# GAP-verified example
S = SymmetricGroup(6)
series = S.lower_central_series()
assert len(series) == 2
assert series[1].is_subgroup(AlternatingGroup(6))
def test_commutator():
# the commutator of the trivial group and the trivial group is trivial
S = SymmetricGroup(3)
triv = PermutationGroup([Permutation([0, 1, 2])])
assert S.commutator(triv, triv).is_subgroup(triv)
# the commutator of the trivial group and any other group is again trivial
A = AlternatingGroup(3)
assert S.commutator(triv, A).is_subgroup(triv)
# the commutator is commutative
for i in (3, 4, 5):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
D = DihedralGroup(i)
assert S.commutator(A, D).is_subgroup(S.commutator(D, A))
# the commutator of an abelian group is trivial
S = SymmetricGroup(7)
A1 = AbelianGroup(2, 5)
A2 = AbelianGroup(3, 4)
triv = PermutationGroup([Permutation([0, 1, 2, 3, 4, 5, 6])])
assert S.commutator(A1, A1).is_subgroup(triv)
assert S.commutator(A2, A2).is_subgroup(triv)
# examples calculated by hand
S = SymmetricGroup(3)
A = AlternatingGroup(3)
assert S.commutator(A, S).is_subgroup(A)
def test_is_nilpotent():
# every abelian group is nilpotent
for i in (1, 2, 3):
C = CyclicGroup(i)
Ab = AbelianGroup(i, i + 2)
assert C.is_nilpotent
assert Ab.is_nilpotent
Ab = AbelianGroup(5, 7, 10)
assert Ab.is_nilpotent
# A_5 is not solvable and thus not nilpotent
assert AlternatingGroup(5).is_nilpotent is False
def test_is_trivial():
for i in range(5):
triv = PermutationGroup([Permutation(list(range(i)))])
assert triv.is_trivial
def test_pointwise_stabilizer():
S = SymmetricGroup(2)
stab = S.pointwise_stabilizer([0])
assert stab.generators == [Permutation(1)]
S = SymmetricGroup(5)
points = []
stab = S
for point in (2, 0, 3, 4, 1):
stab = stab.stabilizer(point)
points.append(point)
assert S.pointwise_stabilizer(points).is_subgroup(stab)
def test_make_perm():
assert cube.pgroup.make_perm(5, seed=list(range(5))) == \
Permutation([4, 7, 6, 5, 0, 3, 2, 1])
assert cube.pgroup.make_perm(7, seed=list(range(7))) == \
Permutation([6, 7, 3, 2, 5, 4, 0, 1])
def test_elements():
p = Permutation(2, 3)
assert PermutationGroup(p).elements == {Permutation(3), Permutation(2, 3)}
def test_is_group():
assert PermutationGroup(Permutation(1,2), Permutation(2,4)).is_group == True
assert SymmetricGroup(4).is_group == True
def test_PermutationGroup():
assert PermutationGroup() == PermutationGroup(Permutation())
assert (PermutationGroup() == 0) is False
def test_coset_transvesal():
G = AlternatingGroup(5)
H = PermutationGroup(Permutation(0,1,2),Permutation(1,2)(3,4))
assert G.coset_transversal(H) == \
[Permutation(4), Permutation(2, 3, 4), Permutation(2, 4, 3),
Permutation(1, 2, 4), Permutation(4)(1, 2, 3), Permutation(1, 3)(2, 4),
Permutation(0, 1, 2, 3, 4), Permutation(0, 1, 2, 4, 3),
Permutation(0, 1, 3, 2, 4), Permutation(0, 2, 4, 1, 3)]
def test_coset_table():
G = PermutationGroup(Permutation(0,1,2,3), Permutation(0,1,2),
Permutation(0,4,2,7), Permutation(5,6), Permutation(0,7));
H = PermutationGroup(Permutation(0,1,2,3), Permutation(0,7))
assert G.coset_table(H) == \
[[0, 0, 0, 0, 1, 2, 3, 3, 0, 0], [4, 5, 2, 5, 6, 0, 7, 7, 1, 1],
[5, 4, 5, 1, 0, 6, 8, 8, 6, 6], [3, 3, 3, 3, 7, 8, 0, 0, 3, 3],
[2, 1, 4, 4, 4, 4, 9, 9, 4, 4], [1, 2, 1, 2, 5, 5, 10, 10, 5, 5],
[6, 6, 6, 6, 2, 1, 11, 11, 2, 2], [9, 10, 8, 10, 11, 3, 1, 1, 7, 7],
[10, 9, 10, 7, 3, 11, 2, 2, 11, 11], [8, 7, 9, 9, 9, 9, 4, 4, 9, 9],
[7, 8, 7, 8, 10, 10, 5, 5, 10, 10], [11, 11, 11, 11, 8, 7, 6, 6, 8, 8]]
def test_subgroup():
G = PermutationGroup(Permutation(0,1,2), Permutation(0,2,3))
H = G.subgroup([Permutation(0,1,3)])
assert H.is_subgroup(G)
def test_generator_product():
G = SymmetricGroup(5)
p = Permutation(0, 2, 3)(1, 4)
gens = G.generator_product(p)
assert all(g in G.strong_gens for g in gens)
w = G.identity
for g in gens:
w = g*w
assert w == p
def test_sylow_subgroup():
P = PermutationGroup(Permutation(1, 5)(2, 4), Permutation(0, 1, 2, 3, 4, 5))
S = P.sylow_subgroup(2)
assert S.order() == 4
P = DihedralGroup(12)
S = P.sylow_subgroup(3)
assert S.order() == 3
P = PermutationGroup(Permutation(1, 5)(2, 4), Permutation(0, 1, 2, 3, 4, 5), Permutation(0, 2))
S = P.sylow_subgroup(3)
assert S.order() == 9
S = P.sylow_subgroup(2)
assert S.order() == 8
P = SymmetricGroup(10)
S = P.sylow_subgroup(2)
assert S.order() == 256
S = P.sylow_subgroup(3)
assert S.order() == 81
S = P.sylow_subgroup(5)
assert S.order() == 25
# the length of the lower central series
# of a p-Sylow subgroup of Sym(n) grows with
# the highest exponent exp of p such
# that n >= p**exp
exp = 1
length = 0
for i in range(2, 9):
P = SymmetricGroup(i)
S = P.sylow_subgroup(2)
ls = S.lower_central_series()
if i // 2**exp > 0:
# length increases with exponent
assert len(ls) > length
length = len(ls)
exp += 1
else:
assert len(ls) == length
G = SymmetricGroup(100)
S = G.sylow_subgroup(3)
assert G.order() % S.order() == 0
assert G.order()/S.order() % 3 > 0
G = AlternatingGroup(100)
S = G.sylow_subgroup(2)
assert G.order() % S.order() == 0
assert G.order()/S.order() % 2 > 0
@slow
def test_presentation():
def _test(P):
G = P.presentation()
return G.order() == P.order()
def _strong_test(P):
G = P.strong_presentation()
chk = len(G.generators) == len(P.strong_gens)
return chk and G.order() == P.order()
P = PermutationGroup(Permutation(0,1,5,2)(3,7,4,6), Permutation(0,3,5,4)(1,6,2,7))
assert _test(P)
P = AlternatingGroup(5)
assert _test(P)
P = SymmetricGroup(5)
assert _test(P)
P = PermutationGroup([Permutation(0,3,1,2), Permutation(3)(0,1), Permutation(0,1)(2,3)])
G = P.strong_presentation()
assert _strong_test(P)
P = DihedralGroup(6)
G = P.strong_presentation()
assert _strong_test(P)
a = Permutation(0,1)(2,3)
b = Permutation(0,2)(3,1)
c = Permutation(4,5)
P = PermutationGroup(c, a, b)
assert _strong_test(P)
def test_polycyclic():
a = Permutation([0, 1, 2])
b = Permutation([2, 1, 0])
G = PermutationGroup([a, b])
assert G.is_polycyclic == True
a = Permutation([1, 2, 3, 4, 0])
b = Permutation([1, 0, 2, 3, 4])
G = PermutationGroup([a, b])
assert G.is_polycyclic == False
def test_elementary():
a = Permutation([1, 5, 2, 0, 3, 6, 4])
G = PermutationGroup([a])
assert G.is_elementary(7) == False
a = Permutation(0, 1)(2, 3)
b = Permutation(0, 2)(3, 1)
G = PermutationGroup([a, b])
assert G.is_elementary(2) == True
c = Permutation(4, 5, 6)
G = PermutationGroup([a, b, c])
assert G.is_elementary(2) == False
G = SymmetricGroup(4).sylow_subgroup(2)
assert G.is_elementary(2) == False
H = AlternatingGroup(4).sylow_subgroup(2)
assert H.is_elementary(2) == True
|
756d6b9f62fb76d589423537fd1cf691f1590b625b70deb952f00101a007efd5
|
from sympy.core.compatibility import range
from sympy.combinatorics.partitions import (Partition, IntegerPartition,
RGS_enum, RGS_unrank, RGS_rank,
random_integer_partition)
from sympy.utilities.pytest import raises
from sympy.utilities.iterables import default_sort_key, partitions
def test_partition():
from sympy.abc import x
raises(ValueError, lambda: Partition(*list(range(3))))
raises(ValueError, lambda: Partition([1, 1, 2]))
a = Partition([1, 2, 3], [4])
b = Partition([1, 2], [3, 4])
c = Partition([x])
l = [a, b, c]
l.sort(key=default_sort_key)
assert l == [c, a, b]
l.sort(key=lambda w: default_sort_key(w, order='rev-lex'))
assert l == [c, a, b]
assert (a == b) is False
assert a <= b
assert (a > b) is False
assert a != b
assert a < b
assert (a + 2).partition == [[1, 2], [3, 4]]
assert (b - 1).partition == [[1, 2, 4], [3]]
assert (a - 1).partition == [[1, 2, 3, 4]]
assert (a + 1).partition == [[1, 2, 4], [3]]
assert (b + 1).partition == [[1, 2], [3], [4]]
assert a.rank == 1
assert b.rank == 3
assert a.RGS == (0, 0, 0, 1)
assert b.RGS == (0, 0, 1, 1)
def test_integer_partition():
# no zeros in partition
raises(ValueError, lambda: IntegerPartition(list(range(3))))
# check fails since 1 + 2 != 100
raises(ValueError, lambda: IntegerPartition(100, list(range(1, 3))))
a = IntegerPartition(8, [1, 3, 4])
b = a.next_lex()
c = IntegerPartition([1, 3, 4])
d = IntegerPartition(8, {1: 3, 3: 1, 2: 1})
assert a == c
assert a.integer == d.integer
assert a.conjugate == [3, 2, 2, 1]
assert (a == b) is False
assert a <= b
assert (a > b) is False
assert a != b
for i in range(1, 11):
next = set()
prev = set()
a = IntegerPartition([i])
ans = {IntegerPartition(p) for p in partitions(i)}
n = len(ans)
for j in range(n):
next.add(a)
a = a.next_lex()
IntegerPartition(i, a.partition) # check it by giving i
for j in range(n):
prev.add(a)
a = a.prev_lex()
IntegerPartition(i, a.partition) # check it by giving i
assert next == ans
assert prev == ans
assert IntegerPartition([1, 2, 3]).as_ferrers() == '###\n##\n#'
assert IntegerPartition([1, 1, 3]).as_ferrers('o') == 'ooo\no\no'
assert str(IntegerPartition([1, 1, 3])) == '[3, 1, 1]'
assert IntegerPartition([1, 1, 3]).partition == [3, 1, 1]
raises(ValueError, lambda: random_integer_partition(-1))
assert random_integer_partition(1) == [1]
assert random_integer_partition(10, seed=[1, 3, 2, 1, 5, 1]
) == [5, 2, 1, 1, 1]
def test_rgs():
raises(ValueError, lambda: RGS_unrank(-1, 3))
raises(ValueError, lambda: RGS_unrank(3, 0))
raises(ValueError, lambda: RGS_unrank(10, 1))
raises(ValueError, lambda: Partition.from_rgs(list(range(3)), list(range(2))))
raises(ValueError, lambda: Partition.from_rgs(list(range(1, 3)), list(range(2))))
assert RGS_enum(-1) == 0
assert RGS_enum(1) == 1
assert RGS_unrank(7, 5) == [0, 0, 1, 0, 2]
assert RGS_unrank(23, 14) == [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2]
assert RGS_rank(RGS_unrank(40, 100)) == 40
|
6b21717d85c7a18973f9fed9938d13beb91529291eeb6ecfb7aec3cf527815a0
|
# -*- coding: utf-8 -*-
from sympy.combinatorics.fp_groups import FpGroup
from sympy.combinatorics.coset_table import (CosetTable,
coset_enumeration_r, coset_enumeration_c)
from sympy.combinatorics.coset_table import modified_coset_enumeration_r
from sympy.combinatorics.free_groups import free_group
from sympy.utilities.pytest import slow
"""
References
==========
[1] Holt, D., Eick, B., O'Brien, E.
"Handbook of Computational Group Theory"
[2] John J. Cannon; Lucien A. Dimino; George Havas; Jane M. Watson
Mathematics of Computation, Vol. 27, No. 123. (Jul., 1973), pp. 463-490.
"Implementation and Analysis of the Todd-Coxeter Algorithm"
"""
def test_scan_1():
# Example 5.1 from [1]
F, x, y = free_group("x, y")
f = FpGroup(F, [x**3, y**3, x**-1*y**-1*x*y])
c = CosetTable(f, [x])
c.scan_and_fill(0, x)
assert c.table == [[0, 0, None, None]]
assert c.p == [0]
assert c.n == 1
assert c.omega == [0]
c.scan_and_fill(0, x**3)
assert c.table == [[0, 0, None, None]]
assert c.p == [0]
assert c.n == 1
assert c.omega == [0]
c.scan_and_fill(0, y**3)
assert c.table == [[0, 0, 1, 2], [None, None, 2, 0], [None, None, 0, 1]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(0, x**-1*y**-1*x*y)
assert c.table == [[0, 0, 1, 2], [None, None, 2, 0], [2, 2, 0, 1]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(1, x**3)
assert c.table == [[0, 0, 1, 2], [3, 4, 2, 0], [2, 2, 0, 1], \
[4, 1, None, None], [1, 3, None, None]]
assert c.p == [0, 1, 2, 3, 4]
assert c.n == 5
assert c.omega == [0, 1, 2, 3, 4]
c.scan_and_fill(1, y**3)
assert c.table == [[0, 0, 1, 2], [3, 4, 2, 0], [2, 2, 0, 1], \
[4, 1, None, None], [1, 3, None, None]]
assert c.p == [0, 1, 2, 3, 4]
assert c.n == 5
assert c.omega == [0, 1, 2, 3, 4]
c.scan_and_fill(1, x**-1*y**-1*x*y)
assert c.table == [[0, 0, 1, 2], [1, 1, 2, 0], [2, 2, 0, 1], \
[None, 1, None, None], [1, 3, None, None]]
assert c.p == [0, 1, 2, 1, 1]
assert c.n == 3
assert c.omega == [0, 1, 2]
# Example 5.2 from [1]
f = FpGroup(F, [x**2, y**3, (x*y)**3])
c = CosetTable(f, [x*y])
c.scan_and_fill(0, x*y)
assert c.table == [[1, None, None, 1], [None, 0, 0, None]]
assert c.p == [0, 1]
assert c.n == 2
assert c.omega == [0, 1]
c.scan_and_fill(0, x**2)
assert c.table == [[1, 1, None, 1], [0, 0, 0, None]]
assert c.p == [0, 1]
assert c.n == 2
assert c.omega == [0, 1]
c.scan_and_fill(0, y**3)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(0, (x*y)**3)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(1, x**2)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(1, y**3)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(1, (x*y)**3)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [3, 4, 1, 0], [None, 2, 4, None], [2, None, None, 3]]
assert c.p == [0, 1, 2, 3, 4]
assert c.n == 5
assert c.omega == [0, 1, 2, 3, 4]
c.scan_and_fill(2, x**2)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [3, 3, 1, 0], [2, 2, 3, 3], [2, None, None, 3]]
assert c.p == [0, 1, 2, 3, 3]
assert c.n == 4
assert c.omega == [0, 1, 2, 3]
@slow
def test_coset_enumeration():
# this test function contains the combined tests for the two strategies
# i.e. HLT and Felsch strategies.
# Example 5.1 from [1]
F, x, y = free_group("x, y")
f = FpGroup(F, [x**3, y**3, x**-1*y**-1*x*y])
C_r = coset_enumeration_r(f, [x])
C_r.compress(); C_r.standardize()
C_c = coset_enumeration_c(f, [x])
C_c.compress(); C_c.standardize()
table1 = [[0, 0, 1, 2], [1, 1, 2, 0], [2, 2, 0, 1]]
assert C_r.table == table1
assert C_c.table == table1
# E₁ from [2] Pg. 474
F, r, s, t = free_group("r, s, t")
E1 = FpGroup(F, [t**-1*r*t*r**-2, r**-1*s*r*s**-2, s**-1*t*s*t**-2])
C_r = coset_enumeration_r(E1, [])
C_r.compress()
C_c = coset_enumeration_c(E1, [])
C_c.compress()
table2 = [[0, 0, 0, 0, 0, 0]]
assert C_r.table == table2
# test for issue #11449
assert C_c.table == table2
# Cox group from [2] Pg. 474
F, a, b = free_group("a, b")
Cox = FpGroup(F, [a**6, b**6, (a*b)**2, (a**2*b**2)**2, (a**3*b**3)**5])
C_r = coset_enumeration_r(Cox, [a])
C_r.compress(); C_r.standardize()
C_c = coset_enumeration_c(Cox, [a])
C_c.compress(); C_c.standardize()
table3 = [[0, 0, 1, 2],
[2, 3, 4, 0],
[5, 1, 0, 6],
[1, 7, 8, 9],
[9, 10, 11, 1],
[12, 2, 9, 13],
[14, 9, 2, 11],
[3, 12, 15, 16],
[16, 17, 18, 3],
[6, 4, 3, 5],
[4, 19, 20, 21],
[21, 22, 6, 4],
[7, 5, 23, 24],
[25, 23, 5, 18],
[19, 6, 22, 26],
[24, 27, 28, 7],
[29, 8, 7, 30],
[8, 31, 32, 33],
[33, 34, 13, 8],
[10, 14, 35, 35],
[35, 36, 37, 10],
[30, 11, 10, 29],
[11, 38, 39, 14],
[13, 39, 38, 12],
[40, 15, 12, 41],
[42, 13, 34, 43],
[44, 35, 14, 45],
[15, 46, 47, 34],
[34, 48, 49, 15],
[50, 16, 21, 51],
[52, 21, 16, 49],
[17, 50, 53, 54],
[54, 55, 56, 17],
[41, 18, 17, 40],
[18, 28, 27, 25],
[26, 20, 19, 19],
[20, 57, 58, 59],
[59, 60, 51, 20],
[22, 52, 61, 23],
[23, 62, 63, 22],
[64, 24, 33, 65],
[48, 33, 24, 61],
[62, 25, 54, 66],
[67, 54, 25, 68],
[57, 26, 59, 69],
[70, 59, 26, 63],
[27, 64, 71, 72],
[72, 73, 68, 27],
[28, 41, 74, 75],
[75, 76, 30, 28],
[31, 29, 77, 78],
[79, 77, 29, 37],
[38, 30, 76, 80],
[78, 81, 82, 31],
[43, 32, 31, 42],
[32, 83, 84, 85],
[85, 86, 65, 32],
[36, 44, 87, 88],
[88, 89, 90, 36],
[45, 37, 36, 44],
[37, 82, 81, 79],
[80, 74, 41, 38],
[39, 42, 91, 92],
[92, 93, 45, 39],
[46, 40, 94, 95],
[96, 94, 40, 56],
[97, 91, 42, 82],
[83, 43, 98, 99],
[100, 98, 43, 47],
[101, 87, 44, 90],
[82, 45, 93, 97],
[95, 102, 103, 46],
[104, 47, 46, 105],
[47, 106, 107, 100],
[61, 108, 109, 48],
[105, 49, 48, 104],
[49, 110, 111, 52],
[51, 111, 110, 50],
[112, 53, 50, 113],
[114, 51, 60, 115],
[116, 61, 52, 117],
[53, 118, 119, 60],
[60, 70, 66, 53],
[55, 67, 120, 121],
[121, 122, 123, 55],
[113, 56, 55, 112],
[56, 103, 102, 96],
[69, 124, 125, 57],
[115, 58, 57, 114],
[58, 126, 127, 128],
[128, 128, 69, 58],
[66, 129, 130, 62],
[117, 63, 62, 116],
[63, 125, 124, 70],
[65, 109, 108, 64],
[131, 71, 64, 132],
[133, 65, 86, 134],
[135, 66, 70, 136],
[68, 130, 129, 67],
[137, 120, 67, 138],
[132, 68, 73, 131],
[139, 69, 128, 140],
[71, 141, 142, 86],
[86, 143, 144, 71],
[145, 72, 75, 146],
[147, 75, 72, 144],
[73, 145, 148, 120],
[120, 149, 150, 73],
[74, 151, 152, 94],
[94, 153, 146, 74],
[76, 147, 154, 77],
[77, 155, 156, 76],
[157, 78, 85, 158],
[143, 85, 78, 154],
[155, 79, 88, 159],
[160, 88, 79, 161],
[151, 80, 92, 162],
[163, 92, 80, 156],
[81, 157, 164, 165],
[165, 166, 161, 81],
[99, 107, 106, 83],
[134, 84, 83, 133],
[84, 167, 168, 169],
[169, 170, 158, 84],
[87, 171, 172, 93],
[93, 163, 159, 87],
[89, 160, 173, 174],
[174, 175, 176, 89],
[90, 90, 89, 101],
[91, 177, 178, 98],
[98, 179, 162, 91],
[180, 95, 100, 181],
[179, 100, 95, 152],
[153, 96, 121, 148],
[182, 121, 96, 183],
[177, 97, 165, 184],
[185, 165, 97, 172],
[186, 99, 169, 187],
[188, 169, 99, 178],
[171, 101, 174, 189],
[190, 174, 101, 176],
[102, 180, 191, 192],
[192, 193, 183, 102],
[103, 113, 194, 195],
[195, 196, 105, 103],
[106, 104, 197, 198],
[199, 197, 104, 109],
[110, 105, 196, 200],
[198, 201, 133, 106],
[107, 186, 202, 203],
[203, 204, 181, 107],
[108, 116, 205, 206],
[206, 207, 132, 108],
[109, 133, 201, 199],
[200, 194, 113, 110],
[111, 114, 208, 209],
[209, 210, 117, 111],
[118, 112, 211, 212],
[213, 211, 112, 123],
[214, 208, 114, 125],
[126, 115, 215, 216],
[217, 215, 115, 119],
[218, 205, 116, 130],
[125, 117, 210, 214],
[212, 219, 220, 118],
[136, 119, 118, 135],
[119, 221, 222, 217],
[122, 182, 223, 224],
[224, 225, 226, 122],
[138, 123, 122, 137],
[123, 220, 219, 213],
[124, 139, 227, 228],
[228, 229, 136, 124],
[216, 222, 221, 126],
[140, 127, 126, 139],
[127, 230, 231, 232],
[232, 233, 140, 127],
[129, 135, 234, 235],
[235, 236, 138, 129],
[130, 132, 207, 218],
[141, 131, 237, 238],
[239, 237, 131, 150],
[167, 134, 240, 241],
[242, 240, 134, 142],
[243, 234, 135, 220],
[221, 136, 229, 244],
[149, 137, 245, 246],
[247, 245, 137, 226],
[220, 138, 236, 243],
[244, 227, 139, 221],
[230, 140, 233, 248],
[238, 249, 250, 141],
[251, 142, 141, 252],
[142, 253, 254, 242],
[154, 255, 256, 143],
[252, 144, 143, 251],
[144, 257, 258, 147],
[146, 258, 257, 145],
[259, 148, 145, 260],
[261, 146, 153, 262],
[263, 154, 147, 264],
[148, 265, 266, 153],
[246, 267, 268, 149],
[260, 150, 149, 259],
[150, 250, 249, 239],
[162, 269, 270, 151],
[262, 152, 151, 261],
[152, 271, 272, 179],
[159, 273, 274, 155],
[264, 156, 155, 263],
[156, 270, 269, 163],
[158, 256, 255, 157],
[275, 164, 157, 276],
[277, 158, 170, 278],
[279, 159, 163, 280],
[161, 274, 273, 160],
[281, 173, 160, 282],
[276, 161, 166, 275],
[283, 162, 179, 284],
[164, 285, 286, 170],
[170, 188, 184, 164],
[166, 185, 189, 173],
[173, 287, 288, 166],
[241, 254, 253, 167],
[278, 168, 167, 277],
[168, 289, 290, 291],
[291, 292, 187, 168],
[189, 293, 294, 171],
[280, 172, 171, 279],
[172, 295, 296, 185],
[175, 190, 297, 297],
[297, 298, 299, 175],
[282, 176, 175, 281],
[176, 294, 293, 190],
[184, 296, 295, 177],
[284, 178, 177, 283],
[178, 300, 301, 188],
[181, 272, 271, 180],
[302, 191, 180, 303],
[304, 181, 204, 305],
[183, 266, 265, 182],
[306, 223, 182, 307],
[303, 183, 193, 302],
[308, 184, 188, 309],
[310, 189, 185, 311],
[187, 301, 300, 186],
[305, 202, 186, 304],
[312, 187, 292, 313],
[314, 297, 190, 315],
[191, 316, 317, 204],
[204, 318, 319, 191],
[320, 192, 195, 321],
[322, 195, 192, 319],
[193, 320, 323, 223],
[223, 324, 325, 193],
[194, 326, 327, 211],
[211, 328, 321, 194],
[196, 322, 329, 197],
[197, 330, 331, 196],
[332, 198, 203, 333],
[318, 203, 198, 329],
[330, 199, 206, 334],
[335, 206, 199, 336],
[326, 200, 209, 337],
[338, 209, 200, 331],
[201, 332, 339, 240],
[240, 340, 336, 201],
[202, 341, 342, 292],
[292, 343, 333, 202],
[205, 344, 345, 210],
[210, 338, 334, 205],
[207, 335, 346, 237],
[237, 347, 348, 207],
[208, 349, 350, 215],
[215, 351, 337, 208],
[352, 212, 217, 353],
[351, 217, 212, 327],
[328, 213, 224, 323],
[354, 224, 213, 355],
[349, 214, 228, 356],
[357, 228, 214, 345],
[358, 216, 232, 359],
[360, 232, 216, 350],
[344, 218, 235, 361],
[362, 235, 218, 348],
[219, 352, 363, 364],
[364, 365, 355, 219],
[222, 358, 366, 367],
[367, 368, 353, 222],
[225, 354, 369, 370],
[370, 371, 372, 225],
[307, 226, 225, 306],
[226, 268, 267, 247],
[227, 373, 374, 233],
[233, 360, 356, 227],
[229, 357, 361, 234],
[234, 375, 376, 229],
[248, 231, 230, 230],
[231, 377, 378, 379],
[379, 380, 359, 231],
[236, 362, 381, 245],
[245, 382, 383, 236],
[384, 238, 242, 385],
[340, 242, 238, 346],
[347, 239, 246, 381],
[386, 246, 239, 387],
[388, 241, 291, 389],
[343, 291, 241, 339],
[375, 243, 364, 390],
[391, 364, 243, 383],
[373, 244, 367, 392],
[393, 367, 244, 376],
[382, 247, 370, 394],
[395, 370, 247, 396],
[377, 248, 379, 397],
[398, 379, 248, 374],
[249, 384, 399, 400],
[400, 401, 387, 249],
[250, 260, 402, 403],
[403, 404, 252, 250],
[253, 251, 405, 406],
[407, 405, 251, 256],
[257, 252, 404, 408],
[406, 409, 277, 253],
[254, 388, 410, 411],
[411, 412, 385, 254],
[255, 263, 413, 414],
[414, 415, 276, 255],
[256, 277, 409, 407],
[408, 402, 260, 257],
[258, 261, 416, 417],
[417, 418, 264, 258],
[265, 259, 419, 420],
[421, 419, 259, 268],
[422, 416, 261, 270],
[271, 262, 423, 424],
[425, 423, 262, 266],
[426, 413, 263, 274],
[270, 264, 418, 422],
[420, 427, 307, 265],
[266, 303, 428, 425],
[267, 386, 429, 430],
[430, 431, 396, 267],
[268, 307, 427, 421],
[269, 283, 432, 433],
[433, 434, 280, 269],
[424, 428, 303, 271],
[272, 304, 435, 436],
[436, 437, 284, 272],
[273, 279, 438, 439],
[439, 440, 282, 273],
[274, 276, 415, 426],
[285, 275, 441, 442],
[443, 441, 275, 288],
[289, 278, 444, 445],
[446, 444, 278, 286],
[447, 438, 279, 294],
[295, 280, 434, 448],
[287, 281, 449, 450],
[451, 449, 281, 299],
[294, 282, 440, 447],
[448, 432, 283, 295],
[300, 284, 437, 452],
[442, 453, 454, 285],
[309, 286, 285, 308],
[286, 455, 456, 446],
[450, 457, 458, 287],
[311, 288, 287, 310],
[288, 454, 453, 443],
[445, 456, 455, 289],
[313, 290, 289, 312],
[290, 459, 460, 461],
[461, 462, 389, 290],
[293, 310, 463, 464],
[464, 465, 315, 293],
[296, 308, 466, 467],
[467, 468, 311, 296],
[298, 314, 469, 470],
[470, 471, 472, 298],
[315, 299, 298, 314],
[299, 458, 457, 451],
[452, 435, 304, 300],
[301, 312, 473, 474],
[474, 475, 309, 301],
[316, 302, 476, 477],
[478, 476, 302, 325],
[341, 305, 479, 480],
[481, 479, 305, 317],
[324, 306, 482, 483],
[484, 482, 306, 372],
[485, 466, 308, 454],
[455, 309, 475, 486],
[487, 463, 310, 458],
[454, 311, 468, 485],
[486, 473, 312, 455],
[459, 313, 488, 489],
[490, 488, 313, 342],
[491, 469, 314, 472],
[458, 315, 465, 487],
[477, 492, 485, 316],
[463, 317, 316, 468],
[317, 487, 493, 481],
[329, 447, 464, 318],
[468, 319, 318, 463],
[319, 467, 448, 322],
[321, 448, 467, 320],
[475, 323, 320, 466],
[432, 321, 328, 437],
[438, 329, 322, 434],
[323, 474, 452, 328],
[483, 494, 486, 324],
[466, 325, 324, 475],
[325, 485, 492, 478],
[337, 422, 433, 326],
[437, 327, 326, 432],
[327, 436, 424, 351],
[334, 426, 439, 330],
[434, 331, 330, 438],
[331, 433, 422, 338],
[333, 464, 447, 332],
[449, 339, 332, 440],
[465, 333, 343, 469],
[413, 334, 338, 418],
[336, 439, 426, 335],
[441, 346, 335, 415],
[440, 336, 340, 449],
[416, 337, 351, 423],
[339, 451, 470, 343],
[346, 443, 450, 340],
[480, 493, 487, 341],
[469, 342, 341, 465],
[342, 491, 495, 490],
[361, 407, 414, 344],
[418, 345, 344, 413],
[345, 417, 408, 357],
[381, 446, 442, 347],
[415, 348, 347, 441],
[348, 414, 407, 362],
[356, 408, 417, 349],
[423, 350, 349, 416],
[350, 425, 420, 360],
[353, 424, 436, 352],
[479, 363, 352, 435],
[428, 353, 368, 476],
[355, 452, 474, 354],
[488, 369, 354, 473],
[435, 355, 365, 479],
[402, 356, 360, 419],
[405, 361, 357, 404],
[359, 420, 425, 358],
[476, 366, 358, 428],
[427, 359, 380, 482],
[444, 381, 362, 409],
[363, 481, 477, 368],
[368, 393, 390, 363],
[365, 391, 394, 369],
[369, 490, 480, 365],
[366, 478, 483, 380],
[380, 398, 392, 366],
[371, 395, 496, 497],
[497, 498, 489, 371],
[473, 372, 371, 488],
[372, 486, 494, 484],
[392, 400, 403, 373],
[419, 374, 373, 402],
[374, 421, 430, 398],
[390, 411, 406, 375],
[404, 376, 375, 405],
[376, 403, 400, 393],
[397, 430, 421, 377],
[482, 378, 377, 427],
[378, 484, 497, 499],
[499, 499, 397, 378],
[394, 461, 445, 382],
[409, 383, 382, 444],
[383, 406, 411, 391],
[385, 450, 443, 384],
[492, 399, 384, 453],
[457, 385, 412, 493],
[387, 442, 446, 386],
[494, 429, 386, 456],
[453, 387, 401, 492],
[389, 470, 451, 388],
[493, 410, 388, 457],
[471, 389, 462, 495],
[412, 390, 393, 399],
[462, 394, 391, 410],
[401, 392, 398, 429],
[396, 445, 461, 395],
[498, 496, 395, 460],
[456, 396, 431, 494],
[431, 397, 499, 496],
[399, 477, 481, 412],
[429, 483, 478, 401],
[410, 480, 490, 462],
[496, 497, 484, 431],
[489, 495, 491, 459],
[495, 460, 459, 471],
[460, 489, 498, 498],
[472, 472, 471, 491]]
C_r.table == table3
C_c.table == table3
# Group denoted by B₂,₄ from [2] Pg. 474
F, a, b = free_group("a, b")
B_2_4 = FpGroup(F, [a**4, b**4, (a*b)**4, (a**-1*b)**4, (a**2*b)**4, \
(a*b**2)**4, (a**2*b**2)**4, (a**-1*b*a*b)**4, (a*b**-1*a*b)**4])
C_r = coset_enumeration_r(B_2_4, [a])
C_c = coset_enumeration_c(B_2_4, [a])
index_r = 0
for i in range(len(C_r.p)):
if C_r.p[i] == i:
index_r += 1
assert index_r == 1024
index_c = 0
for i in range(len(C_c.p)):
if C_c.p[i] == i:
index_c += 1
assert index_c == 1024
# trivial Macdonald group G(2,2) from [2] Pg. 480
M = FpGroup(F, [b**-1*a**-1*b*a*b**-1*a*b*a**-2, a**-1*b**-1*a*b*a**-1*b*a*b**-2])
C_r = coset_enumeration_r(M, [a])
C_r.compress(); C_r.standardize()
C_c = coset_enumeration_c(M, [a])
C_c.compress(); C_c.standardize()
table4 = [[0, 0, 0, 0]]
assert C_r.table == table4
assert C_c.table == table4
def test_look_ahead():
# Section 3.2 [Test Example] Example (d) from [2]
F, a, b, c = free_group("a, b, c")
f = FpGroup(F, [a**11, b**5, c**4, (a*c)**3, b**2*c**-1*b**-1*c, a**4*b**-1*a**-1*b])
H = [c, b, c**2]
table0 = [[1, 2, 0, 0, 0, 0],
[3, 0, 4, 5, 6, 7],
[0, 8, 9, 10, 11, 12],
[5, 1, 10, 13, 14, 15],
[16, 5, 16, 1, 17, 18],
[4, 3, 1, 8, 19, 20],
[12, 21, 22, 23, 24, 1],
[25, 26, 27, 28, 1, 24],
[2, 10, 5, 16, 22, 28],
[10, 13, 13, 2, 29, 30]]
CosetTable.max_stack_size = 10
C_c = coset_enumeration_c(f, H)
C_c.compress(); C_c.standardize()
assert C_c.table[: 10] == table0
def test_modified_methods():
'''
Tests for modified coset table methods.
Example 5.7 from [1] Holt, D., Eick, B., O'Brien
"Handbook of Computational Group Theory".
'''
F, x, y = free_group("x, y")
f = FpGroup(F, [x**3, y**5, (x*y)**2])
H = [x*y, x**-1*y**-1*x*y*x]
C = CosetTable(f, H)
C.modified_define(0, x)
identity = C._grp.identity
a_0 = C._grp.generators[0]
a_1 = C._grp.generators[1]
assert C.P == [[identity, None, None, None],
[None, identity, None, None]]
assert C.table == [[1, None, None, None],
[None, 0, None, None]]
C.modified_define(1, x)
assert C.table == [[1, None, None, None],
[2, 0, None, None],
[None, 1, None, None]]
assert C.P == [[identity, None, None, None],
[identity, identity, None, None],
[None, identity, None, None]]
C.modified_scan(0, x**3, C._grp.identity, fill=False)
assert C.P == [[identity, identity, None, None],
[identity, identity, None, None],
[identity, identity, None, None]]
assert C.table == [[1, 2, None, None],
[2, 0, None, None],
[0, 1, None, None]]
C.modified_scan(0, x*y, C._grp.generators[0], fill=False)
assert C.P == [[identity, identity, None, a_0**-1],
[identity, identity, a_0, None],
[identity, identity, None, None]]
assert C.table == [[1, 2, None, 1],
[2, 0, 0, None],
[0, 1, None, None]]
C.modified_define(2, y**-1)
assert C.table == [[1, 2, None, 1],
[2, 0, 0, None],
[0, 1, None, 3],
[None, None, 2, None]]
assert C.P == [[identity, identity, None, a_0**-1],
[identity, identity, a_0, None],
[identity, identity, None, identity],
[None, None, identity, None]]
C.modified_scan(0, x**-1*y**-1*x*y*x, C._grp.generators[1])
assert C.table == [[1, 2, None, 1],
[2, 0, 0, None],
[0, 1, None, 3],
[3, 3, 2, None]]
assert C.P == [[identity, identity, None, a_0**-1],
[identity, identity, a_0, None],
[identity, identity, None, identity],
[a_1, a_1**-1, identity, None]]
C.modified_scan(2, (x*y)**2, C._grp.identity)
assert C.table == [[1, 2, 3, 1],
[2, 0, 0, None],
[0, 1, None, 3],
[3, 3, 2, 0]]
assert C.P == [[identity, identity, a_1**-1, a_0**-1],
[identity, identity, a_0, None],
[identity, identity, None, identity],
[a_1, a_1**-1, identity, a_1]]
C.modified_define(2, y)
assert C.table == [[1, 2, 3, 1],
[2, 0, 0, None],
[0, 1, 4, 3],
[3, 3, 2, 0],
[None, None, None, 2]]
assert C.P == [[identity, identity, a_1**-1, a_0**-1],
[identity, identity, a_0, None],
[identity, identity, identity, identity],
[a_1, a_1**-1, identity, a_1],
[None, None, None, identity]]
C.modified_scan(0, y**5, C._grp.identity)
assert C.table == [[1, 2, 3, 1], [2, 0, 0, 4], [0, 1, 4, 3], [3, 3, 2, 0], [None, None, 1, 2]]
assert C.P == [[identity, identity, a_1**-1, a_0**-1],
[identity, identity, a_0, a_0*a_1**-1],
[identity, identity, identity, identity],
[a_1, a_1**-1, identity, a_1],
[None, None, a_1*a_0**-1, identity]]
C.modified_scan(1, (x*y)**2, C._grp.identity)
assert C.table == [[1, 2, 3, 1],
[2, 0, 0, 4],
[0, 1, 4, 3],
[3, 3, 2, 0],
[4, 4, 1, 2]]
assert C.P == [[identity, identity, a_1**-1, a_0**-1],
[identity, identity, a_0, a_0*a_1**-1],
[identity, identity, identity, identity],
[a_1, a_1**-1, identity, a_1],
[a_0*a_1**-1, a_1*a_0**-1, a_1*a_0**-1, identity]]
# Modified coset enumeration test
f = FpGroup(F, [x**3, y**3, x**-1*y**-1*x*y])
C = coset_enumeration_r(f, [x])
C_m = modified_coset_enumeration_r(f, [x])
assert C_m.table == C.table
|
a809300e2fbc0b228103ebae30f5cc4d37eeb2cc3dd4abca789508aaa8d92c90
|
from sympy.combinatorics.subsets import Subset, ksubsets
from sympy.utilities.pytest import raises
def test_subset():
a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
assert a.next_binary() == Subset(['b'], ['a', 'b', 'c', 'd'])
assert a.prev_binary() == Subset(['c'], ['a', 'b', 'c', 'd'])
assert a.next_lexicographic() == Subset(['d'], ['a', 'b', 'c', 'd'])
assert a.prev_lexicographic() == Subset(['c'], ['a', 'b', 'c', 'd'])
assert a.next_gray() == Subset(['c'], ['a', 'b', 'c', 'd'])
assert a.prev_gray() == Subset(['d'], ['a', 'b', 'c', 'd'])
assert a.rank_binary == 3
assert a.rank_lexicographic == 14
assert a.rank_gray == 2
assert a.cardinality == 16
assert a.size == 2
assert Subset.bitlist_from_subset(a, ['a', 'b', 'c', 'd']) == '0011'
a = Subset([2, 5, 7], [1, 2, 3, 4, 5, 6, 7])
assert a.next_binary() == Subset([2, 5, 6], [1, 2, 3, 4, 5, 6, 7])
assert a.prev_binary() == Subset([2, 5], [1, 2, 3, 4, 5, 6, 7])
assert a.next_lexicographic() == Subset([2, 6], [1, 2, 3, 4, 5, 6, 7])
assert a.prev_lexicographic() == Subset([2, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7])
assert a.next_gray() == Subset([2, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7])
assert a.prev_gray() == Subset([2, 5], [1, 2, 3, 4, 5, 6, 7])
assert a.rank_binary == 37
assert a.rank_lexicographic == 93
assert a.rank_gray == 57
assert a.cardinality == 128
superset = ['a', 'b', 'c', 'd']
assert Subset.unrank_binary(4, superset).rank_binary == 4
assert Subset.unrank_gray(10, superset).rank_gray == 10
superset = [1, 2, 3, 4, 5, 6, 7, 8, 9]
assert Subset.unrank_binary(33, superset).rank_binary == 33
assert Subset.unrank_gray(25, superset).rank_gray == 25
a = Subset([], ['a', 'b', 'c', 'd'])
i = 1
while a.subset != Subset(['d'], ['a', 'b', 'c', 'd']).subset:
a = a.next_lexicographic()
i = i + 1
assert i == 16
i = 1
while a.subset != Subset([], ['a', 'b', 'c', 'd']).subset:
a = a.prev_lexicographic()
i = i + 1
assert i == 16
raises(ValueError, lambda: Subset(['a', 'b'], ['a']))
raises(ValueError, lambda: Subset(['a'], ['b', 'c']))
raises(ValueError, lambda: Subset.subset_from_bitlist(['a', 'b'], '010'))
def test_ksubsets():
assert list(ksubsets([1, 2, 3], 2)) == [(1, 2), (1, 3), (2, 3)]
assert list(ksubsets([1, 2, 3, 4, 5], 2)) == [(1, 2), (1, 3), (1, 4),
(1, 5), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5)]
|
f4b66aee57c6549f771b7c82f4daf6b8d847e3c33e3c502c5da4e717eac4a539
|
from sympy.concrete import Sum
from sympy.concrete.delta import deltaproduct as dp, deltasummation as ds, _extract_delta
from sympy.core import Eq, S, symbols, oo
from sympy.functions import KroneckerDelta as KD, Piecewise, piecewise_fold
from sympy.logic import And
from sympy.utilities.pytest import raises
i, j, k, l, m = symbols("i j k l m", integer=True, finite=True)
x, y = symbols("x y", commutative=False)
def test_deltaproduct_trivial():
assert dp(x, (j, 1, 0)) == 1
assert dp(x, (j, 1, 3)) == x**3
assert dp(x + y, (j, 1, 3)) == (x + y)**3
assert dp(x*y, (j, 1, 3)) == (x*y)**3
assert dp(KD(i, j), (k, 1, 3)) == KD(i, j)
assert dp(x*KD(i, j), (k, 1, 3)) == x**3*KD(i, j)
assert dp(x*y*KD(i, j), (k, 1, 3)) == (x*y)**3*KD(i, j)
def test_deltaproduct_basic():
assert dp(KD(i, j), (j, 1, 3)) == 0
assert dp(KD(i, j), (j, 1, 1)) == KD(i, 1)
assert dp(KD(i, j), (j, 2, 2)) == KD(i, 2)
assert dp(KD(i, j), (j, 3, 3)) == KD(i, 3)
assert dp(KD(i, j), (j, 1, k)) == KD(i, 1)*KD(k, 1) + KD(k, 0)
assert dp(KD(i, j), (j, k, 3)) == KD(i, 3)*KD(k, 3) + KD(k, 4)
assert dp(KD(i, j), (j, k, l)) == KD(i, l)*KD(k, l) + KD(k, l + 1)
def test_deltaproduct_mul_x_kd():
assert dp(x*KD(i, j), (j, 1, 3)) == 0
assert dp(x*KD(i, j), (j, 1, 1)) == x*KD(i, 1)
assert dp(x*KD(i, j), (j, 2, 2)) == x*KD(i, 2)
assert dp(x*KD(i, j), (j, 3, 3)) == x*KD(i, 3)
assert dp(x*KD(i, j), (j, 1, k)) == x*KD(i, 1)*KD(k, 1) + KD(k, 0)
assert dp(x*KD(i, j), (j, k, 3)) == x*KD(i, 3)*KD(k, 3) + KD(k, 4)
assert dp(x*KD(i, j), (j, k, l)) == x*KD(i, l)*KD(k, l) + KD(k, l + 1)
def test_deltaproduct_mul_add_x_y_kd():
assert dp((x + y)*KD(i, j), (j, 1, 3)) == 0
assert dp((x + y)*KD(i, j), (j, 1, 1)) == (x + y)*KD(i, 1)
assert dp((x + y)*KD(i, j), (j, 2, 2)) == (x + y)*KD(i, 2)
assert dp((x + y)*KD(i, j), (j, 3, 3)) == (x + y)*KD(i, 3)
assert dp((x + y)*KD(i, j), (j, 1, k)) == \
(x + y)*KD(i, 1)*KD(k, 1) + KD(k, 0)
assert dp((x + y)*KD(i, j), (j, k, 3)) == \
(x + y)*KD(i, 3)*KD(k, 3) + KD(k, 4)
assert dp((x + y)*KD(i, j), (j, k, l)) == \
(x + y)*KD(i, l)*KD(k, l) + KD(k, l + 1)
def test_deltaproduct_add_kd_kd():
assert dp(KD(i, k) + KD(j, k), (k, 1, 3)) == 0
assert dp(KD(i, k) + KD(j, k), (k, 1, 1)) == KD(i, 1) + KD(j, 1)
assert dp(KD(i, k) + KD(j, k), (k, 2, 2)) == KD(i, 2) + KD(j, 2)
assert dp(KD(i, k) + KD(j, k), (k, 3, 3)) == KD(i, 3) + KD(j, 3)
assert dp(KD(i, k) + KD(j, k), (k, 1, l)) == KD(l, 0) + \
KD(i, 1)*KD(l, 1) + KD(j, 1)*KD(l, 1) + \
KD(i, 1)*KD(j, 2)*KD(l, 2) + KD(j, 1)*KD(i, 2)*KD(l, 2)
assert dp(KD(i, k) + KD(j, k), (k, l, 3)) == KD(l, 4) + \
KD(i, 3)*KD(l, 3) + KD(j, 3)*KD(l, 3) + \
KD(i, 2)*KD(j, 3)*KD(l, 2) + KD(i, 3)*KD(j, 2)*KD(l, 2)
assert dp(KD(i, k) + KD(j, k), (k, l, m)) == KD(l, m + 1) + \
KD(i, m)*KD(l, m) + KD(j, m)*KD(l, m) + \
KD(i, m)*KD(j, m - 1)*KD(l, m - 1) + KD(i, m - 1)*KD(j, m)*KD(l, m - 1)
def test_deltaproduct_mul_x_add_kd_kd():
assert dp(x*(KD(i, k) + KD(j, k)), (k, 1, 3)) == 0
assert dp(x*(KD(i, k) + KD(j, k)), (k, 1, 1)) == x*(KD(i, 1) + KD(j, 1))
assert dp(x*(KD(i, k) + KD(j, k)), (k, 2, 2)) == x*(KD(i, 2) + KD(j, 2))
assert dp(x*(KD(i, k) + KD(j, k)), (k, 3, 3)) == x*(KD(i, 3) + KD(j, 3))
assert dp(x*(KD(i, k) + KD(j, k)), (k, 1, l)) == KD(l, 0) + \
x*KD(i, 1)*KD(l, 1) + x*KD(j, 1)*KD(l, 1) + \
x**2*KD(i, 1)*KD(j, 2)*KD(l, 2) + x**2*KD(j, 1)*KD(i, 2)*KD(l, 2)
assert dp(x*(KD(i, k) + KD(j, k)), (k, l, 3)) == KD(l, 4) + \
x*KD(i, 3)*KD(l, 3) + x*KD(j, 3)*KD(l, 3) + \
x**2*KD(i, 2)*KD(j, 3)*KD(l, 2) + x**2*KD(i, 3)*KD(j, 2)*KD(l, 2)
assert dp(x*(KD(i, k) + KD(j, k)), (k, l, m)) == KD(l, m + 1) + \
x*KD(i, m)*KD(l, m) + x*KD(j, m)*KD(l, m) + \
x**2*KD(i, m - 1)*KD(j, m)*KD(l, m - 1) + \
x**2*KD(i, m)*KD(j, m - 1)*KD(l, m - 1)
def test_deltaproduct_mul_add_x_y_add_kd_kd():
assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 3)) == 0
assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 1)) == \
(x + y)*(KD(i, 1) + KD(j, 1))
assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 2, 2)) == \
(x + y)*(KD(i, 2) + KD(j, 2))
assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 3, 3)) == \
(x + y)*(KD(i, 3) + KD(j, 3))
assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, 1, l)) == KD(l, 0) + \
(x + y)*KD(i, 1)*KD(l, 1) + (x + y)*KD(j, 1)*KD(l, 1) + \
(x + y)**2*KD(i, 1)*KD(j, 2)*KD(l, 2) + \
(x + y)**2*KD(j, 1)*KD(i, 2)*KD(l, 2)
assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, l, 3)) == KD(l, 4) + \
(x + y)*KD(i, 3)*KD(l, 3) + (x + y)*KD(j, 3)*KD(l, 3) + \
(x + y)**2*KD(i, 2)*KD(j, 3)*KD(l, 2) + \
(x + y)**2*KD(i, 3)*KD(j, 2)*KD(l, 2)
assert dp((x + y)*(KD(i, k) + KD(j, k)), (k, l, m)) == KD(l, m + 1) + \
(x + y)*KD(i, m)*KD(l, m) + (x + y)*KD(j, m)*KD(l, m) + \
(x + y)**2*KD(i, m - 1)*KD(j, m)*KD(l, m - 1) + \
(x + y)**2*KD(i, m)*KD(j, m - 1)*KD(l, m - 1)
def test_deltaproduct_add_mul_x_y_mul_x_kd():
assert dp(x*y + x*KD(i, j), (j, 1, 3)) == (x*y)**3 + \
x*(x*y)**2*KD(i, 1) + (x*y)*x*(x*y)*KD(i, 2) + (x*y)**2*x*KD(i, 3)
assert dp(x*y + x*KD(i, j), (j, 1, 1)) == x*y + x*KD(i, 1)
assert dp(x*y + x*KD(i, j), (j, 2, 2)) == x*y + x*KD(i, 2)
assert dp(x*y + x*KD(i, j), (j, 3, 3)) == x*y + x*KD(i, 3)
assert dp(x*y + x*KD(i, j), (j, 1, k)) == \
(x*y)**k + Piecewise(
((x*y)**(i - 1)*x*(x*y)**(k - i), And(S(1) <= i, i <= k)),
(0, True)
)
assert dp(x*y + x*KD(i, j), (j, k, 3)) == \
(x*y)**(-k + 4) + Piecewise(
((x*y)**(i - k)*x*(x*y)**(3 - i), And(k <= i, i <= 3)),
(0, True)
)
assert dp(x*y + x*KD(i, j), (j, k, l)) == \
(x*y)**(-k + l + 1) + Piecewise(
((x*y)**(i - k)*x*(x*y)**(l - i), And(k <= i, i <= l)),
(0, True)
)
def test_deltaproduct_mul_x_add_y_kd():
assert dp(x*(y + KD(i, j)), (j, 1, 3)) == (x*y)**3 + \
x*(x*y)**2*KD(i, 1) + (x*y)*x*(x*y)*KD(i, 2) + (x*y)**2*x*KD(i, 3)
assert dp(x*(y + KD(i, j)), (j, 1, 1)) == x*(y + KD(i, 1))
assert dp(x*(y + KD(i, j)), (j, 2, 2)) == x*(y + KD(i, 2))
assert dp(x*(y + KD(i, j)), (j, 3, 3)) == x*(y + KD(i, 3))
assert dp(x*(y + KD(i, j)), (j, 1, k)) == \
(x*y)**k + Piecewise(
((x*y)**(i - 1)*x*(x*y)**(k - i), And(S(1) <= i, i <= k)),
(0, True)
)
assert dp(x*(y + KD(i, j)), (j, k, 3)) == \
(x*y)**(-k + 4) + Piecewise(
((x*y)**(i - k)*x*(x*y)**(3 - i), And(k <= i, i <= 3)),
(0, True)
)
assert dp(x*(y + KD(i, j)), (j, k, l)) == \
(x*y)**(-k + l + 1) + Piecewise(
((x*y)**(i - k)*x*(x*y)**(l - i), And(k <= i, i <= l)),
(0, True)
)
def test_deltaproduct_mul_x_add_y_twokd():
assert dp(x*(y + 2*KD(i, j)), (j, 1, 3)) == (x*y)**3 + \
2*x*(x*y)**2*KD(i, 1) + 2*x*y*x*x*y*KD(i, 2) + 2*(x*y)**2*x*KD(i, 3)
assert dp(x*(y + 2*KD(i, j)), (j, 1, 1)) == x*(y + 2*KD(i, 1))
assert dp(x*(y + 2*KD(i, j)), (j, 2, 2)) == x*(y + 2*KD(i, 2))
assert dp(x*(y + 2*KD(i, j)), (j, 3, 3)) == x*(y + 2*KD(i, 3))
assert dp(x*(y + 2*KD(i, j)), (j, 1, k)) == \
(x*y)**k + Piecewise(
(2*(x*y)**(i - 1)*x*(x*y)**(k - i), And(S(1) <= i, i <= k)),
(0, True)
)
assert dp(x*(y + 2*KD(i, j)), (j, k, 3)) == \
(x*y)**(-k + 4) + Piecewise(
(2*(x*y)**(i - k)*x*(x*y)**(3 - i), And(k <= i, i <= 3)),
(0, True)
)
assert dp(x*(y + 2*KD(i, j)), (j, k, l)) == \
(x*y)**(-k + l + 1) + Piecewise(
(2*(x*y)**(i - k)*x*(x*y)**(l - i), And(k <= i, i <= l)),
(0, True)
)
def test_deltaproduct_mul_add_x_y_add_y_kd():
assert dp((x + y)*(y + KD(i, j)), (j, 1, 3)) == ((x + y)*y)**3 + \
(x + y)*((x + y)*y)**2*KD(i, 1) + \
(x + y)*y*(x + y)**2*y*KD(i, 2) + \
((x + y)*y)**2*(x + y)*KD(i, 3)
assert dp((x + y)*(y + KD(i, j)), (j, 1, 1)) == (x + y)*(y + KD(i, 1))
assert dp((x + y)*(y + KD(i, j)), (j, 2, 2)) == (x + y)*(y + KD(i, 2))
assert dp((x + y)*(y + KD(i, j)), (j, 3, 3)) == (x + y)*(y + KD(i, 3))
assert dp((x + y)*(y + KD(i, j)), (j, 1, k)) == \
((x + y)*y)**k + Piecewise(
(((x + y)*y)**(i - 1)*(x + y)*((x + y)*y)**(k - i),
And(S(1) <= i, i <= k)),
(0, True)
)
assert dp((x + y)*(y + KD(i, j)), (j, k, 3)) == \
((x + y)*y)**(-k + 4) + Piecewise(
(((x + y)*y)**(i - k)*(x + y)*((x + y)*y)**(3 - i),
And(k <= i, i <= 3)),
(0, True)
)
assert dp((x + y)*(y + KD(i, j)), (j, k, l)) == \
((x + y)*y)**(-k + l + 1) + Piecewise(
(((x + y)*y)**(i - k)*(x + y)*((x + y)*y)**(l - i),
And(k <= i, i <= l)),
(0, True)
)
def test_deltaproduct_mul_add_x_kd_add_y_kd():
assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 1, 3)) == \
KD(i, 1)*(KD(i, k) + x)*((KD(i, k) + x)*y)**2 + \
KD(i, 2)*(KD(i, k) + x)*y*(KD(i, k) + x)**2*y + \
KD(i, 3)*((KD(i, k) + x)*y)**2*(KD(i, k) + x) + \
((KD(i, k) + x)*y)**3
assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 1, 1)) == \
(x + KD(i, k))*(y + KD(i, 1))
assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 2, 2)) == \
(x + KD(i, k))*(y + KD(i, 2))
assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 3, 3)) == \
(x + KD(i, k))*(y + KD(i, 3))
assert dp((x + KD(i, k))*(y + KD(i, j)), (j, 1, k)) == \
((x + KD(i, k))*y)**k + Piecewise(
(((x + KD(i, k))*y)**(i - 1)*(x + KD(i, k))*
((x + KD(i, k))*y)**(-i + k), And(S(1) <= i, i <= k)),
(0, True)
)
assert dp((x + KD(i, k))*(y + KD(i, j)), (j, k, 3)) == \
((x + KD(i, k))*y)**(4 - k) + Piecewise(
(((x + KD(i, k))*y)**(i - k)*(x + KD(i, k))*
((x + KD(i, k))*y)**(-i + 3), And(k <= i, i <= 3)),
(0, True)
)
assert dp((x + KD(i, k))*(y + KD(i, j)), (j, k, l)) == \
((x + KD(i, k))*y)**(-k + l + 1) + Piecewise(
(((x + KD(i, k))*y)**(i - k)*(x + KD(i, k))*
((x + KD(i, k))*y)**(-i + l), And(k <= i, i <= l)),
(0, True)
)
def test_deltasummation_trivial():
assert ds(x, (j, 1, 0)) == 0
assert ds(x, (j, 1, 3)) == 3*x
assert ds(x + y, (j, 1, 3)) == 3*(x + y)
assert ds(x*y, (j, 1, 3)) == 3*x*y
assert ds(KD(i, j), (k, 1, 3)) == 3*KD(i, j)
assert ds(x*KD(i, j), (k, 1, 3)) == 3*x*KD(i, j)
assert ds(x*y*KD(i, j), (k, 1, 3)) == 3*x*y*KD(i, j)
def test_deltasummation_basic_numerical():
n = symbols('n', integer=True, nonzero=True)
assert ds(KD(n, 0), (n, 1, 3)) == 0
# return unevaluated, until it gets implemented
assert ds(KD(i**2, j**2), (j, -oo, oo)) == \
Sum(KD(i**2, j**2), (j, -oo, oo))
assert Piecewise((KD(i, k), And(S(1) <= i, i <= 3)), (0, True)) == \
ds(KD(i, j)*KD(j, k), (j, 1, 3)) == \
ds(KD(j, k)*KD(i, j), (j, 1, 3))
assert ds(KD(i, k), (k, -oo, oo)) == 1
assert ds(KD(i, k), (k, 0, oo)) == Piecewise((1, S(0) <= i), (0, True))
assert ds(KD(i, k), (k, 1, 3)) == \
Piecewise((1, And(S(1) <= i, i <= 3)), (0, True))
assert ds(k*KD(i, j)*KD(j, k), (k, -oo, oo)) == j*KD(i, j)
assert ds(j*KD(i, j), (j, -oo, oo)) == i
assert ds(i*KD(i, j), (i, -oo, oo)) == j
assert ds(x, (i, 1, 3)) == 3*x
assert ds((i + j)*KD(i, j), (j, -oo, oo)) == 2*i
def test_deltasummation_basic_symbolic():
assert ds(KD(i, j), (j, 1, 3)) == \
Piecewise((1, And(S(1) <= i, i <= 3)), (0, True))
assert ds(KD(i, j), (j, 1, 1)) == Piecewise((1, Eq(i, 1)), (0, True))
assert ds(KD(i, j), (j, 2, 2)) == Piecewise((1, Eq(i, 2)), (0, True))
assert ds(KD(i, j), (j, 3, 3)) == Piecewise((1, Eq(i, 3)), (0, True))
assert ds(KD(i, j), (j, 1, k)) == \
Piecewise((1, And(S(1) <= i, i <= k)), (0, True))
assert ds(KD(i, j), (j, k, 3)) == \
Piecewise((1, And(k <= i, i <= 3)), (0, True))
assert ds(KD(i, j), (j, k, l)) == \
Piecewise((1, And(k <= i, i <= l)), (0, True))
def test_deltasummation_mul_x_kd():
assert ds(x*KD(i, j), (j, 1, 3)) == \
Piecewise((x, And(S(1) <= i, i <= 3)), (0, True))
assert ds(x*KD(i, j), (j, 1, 1)) == Piecewise((x, Eq(i, 1)), (0, True))
assert ds(x*KD(i, j), (j, 2, 2)) == Piecewise((x, Eq(i, 2)), (0, True))
assert ds(x*KD(i, j), (j, 3, 3)) == Piecewise((x, Eq(i, 3)), (0, True))
assert ds(x*KD(i, j), (j, 1, k)) == \
Piecewise((x, And(S(1) <= i, i <= k)), (0, True))
assert ds(x*KD(i, j), (j, k, 3)) == \
Piecewise((x, And(k <= i, i <= 3)), (0, True))
assert ds(x*KD(i, j), (j, k, l)) == \
Piecewise((x, And(k <= i, i <= l)), (0, True))
def test_deltasummation_mul_add_x_y_kd():
assert ds((x + y)*KD(i, j), (j, 1, 3)) == \
Piecewise((x + y, And(S(1) <= i, i <= 3)), (0, True))
assert ds((x + y)*KD(i, j), (j, 1, 1)) == \
Piecewise((x + y, Eq(i, 1)), (0, True))
assert ds((x + y)*KD(i, j), (j, 2, 2)) == \
Piecewise((x + y, Eq(i, 2)), (0, True))
assert ds((x + y)*KD(i, j), (j, 3, 3)) == \
Piecewise((x + y, Eq(i, 3)), (0, True))
assert ds((x + y)*KD(i, j), (j, 1, k)) == \
Piecewise((x + y, And(S(1) <= i, i <= k)), (0, True))
assert ds((x + y)*KD(i, j), (j, k, 3)) == \
Piecewise((x + y, And(k <= i, i <= 3)), (0, True))
assert ds((x + y)*KD(i, j), (j, k, l)) == \
Piecewise((x + y, And(k <= i, i <= l)), (0, True))
def test_deltasummation_add_kd_kd():
assert ds(KD(i, k) + KD(j, k), (k, 1, 3)) == piecewise_fold(
Piecewise((1, And(S(1) <= i, i <= 3)), (0, True)) +
Piecewise((1, And(S(1) <= j, j <= 3)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, 1, 1)) == piecewise_fold(
Piecewise((1, Eq(i, 1)), (0, True)) +
Piecewise((1, Eq(j, 1)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, 2, 2)) == piecewise_fold(
Piecewise((1, Eq(i, 2)), (0, True)) +
Piecewise((1, Eq(j, 2)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, 3, 3)) == piecewise_fold(
Piecewise((1, Eq(i, 3)), (0, True)) +
Piecewise((1, Eq(j, 3)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, 1, l)) == piecewise_fold(
Piecewise((1, And(S(1) <= i, i <= l)), (0, True)) +
Piecewise((1, And(S(1) <= j, j <= l)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, l, 3)) == piecewise_fold(
Piecewise((1, And(l <= i, i <= 3)), (0, True)) +
Piecewise((1, And(l <= j, j <= 3)), (0, True)))
assert ds(KD(i, k) + KD(j, k), (k, l, m)) == piecewise_fold(
Piecewise((1, And(l <= i, i <= m)), (0, True)) +
Piecewise((1, And(l <= j, j <= m)), (0, True)))
def test_deltasummation_add_mul_x_kd_kd():
assert ds(x*KD(i, k) + KD(j, k), (k, 1, 3)) == piecewise_fold(
Piecewise((x, And(S(1) <= i, i <= 3)), (0, True)) +
Piecewise((1, And(S(1) <= j, j <= 3)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, 1, 1)) == piecewise_fold(
Piecewise((x, Eq(i, 1)), (0, True)) +
Piecewise((1, Eq(j, 1)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, 2, 2)) == piecewise_fold(
Piecewise((x, Eq(i, 2)), (0, True)) +
Piecewise((1, Eq(j, 2)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, 3, 3)) == piecewise_fold(
Piecewise((x, Eq(i, 3)), (0, True)) +
Piecewise((1, Eq(j, 3)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, 1, l)) == piecewise_fold(
Piecewise((x, And(S(1) <= i, i <= l)), (0, True)) +
Piecewise((1, And(S(1) <= j, j <= l)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, l, 3)) == piecewise_fold(
Piecewise((x, And(l <= i, i <= 3)), (0, True)) +
Piecewise((1, And(l <= j, j <= 3)), (0, True)))
assert ds(x*KD(i, k) + KD(j, k), (k, l, m)) == piecewise_fold(
Piecewise((x, And(l <= i, i <= m)), (0, True)) +
Piecewise((1, And(l <= j, j <= m)), (0, True)))
def test_deltasummation_mul_x_add_kd_kd():
assert ds(x*(KD(i, k) + KD(j, k)), (k, 1, 3)) == piecewise_fold(
Piecewise((x, And(S(1) <= i, i <= 3)), (0, True)) +
Piecewise((x, And(S(1) <= j, j <= 3)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, 1, 1)) == piecewise_fold(
Piecewise((x, Eq(i, 1)), (0, True)) +
Piecewise((x, Eq(j, 1)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, 2, 2)) == piecewise_fold(
Piecewise((x, Eq(i, 2)), (0, True)) +
Piecewise((x, Eq(j, 2)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, 3, 3)) == piecewise_fold(
Piecewise((x, Eq(i, 3)), (0, True)) +
Piecewise((x, Eq(j, 3)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, 1, l)) == piecewise_fold(
Piecewise((x, And(S(1) <= i, i <= l)), (0, True)) +
Piecewise((x, And(S(1) <= j, j <= l)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, l, 3)) == piecewise_fold(
Piecewise((x, And(l <= i, i <= 3)), (0, True)) +
Piecewise((x, And(l <= j, j <= 3)), (0, True)))
assert ds(x*(KD(i, k) + KD(j, k)), (k, l, m)) == piecewise_fold(
Piecewise((x, And(l <= i, i <= m)), (0, True)) +
Piecewise((x, And(l <= j, j <= m)), (0, True)))
def test_deltasummation_mul_add_x_y_add_kd_kd():
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 3)) == piecewise_fold(
Piecewise((x + y, And(S(1) <= i, i <= 3)), (0, True)) +
Piecewise((x + y, And(S(1) <= j, j <= 3)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 1, 1)) == piecewise_fold(
Piecewise((x + y, Eq(i, 1)), (0, True)) +
Piecewise((x + y, Eq(j, 1)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 2, 2)) == piecewise_fold(
Piecewise((x + y, Eq(i, 2)), (0, True)) +
Piecewise((x + y, Eq(j, 2)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 3, 3)) == piecewise_fold(
Piecewise((x + y, Eq(i, 3)), (0, True)) +
Piecewise((x + y, Eq(j, 3)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, 1, l)) == piecewise_fold(
Piecewise((x + y, And(S(1) <= i, i <= l)), (0, True)) +
Piecewise((x + y, And(S(1) <= j, j <= l)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, l, 3)) == piecewise_fold(
Piecewise((x + y, And(l <= i, i <= 3)), (0, True)) +
Piecewise((x + y, And(l <= j, j <= 3)), (0, True)))
assert ds((x + y)*(KD(i, k) + KD(j, k)), (k, l, m)) == piecewise_fold(
Piecewise((x + y, And(l <= i, i <= m)), (0, True)) +
Piecewise((x + y, And(l <= j, j <= m)), (0, True)))
def test_deltasummation_add_mul_x_y_mul_x_kd():
assert ds(x*y + x*KD(i, j), (j, 1, 3)) == \
Piecewise((3*x*y + x, And(S(1) <= i, i <= 3)), (3*x*y, True))
assert ds(x*y + x*KD(i, j), (j, 1, 1)) == \
Piecewise((x*y + x, Eq(i, 1)), (x*y, True))
assert ds(x*y + x*KD(i, j), (j, 2, 2)) == \
Piecewise((x*y + x, Eq(i, 2)), (x*y, True))
assert ds(x*y + x*KD(i, j), (j, 3, 3)) == \
Piecewise((x*y + x, Eq(i, 3)), (x*y, True))
assert ds(x*y + x*KD(i, j), (j, 1, k)) == \
Piecewise((k*x*y + x, And(S(1) <= i, i <= k)), (k*x*y, True))
assert ds(x*y + x*KD(i, j), (j, k, 3)) == \
Piecewise(((4 - k)*x*y + x, And(k <= i, i <= 3)), ((4 - k)*x*y, True))
assert ds(x*y + x*KD(i, j), (j, k, l)) == Piecewise(
((l - k + 1)*x*y + x, And(k <= i, i <= l)), ((l - k + 1)*x*y, True))
def test_deltasummation_mul_x_add_y_kd():
assert ds(x*(y + KD(i, j)), (j, 1, 3)) == \
Piecewise((3*x*y + x, And(S(1) <= i, i <= 3)), (3*x*y, True))
assert ds(x*(y + KD(i, j)), (j, 1, 1)) == \
Piecewise((x*y + x, Eq(i, 1)), (x*y, True))
assert ds(x*(y + KD(i, j)), (j, 2, 2)) == \
Piecewise((x*y + x, Eq(i, 2)), (x*y, True))
assert ds(x*(y + KD(i, j)), (j, 3, 3)) == \
Piecewise((x*y + x, Eq(i, 3)), (x*y, True))
assert ds(x*(y + KD(i, j)), (j, 1, k)) == \
Piecewise((k*x*y + x, And(S(1) <= i, i <= k)), (k*x*y, True))
assert ds(x*(y + KD(i, j)), (j, k, 3)) == \
Piecewise(((4 - k)*x*y + x, And(k <= i, i <= 3)), ((4 - k)*x*y, True))
assert ds(x*(y + KD(i, j)), (j, k, l)) == Piecewise(
((l - k + 1)*x*y + x, And(k <= i, i <= l)), ((l - k + 1)*x*y, True))
def test_deltasummation_mul_x_add_y_twokd():
assert ds(x*(y + 2*KD(i, j)), (j, 1, 3)) == \
Piecewise((3*x*y + 2*x, And(S(1) <= i, i <= 3)), (3*x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, 1, 1)) == \
Piecewise((x*y + 2*x, Eq(i, 1)), (x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, 2, 2)) == \
Piecewise((x*y + 2*x, Eq(i, 2)), (x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, 3, 3)) == \
Piecewise((x*y + 2*x, Eq(i, 3)), (x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, 1, k)) == \
Piecewise((k*x*y + 2*x, And(S(1) <= i, i <= k)), (k*x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, k, 3)) == Piecewise(
((4 - k)*x*y + 2*x, And(k <= i, i <= 3)), ((4 - k)*x*y, True))
assert ds(x*(y + 2*KD(i, j)), (j, k, l)) == Piecewise(
((l - k + 1)*x*y + 2*x, And(k <= i, i <= l)), ((l - k + 1)*x*y, True))
def test_deltasummation_mul_add_x_y_add_y_kd():
assert ds((x + y)*(y + KD(i, j)), (j, 1, 3)) == Piecewise(
(3*(x + y)*y + x + y, And(S(1) <= i, i <= 3)), (3*(x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, 1, 1)) == \
Piecewise(((x + y)*y + x + y, Eq(i, 1)), ((x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, 2, 2)) == \
Piecewise(((x + y)*y + x + y, Eq(i, 2)), ((x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, 3, 3)) == \
Piecewise(((x + y)*y + x + y, Eq(i, 3)), ((x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, 1, k)) == Piecewise(
(k*(x + y)*y + x + y, And(S(1) <= i, i <= k)), (k*(x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, k, 3)) == Piecewise(
((4 - k)*(x + y)*y + x + y, And(k <= i, i <= 3)),
((4 - k)*(x + y)*y, True))
assert ds((x + y)*(y + KD(i, j)), (j, k, l)) == Piecewise(
((l - k + 1)*(x + y)*y + x + y, And(k <= i, i <= l)),
((l - k + 1)*(x + y)*y, True))
def test_deltasummation_mul_add_x_kd_add_y_kd():
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 1, 3)) == piecewise_fold(
Piecewise((KD(i, k) + x, And(S(1) <= i, i <= 3)), (0, True)) +
3*(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 1, 1)) == piecewise_fold(
Piecewise((KD(i, k) + x, Eq(i, 1)), (0, True)) +
(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 2, 2)) == piecewise_fold(
Piecewise((KD(i, k) + x, Eq(i, 2)), (0, True)) +
(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 3, 3)) == piecewise_fold(
Piecewise((KD(i, k) + x, Eq(i, 3)), (0, True)) +
(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, 1, k)) == piecewise_fold(
Piecewise((KD(i, k) + x, And(S(1) <= i, i <= k)), (0, True)) +
k*(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, k, 3)) == piecewise_fold(
Piecewise((KD(i, k) + x, And(k <= i, i <= 3)), (0, True)) +
(4 - k)*(KD(i, k) + x)*y)
assert ds((x + KD(i, k))*(y + KD(i, j)), (j, k, l)) == piecewise_fold(
Piecewise((KD(i, k) + x, And(k <= i, i <= l)), (0, True)) +
(l - k + 1)*(KD(i, k) + x)*y)
def test_extract_delta():
raises(ValueError, lambda: _extract_delta(KD(i, j) + KD(k, l), i))
|
87976fe50a3d53c9bebef01cf3115cfcf9ba3b3cda0e9753677340fe1d2fb2aa
|
from sympy.concrete.guess import (
find_simple_recurrence_vector,
find_simple_recurrence,
rationalize,
guess_generating_function_rational,
guess_generating_function,
guess
)
from sympy import (Function, Symbol, sympify, Rational, symbols, S,
fibonacci, factorial, exp, Product, RisingFactorial)
def test_find_simple_recurrence_vector():
assert find_simple_recurrence_vector(
[fibonacci(k) for k in range(12)]) == [1, -1, -1]
def test_find_simple_recurrence():
a = Function('a')
n = Symbol('n')
assert find_simple_recurrence([fibonacci(k) for k in range(12)]) == (
-a(n) - a(n + 1) + a(n + 2))
f = Function('a')
i = Symbol('n')
a = [1, 1, 1]
for k in range(15): a.append(5*a[-1]-3*a[-2]+8*a[-3])
assert find_simple_recurrence(a, A=f, N=i) == (
-8*f(i) + 3*f(i + 1) - 5*f(i + 2) + f(i + 3))
assert find_simple_recurrence([0, 2, 15, 74, 12, 3, 0,
1, 2, 85, 4, 5, 63]) == 0
def test_rationalize():
from mpmath import cos, pi, mpf
assert rationalize(cos(pi/3)) == Rational(1, 2)
assert rationalize(mpf("0.333333333333333")) == Rational(1, 3)
assert rationalize(mpf("-0.333333333333333")) == Rational(-1, 3)
assert rationalize(pi, maxcoeff = 250) == Rational(355, 113)
def test_guess_generating_function_rational():
x = Symbol('x')
assert guess_generating_function_rational([fibonacci(k)
for k in range(5, 15)]) == ((3*x + 5)/(-x**2 - x + 1))
def test_guess_generating_function():
x = Symbol('x')
assert guess_generating_function([fibonacci(k)
for k in range(5, 15)])['ogf'] == ((3*x + 5)/(-x**2 - x + 1))
assert guess_generating_function(
[1, 2, 5, 14, 41, 124, 383, 1200, 3799, 12122, 38919])['ogf'] == (
(1/(x**4 + 2*x**2 - 4*x + 1))**Rational(1, 2))
assert guess_generating_function(sympify(
"[3/2, 11/2, 0, -121/2, -363/2, 121, 4719/2, 11495/2, -8712, -178717/2]")
)['ogf'] == (x + Rational(3, 2))/(11*x**2 - 3*x + 1)
assert guess_generating_function([factorial(k) for k in range(12)],
types=['egf'])['egf'] == 1/(-x + 1)
assert guess_generating_function([k+1 for k in range(12)],
types=['egf']) == {'egf': (x + 1)*exp(x), 'lgdegf': (x + 2)/(x + 1)}
def test_guess():
i0, i1 = symbols('i0 i1')
assert guess([1, 2, 6, 24, 120], evaluate=False) == [Product(i1 + 1, (i1, 1, i0 - 1))]
assert guess([1, 2, 6, 24, 120]) == [RisingFactorial(2, i0 - 1)]
assert guess([1, 2, 7, 42, 429, 7436, 218348, 10850216], niter=4) == [
2**(i0 - 1)*(S(27)/16)**(i0**2/2 - 3*i0/2 +
1)*Product(RisingFactorial(S(5)/3, i1 - 1)*RisingFactorial(S(7)/3, i1
- 1)/(RisingFactorial(S(3)/2, i1 - 1)*RisingFactorial(S(5)/2, i1 -
1)), (i1, 1, i0 - 1))]
|
bc64d511d50afdf6ff192fec77be933efce04b8a8612fd56b75380f1e1c96973
|
"""Tests for Gosper's algorithm for hypergeometric summation. """
from sympy import binomial, factorial, gamma, Poly, S, simplify, sqrt, exp, \
log, Symbol, pi
from sympy.abc import a, b, j, k, m, n, r, x
from sympy.concrete.gosper import gosper_normal, gosper_sum, gosper_term
def test_gosper_normal():
eq = 4*n + 5, 2*(4*n + 1)*(2*n + 3), n
assert gosper_normal(*eq) == \
(Poly(S(1)/4, n), Poly(n + S(3)/2), Poly(n + S(1)/4))
assert gosper_normal(*eq, polys=False) == \
(S(1)/4, n + S(3)/2, n + S(1)/4)
def test_gosper_term():
assert gosper_term((4*k + 1)*factorial(
k)/factorial(2*k + 1), k) == (-k - S(1)/2)/(k + S(1)/4)
def test_gosper_sum():
assert gosper_sum(1, (k, 0, n)) == 1 + n
assert gosper_sum(k, (k, 0, n)) == n*(1 + n)/2
assert gosper_sum(k**2, (k, 0, n)) == n*(1 + n)*(1 + 2*n)/6
assert gosper_sum(k**3, (k, 0, n)) == n**2*(1 + n)**2/4
assert gosper_sum(2**k, (k, 0, n)) == 2*2**n - 1
assert gosper_sum(factorial(k), (k, 0, n)) is None
assert gosper_sum(binomial(n, k), (k, 0, n)) is None
assert gosper_sum(factorial(k)/k**2, (k, 0, n)) is None
assert gosper_sum((k - 3)*factorial(k), (k, 0, n)) is None
assert gosper_sum(k*factorial(k), k) == factorial(k)
assert gosper_sum(
k*factorial(k), (k, 0, n)) == n*factorial(n) + factorial(n) - 1
assert gosper_sum((-1)**k*binomial(n, k), (k, 0, n)) == 0
assert gosper_sum((
-1)**k*binomial(n, k), (k, 0, m)) == -(-1)**m*(m - n)*binomial(n, m)/n
assert gosper_sum((4*k + 1)*factorial(k)/factorial(2*k + 1), (k, 0, n)) == \
(2*factorial(2*n + 1) - factorial(n))/factorial(2*n + 1)
# issue 6033:
assert gosper_sum(
n*(n + a + b)*a**n*b**n/(factorial(n + a)*factorial(n + b)), \
(n, 0, m)).simplify() == -exp(m*log(a) + m*log(b))*gamma(a + 1) \
*gamma(b + 1)/(gamma(a)*gamma(b)*gamma(a + m + 1)*gamma(b + m + 1)) \
+ 1/(gamma(a)*gamma(b))
def test_gosper_sum_indefinite():
assert gosper_sum(k, k) == k*(k - 1)/2
assert gosper_sum(k**2, k) == k*(k - 1)*(2*k - 1)/6
assert gosper_sum(1/(k*(k + 1)), k) == -1/k
assert gosper_sum(-(27*k**4 + 158*k**3 + 430*k**2 + 678*k + 445)*gamma(2*k
+ 4)/(3*(3*k + 7)*gamma(3*k + 6)), k) == \
(3*k + 5)*(k**2 + 2*k + 5)*gamma(2*k + 4)/gamma(3*k + 6)
def test_gosper_sum_parametric():
assert gosper_sum(binomial(S(1)/2, m - j + 1)*binomial(S(1)/2, m + j), (j, 1, n)) == \
n*(1 + m - n)*(-1 + 2*m + 2*n)*binomial(S(1)/2, 1 + m - n)* \
binomial(S(1)/2, m + n)/(m*(1 + 2*m))
def test_gosper_sum_algebraic():
assert gosper_sum(
n**2 + sqrt(2), (n, 0, m)) == (m + 1)*(2*m**2 + m + 6*sqrt(2))/6
def test_gosper_sum_iterated():
f1 = binomial(2*k, k)/4**k
f2 = (1 + 2*n)*binomial(2*n, n)/4**n
f3 = (1 + 2*n)*(3 + 2*n)*binomial(2*n, n)/(3*4**n)
f4 = (1 + 2*n)*(3 + 2*n)*(5 + 2*n)*binomial(2*n, n)/(15*4**n)
f5 = (1 + 2*n)*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)*binomial(2*n, n)/(105*4**n)
assert gosper_sum(f1, (k, 0, n)) == f2
assert gosper_sum(f2, (n, 0, n)) == f3
assert gosper_sum(f3, (n, 0, n)) == f4
assert gosper_sum(f4, (n, 0, n)) == f5
# the AeqB tests test expressions given in
# www.math.upenn.edu/~wilf/AeqB.pdf
def test_gosper_sum_AeqB_part1():
f1a = n**4
f1b = n**3*2**n
f1c = 1/(n**2 + sqrt(5)*n - 1)
f1d = n**4*4**n/binomial(2*n, n)
f1e = factorial(3*n)/(factorial(n)*factorial(n + 1)*factorial(n + 2)*27**n)
f1f = binomial(2*n, n)**2/((n + 1)*4**(2*n))
f1g = (4*n - 1)*binomial(2*n, n)**2/((2*n - 1)**2*4**(2*n))
f1h = n*factorial(n - S(1)/2)**2/factorial(n + 1)**2
g1a = m*(m + 1)*(2*m + 1)*(3*m**2 + 3*m - 1)/30
g1b = 26 + 2**(m + 1)*(m**3 - 3*m**2 + 9*m - 13)
g1c = (m + 1)*(m*(m**2 - 7*m + 3)*sqrt(5) - (
3*m**3 - 7*m**2 + 19*m - 6))/(2*m**3*sqrt(5) + m**4 + 5*m**2 - 1)/6
g1d = -S(2)/231 + 2*4**m*(m + 1)*(63*m**4 + 112*m**3 + 18*m**2 -
22*m + 3)/(693*binomial(2*m, m))
g1e = -S(9)/2 + (81*m**2 + 261*m + 200)*factorial(
3*m + 2)/(40*27**m*factorial(m)*factorial(m + 1)*factorial(m + 2))
g1f = (2*m + 1)**2*binomial(2*m, m)**2/(4**(2*m)*(m + 1))
g1g = -binomial(2*m, m)**2/4**(2*m)
g1h = 4*pi -(2*m + 1)**2*(3*m + 4)*factorial(m - S(1)/2)**2/factorial(m + 1)**2
g = gosper_sum(f1a, (n, 0, m))
assert g is not None and simplify(g - g1a) == 0
g = gosper_sum(f1b, (n, 0, m))
assert g is not None and simplify(g - g1b) == 0
g = gosper_sum(f1c, (n, 0, m))
assert g is not None and simplify(g - g1c) == 0
g = gosper_sum(f1d, (n, 0, m))
assert g is not None and simplify(g - g1d) == 0
g = gosper_sum(f1e, (n, 0, m))
assert g is not None and simplify(g - g1e) == 0
g = gosper_sum(f1f, (n, 0, m))
assert g is not None and simplify(g - g1f) == 0
g = gosper_sum(f1g, (n, 0, m))
assert g is not None and simplify(g - g1g) == 0
g = gosper_sum(f1h, (n, 0, m))
# need to call rewrite(gamma) here because we have terms involving
# factorial(1/2)
assert g is not None and simplify(g - g1h).rewrite(gamma) == 0
def test_gosper_sum_AeqB_part2():
f2a = n**2*a**n
f2b = (n - r/2)*binomial(r, n)
f2c = factorial(n - 1)**2/(factorial(n - x)*factorial(n + x))
g2a = -a*(a + 1)/(a - 1)**3 + a**(
m + 1)*(a**2*m**2 - 2*a*m**2 + m**2 - 2*a*m + 2*m + a + 1)/(a - 1)**3
g2b = (m - r)*binomial(r, m)/2
ff = factorial(1 - x)*factorial(1 + x)
g2c = 1/ff*(
1 - 1/x**2) + factorial(m)**2/(x**2*factorial(m - x)*factorial(m + x))
g = gosper_sum(f2a, (n, 0, m))
assert g is not None and simplify(g - g2a) == 0
g = gosper_sum(f2b, (n, 0, m))
assert g is not None and simplify(g - g2b) == 0
g = gosper_sum(f2c, (n, 1, m))
assert g is not None and simplify(g - g2c) == 0
def test_gosper_nan():
a = Symbol('a', positive=True)
b = Symbol('b', positive=True)
n = Symbol('n', integer=True)
m = Symbol('m', integer=True)
f2d = n*(n + a + b)*a**n*b**n/(factorial(n + a)*factorial(n + b))
g2d = 1/(factorial(a - 1)*factorial(
b - 1)) - a**(m + 1)*b**(m + 1)/(factorial(a + m)*factorial(b + m))
g = gosper_sum(f2d, (n, 0, m))
assert simplify(g - g2d) == 0
def test_gosper_sum_AeqB_part3():
f3a = 1/n**4
f3b = (6*n + 3)/(4*n**4 + 8*n**3 + 8*n**2 + 4*n + 3)
f3c = 2**n*(n**2 - 2*n - 1)/(n**2*(n + 1)**2)
f3d = n**2*4**n/((n + 1)*(n + 2))
f3e = 2**n/(n + 1)
f3f = 4*(n - 1)*(n**2 - 2*n - 1)/(n**2*(n + 1)**2*(n - 2)**2*(n - 3)**2)
f3g = (n**4 - 14*n**2 - 24*n - 9)*2**n/(n**2*(n + 1)**2*(n + 2)**2*
(n + 3)**2)
# g3a -> no closed form
g3b = m*(m + 2)/(2*m**2 + 4*m + 3)
g3c = 2**m/m**2 - 2
g3d = S(2)/3 + 4**(m + 1)*(m - 1)/(m + 2)/3
# g3e -> no closed form
g3f = -(-S(1)/16 + 1/((m - 2)**2*(m + 1)**2)) # the AeqB key is wrong
g3g = -S(2)/9 + 2**(m + 1)/((m + 1)**2*(m + 3)**2)
g = gosper_sum(f3a, (n, 1, m))
assert g is None
g = gosper_sum(f3b, (n, 1, m))
assert g is not None and simplify(g - g3b) == 0
g = gosper_sum(f3c, (n, 1, m - 1))
assert g is not None and simplify(g - g3c) == 0
g = gosper_sum(f3d, (n, 1, m))
assert g is not None and simplify(g - g3d) == 0
g = gosper_sum(f3e, (n, 0, m - 1))
assert g is None
g = gosper_sum(f3f, (n, 4, m))
assert g is not None and simplify(g - g3f) == 0
g = gosper_sum(f3g, (n, 1, m))
assert g is not None and simplify(g - g3g) == 0
|
10d2d664ffed43098ce1e5172e5331465de07285044e27e73756dbba152fc00f
|
from sympy import (
Abs, And, binomial, Catalan, cos, Derivative, E, Eq, exp, EulerGamma,
factorial, Function, harmonic, I, Integral, KroneckerDelta, log,
nan, oo, pi, Piecewise, Product, product, Rational, S, simplify,
sin, sqrt, Sum, summation, Symbol, symbols, sympify, zeta, gamma, Le,
Indexed, Idx, IndexedBase, prod, Dummy, lowergamma)
from sympy.abc import a, b, c, d, k, m, x, y, z
from sympy.concrete.summations import telescopic
from sympy.concrete.expr_with_intlimits import ReorderError
from sympy.utilities.pytest import XFAIL, raises, slow
from sympy.matrices import Matrix
from sympy.core.mod import Mod
from sympy.core.compatibility import range
n = Symbol('n', integer=True)
def test_karr_convention():
# Test the Karr summation convention that we want to hold.
# See his paper "Summation in Finite Terms" for a detailed
# reasoning why we really want exactly this definition.
# The convention is described on page 309 and essentially
# in section 1.4, definition 3:
#
# \sum_{m <= i < n} f(i) 'has the obvious meaning' for m < n
# \sum_{m <= i < n} f(i) = 0 for m = n
# \sum_{m <= i < n} f(i) = - \sum_{n <= i < m} f(i) for m > n
#
# It is important to note that he defines all sums with
# the upper limit being *exclusive*.
# In contrast, sympy and the usual mathematical notation has:
#
# sum_{i = a}^b f(i) = f(a) + f(a+1) + ... + f(b-1) + f(b)
#
# with the upper limit *inclusive*. So translating between
# the two we find that:
#
# \sum_{m <= i < n} f(i) = \sum_{i = m}^{n-1} f(i)
#
# where we intentionally used two different ways to typeset the
# sum and its limits.
i = Symbol("i", integer=True)
k = Symbol("k", integer=True)
j = Symbol("j", integer=True)
# A simple example with a concrete summand and symbolic limits.
# The normal sum: m = k and n = k + j and therefore m < n:
m = k
n = k + j
a = m
b = n - 1
S1 = Sum(i**2, (i, a, b)).doit()
# The reversed sum: m = k + j and n = k and therefore m > n:
m = k + j
n = k
a = m
b = n - 1
S2 = Sum(i**2, (i, a, b)).doit()
assert simplify(S1 + S2) == 0
# Test the empty sum: m = k and n = k and therefore m = n:
m = k
n = k
a = m
b = n - 1
Sz = Sum(i**2, (i, a, b)).doit()
assert Sz == 0
# Another example this time with an unspecified summand and
# numeric limits. (We can not do both tests in the same example.)
f = Function("f")
# The normal sum with m < n:
m = 2
n = 11
a = m
b = n - 1
S1 = Sum(f(i), (i, a, b)).doit()
# The reversed sum with m > n:
m = 11
n = 2
a = m
b = n - 1
S2 = Sum(f(i), (i, a, b)).doit()
assert simplify(S1 + S2) == 0
# Test the empty sum with m = n:
m = 5
n = 5
a = m
b = n - 1
Sz = Sum(f(i), (i, a, b)).doit()
assert Sz == 0
e = Piecewise((exp(-i), Mod(i, 2) > 0), (0, True))
s = Sum(e, (i, 0, 11))
assert s.n(3) == s.doit().n(3)
def test_karr_proposition_2a():
# Test Karr, page 309, proposition 2, part a
i = Symbol("i", integer=True)
u = Symbol("u", integer=True)
v = Symbol("v", integer=True)
def test_the_sum(m, n):
# g
g = i**3 + 2*i**2 - 3*i
# f = Delta g
f = simplify(g.subs(i, i+1) - g)
# The sum
a = m
b = n - 1
S = Sum(f, (i, a, b)).doit()
# Test if Sum_{m <= i < n} f(i) = g(n) - g(m)
assert simplify(S - (g.subs(i, n) - g.subs(i, m))) == 0
# m < n
test_the_sum(u, u+v)
# m = n
test_the_sum(u, u )
# m > n
test_the_sum(u+v, u )
def test_karr_proposition_2b():
# Test Karr, page 309, proposition 2, part b
i = Symbol("i", integer=True)
u = Symbol("u", integer=True)
v = Symbol("v", integer=True)
w = Symbol("w", integer=True)
def test_the_sum(l, n, m):
# Summand
s = i**3
# First sum
a = l
b = n - 1
S1 = Sum(s, (i, a, b)).doit()
# Second sum
a = l
b = m - 1
S2 = Sum(s, (i, a, b)).doit()
# Third sum
a = m
b = n - 1
S3 = Sum(s, (i, a, b)).doit()
# Test if S1 = S2 + S3 as required
assert S1 - (S2 + S3) == 0
# l < m < n
test_the_sum(u, u+v, u+v+w)
# l < m = n
test_the_sum(u, u+v, u+v )
# l < m > n
test_the_sum(u, u+v+w, v )
# l = m < n
test_the_sum(u, u, u+v )
# l = m = n
test_the_sum(u, u, u )
# l = m > n
test_the_sum(u+v, u+v, u )
# l > m < n
test_the_sum(u+v, u, u+w )
# l > m = n
test_the_sum(u+v, u, u )
# l > m > n
test_the_sum(u+v+w, u+v, u )
def test_arithmetic_sums():
assert summation(1, (n, a, b)) == b - a + 1
assert Sum(S.NaN, (n, a, b)) is S.NaN
assert Sum(x, (n, a, a)).doit() == x
assert Sum(x, (x, a, a)).doit() == a
assert Sum(x, (n, 1, a)).doit() == a*x
lo, hi = 1, 2
s1 = Sum(n, (n, lo, hi))
s2 = Sum(n, (n, hi, lo))
assert s1 != s2
assert s1.doit() == 3 and s2.doit() == 0
lo, hi = x, x + 1
s1 = Sum(n, (n, lo, hi))
s2 = Sum(n, (n, hi, lo))
assert s1 != s2
assert s1.doit() == 2*x + 1 and s2.doit() == 0
assert Sum(Integral(x, (x, 1, y)) + x, (x, 1, 2)).doit() == \
y**2 + 2
assert summation(1, (n, 1, 10)) == 10
assert summation(2*n, (n, 0, 10**10)) == 100000000010000000000
assert summation(4*n*m, (n, a, 1), (m, 1, d)).expand() == \
2*d + 2*d**2 + a*d + a*d**2 - d*a**2 - a**2*d**2
assert summation(cos(n), (n, -2, 1)) == cos(-2) + cos(-1) + cos(0) + cos(1)
assert summation(cos(n), (n, x, x + 2)) == cos(x) + cos(x + 1) + cos(x + 2)
assert isinstance(summation(cos(n), (n, x, x + S.Half)), Sum)
assert summation(k, (k, 0, oo)) == oo
def test_polynomial_sums():
assert summation(n**2, (n, 3, 8)) == 199
assert summation(n, (n, a, b)) == \
((a + b)*(b - a + 1)/2).expand()
assert summation(n**2, (n, 1, b)) == \
((2*b**3 + 3*b**2 + b)/6).expand()
assert summation(n**3, (n, 1, b)) == \
((b**4 + 2*b**3 + b**2)/4).expand()
assert summation(n**6, (n, 1, b)) == \
((6*b**7 + 21*b**6 + 21*b**5 - 7*b**3 + b)/42).expand()
def test_geometric_sums():
assert summation(pi**n, (n, 0, b)) == (1 - pi**(b + 1)) / (1 - pi)
assert summation(2 * 3**n, (n, 0, b)) == 3**(b + 1) - 1
assert summation(Rational(1, 2)**n, (n, 1, oo)) == 1
assert summation(2**n, (n, 0, b)) == 2**(b + 1) - 1
assert summation(2**n, (n, 1, oo)) == oo
assert summation(2**(-n), (n, 1, oo)) == 1
assert summation(3**(-n), (n, 4, oo)) == Rational(1, 54)
assert summation(2**(-4*n + 3), (n, 1, oo)) == Rational(8, 15)
assert summation(2**(n + 1), (n, 1, b)).expand() == 4*(2**b - 1)
# issue 6664:
assert summation(x**n, (n, 0, oo)) == \
Piecewise((1/(-x + 1), Abs(x) < 1), (Sum(x**n, (n, 0, oo)), True))
assert summation(-2**n, (n, 0, oo)) == -oo
assert summation(I**n, (n, 0, oo)) == Sum(I**n, (n, 0, oo))
# issue 6802:
assert summation((-1)**(2*x + 2), (x, 0, n)) == n + 1
assert summation((-2)**(2*x + 2), (x, 0, n)) == 4*4**(n + 1)/S(3) - S(4)/3
assert summation((-1)**x, (x, 0, n)) == -(-1)**(n + 1)/S(2) + S(1)/2
assert summation(y**x, (x, a, b)) == \
Piecewise((-a + b + 1, Eq(y, 1)), ((y**a - y**(b + 1))/(-y + 1), True))
assert summation((-2)**(y*x + 2), (x, 0, n)) == \
4*Piecewise((n + 1, Eq((-2)**y, 1)),
((-(-2)**(y*(n + 1)) + 1)/(-(-2)**y + 1), True))
# issue 8251:
assert summation((1/(n + 1)**2)*n**2, (n, 0, oo)) == oo
#issue 9908:
assert Sum(1/(n**3 - 1), (n, -oo, -2)).doit() == summation(1/(n**3 - 1), (n, -oo, -2))
#issue 11642:
result = Sum(0.5**n, (n, 1, oo)).doit()
assert result == 1
assert result.is_Float
result = Sum(0.25**n, (n, 1, oo)).doit()
assert result == S(1)/3
assert result.is_Float
result = Sum(0.99999**n, (n, 1, oo)).doit()
assert result == 99999
assert result.is_Float
result = Sum(Rational(1, 2)**n, (n, 1, oo)).doit()
assert result == 1
assert not result.is_Float
result = Sum(Rational(3, 5)**n, (n, 1, oo)).doit()
assert result == S(3)/2
assert not result.is_Float
assert Sum(1.0**n, (n, 1, oo)).doit() == oo
assert Sum(2.43**n, (n, 1, oo)).doit() == oo
# Issue 13979:
i, k, q = symbols('i k q', integer=True)
result = summation(
exp(-2*I*pi*k*i/n) * exp(2*I*pi*q*i/n) / n, (i, 0, n - 1)
)
assert result.simplify() == Piecewise(
(1, Eq(exp(2*I*pi*(-k + q)/n), 1)), (0, True)
)
def test_harmonic_sums():
assert summation(1/k, (k, 0, n)) == Sum(1/k, (k, 0, n))
assert summation(1/k, (k, 1, n)) == harmonic(n)
assert summation(n/k, (k, 1, n)) == n*harmonic(n)
assert summation(1/k, (k, 5, n)) == harmonic(n) - harmonic(4)
def test_composite_sums():
f = Rational(1, 2)*(7 - 6*n + Rational(1, 7)*n**3)
s = summation(f, (n, a, b))
assert not isinstance(s, Sum)
A = 0
for i in range(-3, 5):
A += f.subs(n, i)
B = s.subs(a, -3).subs(b, 4)
assert A == B
def test_hypergeometric_sums():
assert summation(
binomial(2*k, k)/4**k, (k, 0, n)) == (1 + 2*n)*binomial(2*n, n)/4**n
def test_other_sums():
f = m**2 + m*exp(m)
g = 3*exp(S(3)/2)/2 + exp(S(1)/2)/2 - exp(-S(1)/2)/2 - 3*exp(-S(3)/2)/2 + 5
assert summation(f, (m, -S(3)/2, S(3)/2)).expand() == g
assert summation(f, (m, -1.5, 1.5)).evalf().epsilon_eq(g.evalf(), 1e-10)
fac = factorial
def NS(e, n=15, **options):
return str(sympify(e).evalf(n, **options))
def test_evalf_fast_series():
# Euler transformed series for sqrt(1+x)
assert NS(Sum(
fac(2*n + 1)/fac(n)**2/2**(3*n + 1), (n, 0, oo)), 100) == NS(sqrt(2), 100)
# Some series for exp(1)
estr = NS(E, 100)
assert NS(Sum(1/fac(n), (n, 0, oo)), 100) == estr
assert NS(1/Sum((1 - 2*n)/fac(2*n), (n, 0, oo)), 100) == estr
assert NS(Sum((2*n + 1)/fac(2*n), (n, 0, oo)), 100) == estr
assert NS(Sum((4*n + 3)/2**(2*n + 1)/fac(2*n + 1), (n, 0, oo))**2, 100) == estr
pistr = NS(pi, 100)
# Ramanujan series for pi
assert NS(9801/sqrt(8)/Sum(fac(
4*n)*(1103 + 26390*n)/fac(n)**4/396**(4*n), (n, 0, oo)), 100) == pistr
assert NS(1/Sum(
binomial(2*n, n)**3 * (42*n + 5)/2**(12*n + 4), (n, 0, oo)), 100) == pistr
# Machin's formula for pi
assert NS(16*Sum((-1)**n/(2*n + 1)/5**(2*n + 1), (n, 0, oo)) -
4*Sum((-1)**n/(2*n + 1)/239**(2*n + 1), (n, 0, oo)), 100) == pistr
# Apery's constant
astr = NS(zeta(3), 100)
P = 126392*n**5 + 412708*n**4 + 531578*n**3 + 336367*n**2 + 104000* \
n + 12463
assert NS(Sum((-1)**n * P / 24 * (fac(2*n + 1)*fac(2*n)*fac(
n))**3 / fac(3*n + 2) / fac(4*n + 3)**3, (n, 0, oo)), 100) == astr
assert NS(Sum((-1)**n * (205*n**2 + 250*n + 77)/64 * fac(n)**10 /
fac(2*n + 1)**5, (n, 0, oo)), 100) == astr
def test_evalf_fast_series_issue_4021():
# Catalan's constant
assert NS(Sum((-1)**(n - 1)*2**(8*n)*(40*n**2 - 24*n + 3)*fac(2*n)**3*
fac(n)**2/n**3/(2*n - 1)/fac(4*n)**2, (n, 1, oo))/64, 100) == \
NS(Catalan, 100)
astr = NS(zeta(3), 100)
assert NS(5*Sum(
(-1)**(n - 1)*fac(n)**2 / n**3 / fac(2*n), (n, 1, oo))/2, 100) == astr
assert NS(Sum((-1)**(n - 1)*(56*n**2 - 32*n + 5) / (2*n - 1)**2 * fac(n - 1)
**3 / fac(3*n), (n, 1, oo))/4, 100) == astr
def test_evalf_slow_series():
assert NS(Sum((-1)**n / n, (n, 1, oo)), 15) == NS(-log(2), 15)
assert NS(Sum((-1)**n / n, (n, 1, oo)), 50) == NS(-log(2), 50)
assert NS(Sum(1/n**2, (n, 1, oo)), 15) == NS(pi**2/6, 15)
assert NS(Sum(1/n**2, (n, 1, oo)), 100) == NS(pi**2/6, 100)
assert NS(Sum(1/n**2, (n, 1, oo)), 500) == NS(pi**2/6, 500)
assert NS(Sum((-1)**n / (2*n + 1)**3, (n, 0, oo)), 15) == NS(pi**3/32, 15)
assert NS(Sum((-1)**n / (2*n + 1)**3, (n, 0, oo)), 50) == NS(pi**3/32, 50)
def test_euler_maclaurin():
# Exact polynomial sums with E-M
def check_exact(f, a, b, m, n):
A = Sum(f, (k, a, b))
s, e = A.euler_maclaurin(m, n)
assert (e == 0) and (s.expand() == A.doit())
check_exact(k**4, a, b, 0, 2)
check_exact(k**4 + 2*k, a, b, 1, 2)
check_exact(k**4 + k**2, a, b, 1, 5)
check_exact(k**5, 2, 6, 1, 2)
check_exact(k**5, 2, 6, 1, 3)
assert Sum(x-1, (x, 0, 2)).euler_maclaurin(m=30, n=30, eps=2**-15) == (0, 0)
# Not exact
assert Sum(k**6, (k, a, b)).euler_maclaurin(0, 2)[1] != 0
# Numerical test
for m, n in [(2, 4), (2, 20), (10, 20), (18, 20)]:
A = Sum(1/k**3, (k, 1, oo))
s, e = A.euler_maclaurin(m, n)
assert abs((s - zeta(3)).evalf()) < e.evalf()
raises(ValueError, lambda: Sum(1, (x, 0, 1), (k, 0, 1)).euler_maclaurin())
@slow
def test_evalf_euler_maclaurin():
assert NS(Sum(1/k**k, (k, 1, oo)), 15) == '1.29128599706266'
assert NS(Sum(1/k**k, (k, 1, oo)),
50) == '1.2912859970626635404072825905956005414986193682745'
assert NS(Sum(1/k - log(1 + 1/k), (k, 1, oo)), 15) == NS(EulerGamma, 15)
assert NS(Sum(1/k - log(1 + 1/k), (k, 1, oo)), 50) == NS(EulerGamma, 50)
assert NS(Sum(log(k)/k**2, (k, 1, oo)), 15) == '0.937548254315844'
assert NS(Sum(log(k)/k**2, (k, 1, oo)),
50) == '0.93754825431584375370257409456786497789786028861483'
assert NS(Sum(1/k, (k, 1000000, 2000000)), 15) == '0.693147930560008'
assert NS(Sum(1/k, (k, 1000000, 2000000)),
50) == '0.69314793056000780941723211364567656807940638436025'
def test_evalf_symbolic():
f, g = symbols('f g', cls=Function)
# issue 6328
expr = Sum(f(x), (x, 1, 3)) + Sum(g(x), (x, 1, 3))
assert expr.evalf() == expr
def test_evalf_issue_3273():
assert Sum(0, (k, 1, oo)).evalf() == 0
def test_simple_products():
assert Product(S.NaN, (x, 1, 3)) is S.NaN
assert product(S.NaN, (x, 1, 3)) is S.NaN
assert Product(x, (n, a, a)).doit() == x
assert Product(x, (x, a, a)).doit() == a
assert Product(x, (y, 1, a)).doit() == x**a
lo, hi = 1, 2
s1 = Product(n, (n, lo, hi))
s2 = Product(n, (n, hi, lo))
assert s1 != s2
# This IS correct according to Karr product convention
assert s1.doit() == 2
assert s2.doit() == 1
lo, hi = x, x + 1
s1 = Product(n, (n, lo, hi))
s2 = Product(n, (n, hi, lo))
s3 = 1 / Product(n, (n, hi + 1, lo - 1))
assert s1 != s2
# This IS correct according to Karr product convention
assert s1.doit() == x*(x + 1)
assert s2.doit() == 1
assert s3.doit() == x*(x + 1)
assert Product(Integral(2*x, (x, 1, y)) + 2*x, (x, 1, 2)).doit() == \
(y**2 + 1)*(y**2 + 3)
assert product(2, (n, a, b)) == 2**(b - a + 1)
assert product(n, (n, 1, b)) == factorial(b)
assert product(n**3, (n, 1, b)) == factorial(b)**3
assert product(3**(2 + n), (n, a, b)) \
== 3**(2*(1 - a + b) + b/2 + (b**2)/2 + a/2 - (a**2)/2)
assert product(cos(n), (n, 3, 5)) == cos(3)*cos(4)*cos(5)
assert product(cos(n), (n, x, x + 2)) == cos(x)*cos(x + 1)*cos(x + 2)
assert isinstance(product(cos(n), (n, x, x + S.Half)), Product)
# If Product managed to evaluate this one, it most likely got it wrong!
assert isinstance(Product(n**n, (n, 1, b)), Product)
def test_rational_products():
assert simplify(product(1 + 1/n, (n, a, b))) == (1 + b)/a
assert simplify(product(n + 1, (n, a, b))) == gamma(2 + b)/gamma(1 + a)
assert simplify(product((n + 1)/(n - 1), (n, a, b))) == b*(1 + b)/(a*(a - 1))
assert simplify(product(n/(n + 1)/(n + 2), (n, a, b))) == \
a*gamma(a + 2)/(b + 1)/gamma(b + 3)
assert simplify(product(n*(n + 1)/(n - 1)/(n - 2), (n, a, b))) == \
b**2*(b - 1)*(1 + b)/(a - 1)**2/(a*(a - 2))
def test_wallis_product():
# Wallis product, given in two different forms to ensure that Product
# can factor simple rational expressions
A = Product(4*n**2 / (4*n**2 - 1), (n, 1, b))
B = Product((2*n)*(2*n)/(2*n - 1)/(2*n + 1), (n, 1, b))
R = pi*gamma(b + 1)**2/(2*gamma(b + S(1)/2)*gamma(b + S(3)/2))
assert simplify(A.doit()) == R
assert simplify(B.doit()) == R
# This one should eventually also be doable (Euler's product formula for sin)
# assert Product(1+x/n**2, (n, 1, b)) == ...
def test_telescopic_sums():
#checks also input 2 of comment 1 issue 4127
assert Sum(1/k - 1/(k + 1), (k, 1, n)).doit() == 1 - 1/(1 + n)
f = Function("f")
assert Sum(
f(k) - f(k + 2), (k, m, n)).doit() == -f(1 + n) - f(2 + n) + f(m) + f(1 + m)
assert Sum(cos(k) - cos(k + 3), (k, 1, n)).doit() == -cos(1 + n) - \
cos(2 + n) - cos(3 + n) + cos(1) + cos(2) + cos(3)
# dummy variable shouldn't matter
assert telescopic(1/m, -m/(1 + m), (m, n - 1, n)) == \
telescopic(1/k, -k/(1 + k), (k, n - 1, n))
assert Sum(1/x/(x - 1), (x, a, b)).doit() == -((a - b - 1)/(b*(a - 1)))
def test_sum_reconstruct():
s = Sum(n**2, (n, -1, 1))
assert s == Sum(*s.args)
raises(ValueError, lambda: Sum(x, x))
raises(ValueError, lambda: Sum(x, (x, 1)))
def test_limit_subs():
for F in (Sum, Product, Integral):
assert F(a*exp(a), (a, -2, 2)) == F(a*exp(a), (a, -b, b)).subs(b, 2)
assert F(a, (a, F(b, (b, 1, 2)), 4)).subs(F(b, (b, 1, 2)), c) == \
F(a, (a, c, 4))
assert F(x, (x, 1, x + y)).subs(x, 1) == F(x, (x, 1, y + 1))
def test_function_subs():
f = Function("f")
S = Sum(x*f(y),(x,0,oo),(y,0,oo))
assert S.subs(f(y),y) == Sum(x*y,(x,0,oo),(y,0,oo))
assert S.subs(f(x),x) == S
raises(ValueError, lambda: S.subs(f(y),x+y) )
S = Sum(x*log(y),(x,0,oo),(y,0,oo))
assert S.subs(log(y),y) == S
S = Sum(x*f(y),(x,0,oo),(y,0,oo))
assert S.subs(f(y),y) == Sum(x*y,(x,0,oo),(y,0,oo))
def test_equality():
# if this fails remove special handling below
raises(ValueError, lambda: Sum(x, x))
r = symbols('x', real=True)
for F in (Sum, Product, Integral):
try:
assert F(x, x) != F(y, y)
assert F(x, (x, 1, 2)) != F(x, x)
assert F(x, (x, x)) != F(x, x) # or else they print the same
assert F(1, x) != F(1, y)
except ValueError:
pass
assert F(a, (x, 1, 2)) != F(a, (x, 1, 3)) # diff limit
assert F(a, (x, 1, x)) != F(a, (y, 1, y))
assert F(a, (x, 1, 2)) != F(b, (x, 1, 2)) # diff expression
assert F(x, (x, 1, 2)) != F(r, (r, 1, 2)) # diff assumptions
assert F(1, (x, 1, x)) != F(1, (y, 1, x)) # only dummy is diff
assert F(1, (x, 1, x)).dummy_eq(F(1, (y, 1, x)))
# issue 5265
assert Sum(x, (x, 1, x)).subs(x, a) == Sum(x, (x, 1, a))
def test_Sum_doit():
f = Function('f')
assert Sum(n*Integral(a**2), (n, 0, 2)).doit() == a**3
assert Sum(n*Integral(a**2), (n, 0, 2)).doit(deep=False) == \
3*Integral(a**2)
assert summation(n*Integral(a**2), (n, 0, 2)) == 3*Integral(a**2)
# test nested sum evaluation
s = Sum( Sum( Sum(2,(z,1,n+1)), (y,x+1,n)), (x,1,n))
assert 0 == (s.doit() - n*(n+1)*(n-1)).factor()
assert Sum(KroneckerDelta(m, n), (m, -oo, oo)).doit() == Piecewise((1, And(-oo < n, n < oo)), (0, True))
assert Sum(x*KroneckerDelta(m, n), (m, -oo, oo)).doit() == Piecewise((x, And(-oo < n, n < oo)), (0, True))
assert Sum(Sum(KroneckerDelta(m, n), (m, 1, 3)), (n, 1, 3)).doit() == 3
assert Sum(Sum(KroneckerDelta(k, m), (m, 1, 3)), (n, 1, 3)).doit() == \
3 * Piecewise((1, And(S(1) <= k, k <= 3)), (0, True))
assert Sum(f(n) * Sum(KroneckerDelta(m, n), (m, 0, oo)), (n, 1, 3)).doit() == \
f(1) + f(2) + f(3)
assert Sum(f(n) * Sum(KroneckerDelta(m, n), (m, 0, oo)), (n, 1, oo)).doit() == \
Sum(Piecewise((f(n), And(Le(0, n), n < oo)), (0, True)), (n, 1, oo))
l = Symbol('l', integer=True, positive=True)
assert Sum(f(l) * Sum(KroneckerDelta(m, l), (m, 0, oo)), (l, 1, oo)).doit() == \
Sum(f(l), (l, 1, oo))
# issue 2597
nmax = symbols('N', integer=True, positive=True)
pw = Piecewise((1, And(S(1) <= n, n <= nmax)), (0, True))
assert Sum(pw, (n, 1, nmax)).doit() == Sum(pw, (n, 1, nmax))
q, s = symbols('q, s')
assert summation(1/n**(2*s), (n, 1, oo)) == Piecewise((zeta(2*s), 2*s > 1),
(Sum(n**(-2*s), (n, 1, oo)), True))
assert summation(1/(n+1)**s, (n, 0, oo)) == Piecewise((zeta(s), s > 1),
(Sum((n + 1)**(-s), (n, 0, oo)), True))
assert summation(1/(n+q)**s, (n, 0, oo)) == Piecewise(
(zeta(s, q), And(q > 0, s > 1)),
(Sum((n + q)**(-s), (n, 0, oo)), True))
assert summation(1/(n+q)**s, (n, q, oo)) == Piecewise(
(zeta(s, 2*q), And(2*q > 0, s > 1)),
(Sum((n + q)**(-s), (n, q, oo)), True))
assert summation(1/n**2, (n, 1, oo)) == zeta(2)
assert summation(1/n**s, (n, 0, oo)) == Sum(n**(-s), (n, 0, oo))
def test_Product_doit():
assert Product(n*Integral(a**2), (n, 1, 3)).doit() == 2 * a**9 / 9
assert Product(n*Integral(a**2), (n, 1, 3)).doit(deep=False) == \
6*Integral(a**2)**3
assert product(n*Integral(a**2), (n, 1, 3)) == 6*Integral(a**2)**3
def test_Sum_interface():
assert isinstance(Sum(0, (n, 0, 2)), Sum)
assert Sum(nan, (n, 0, 2)) is nan
assert Sum(nan, (n, 0, oo)) is nan
assert Sum(0, (n, 0, 2)).doit() == 0
assert isinstance(Sum(0, (n, 0, oo)), Sum)
assert Sum(0, (n, 0, oo)).doit() == 0
raises(ValueError, lambda: Sum(1))
raises(ValueError, lambda: summation(1))
def test_diff():
assert Sum(x, (x, 1, 2)).diff(x) == 0
assert Sum(x*y, (x, 1, 2)).diff(x) == 0
assert Sum(x*y, (y, 1, 2)).diff(x) == Sum(y, (y, 1, 2))
e = Sum(x*y, (x, 1, a))
assert e.diff(a) == Derivative(e, a)
assert Sum(x*y, (x, 1, 3), (a, 2, 5)).diff(y).doit() == \
Sum(x*y, (x, 1, 3), (a, 2, 5)).doit().diff(y) == 24
assert Sum(x, (x, 1, 2)).diff(y) == 0
def test_hypersum():
from sympy import sin
assert simplify(summation(x**n/fac(n), (n, 1, oo))) == -1 + exp(x)
assert summation((-1)**n * x**(2*n) / fac(2*n), (n, 0, oo)) == cos(x)
assert simplify(summation((-1)**n*x**(2*n + 1) /
factorial(2*n + 1), (n, 3, oo))) == -x + sin(x) + x**3/6 - x**5/120
assert summation(1/(n + 2)**3, (n, 1, oo)) == -S(9)/8 + zeta(3)
assert summation(1/n**4, (n, 1, oo)) == pi**4/90
s = summation(x**n*n, (n, -oo, 0))
assert s.is_Piecewise
assert s.args[0].args[0] == -1/(x*(1 - 1/x)**2)
assert s.args[0].args[1] == (abs(1/x) < 1)
m = Symbol('n', integer=True, positive=True)
assert summation(binomial(m, k), (k, 0, m)) == 2**m
def test_issue_4170():
assert summation(1/factorial(k), (k, 0, oo)) == E
def test_is_commutative():
from sympy.physics.secondquant import NO, F, Fd
m = Symbol('m', commutative=False)
for f in (Sum, Product, Integral):
assert f(z, (z, 1, 1)).is_commutative is True
assert f(z*y, (z, 1, 6)).is_commutative is True
assert f(m*x, (x, 1, 2)).is_commutative is False
assert f(NO(Fd(x)*F(y))*z, (z, 1, 2)).is_commutative is False
def test_is_zero():
for func in [Sum, Product]:
assert func(0, (x, 1, 1)).is_zero is True
assert func(x, (x, 1, 1)).is_zero is None
def test_is_number():
# is number should not rely on evaluation or assumptions,
# it should be equivalent to `not foo.free_symbols`
assert Sum(1, (x, 1, 1)).is_number is True
assert Sum(1, (x, 1, x)).is_number is False
assert Sum(0, (x, y, z)).is_number is False
assert Sum(x, (y, 1, 2)).is_number is False
assert Sum(x, (y, 1, 1)).is_number is False
assert Sum(x, (x, 1, 2)).is_number is True
assert Sum(x*y, (x, 1, 2), (y, 1, 3)).is_number is True
assert Product(2, (x, 1, 1)).is_number is True
assert Product(2, (x, 1, y)).is_number is False
assert Product(0, (x, y, z)).is_number is False
assert Product(1, (x, y, z)).is_number is False
assert Product(x, (y, 1, x)).is_number is False
assert Product(x, (y, 1, 2)).is_number is False
assert Product(x, (y, 1, 1)).is_number is False
assert Product(x, (x, 1, 2)).is_number is True
def test_free_symbols():
for func in [Sum, Product]:
assert func(1, (x, 1, 2)).free_symbols == set()
assert func(0, (x, 1, y)).free_symbols == {y}
assert func(2, (x, 1, y)).free_symbols == {y}
assert func(x, (x, 1, 2)).free_symbols == set()
assert func(x, (x, 1, y)).free_symbols == {y}
assert func(x, (y, 1, y)).free_symbols == {x, y}
assert func(x, (y, 1, 2)).free_symbols == {x}
assert func(x, (y, 1, 1)).free_symbols == {x}
assert func(x, (y, 1, z)).free_symbols == {x, z}
assert func(x, (x, 1, y), (y, 1, 2)).free_symbols == set()
assert func(x, (x, 1, y), (y, 1, z)).free_symbols == {z}
assert func(x, (x, 1, y), (y, 1, y)).free_symbols == {y}
assert func(x, (y, 1, y), (y, 1, z)).free_symbols == {x, z}
assert Sum(1, (x, 1, y)).free_symbols == {y}
# free_symbols answers whether the object *as written* has free symbols,
# not whether the evaluated expression has free symbols
assert Product(1, (x, 1, y)).free_symbols == {y}
def test_conjugate_transpose():
A, B = symbols("A B", commutative=False)
p = Sum(A*B**n, (n, 1, 3))
assert p.adjoint().doit() == p.doit().adjoint()
assert p.conjugate().doit() == p.doit().conjugate()
assert p.transpose().doit() == p.doit().transpose()
def test_issue_4171():
assert summation(factorial(2*k + 1)/factorial(2*k), (k, 0, oo)) == oo
assert summation(2*k + 1, (k, 0, oo)) == oo
def test_issue_6273():
assert Sum(x, (x, 1, n)).n(2, subs={n: 1}) == 1
def test_issue_6274():
assert Sum(x, (x, 1, 0)).doit() == 0
assert NS(Sum(x, (x, 1, 0))) == '0'
assert Sum(n, (n, 10, 5)).doit() == -30
assert NS(Sum(n, (n, 10, 5))) == '-30.0000000000000'
def test_simplify():
y, t, v = symbols('y, t, v')
assert simplify(Sum(x*y, (x, n, m), (y, a, k)) + \
Sum(y, (x, n, m), (y, a, k))) == Sum(y * (x + 1), (x, n, m), (y, a, k))
assert simplify(Sum(x, (x, n, m)) + Sum(x, (x, m + 1, a))) == \
Sum(x, (x, n, a))
assert simplify(Sum(x, (x, k + 1, a)) + Sum(x, (x, n, k))) == \
Sum(x, (x, n, a))
assert simplify(Sum(x, (x, k + 1, a)) + Sum(x + 1, (x, n, k))) == \
Sum(x, (x, n, a)) + Sum(1, (x, n, k))
assert simplify(Sum(x, (x, 0, 3)) * 3 + 3 * Sum(x, (x, 4, 6)) + \
4 * Sum(z, (z, 0, 1))) == 4*Sum(z, (z, 0, 1)) + 3*Sum(x, (x, 0, 6))
assert simplify(3*Sum(x**2, (x, a, b)) + Sum(x, (x, a, b))) == \
Sum(x*(3*x + 1), (x, a, b))
assert simplify(Sum(x**3, (x, n, k)) * 3 + 3 * Sum(x, (x, n, k)) + \
4 * y * Sum(z, (z, n, k))) + 1 == \
4*y*Sum(z, (z, n, k)) + 3*Sum(x**3 + x, (x, n, k)) + 1
assert simplify(Sum(x, (x, a, b)) + 1 + Sum(x, (x, b + 1, c))) == \
1 + Sum(x, (x, a, c))
assert simplify(Sum(x, (t, a, b)) + Sum(y, (t, a, b)) + \
Sum(x, (t, b+1, c))) == x * Sum(1, (t, a, c)) + y * Sum(1, (t, a, b))
assert simplify(Sum(x, (t, a, b)) + Sum(x, (t, b+1, c)) + \
Sum(y, (t, a, b))) == x * Sum(1, (t, a, c)) + y * Sum(1, (t, a, b))
assert simplify(Sum(x, (t, a, b)) + 2 * Sum(x, (t, b+1, c))) == \
simplify(Sum(x, (t, a, b)) + Sum(x, (t, b+1, c)) + Sum(x, (t, b+1, c)))
assert simplify(Sum(x, (x, a, b))*Sum(x**2, (x, a, b))) == \
Sum(x, (x, a, b)) * Sum(x**2, (x, a, b))
assert simplify(Sum(x, (t, a, b)) + Sum(y, (t, a, b)) + Sum(z, (t, a, b))) \
== (x + y + z) * Sum(1, (t, a, b)) # issue 8596
assert simplify(Sum(x, (t, a, b)) + Sum(y, (t, a, b)) + Sum(z, (t, a, b)) + \
Sum(v, (t, a, b))) == (x + y + z + v) * Sum(1, (t, a, b)) # issue 8596
assert simplify(Sum(x * y, (x, a, b)) / (3 * y)) == \
(Sum(x, (x, a, b)) / 3)
assert simplify(Sum(Function('f')(x) * y * z, (x, a, b)) / (y * z)) \
== Sum(Function('f')(x), (x, a, b))
assert simplify(Sum(c * x, (x, a, b)) - c * Sum(x, (x, a, b))) == 0
assert simplify(c * (Sum(x, (x, a, b)) + y)) == c * (y + Sum(x, (x, a, b)))
assert simplify(c * (Sum(x, (x, a, b)) + y * Sum(x, (x, a, b)))) == \
c * (y + 1) * Sum(x, (x, a, b))
assert simplify(Sum(Sum(c * x, (x, a, b)), (y, a, b))) == \
c * Sum(x, (x, a, b), (y, a, b))
assert simplify(Sum((3 + y) * Sum(c * x, (x, a, b)), (y, a, b))) == \
c * Sum((3 + y), (y, a, b)) * Sum(x, (x, a, b))
assert simplify(Sum((3 + t) * Sum(c * t, (x, a, b)), (y, a, b))) == \
c*t*(t + 3)*Sum(1, (x, a, b))*Sum(1, (y, a, b))
assert simplify(Sum(Sum(d * t, (x, a, b - 1)) + \
Sum(d * t, (x, b, c)), (t, a, b))) == \
d * Sum(1, (x, a, c)) * Sum(t, (t, a, b))
def test_change_index():
b, v = symbols('b, v', integer = True)
assert Sum(x, (x, a, b)).change_index(x, x + 1, y) == \
Sum(y - 1, (y, a + 1, b + 1))
assert Sum(x**2, (x, a, b)).change_index( x, x - 1) == \
Sum((x+1)**2, (x, a - 1, b - 1))
assert Sum(x**2, (x, a, b)).change_index( x, -x, y) == \
Sum((-y)**2, (y, -b, -a))
assert Sum(x, (x, a, b)).change_index( x, -x - 1) == \
Sum(-x - 1, (x, -b - 1, -a - 1))
assert Sum(x*y, (x, a, b), (y, c, d)).change_index( x, x - 1, z) == \
Sum((z + 1)*y, (z, a - 1, b - 1), (y, c, d))
assert Sum(x, (x, a, b)).change_index( x, x + v) == \
Sum(-v + x, (x, a + v, b + v))
assert Sum(x, (x, a, b)).change_index( x, -x - v) == \
Sum(-v - x, (x, -b - v, -a - v))
def test_reorder():
b, y, c, d, z = symbols('b, y, c, d, z', integer = True)
assert Sum(x*y, (x, a, b), (y, c, d)).reorder((0, 1)) == \
Sum(x*y, (y, c, d), (x, a, b))
assert Sum(x, (x, a, b), (x, c, d)).reorder((0, 1)) == \
Sum(x, (x, c, d), (x, a, b))
assert Sum(x*y + z, (x, a, b), (z, m, n), (y, c, d)).reorder(\
(2, 0), (0, 1)) == Sum(x*y + z, (z, m, n), (y, c, d), (x, a, b))
assert Sum(x*y*z, (x, a, b), (y, c, d), (z, m, n)).reorder(\
(0, 1), (1, 2), (0, 2)) == Sum(x*y*z, (x, a, b), (z, m, n), (y, c, d))
assert Sum(x*y*z, (x, a, b), (y, c, d), (z, m, n)).reorder(\
(x, y), (y, z), (x, z)) == Sum(x*y*z, (x, a, b), (z, m, n), (y, c, d))
assert Sum(x*y, (x, a, b), (y, c, d)).reorder((x, 1)) == \
Sum(x*y, (y, c, d), (x, a, b))
assert Sum(x*y, (x, a, b), (y, c, d)).reorder((y, x)) == \
Sum(x*y, (y, c, d), (x, a, b))
def test_reverse_order():
assert Sum(x, (x, 0, 3)).reverse_order(0) == Sum(-x, (x, 4, -1))
assert Sum(x*y, (x, 1, 5), (y, 0, 6)).reverse_order(0, 1) == \
Sum(x*y, (x, 6, 0), (y, 7, -1))
assert Sum(x, (x, 1, 2)).reverse_order(0) == Sum(-x, (x, 3, 0))
assert Sum(x, (x, 1, 3)).reverse_order(0) == Sum(-x, (x, 4, 0))
assert Sum(x, (x, 1, a)).reverse_order(0) == Sum(-x, (x, a + 1, 0))
assert Sum(x, (x, a, 5)).reverse_order(0) == Sum(-x, (x, 6, a - 1))
assert Sum(x, (x, a + 1, a + 5)).reverse_order(0) == \
Sum(-x, (x, a + 6, a))
assert Sum(x, (x, a + 1, a + 2)).reverse_order(0) == \
Sum(-x, (x, a + 3, a))
assert Sum(x, (x, a + 1, a + 1)).reverse_order(0) == \
Sum(-x, (x, a + 2, a))
assert Sum(x, (x, a, b)).reverse_order(0) == Sum(-x, (x, b + 1, a - 1))
assert Sum(x, (x, a, b)).reverse_order(x) == Sum(-x, (x, b + 1, a - 1))
assert Sum(x*y, (x, a, b), (y, 2, 5)).reverse_order(x, 1) == \
Sum(x*y, (x, b + 1, a - 1), (y, 6, 1))
assert Sum(x*y, (x, a, b), (y, 2, 5)).reverse_order(y, x) == \
Sum(x*y, (x, b + 1, a - 1), (y, 6, 1))
def test_issue_7097():
assert sum(x**n/n for n in range(1, 401)) == summation(x**n/n, (n, 1, 400))
def test_factor_expand_subs():
# test factoring
assert Sum(4 * x, (x, 1, y)).factor() == 4 * Sum(x, (x, 1, y))
assert Sum(x * a, (x, 1, y)).factor() == a * Sum(x, (x, 1, y))
assert Sum(4 * x * a, (x, 1, y)).factor() == 4 * a * Sum(x, (x, 1, y))
assert Sum(4 * x * y, (x, 1, y)).factor() == 4 * y * Sum(x, (x, 1, y))
# test expand
assert Sum(x+1,(x,1,y)).expand() == Sum(x,(x,1,y)) + Sum(1,(x,1,y))
assert Sum(x+a*x**2,(x,1,y)).expand() == Sum(x,(x,1,y)) + Sum(a*x**2,(x,1,y))
assert Sum(x**(n + 1)*(n + 1), (n, -1, oo)).expand() \
== Sum(x*x**n, (n, -1, oo)) + Sum(n*x*x**n, (n, -1, oo))
assert Sum(x**(n + 1)*(n + 1), (n, -1, oo)).expand(power_exp=False) \
== Sum(n*x**(n+1), (n, -1, oo)) + Sum(x**(n+1), (n, -1, oo))
assert Sum(a*n+a*n**2,(n,0,4)).expand() \
== Sum(a*n,(n,0,4)) + Sum(a*n**2,(n,0,4))
assert Sum(x**a*x**n,(x,0,3)) \
== Sum(x**(a+n),(x,0,3)).expand(power_exp=True)
assert Sum(x**(a+n),(x,0,3)) \
== Sum(x**(a+n),(x,0,3)).expand(power_exp=False)
# test subs
assert Sum(1/(1+a*x**2),(x,0,3)).subs([(a,3)]) == Sum(1/(1+3*x**2),(x,0,3))
assert Sum(x*y,(x,0,y),(y,0,x)).subs([(x,3)]) == Sum(x*y,(x,0,y),(y,0,3))
assert Sum(x,(x,1,10)).subs([(x,y-2)]) == Sum(x,(x,1,10))
assert Sum(1/x,(x,1,10)).subs([(x,(3+n)**3)]) == Sum(1/x,(x,1,10))
assert Sum(1/x,(x,1,10)).subs([(x,3*x-2)]) == Sum(1/x,(x,1,10))
def test_distribution_over_equality():
f = Function('f')
assert Product(Eq(x*2, f(x)), (x, 1, 3)).doit() == Eq(48, f(1)*f(2)*f(3))
assert Sum(Eq(f(x), x**2), (x, 0, y)) == \
Eq(Sum(f(x), (x, 0, y)), Sum(x**2, (x, 0, y)))
def test_issue_2787():
n, k = symbols('n k', positive=True, integer=True)
p = symbols('p', positive=True)
binomial_dist = binomial(n, k)*p**k*(1 - p)**(n - k)
s = Sum(binomial_dist*k, (k, 0, n))
res = s.doit().simplify()
assert res == Piecewise(
(n*p, p/Abs(p - 1) <= 1),
((-p + 1)**n*Sum(k*p**k*(-p + 1)**(-k)*binomial(n, k), (k, 0, n)),
True))
def test_issue_4668():
assert summation(1/n, (n, 2, oo)) == oo
def test_matrix_sum():
A = Matrix([[0,1],[n,0]])
assert Sum(A,(n,0,3)).doit() == Matrix([[0, 4], [6, 0]])
def test_indexed_idx_sum():
i = symbols('i', cls=Idx)
r = Indexed('r', i)
assert Sum(r, (i, 0, 3)).doit() == sum([r.xreplace({i: j}) for j in range(4)])
assert Product(r, (i, 0, 3)).doit() == prod([r.xreplace({i: j}) for j in range(4)])
j = symbols('j', integer=True)
assert Sum(r, (i, j, j+2)).doit() == sum([r.xreplace({i: j+k}) for k in range(3)])
assert Product(r, (i, j, j+2)).doit() == prod([r.xreplace({i: j+k}) for k in range(3)])
k = Idx('k', range=(1, 3))
A = IndexedBase('A')
assert Sum(A[k], k).doit() == sum([A[Idx(j, (1, 3))] for j in range(1, 4)])
assert Product(A[k], k).doit() == prod([A[Idx(j, (1, 3))] for j in range(1, 4)])
raises(ValueError, lambda: Sum(A[k], (k, 1, 4)))
raises(ValueError, lambda: Sum(A[k], (k, 0, 3)))
raises(ValueError, lambda: Sum(A[k], (k, 2, oo)))
raises(ValueError, lambda: Product(A[k], (k, 1, 4)))
raises(ValueError, lambda: Product(A[k], (k, 0, 3)))
raises(ValueError, lambda: Product(A[k], (k, 2, oo)))
def test_is_convergent():
# divergence tests --
assert Sum(n/(2*n + 1), (n, 1, oo)).is_convergent() is S.false
assert Sum(factorial(n)/5**n, (n, 1, oo)).is_convergent() is S.false
assert Sum(3**(-2*n - 1)*n**n, (n, 1, oo)).is_convergent() is S.false
assert Sum((-1)**n*n, (n, 3, oo)).is_convergent() is S.false
assert Sum((-1)**n, (n, 1, oo)).is_convergent() is S.false
assert Sum(log(1/n), (n, 2, oo)).is_convergent() is S.false
# root test --
assert Sum((-12)**n/n, (n, 1, oo)).is_convergent() is S.false
# integral test --
# p-series test --
assert Sum(1/(n**2 + 1), (n, 1, oo)).is_convergent() is S.true
assert Sum(1/n**(S(6)/5), (n, 1, oo)).is_convergent() is S.true
assert Sum(2/(n*sqrt(n - 1)), (n, 2, oo)).is_convergent() is S.true
assert Sum(1/(sqrt(n)*sqrt(n)), (n, 2, oo)).is_convergent() is S.false
# comparison test --
assert Sum(1/(n + log(n)), (n, 1, oo)).is_convergent() is S.false
assert Sum(1/(n**2*log(n)), (n, 2, oo)).is_convergent() is S.true
assert Sum(1/(n*log(n)), (n, 2, oo)).is_convergent() is S.false
assert Sum(2/(n*log(n)*log(log(n))**2), (n, 5, oo)).is_convergent() is S.true
assert Sum(2/(n*log(n)**2), (n, 2, oo)).is_convergent() is S.true
assert Sum((n - 1)/(n**2*log(n)**3), (n, 2, oo)).is_convergent() is S.true
assert Sum(1/(n*log(n)*log(log(n))), (n, 5, oo)).is_convergent() is S.false
assert Sum((n - 1)/(n*log(n)**3), (n, 3, oo)).is_convergent() is S.false
assert Sum(2/(n**2*log(n)), (n, 2, oo)).is_convergent() is S.true
assert Sum(1/(n*sqrt(log(n))*log(log(n))), (n, 100, oo)).is_convergent() is S.false
assert Sum(log(log(n))/(n*log(n)**2), (n, 100, oo)).is_convergent() is S.true
assert Sum(log(n)/n**2, (n, 5, oo)).is_convergent() is S.true
# alternating series tests --
assert Sum((-1)**(n - 1)/(n**2 - 1), (n, 3, oo)).is_convergent() is S.true
# with -negativeInfinite Limits
assert Sum(1/(n**2 + 1), (n, -oo, 1)).is_convergent() is S.true
assert Sum(1/(n - 1), (n, -oo, -1)).is_convergent() is S.false
assert Sum(1/(n**2 - 1), (n, -oo, -5)).is_convergent() is S.true
assert Sum(1/(n**2 - 1), (n, -oo, 2)).is_convergent() is S.true
assert Sum(1/(n**2 - 1), (n, -oo, oo)).is_convergent() is S.true
# piecewise functions
f = Piecewise((n**(-2), n <= 1), (n**2, n > 1))
assert Sum(f, (n, 1, oo)).is_convergent() is S.false
assert Sum(f, (n, -oo, oo)).is_convergent() is S.false
#assert Sum(f, (n, -oo, 1)).is_convergent() is S.true
# integral test
assert Sum(log(n)/n**3, (n, 1, oo)).is_convergent() is S.true
assert Sum(-log(n)/n**3, (n, 1, oo)).is_convergent() is S.true
# the following function has maxima located at (x, y) =
# (1.2, 0.43), (3.0, -0.25) and (6.8, 0.050)
eq = (x - 2)*(x**2 - 6*x + 4)*exp(-x)
assert Sum(eq, (x, 1, oo)).is_convergent() is S.true
assert Sum(eq, (x, 1, 2)).is_convergent() is S.true
assert Sum(1/(x**3), (x, 1, oo)).is_convergent() is S.true
assert Sum(1/(x**(S(1)/2)), (x, 1, oo)).is_convergent() is S.false
def test_is_absolutely_convergent():
assert Sum((-1)**n, (n, 1, oo)).is_absolutely_convergent() is S.false
assert Sum((-1)**n/n**2, (n, 1, oo)).is_absolutely_convergent() is S.true
@XFAIL
def test_convergent_failing():
# dirichlet tests
assert Sum(sin(n)/n, (n, 1, oo)).is_convergent() is S.true
assert Sum(sin(2*n)/n, (n, 1, oo)).is_convergent() is S.true
def test_issue_6966():
i, k, m = symbols('i k m', integer=True)
z_i, q_i = symbols('z_i q_i')
a_k = Sum(-q_i*z_i/k,(i,1,m))
b_k = a_k.diff(z_i)
assert isinstance(b_k, Sum)
assert b_k == Sum(-q_i/k,(i,1,m))
def test_issue_10156():
cx = Sum(2*y**2*x, (x, 1,3))
e = 2*y*Sum(2*cx*x**2, (x, 1, 9))
assert e.factor() == \
8*y**3*Sum(x, (x, 1, 3))*Sum(x**2, (x, 1, 9))
def test_issue_14129():
assert Sum( k*x**k, (k, 0, n-1)).doit() == \
Piecewise((n**2/2 - n/2, Eq(x, 1)), ((n*x*x**n -
n*x**n - x*x**n + x)/(x - 1)**2, True))
assert Sum( x**k, (k, 0, n-1)).doit() == \
Piecewise((n, Eq(x, 1)), ((-x**n + 1)/(-x + 1), True))
assert Sum( k*(x/y+x)**k, (k, 0, n-1)).doit() == \
Piecewise((n*(n - 1)/2, Eq(x, y/(y + 1))),
(x*(y + 1)*(n*x*y*(x + x/y)**n/(x + x/y)
+ n*x*(x + x/y)**n/(x + x/y) - n*y*(x
+ x/y)**n/(x + x/y) - x*y*(x + x/y)**n/(x
+ x/y) - x*(x + x/y)**n/(x + x/y) + y)/(x*y
+ x - y)**2, True))
def test_issue_14112():
assert Sum((-1)**n/sqrt(n), (n, 1, oo)).is_absolutely_convergent() is S.false
assert Sum((-1)**(2*n)/n, (n, 1, oo)).is_convergent() is S.false
assert Sum((-2)**n + (-3)**n, (n, 1, oo)).is_convergent() is S.false
def test_sin_times_absolutely_convergent():
assert Sum(sin(n) / n**3, (n, 1, oo)).is_convergent() is S.true
assert Sum(sin(n) * log(n) / n**3, (n, 1, oo)).is_convergent() is S.true
def test_issue_14111():
assert Sum(1/log(log(n)), (n, 22, oo)).is_convergent() is S.false
def test_issue_14484():
raises(NotImplementedError, lambda: Sum(sin(n)/log(log(n)), (n, 22, oo)).is_convergent())
def test_issue_14640():
i, n = symbols("i n", integer=True)
a, b, c = symbols("a b c")
assert Sum(a**-i/(a - b), (i, 0, n)).doit() == Sum(
1/(a*a**i - a**i*b), (i, 0, n)).doit() == Piecewise(
(n + 1, Eq(1/a, 1)),
((-a**(-n - 1) + 1)/(1 - 1/a), True))/(a - b)
assert Sum((b*a**i - c*a**i)**-2, (i, 0, n)).doit() == Piecewise(
(n + 1, Eq(a**(-2), 1)),
((-a**(-2*n - 2) + 1)/(1 - 1/a**2), True))/(b - c)**2
s = Sum(i*(a**(n - i) - b**(n - i))/(a - b), (i, 0, n)).doit()
assert not s.has(Sum)
assert s.subs({a: 2, b: 3, n: 5}) == 122
def test_issue_15943():
assert Sum(binomial(n, k)*factorial(n - k), (k, 0, n)).doit() == -E*(
n + 1)*gamma(n + 1)*lowergamma(n + 1, 1)/gamma(n + 2
) + E*gamma(n + 1)
def test_Sum_dummy_eq():
assert not Sum(x, (x, a, b)).dummy_eq(1)
assert not Sum(x, (x, a, b)).dummy_eq(Sum(x, (x, a, b), (a, 1, 2)))
assert not Sum(x, (x, a, b)).dummy_eq(Sum(x, (x, a, c)))
assert Sum(x, (x, a, b)).dummy_eq(Sum(x, (x, a, b)))
d = Dummy()
assert Sum(x, (x, a, d)).dummy_eq(Sum(x, (x, a, c)), c)
assert not Sum(x, (x, a, d)).dummy_eq(Sum(x, (x, a, c)))
assert Sum(x, (x, a, c)).dummy_eq(Sum(y, (y, a, c)))
assert Sum(x, (x, a, d)).dummy_eq(Sum(y, (y, a, c)), c)
assert not Sum(x, (x, a, d)).dummy_eq(Sum(y, (y, a, c)))
def test_issue_15852():
assert summation(x**y*y, (y, -oo, oo)).doit() == Sum(x**y*y, (y, -oo, oo))
def test_exceptions():
S = Sum(x, (x, a, b))
raises(ValueError, lambda: S.change_index(x, x**2, y))
S = Sum(x, (x, a, b), (x, 1, 4))
raises(ValueError, lambda: S.index(x))
S = Sum(x, (x, a, b), (y, 1, 4))
raises(ValueError, lambda: S.reorder([x]))
S = Sum(x, (x, y, b), (y, 1, 4))
raises(ReorderError, lambda: S.reorder_limit(0, 1))
S = Sum(x*y, (x, a, b), (y, 1, 4))
raises(NotImplementedError, lambda: S.is_convergent())
|
40bc9dabcf48260fa7c6e09bacf373e309d01ccb999cc80595c81d6caf868700
|
# This testfile tests SymPy <-> Sage compatibility
#
# Execute this test inside Sage, e.g. with:
# sage -python bin/test sympy/external/tests/test_sage.py
#
# This file can be tested by Sage itself by:
# sage -t sympy/external/tests/test_sage.py
# and if all tests pass, it should be copied (verbatim) to Sage, so that it is
# automatically doctested by Sage. Note that this second method imports the
# version of SymPy in Sage, whereas the -python method imports the local version
# of SymPy (both use the local version of the tests, however).
#
# Don't test any SymPy features here. Just pure interaction with Sage.
# Always write regular SymPy tests for anything, that can be tested in pure
# Python (without Sage). Here we test everything, that a user may need when
# using SymPy with Sage.
import os
import re
import sys
from sympy.external import import_module
sage = import_module('sage.all', __import__kwargs={'fromlist': ['all']})
if not sage:
#bin/test will not execute any tests now
disabled = True
import sympy
from sympy.utilities.pytest import XFAIL
def is_trivially_equal(lhs, rhs):
"""
True if lhs and rhs are trivially equal.
Use this for comparison of Sage expressions. Otherwise you
may start the whole proof machinery which may not exist at
the time of testing.
"""
assert (lhs - rhs).is_trivial_zero()
def check_expression(expr, var_symbols, only_from_sympy=False):
"""
Does eval(expr) both in Sage and SymPy and does other checks.
"""
# evaluate the expression in the context of Sage:
if var_symbols:
sage.var(var_symbols)
a = globals().copy()
# safety checks...
a.update(sage.__dict__)
assert "sin" in a
is_different = False
try:
e_sage = eval(expr, a)
assert not isinstance(e_sage, sympy.Basic)
except (NameError, TypeError):
is_different = True
pass
# evaluate the expression in the context of SymPy:
if var_symbols:
sympy_vars = sympy.var(var_symbols)
b = globals().copy()
b.update(sympy.__dict__)
assert "sin" in b
b.update(sympy.__dict__)
e_sympy = eval(expr, b)
assert isinstance(e_sympy, sympy.Basic)
# Sympy func may have specific _sage_ method
if is_different:
_sage_method = getattr(e_sympy.func, "_sage_")
e_sage = _sage_method(sympy.S(e_sympy))
# Do the actual checks:
if not only_from_sympy:
assert sympy.S(e_sage) == e_sympy
is_trivially_equal(e_sage, sage.SR(e_sympy))
def test_basics():
check_expression("x", "x")
check_expression("x**2", "x")
check_expression("x**2+y**3", "x y")
check_expression("1/(x+y)**2-x**3/4", "x y")
def test_complex():
check_expression("I", "")
check_expression("23+I*4", "x")
@XFAIL
def test_complex_fail():
# Sage doesn't properly implement _sympy_ on I
check_expression("I*y", "y")
check_expression("x+I*y", "x y")
def test_integer():
check_expression("4*x", "x")
check_expression("-4*x", "x")
def test_real():
check_expression("1.123*x", "x")
check_expression("-18.22*x", "x")
def test_E():
assert sympy.sympify(sage.e) == sympy.E
is_trivially_equal(sage.e, sage.SR(sympy.E))
def test_pi():
assert sympy.sympify(sage.pi) == sympy.pi
is_trivially_equal(sage.pi, sage.SR(sympy.pi))
def test_euler_gamma():
assert sympy.sympify(sage.euler_gamma) == sympy.EulerGamma
is_trivially_equal(sage.euler_gamma, sage.SR(sympy.EulerGamma))
def test_oo():
assert sympy.sympify(sage.oo) == sympy.oo
assert sage.oo == sage.SR(sympy.oo).pyobject()
assert sympy.sympify(-sage.oo) == -sympy.oo
assert -sage.oo == sage.SR(-sympy.oo).pyobject()
#assert sympy.sympify(sage.UnsignedInfinityRing.gen()) == sympy.zoo
#assert sage.UnsignedInfinityRing.gen() == sage.SR(sympy.zoo)
def test_NaN():
assert sympy.sympify(sage.NaN) == sympy.nan
is_trivially_equal(sage.NaN, sage.SR(sympy.nan))
def test_Catalan():
assert sympy.sympify(sage.catalan) == sympy.Catalan
is_trivially_equal(sage.catalan, sage.SR(sympy.Catalan))
def test_GoldenRation():
assert sympy.sympify(sage.golden_ratio) == sympy.GoldenRatio
is_trivially_equal(sage.golden_ratio, sage.SR(sympy.GoldenRatio))
def test_functions():
# Test at least one Function without own _sage_ method
assert not "_sage_" in sympy.factorial.__dict__
check_expression("factorial(x)", "x")
check_expression("sin(x)", "x")
check_expression("cos(x)", "x")
check_expression("tan(x)", "x")
check_expression("cot(x)", "x")
check_expression("asin(x)", "x")
check_expression("acos(x)", "x")
check_expression("atan(x)", "x")
check_expression("atan2(y, x)", "x, y")
check_expression("acot(x)", "x")
check_expression("sinh(x)", "x")
check_expression("cosh(x)", "x")
check_expression("tanh(x)", "x")
check_expression("coth(x)", "x")
check_expression("asinh(x)", "x")
check_expression("acosh(x)", "x")
check_expression("atanh(x)", "x")
check_expression("acoth(x)", "x")
check_expression("exp(x)", "x")
check_expression("gamma(x)", "x")
check_expression("log(x)", "x")
check_expression("re(x)", "x")
check_expression("im(x)", "x")
check_expression("sign(x)", "x")
check_expression("abs(x)", "x")
check_expression("arg(x)", "x")
check_expression("conjugate(x)", "x")
# The following tests differently named functions
check_expression("besselj(y, x)", "x, y")
check_expression("bessely(y, x)", "x, y")
check_expression("besseli(y, x)", "x, y")
check_expression("besselk(y, x)", "x, y")
check_expression("DiracDelta(x)", "x")
check_expression("KroneckerDelta(x, y)", "x, y")
check_expression("expint(y, x)", "x, y")
check_expression("Si(x)", "x")
check_expression("Ci(x)", "x")
check_expression("Shi(x)", "x")
check_expression("Chi(x)", "x")
check_expression("loggamma(x)", "x")
check_expression("Ynm(n,m,x,y)", "n, m, x, y")
check_expression("hyper((n,m),(m,n),x)", "n, m, x")
check_expression("uppergamma(y, x)", "x, y")
def test_issue_4023():
sage.var("a x")
log = sage.log
i = sympy.integrate(log(x)/a, (x, a, a + 1))
i2 = sympy.simplify(i)
s = sage.SR(i2)
is_trivially_equal(s, -log(a) + log(a + 1) + log(a + 1)/a - 1/a)
def test_integral():
#test Sympy-->Sage
check_expression("Integral(x, (x,))", "x", only_from_sympy=True)
check_expression("Integral(x, (x, 0, 1))", "x", only_from_sympy=True)
check_expression("Integral(x*y, (x,), (y, ))", "x,y", only_from_sympy=True)
check_expression("Integral(x*y, (x,), (y, 0, 1))", "x,y", only_from_sympy=True)
check_expression("Integral(x*y, (x, 0, 1), (y,))", "x,y", only_from_sympy=True)
check_expression("Integral(x*y, (x, 0, 1), (y, 0, 1))", "x,y", only_from_sympy=True)
check_expression("Integral(x*y*z, (x, 0, 1), (y, 0, 1), (z, 0, 1))", "x,y,z", only_from_sympy=True)
@XFAIL
def test_integral_failing():
# Note: sage may attempt to turn this into Integral(x, (x, x, 0))
check_expression("Integral(x, (x, 0))", "x", only_from_sympy=True)
check_expression("Integral(x*y, (x,), (y, 0))", "x,y", only_from_sympy=True)
check_expression("Integral(x*y, (x, 0, 1), (y, 0))", "x,y", only_from_sympy=True)
def test_undefined_function():
f = sympy.Function('f')
sf = sage.function('f')
x = sympy.symbols('x')
sx = sage.var('x')
is_trivially_equal(sf(sx), f(x)._sage_())
#assert bool(f == sympy.sympify(sf))
def test_abstract_function():
from sage.symbolic.expression import Expression
x,y = sympy.symbols('x y')
f = sympy.Function('f')
expr = f(x,y)
sexpr = expr._sage_()
assert isinstance(sexpr,Expression), "converted expression %r is not sage expression" % sexpr
# This test has to be uncommented in the future: it depends on the sage ticket #22802 (https://trac.sagemath.org/ticket/22802)
# invexpr = sexpr._sympy_()
# assert invexpr == expr, "inverse coversion %r is not correct " % invexpr
# This string contains Sage doctests, that execute all the functions above.
# When you add a new function, please add it here as well.
"""
TESTS::
sage: from sympy.external.tests.test_sage import *
sage: test_basics()
sage: test_basics()
sage: test_complex()
sage: test_integer()
sage: test_real()
sage: test_E()
sage: test_pi()
sage: test_euler_gamma()
sage: test_oo()
sage: test_NaN()
sage: test_Catalan()
sage: test_GoldenRation()
sage: test_functions()
sage: test_issue_4023()
sage: test_integral()
sage: test_undefined_function()
sage: test_abstract_function()
Sage has no symbolic Lucas function at the moment::
sage: check_expression("lucas(x)", "x")
Traceback (most recent call last):
...
AttributeError...
"""
|
b27c9ef772dca63f9f487056afb102ad8cffb8368b412154b05ad0b6822d9df5
|
# This testfile tests SymPy <-> NumPy compatibility
# Don't test any SymPy features here. Just pure interaction with NumPy.
# Always write regular SymPy tests for anything, that can be tested in pure
# Python (without numpy). Here we test everything, that a user may need when
# using SymPy with NumPy
from sympy.external import import_module
numpy = import_module('numpy')
if numpy:
array, matrix, ndarray = numpy.array, numpy.matrix, numpy.ndarray
else:
#bin/test will not execute any tests now
disabled = True
from sympy import (Rational, Symbol, list2numpy, matrix2numpy, sin, Float,
Matrix, lambdify, symarray, symbols, Integer)
import sympy
import mpmath
from sympy.abc import x, y, z
from sympy.utilities.decorator import conserve_mpmath_dps
from sympy.utilities.pytest import raises
# first, systematically check, that all operations are implemented and don't
# raise an exception
def test_systematic_basic():
def s(sympy_object, numpy_array):
x = sympy_object + numpy_array
x = numpy_array + sympy_object
x = sympy_object - numpy_array
x = numpy_array - sympy_object
x = sympy_object * numpy_array
x = numpy_array * sympy_object
x = sympy_object / numpy_array
x = numpy_array / sympy_object
x = sympy_object ** numpy_array
x = numpy_array ** sympy_object
x = Symbol("x")
y = Symbol("y")
sympy_objs = [
Rational(2, 3),
Float("1.3"),
x,
y,
pow(x, y)*y,
Integer(5),
Float(5.5),
]
numpy_objs = [
array([1]),
array([3, 8, -1]),
array([x, x**2, Rational(5)]),
array([x/y*sin(y), 5, Rational(5)]),
]
for x in sympy_objs:
for y in numpy_objs:
s(x, y)
# now some random tests, that test particular problems and that also
# check that the results of the operations are correct
def test_basics():
one = Rational(1)
zero = Rational(0)
assert array(1) == array(one)
assert array([one]) == array([one])
assert array([x]) == array([x])
assert array(x) == array(Symbol("x"))
assert array(one + x) == array(1 + x)
X = array([one, zero, zero])
assert (X == array([one, zero, zero])).all()
assert (X == array([one, 0, 0])).all()
def test_arrays():
one = Rational(1)
zero = Rational(0)
X = array([one, zero, zero])
Y = one*X
X = array([Symbol("a") + Rational(1, 2)])
Y = X + X
assert Y == array([1 + 2*Symbol("a")])
Y = Y + 1
assert Y == array([2 + 2*Symbol("a")])
Y = X - X
assert Y == array([0])
def test_conversion1():
a = list2numpy([x**2, x])
#looks like an array?
assert isinstance(a, ndarray)
assert a[0] == x**2
assert a[1] == x
assert len(a) == 2
#yes, it's the array
def test_conversion2():
a = 2*list2numpy([x**2, x])
b = list2numpy([2*x**2, 2*x])
assert (a == b).all()
one = Rational(1)
zero = Rational(0)
X = list2numpy([one, zero, zero])
Y = one*X
X = list2numpy([Symbol("a") + Rational(1, 2)])
Y = X + X
assert Y == array([1 + 2*Symbol("a")])
Y = Y + 1
assert Y == array([2 + 2*Symbol("a")])
Y = X - X
assert Y == array([0])
def test_list2numpy():
assert (array([x**2, x]) == list2numpy([x**2, x])).all()
def test_Matrix1():
m = Matrix([[x, x**2], [5, 2/x]])
assert (array(m.subs(x, 2)) == array([[2, 4], [5, 1]])).all()
m = Matrix([[sin(x), x**2], [5, 2/x]])
assert (array(m.subs(x, 2)) == array([[sin(2), 4], [5, 1]])).all()
def test_Matrix2():
m = Matrix([[x, x**2], [5, 2/x]])
assert (matrix(m.subs(x, 2)) == matrix([[2, 4], [5, 1]])).all()
m = Matrix([[sin(x), x**2], [5, 2/x]])
assert (matrix(m.subs(x, 2)) == matrix([[sin(2), 4], [5, 1]])).all()
def test_Matrix3():
a = array([[2, 4], [5, 1]])
assert Matrix(a) == Matrix([[2, 4], [5, 1]])
assert Matrix(a) != Matrix([[2, 4], [5, 2]])
a = array([[sin(2), 4], [5, 1]])
assert Matrix(a) == Matrix([[sin(2), 4], [5, 1]])
assert Matrix(a) != Matrix([[sin(0), 4], [5, 1]])
def test_Matrix4():
a = matrix([[2, 4], [5, 1]])
assert Matrix(a) == Matrix([[2, 4], [5, 1]])
assert Matrix(a) != Matrix([[2, 4], [5, 2]])
a = matrix([[sin(2), 4], [5, 1]])
assert Matrix(a) == Matrix([[sin(2), 4], [5, 1]])
assert Matrix(a) != Matrix([[sin(0), 4], [5, 1]])
def test_Matrix_sum():
M = Matrix([[1, 2, 3], [x, y, x], [2*y, -50, z*x]])
m = matrix([[2, 3, 4], [x, 5, 6], [x, y, z**2]])
assert M + m == Matrix([[3, 5, 7], [2*x, y + 5, x + 6], [2*y + x, y - 50, z*x + z**2]])
assert m + M == Matrix([[3, 5, 7], [2*x, y + 5, x + 6], [2*y + x, y - 50, z*x + z**2]])
assert M + m == M.add(m)
def test_Matrix_mul():
M = Matrix([[1, 2, 3], [x, y, x]])
m = matrix([[2, 4], [x, 6], [x, z**2]])
assert M*m == Matrix([
[ 2 + 5*x, 16 + 3*z**2],
[2*x + x*y + x**2, 4*x + 6*y + x*z**2],
])
assert m*M == Matrix([
[ 2 + 4*x, 4 + 4*y, 6 + 4*x],
[ 7*x, 2*x + 6*y, 9*x],
[x + x*z**2, 2*x + y*z**2, 3*x + x*z**2],
])
a = array([2])
assert a[0] * M == 2 * M
assert M * a[0] == 2 * M
def test_Matrix_array():
class matarray(object):
def __array__(self):
from numpy import array
return array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
matarr = matarray()
assert Matrix(matarr) == Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
def test_matrix2numpy():
a = matrix2numpy(Matrix([[1, x**2], [3*sin(x), 0]]))
assert isinstance(a, ndarray)
assert a.shape == (2, 2)
assert a[0, 0] == 1
assert a[0, 1] == x**2
assert a[1, 0] == 3*sin(x)
assert a[1, 1] == 0
def test_matrix2numpy_conversion():
a = Matrix([[1, 2, sin(x)], [x**2, x, Rational(1, 2)]])
b = array([[1, 2, sin(x)], [x**2, x, Rational(1, 2)]])
assert (matrix2numpy(a) == b).all()
assert matrix2numpy(a).dtype == numpy.dtype('object')
c = matrix2numpy(Matrix([[1, 2], [10, 20]]), dtype='int8')
d = matrix2numpy(Matrix([[1, 2], [10, 20]]), dtype='float64')
assert c.dtype == numpy.dtype('int8')
assert d.dtype == numpy.dtype('float64')
def test_issue_3728():
assert (Rational(1, 2)*array([2*x, 0]) == array([x, 0])).all()
assert (Rational(1, 2) + array(
[2*x, 0]) == array([2*x + Rational(1, 2), Rational(1, 2)])).all()
assert (Float("0.5")*array([2*x, 0]) == array([Float("1.0")*x, 0])).all()
assert (Float("0.5") + array(
[2*x, 0]) == array([2*x + Float("0.5"), Float("0.5")])).all()
@conserve_mpmath_dps
def test_lambdify():
mpmath.mp.dps = 16
sin02 = mpmath.mpf("0.198669330795061215459412627")
f = lambdify(x, sin(x), "numpy")
prec = 1e-15
assert -prec < f(0.2) - sin02 < prec
with raises(AttributeError):
f(x) # if this succeeds, it can't be a numpy function
def test_lambdify_matrix():
f = lambdify(x, Matrix([[x, 2*x], [1, 2]]), [{'ImmutableMatrix': numpy.array}, "numpy"])
assert (f(1) == array([[1, 2], [1, 2]])).all()
def test_lambdify_matrix_multi_input():
M = sympy.Matrix([[x**2, x*y, x*z],
[y*x, y**2, y*z],
[z*x, z*y, z**2]])
f = lambdify((x, y, z), M, [{'ImmutableMatrix': numpy.array}, "numpy"])
xh, yh, zh = 1.0, 2.0, 3.0
expected = array([[xh**2, xh*yh, xh*zh],
[yh*xh, yh**2, yh*zh],
[zh*xh, zh*yh, zh**2]])
actual = f(xh, yh, zh)
assert numpy.allclose(actual, expected)
def test_lambdify_matrix_vec_input():
X = sympy.DeferredVector('X')
M = Matrix([
[X[0]**2, X[0]*X[1], X[0]*X[2]],
[X[1]*X[0], X[1]**2, X[1]*X[2]],
[X[2]*X[0], X[2]*X[1], X[2]**2]])
f = lambdify(X, M, [{'ImmutableMatrix': numpy.array}, "numpy"])
Xh = array([1.0, 2.0, 3.0])
expected = array([[Xh[0]**2, Xh[0]*Xh[1], Xh[0]*Xh[2]],
[Xh[1]*Xh[0], Xh[1]**2, Xh[1]*Xh[2]],
[Xh[2]*Xh[0], Xh[2]*Xh[1], Xh[2]**2]])
actual = f(Xh)
assert numpy.allclose(actual, expected)
def test_lambdify_transl():
from sympy.utilities.lambdify import NUMPY_TRANSLATIONS
for sym, mat in NUMPY_TRANSLATIONS.items():
assert sym in sympy.__dict__
assert mat in numpy.__dict__
def test_symarray():
"""Test creation of numpy arrays of sympy symbols."""
import numpy as np
import numpy.testing as npt
syms = symbols('_0,_1,_2')
s1 = symarray("", 3)
s2 = symarray("", 3)
npt.assert_array_equal(s1, np.array(syms, dtype=object))
assert s1[0] == s2[0]
a = symarray('a', 3)
b = symarray('b', 3)
assert not(a[0] == b[0])
asyms = symbols('a_0,a_1,a_2')
npt.assert_array_equal(a, np.array(asyms, dtype=object))
# Multidimensional checks
a2d = symarray('a', (2, 3))
assert a2d.shape == (2, 3)
a00, a12 = symbols('a_0_0,a_1_2')
assert a2d[0, 0] == a00
assert a2d[1, 2] == a12
a3d = symarray('a', (2, 3, 2))
assert a3d.shape == (2, 3, 2)
a000, a120, a121 = symbols('a_0_0_0,a_1_2_0,a_1_2_1')
assert a3d[0, 0, 0] == a000
assert a3d[1, 2, 0] == a120
assert a3d[1, 2, 1] == a121
def test_vectorize():
assert (numpy.vectorize(
sin)([1, 2, 3]) == numpy.array([sin(1), sin(2), sin(3)])).all()
|
9aff999df94f64516906ac7bdf84e932e1f202e1d242a488b3c2cc529bff3bd8
|
from sympy import (symbols, Symbol, oo, Sum, harmonic, Add, S, binomial,
factorial, log, fibonacci, sin, cos, pi, I, sqrt)
from sympy.series.limitseq import limit_seq
from sympy.series.limitseq import difference_delta as dd
from sympy.utilities.pytest import raises, XFAIL
from sympy.calculus.util import AccumulationBounds
n, m, k = symbols('n m k', integer=True)
def test_difference_delta():
e = n*(n + 1)
e2 = e * k
assert dd(e) == 2*n + 2
assert dd(e2, n, 2) == k*(4*n + 6)
raises(ValueError, lambda: dd(e2))
raises(ValueError, lambda: dd(e2, n, oo))
def test_difference_delta__Sum():
e = Sum(1/k, (k, 1, n))
assert dd(e, n) == 1/(n + 1)
assert dd(e, n, 5) == Add(*[1/(i + n + 1) for i in range(5)])
e = Sum(1/k, (k, 1, 3*n))
assert dd(e, n) == Add(*[1/(i + 3*n + 1) for i in range(3)])
e = n * Sum(1/k, (k, 1, n))
assert dd(e, n) == 1 + Sum(1/k, (k, 1, n))
e = Sum(1/k, (k, 1, n), (m, 1, n))
assert dd(e, n) == harmonic(n)
def test_difference_delta__Add():
e = n + n*(n + 1)
assert dd(e, n) == 2*n + 3
assert dd(e, n, 2) == 4*n + 8
e = n + Sum(1/k, (k, 1, n))
assert dd(e, n) == 1 + 1/(n + 1)
assert dd(e, n, 5) == 5 + Add(*[1/(i + n + 1) for i in range(5)])
def test_difference_delta__Pow():
e = 4**n
assert dd(e, n) == 3*4**n
assert dd(e, n, 2) == 15*4**n
e = 4**(2*n)
assert dd(e, n) == 15*4**(2*n)
assert dd(e, n, 2) == 255*4**(2*n)
e = n**4
assert dd(e, n) == (n + 1)**4 - n**4
e = n**n
assert dd(e, n) == (n + 1)**(n + 1) - n**n
def test_limit_seq():
e = binomial(2*n, n) / Sum(binomial(2*k, k), (k, 1, n))
assert limit_seq(e) == S(3) / 4
assert limit_seq(e, m) == e
e = (5*n**3 + 3*n**2 + 4) / (3*n**3 + 4*n - 5)
assert limit_seq(e, n) == S(5) / 3
e = (harmonic(n) * Sum(harmonic(k), (k, 1, n))) / (n * harmonic(2*n)**2)
assert limit_seq(e, n) == 1
e = Sum(k**2 * Sum(2**m/m, (m, 1, k)), (k, 1, n)) / (2**n*n)
assert limit_seq(e, n) == 4
e = (Sum(binomial(3*k, k) * binomial(5*k, k), (k, 1, n)) /
(binomial(3*n, n) * binomial(5*n, n)))
assert limit_seq(e, n) == S(84375) / 83351
e = Sum(harmonic(k)**2/k, (k, 1, 2*n)) / harmonic(n)**3
assert limit_seq(e, n) == S(1) / 3
raises(ValueError, lambda: limit_seq(e * m))
def test_alternating_sign():
assert limit_seq((-1)**n/n**2, n) == 0
assert limit_seq((-2)**(n+1)/(n + 3**n), n) == 0
assert limit_seq((2*n + (-1)**n)/(n + 1), n) == 2
assert limit_seq(sin(pi*n), n) == 0
assert limit_seq(cos(2*pi*n), n) == 1
assert limit_seq((S(-1)/5)**n, n) == 0
assert limit_seq((-S(1)/5)**n, n) == 0
assert limit_seq((I/3)**n, n) == 0
assert limit_seq(sqrt(n)*(I/2)**n, n) == 0
assert limit_seq(n**7*(I/3)**n, n) == 0
assert limit_seq(n/(n + 1) + (I/2)**n, n) == 1
def test_accum_bounds():
assert limit_seq((-1)**n, n) == AccumulationBounds(-1, 1)
assert limit_seq(cos(pi*n), n) == AccumulationBounds(-1, 1)
assert limit_seq(sin(pi*n/2)**2, n) == AccumulationBounds(0, 1)
assert limit_seq(2*(-3)**n/(n + 3**n), n) == AccumulationBounds(-2, 2)
assert limit_seq(3*n/(n + 1) + 2*(-1)**n, n) == AccumulationBounds(1, 5)
def test_limitseq_sum():
from sympy.abc import x, y, z
assert limit_seq(Sum(1/x, (x, 1, y)) - log(y), y) == S.EulerGamma
assert limit_seq(Sum(1/x, (x, 1, y)) - 1/y, y) == S.Infinity
assert (limit_seq(binomial(2*x, x) / Sum(binomial(2*y, y), (y, 1, x)), x) ==
S(3) / 4)
assert (limit_seq(Sum(y**2 * Sum(2**z/z, (z, 1, y)), (y, 1, x)) /
(2**x*x), x) == 4)
def test_issue_10382():
n = Symbol('n', integer=True)
assert limit_seq(fibonacci(n+1)/fibonacci(n), n) == S.GoldenRatio
@XFAIL
def test_limit_seq_fail():
# improve Summation algorithm or add ad-hoc criteria
e = (harmonic(n)**3 * Sum(1/harmonic(k), (k, 1, n)) /
(n * Sum(harmonic(k)/k, (k, 1, n))))
assert limit_seq(e, n) == 2
# No unique dominant term
e = (Sum(2**k * binomial(2*k, k) / k**2, (k, 1, n)) /
(Sum(2**k/k*2, (k, 1, n)) * Sum(binomial(2*k, k), (k, 1, n))))
assert limit_seq(e, n) == S(3) / 7
# Simplifications of summations needs to be improved.
e = n**3*Sum(2**k/k**2, (k, 1, n))**2 / (2**n * Sum(2**k/k, (k, 1, n)))
assert limit_seq(e, n) == 2
e = (harmonic(n) * Sum(2**k/k, (k, 1, n)) /
(n * Sum(2**k*harmonic(k)/k**2, (k, 1, n))))
assert limit_seq(e, n) == 1
e = (Sum(2**k*factorial(k) / k**2, (k, 1, 2*n)) /
(Sum(4**k/k**2, (k, 1, n)) * Sum(factorial(k), (k, 1, 2*n))))
assert limit_seq(e, n) == S(3) / 16
|
2cb89ed506d8d1a9131cad8a20c88db80134c33b6c7e2971ebbcce60db8f814d
|
from sympy import (S, Tuple, symbols, Interval, EmptySequence, oo, SeqPer,
SeqFormula, sequence, SeqAdd, SeqMul, Indexed, Idx, sqrt,
fibonacci, tribonacci, sin, cos, exp)
from sympy.series.sequences import SeqExpr, SeqExprOp
from sympy.utilities.pytest import raises, slow
x, y, z = symbols('x y z')
n, m = symbols('n m')
def test_EmptySequence():
assert isinstance(S.EmptySequence, EmptySequence)
assert S.EmptySequence.interval is S.EmptySet
assert S.EmptySequence.length is S.Zero
assert list(S.EmptySequence) == []
def test_SeqExpr():
s = SeqExpr((1, n, y), (x, 0, 10))
assert isinstance(s, SeqExpr)
assert s.gen == (1, n, y)
assert s.interval == Interval(0, 10)
assert s.start == 0
assert s.stop == 10
assert s.length == 11
assert s.variables == (x,)
assert SeqExpr((1, 2, 3), (x, 0, oo)).length is oo
def test_SeqPer():
s = SeqPer((1, n, 3), (x, 0, 5))
assert isinstance(s, SeqPer)
assert s.periodical == Tuple(1, n, 3)
assert s.period == 3
assert s.coeff(3) == 1
assert s.free_symbols == {n}
assert list(s) == [1, n, 3, 1, n, 3]
assert s[:] == [1, n, 3, 1, n, 3]
assert SeqPer((1, n, 3), (x, -oo, 0))[0:6] == [1, n, 3, 1, n, 3]
raises(ValueError, lambda: SeqPer((1, 2, 3), (0, 1, 2)))
raises(ValueError, lambda: SeqPer((1, 2, 3), (x, -oo, oo)))
raises(ValueError, lambda: SeqPer(n**2, (0, oo)))
assert SeqPer((n, n**2, n**3), (m, 0, oo))[:6] == \
[n, n**2, n**3, n, n**2, n**3]
assert SeqPer((n, n**2, n**3), (n, 0, oo))[:6] == [0, 1, 8, 3, 16, 125]
assert SeqPer((n, m), (n, 0, oo))[:6] == [0, m, 2, m, 4, m]
def test_SeqFormula():
s = SeqFormula(n**2, (n, 0, 5))
assert isinstance(s, SeqFormula)
assert s.formula == n**2
assert s.coeff(3) == 9
assert list(s) == [i**2 for i in range(6)]
assert s[:] == [i**2 for i in range(6)]
assert SeqFormula(n**2, (n, -oo, 0))[0:6] == [i**2 for i in range(6)]
assert SeqFormula(n**2, (0, oo)) == SeqFormula(n**2, (n, 0, oo))
assert SeqFormula(n**2, (0, m)).subs(m, x) == SeqFormula(n**2, (0, x))
assert SeqFormula(m*n**2, (n, 0, oo)).subs(m, x) == \
SeqFormula(x*n**2, (n, 0, oo))
raises(ValueError, lambda: SeqFormula(n**2, (0, 1, 2)))
raises(ValueError, lambda: SeqFormula(n**2, (n, -oo, oo)))
raises(ValueError, lambda: SeqFormula(m*n**2, (0, oo)))
seq = SeqFormula(x*(y**2 + z), (z, 1, 100))
assert seq.expand() == SeqFormula(x*y**2 + x*z, (z, 1, 100))
seq = SeqFormula(sin(x*(y**2 + z)),(z, 1, 100))
assert seq.expand(trig=True) == SeqFormula(sin(x*y**2)*cos(x*z) + sin(x*z)*cos(x*y**2), (z, 1, 100))
assert seq.expand() == SeqFormula(sin(x*y**2 + x*z), (z, 1, 100))
assert seq.expand(trig=False) == SeqFormula(sin(x*y**2 + x*z), (z, 1, 100))
seq = SeqFormula(exp(x*(y**2 + z)), (z, 1, 100))
assert seq.expand() == SeqFormula(exp(x*y**2)*exp(x*z), (z, 1, 100))
assert seq.expand(power_exp=False) == SeqFormula(exp(x*y**2 + x*z), (z, 1, 100))
assert seq.expand(mul=False, power_exp=False) == SeqFormula(exp(x*(y**2 + z)), (z, 1, 100))
def test_sequence():
form = SeqFormula(n**2, (n, 0, 5))
per = SeqPer((1, 2, 3), (n, 0, 5))
inter = SeqFormula(n**2)
assert sequence(n**2, (n, 0, 5)) == form
assert sequence((1, 2, 3), (n, 0, 5)) == per
assert sequence(n**2) == inter
def test_SeqExprOp():
form = SeqFormula(n**2, (n, 0, 10))
per = SeqPer((1, 2, 3), (m, 5, 10))
s = SeqExprOp(form, per)
assert s.gen == (n**2, (1, 2, 3))
assert s.interval == Interval(5, 10)
assert s.start == 5
assert s.stop == 10
assert s.length == 6
assert s.variables == (n, m)
def test_SeqAdd():
per = SeqPer((1, 2, 3), (n, 0, oo))
form = SeqFormula(n**2)
per_bou = SeqPer((1, 2), (n, 1, 5))
form_bou = SeqFormula(n**2, (6, 10))
form_bou2 = SeqFormula(n**2, (1, 5))
assert SeqAdd() == S.EmptySequence
assert SeqAdd(S.EmptySequence) == S.EmptySequence
assert SeqAdd(per) == per
assert SeqAdd(per, S.EmptySequence) == per
assert SeqAdd(per_bou, form_bou) == S.EmptySequence
s = SeqAdd(per_bou, form_bou2, evaluate=False)
assert s.args == (form_bou2, per_bou)
assert s[:] == [2, 6, 10, 18, 26]
assert list(s) == [2, 6, 10, 18, 26]
assert isinstance(SeqAdd(per, per_bou, evaluate=False), SeqAdd)
s1 = SeqAdd(per, per_bou)
assert isinstance(s1, SeqPer)
assert s1 == SeqPer((2, 4, 4, 3, 3, 5), (n, 1, 5))
s2 = SeqAdd(form, form_bou)
assert isinstance(s2, SeqFormula)
assert s2 == SeqFormula(2*n**2, (6, 10))
assert SeqAdd(form, form_bou, per) == \
SeqAdd(per, SeqFormula(2*n**2, (6, 10)))
assert SeqAdd(form, SeqAdd(form_bou, per)) == \
SeqAdd(per, SeqFormula(2*n**2, (6, 10)))
assert SeqAdd(per, SeqAdd(form, form_bou), evaluate=False) == \
SeqAdd(per, SeqFormula(2*n**2, (6, 10)))
assert SeqAdd(SeqPer((1, 2), (n, 0, oo)), SeqPer((1, 2), (m, 0, oo))) == \
SeqPer((2, 4), (n, 0, oo))
def test_SeqMul():
per = SeqPer((1, 2, 3), (n, 0, oo))
form = SeqFormula(n**2)
per_bou = SeqPer((1, 2), (n, 1, 5))
form_bou = SeqFormula(n**2, (n, 6, 10))
form_bou2 = SeqFormula(n**2, (1, 5))
assert SeqMul() == S.EmptySequence
assert SeqMul(S.EmptySequence) == S.EmptySequence
assert SeqMul(per) == per
assert SeqMul(per, S.EmptySequence) == S.EmptySequence
assert SeqMul(per_bou, form_bou) == S.EmptySequence
s = SeqMul(per_bou, form_bou2, evaluate=False)
assert s.args == (form_bou2, per_bou)
assert s[:] == [1, 8, 9, 32, 25]
assert list(s) == [1, 8, 9, 32, 25]
assert isinstance(SeqMul(per, per_bou, evaluate=False), SeqMul)
s1 = SeqMul(per, per_bou)
assert isinstance(s1, SeqPer)
assert s1 == SeqPer((1, 4, 3, 2, 2, 6), (n, 1, 5))
s2 = SeqMul(form, form_bou)
assert isinstance(s2, SeqFormula)
assert s2 == SeqFormula(n**4, (6, 10))
assert SeqMul(form, form_bou, per) == \
SeqMul(per, SeqFormula(n**4, (6, 10)))
assert SeqMul(form, SeqMul(form_bou, per)) == \
SeqMul(per, SeqFormula(n**4, (6, 10)))
assert SeqMul(per, SeqMul(form, form_bou2,
evaluate=False), evaluate=False) == \
SeqMul(form, per, form_bou2, evaluate=False)
assert SeqMul(SeqPer((1, 2), (n, 0, oo)), SeqPer((1, 2), (n, 0, oo))) == \
SeqPer((1, 4), (n, 0, oo))
def test_add():
per = SeqPer((1, 2), (n, 0, oo))
form = SeqFormula(n**2)
assert per + (SeqPer((2, 3))) == SeqPer((3, 5), (n, 0, oo))
assert form + SeqFormula(n**3) == SeqFormula(n**2 + n**3)
assert per + form == SeqAdd(per, form)
raises(TypeError, lambda: per + n)
raises(TypeError, lambda: n + per)
def test_sub():
per = SeqPer((1, 2), (n, 0, oo))
form = SeqFormula(n**2)
assert per - (SeqPer((2, 3))) == SeqPer((-1, -1), (n, 0, oo))
assert form - (SeqFormula(n**3)) == SeqFormula(n**2 - n**3)
assert per - form == SeqAdd(per, -form)
raises(TypeError, lambda: per - n)
raises(TypeError, lambda: n - per)
def test_mul__coeff_mul():
assert SeqPer((1, 2), (n, 0, oo)).coeff_mul(2) == SeqPer((2, 4), (n, 0, oo))
assert SeqFormula(n**2).coeff_mul(2) == SeqFormula(2*n**2)
assert S.EmptySequence.coeff_mul(100) == S.EmptySequence
assert SeqPer((1, 2), (n, 0, oo)) * (SeqPer((2, 3))) == \
SeqPer((2, 6), (n, 0, oo))
assert SeqFormula(n**2) * SeqFormula(n**3) == SeqFormula(n**5)
assert S.EmptySequence * SeqFormula(n**2) == S.EmptySequence
assert SeqFormula(n**2) * S.EmptySequence == S.EmptySequence
raises(TypeError, lambda: sequence(n**2) * n)
raises(TypeError, lambda: n * sequence(n**2))
def test_neg():
assert -SeqPer((1, -2), (n, 0, oo)) == SeqPer((-1, 2), (n, 0, oo))
assert -SeqFormula(n**2) == SeqFormula(-n**2)
def test_operations():
per = SeqPer((1, 2), (n, 0, oo))
per2 = SeqPer((2, 4), (n, 0, oo))
form = SeqFormula(n**2)
form2 = SeqFormula(n**3)
assert per + form + form2 == SeqAdd(per, form, form2)
assert per + form - form2 == SeqAdd(per, form, -form2)
assert per + form - S.EmptySequence == SeqAdd(per, form)
assert per + per2 + form == SeqAdd(SeqPer((3, 6), (n, 0, oo)), form)
assert S.EmptySequence - per == -per
assert form + form == SeqFormula(2*n**2)
assert per * form * form2 == SeqMul(per, form, form2)
assert form * form == SeqFormula(n**4)
assert form * -form == SeqFormula(-n**4)
assert form * (per + form2) == SeqMul(form, SeqAdd(per, form2))
assert form * (per + per) == SeqMul(form, per2)
assert form.coeff_mul(m) == SeqFormula(m*n**2, (n, 0, oo))
assert per.coeff_mul(m) == SeqPer((m, 2*m), (n, 0, oo))
def test_Idx_limits():
i = symbols('i', cls=Idx)
r = Indexed('r', i)
assert SeqFormula(r, (i, 0, 5))[:] == [r.subs(i, j) for j in range(6)]
assert SeqPer((1, 2), (i, 0, 5))[:] == [1, 2, 1, 2, 1, 2]
@slow
def test_find_linear_recurrence():
assert sequence((0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55), \
(n, 0, 10)).find_linear_recurrence(11) == [1, 1]
assert sequence((1, 2, 4, 7, 28, 128, 582, 2745, 13021, 61699, 292521, \
1387138), (n, 0, 11)).find_linear_recurrence(12) == [5, -2, 6, -11]
assert sequence(x*n**3+y*n, (n, 0, oo)).find_linear_recurrence(10) \
== [4, -6, 4, -1]
assert sequence(x**n, (n,0,20)).find_linear_recurrence(21) == [x]
assert sequence((1,2,3)).find_linear_recurrence(10, 5) == [0, 0, 1]
assert sequence(((1 + sqrt(5))/2)**n + \
(-(1 + sqrt(5))/2)**(-n)).find_linear_recurrence(10) == [1, 1]
assert sequence(x*((1 + sqrt(5))/2)**n + y*(-(1 + sqrt(5))/2)**(-n), \
(n,0,oo)).find_linear_recurrence(10) == [1, 1]
assert sequence((1,2,3,4,6),(n, 0, 4)).find_linear_recurrence(5) == []
assert sequence((2,3,4,5,6,79),(n, 0, 5)).find_linear_recurrence(6,gfvar=x) \
== ([], None)
assert sequence((2,3,4,5,8,30),(n, 0, 5)).find_linear_recurrence(6,gfvar=x) \
== ([S(19)/2, -20, S(27)/2], (-31*x**2 + 32*x - 4)/(27*x**3 - 40*x**2 + 19*x -2))
assert sequence(fibonacci(n)).find_linear_recurrence(30,gfvar=x) \
== ([1, 1], -x/(x**2 + x - 1))
assert sequence(tribonacci(n)).find_linear_recurrence(30,gfvar=x) \
== ([1, 1, 1], -x/(x**3 + x**2 + x - 1))
|
12e132086f960306466aff67316bbd79583c4722e05cae6d582d23c268341546
|
from sympy import (symbols, factorial, sqrt, Rational, atan, I, log, fps, O,
Sum, oo, S, pi, cos, sin, Function, exp, Derivative, asin,
airyai, acos, acosh, gamma, erf, asech, Add, Integral, Mul,
integrate)
from sympy.series.formal import (rational_algorithm, FormalPowerSeries,
rational_independent, simpleDE, exp_re,
hyper_re)
from sympy.utilities.pytest import raises, XFAIL, slow
x, y, z = symbols('x y z')
n, m, k = symbols('n m k', integer=True)
f, r = Function('f'), Function('r')
def test_rational_algorithm():
f = 1 / ((x - 1)**2 * (x - 2))
assert rational_algorithm(f, x, k) == \
(-2**(-k - 1) + 1 - (factorial(k + 1) / factorial(k)), 0, 0)
f = (1 + x + x**2 + x**3) / ((x - 1) * (x - 2))
assert rational_algorithm(f, x, k) == \
(-15*2**(-k - 1) + 4, x + 4, 0)
f = z / (y*m - m*x - y*x + x**2)
assert rational_algorithm(f, x, k) == \
(((-y**(-k - 1)*z) / (y - m)) + ((m**(-k - 1)*z) / (y - m)), 0, 0)
f = x / (1 - x - x**2)
assert rational_algorithm(f, x, k) is None
assert rational_algorithm(f, x, k, full=True) == \
(((-Rational(1, 2) + sqrt(5)/2)**(-k - 1) *
(-sqrt(5)/10 + Rational(1, 2))) +
((-sqrt(5)/2 - Rational(1, 2))**(-k - 1) *
(sqrt(5)/10 + Rational(1, 2))), 0, 0)
f = 1 / (x**2 + 2*x + 2)
assert rational_algorithm(f, x, k) is None
assert rational_algorithm(f, x, k, full=True) == \
((I*(-1 + I)**(-k - 1)) / 2 - (I*(-1 - I)**(-k - 1)) / 2, 0, 0)
f = log(1 + x)
assert rational_algorithm(f, x, k) == \
(-(-1)**(-k) / k, 0, 1)
f = atan(x)
assert rational_algorithm(f, x, k) is None
assert rational_algorithm(f, x, k, full=True) == \
(((I*I**(-k)) / 2 - (I*(-I)**(-k)) / 2) / k, 0, 1)
f = x*atan(x) - log(1 + x**2) / 2
assert rational_algorithm(f, x, k) is None
assert rational_algorithm(f, x, k, full=True) == \
(((I*I**(-k + 1)) / 2 - (I*(-I)**(-k + 1)) / 2) /
(k*(k - 1)), 0, 2)
f = log((1 + x) / (1 - x)) / 2 - atan(x)
assert rational_algorithm(f, x, k) is None
assert rational_algorithm(f, x, k, full=True) == \
((-(-1)**(-k) / 2 - (I*I**(-k)) / 2 + (I*(-I)**(-k)) / 2 +
Rational(1, 2)) / k, 0, 1)
assert rational_algorithm(cos(x), x, k) is None
def test_rational_independent():
ri = rational_independent
assert ri([], x) == []
assert ri([cos(x), sin(x)], x) == [cos(x), sin(x)]
assert ri([x**2, sin(x), x*sin(x), x**3], x) == \
[x**3 + x**2, x*sin(x) + sin(x)]
assert ri([S.One, x*log(x), log(x), sin(x)/x, cos(x), sin(x), x], x) == \
[x + 1, x*log(x) + log(x), sin(x)/x + sin(x), cos(x)]
def test_simpleDE():
# Tests just the first valid DE
for DE in simpleDE(exp(x), x, f):
assert DE == (-f(x) + Derivative(f(x), x), 1)
break
for DE in simpleDE(sin(x), x, f):
assert DE == (f(x) + Derivative(f(x), x, x), 2)
break
for DE in simpleDE(log(1 + x), x, f):
assert DE == ((x + 1)*Derivative(f(x), x, 2) + Derivative(f(x), x), 2)
break
for DE in simpleDE(asin(x), x, f):
assert DE == (x*Derivative(f(x), x) + (x**2 - 1)*Derivative(f(x), x, x),
2)
break
for DE in simpleDE(exp(x)*sin(x), x, f):
assert DE == (2*f(x) - 2*Derivative(f(x)) + Derivative(f(x), x, x), 2)
break
for DE in simpleDE(((1 + x)/(1 - x))**n, x, f):
assert DE == (2*n*f(x) + (x**2 - 1)*Derivative(f(x), x), 1)
break
for DE in simpleDE(airyai(x), x, f):
assert DE == (-x*f(x) + Derivative(f(x), x, x), 2)
break
def test_exp_re():
d = -f(x) + Derivative(f(x), x)
assert exp_re(d, r, k) == -r(k) + r(k + 1)
d = f(x) + Derivative(f(x), x, x)
assert exp_re(d, r, k) == r(k) + r(k + 2)
d = f(x) + Derivative(f(x), x) + Derivative(f(x), x, x)
assert exp_re(d, r, k) == r(k) + r(k + 1) + r(k + 2)
d = Derivative(f(x), x) + Derivative(f(x), x, x)
assert exp_re(d, r, k) == r(k) + r(k + 1)
d = Derivative(f(x), x, 3) + Derivative(f(x), x, 4) + Derivative(f(x))
assert exp_re(d, r, k) == r(k) + r(k + 2) + r(k + 3)
def test_hyper_re():
d = f(x) + Derivative(f(x), x, x)
assert hyper_re(d, r, k) == r(k) + (k+1)*(k+2)*r(k + 2)
d = -x*f(x) + Derivative(f(x), x, x)
assert hyper_re(d, r, k) == (k + 2)*(k + 3)*r(k + 3) - r(k)
d = 2*f(x) - 2*Derivative(f(x), x) + Derivative(f(x), x, x)
assert hyper_re(d, r, k) == \
(-2*k - 2)*r(k + 1) + (k + 1)*(k + 2)*r(k + 2) + 2*r(k)
d = 2*n*f(x) + (x**2 - 1)*Derivative(f(x), x)
assert hyper_re(d, r, k) == \
k*r(k) + 2*n*r(k + 1) + (-k - 2)*r(k + 2)
d = (x**10 + 4)*Derivative(f(x), x) + x*(x**10 - 1)*Derivative(f(x), x, x)
assert hyper_re(d, r, k) == \
(k*(k - 1) + k)*r(k) + (4*k - (k + 9)*(k + 10) + 40)*r(k + 10)
d = ((x**2 - 1)*Derivative(f(x), x, 3) + 3*x*Derivative(f(x), x, x) +
Derivative(f(x), x))
assert hyper_re(d, r, k) == \
((k*(k - 2)*(k - 1) + 3*k*(k - 1) + k)*r(k) +
(-k*(k + 1)*(k + 2))*r(k + 2))
def test_fps():
assert fps(1) == 1
assert fps(2, x) == 2
assert fps(2, x, dir='+') == 2
assert fps(2, x, dir='-') == 2
assert fps(x**2 + x + 1) == x**2 + x + 1
assert fps(1/x + 1/x**2) == 1/x + 1/x**2
assert fps(log(1 + x), hyper=False, rational=False) == log(1 + x)
f = fps(log(1 + x))
assert isinstance(f, FormalPowerSeries)
assert f.function == log(1 + x)
assert f.subs(x, y) == f
assert f[:5] == [0, x, -x**2/2, x**3/3, -x**4/4]
assert f.as_leading_term(x) == x
assert f.polynomial(6) == x - x**2/2 + x**3/3 - x**4/4 + x**5/5
k = f.ak.variables[0]
assert f.infinite == Sum((-(-1)**(-k)*x**k)/k, (k, 1, oo))
ft, s = f.truncate(n=None), f[:5]
for i, t in enumerate(ft):
if i == 5:
break
assert s[i] == t
f = sin(x).fps(x)
assert isinstance(f, FormalPowerSeries)
assert f.truncate() == x - x**3/6 + x**5/120 + O(x**6)
raises(NotImplementedError, lambda: fps(y*x))
raises(ValueError, lambda: fps(x, dir=0))
@slow
def test_fps__rational():
assert fps(1/x) == (1/x)
assert fps((x**2 + x + 1) / x**3, dir=-1) == (x**2 + x + 1) / x**3
f = 1 / ((x - 1)**2 * (x - 2))
assert fps(f, x).truncate() == \
(-Rational(1, 2) - 5*x/4 - 17*x**2/8 - 49*x**3/16 - 129*x**4/32 -
321*x**5/64 + O(x**6))
f = (1 + x + x**2 + x**3) / ((x - 1) * (x - 2))
assert fps(f, x).truncate() == \
(Rational(1, 2) + 5*x/4 + 17*x**2/8 + 49*x**3/16 + 113*x**4/32 +
241*x**5/64 + O(x**6))
f = x / (1 - x - x**2)
assert fps(f, x, full=True).truncate() == \
x + x**2 + 2*x**3 + 3*x**4 + 5*x**5 + O(x**6)
f = 1 / (x**2 + 2*x + 2)
assert fps(f, x, full=True).truncate() == \
Rational(1, 2) - x/2 + x**2/4 - x**4/8 + x**5/8 + O(x**6)
f = log(1 + x)
assert fps(f, x).truncate() == \
x - x**2/2 + x**3/3 - x**4/4 + x**5/5 + O(x**6)
assert fps(f, x, dir=1).truncate() == fps(f, x, dir=-1).truncate()
assert fps(f, x, 2).truncate() == \
(log(3) - Rational(2, 3) - (x - 2)**2/18 + (x - 2)**3/81 -
(x - 2)**4/324 + (x - 2)**5/1215 + x/3 + O((x - 2)**6, (x, 2)))
assert fps(f, x, 2, dir=-1).truncate() == \
(log(3) - Rational(2, 3) - (-x + 2)**2/18 - (-x + 2)**3/81 -
(-x + 2)**4/324 - (-x + 2)**5/1215 + x/3 + O((x - 2)**6, (x, 2)))
f = atan(x)
assert fps(f, x, full=True).truncate() == x - x**3/3 + x**5/5 + O(x**6)
assert fps(f, x, full=True, dir=1).truncate() == \
fps(f, x, full=True, dir=-1).truncate()
assert fps(f, x, 2, full=True).truncate() == \
(atan(2) - Rational(2, 5) - 2*(x - 2)**2/25 + 11*(x - 2)**3/375 -
6*(x - 2)**4/625 + 41*(x - 2)**5/15625 + x/5 + O((x - 2)**6, (x, 2)))
assert fps(f, x, 2, full=True, dir=-1).truncate() == \
(atan(2) - Rational(2, 5) - 2*(-x + 2)**2/25 - 11*(-x + 2)**3/375 -
6*(-x + 2)**4/625 - 41*(-x + 2)**5/15625 + x/5 + O((x - 2)**6, (x, 2)))
f = x*atan(x) - log(1 + x**2) / 2
assert fps(f, x, full=True).truncate() == x**2/2 - x**4/12 + O(x**6)
f = log((1 + x) / (1 - x)) / 2 - atan(x)
assert fps(f, x, full=True).truncate(n=10) == 2*x**3/3 + 2*x**7/7 + O(x**10)
@slow
def test_fps__hyper():
f = sin(x)
assert fps(f, x).truncate() == x - x**3/6 + x**5/120 + O(x**6)
f = cos(x)
assert fps(f, x).truncate() == 1 - x**2/2 + x**4/24 + O(x**6)
f = exp(x)
assert fps(f, x).truncate() == \
1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + O(x**6)
f = atan(x)
assert fps(f, x).truncate() == x - x**3/3 + x**5/5 + O(x**6)
f = exp(acos(x))
assert fps(f, x).truncate() == \
(exp(pi/2) - x*exp(pi/2) + x**2*exp(pi/2)/2 - x**3*exp(pi/2)/3 +
5*x**4*exp(pi/2)/24 - x**5*exp(pi/2)/6 + O(x**6))
f = exp(acosh(x))
assert fps(f, x).truncate() == I + x - I*x**2/2 - I*x**4/8 + O(x**6)
f = atan(1/x)
assert fps(f, x).truncate() == pi/2 - x + x**3/3 - x**5/5 + O(x**6)
f = x*atan(x) - log(1 + x**2) / 2
assert fps(f, x, rational=False).truncate() == x**2/2 - x**4/12 + O(x**6)
f = log(1 + x)
assert fps(f, x, rational=False).truncate() == \
x - x**2/2 + x**3/3 - x**4/4 + x**5/5 + O(x**6)
f = airyai(x**2)
assert fps(f, x).truncate() == \
(3**Rational(5, 6)*gamma(Rational(1, 3))/(6*pi) -
3**Rational(2, 3)*x**2/(3*gamma(Rational(1, 3))) + O(x**6))
f = exp(x)*sin(x)
assert fps(f, x).truncate() == x + x**2 + x**3/3 - x**5/30 + O(x**6)
f = exp(x)*sin(x)/x
assert fps(f, x).truncate() == 1 + x + x**2/3 - x**4/30 - x**5/90 + O(x**6)
f = sin(x) * cos(x)
assert fps(f, x).truncate() == x - 2*x**3/3 + 2*x**5/15 + O(x**6)
def test_fps_shift():
f = x**-5*sin(x)
assert fps(f, x).truncate() == \
1/x**4 - 1/(6*x**2) + S.One/120 - x**2/5040 + x**4/362880 + O(x**6)
f = x**2*atan(x)
assert fps(f, x, rational=False).truncate() == \
x**3 - x**5/3 + O(x**6)
f = cos(sqrt(x))*x
assert fps(f, x).truncate() == \
x - x**2/2 + x**3/24 - x**4/720 + x**5/40320 + O(x**6)
f = x**2*cos(sqrt(x))
assert fps(f, x).truncate() == \
x**2 - x**3/2 + x**4/24 - x**5/720 + O(x**6)
def test_fps__Add_expr():
f = x*atan(x) - log(1 + x**2) / 2
assert fps(f, x).truncate() == x**2/2 - x**4/12 + O(x**6)
f = sin(x) + cos(x) - exp(x) + log(1 + x)
assert fps(f, x).truncate() == x - 3*x**2/2 - x**4/4 + x**5/5 + O(x**6)
f = 1/x + sin(x)
assert fps(f, x).truncate() == 1/x + x - x**3/6 + x**5/120 + O(x**6)
f = sin(x) - cos(x) + 1/(x - 1)
assert fps(f, x).truncate() == \
-2 - x**2/2 - 7*x**3/6 - 25*x**4/24 - 119*x**5/120 + O(x**6)
def test_fps__asymptotic():
f = exp(x)
assert fps(f, x, oo) == f
assert fps(f, x, -oo).truncate() == O(1/x**6, (x, oo))
f = erf(x)
assert fps(f, x, oo).truncate() == 1 + O(1/x**6, (x, oo))
assert fps(f, x, -oo).truncate() == -1 + O(1/x**6, (x, oo))
f = atan(x)
assert fps(f, x, oo, full=True).truncate() == \
-1/(5*x**5) + 1/(3*x**3) - 1/x + pi/2 + O(1/x**6, (x, oo))
assert fps(f, x, -oo, full=True).truncate() == \
-1/(5*x**5) + 1/(3*x**3) - 1/x - pi/2 + O(1/x**6, (x, oo))
f = log(1 + x)
assert fps(f, x, oo) != \
(-1/(5*x**5) - 1/(4*x**4) + 1/(3*x**3) - 1/(2*x**2) + 1/x - log(1/x) +
O(1/x**6, (x, oo)))
assert fps(f, x, -oo) != \
(-1/(5*x**5) - 1/(4*x**4) + 1/(3*x**3) - 1/(2*x**2) + 1/x + I*pi -
log(-1/x) + O(1/x**6, (x, oo)))
def test_fps__fractional():
f = sin(sqrt(x)) / x
assert fps(f, x).truncate() == \
(1/sqrt(x) - sqrt(x)/6 + x**Rational(3, 2)/120 -
x**Rational(5, 2)/5040 + x**Rational(7, 2)/362880 -
x**Rational(9, 2)/39916800 + x**Rational(11, 2)/6227020800 + O(x**6))
f = sin(sqrt(x)) * x
assert fps(f, x).truncate() == \
(x**Rational(3, 2) - x**Rational(5, 2)/6 + x**Rational(7, 2)/120 -
x**Rational(9, 2)/5040 + x**Rational(11, 2)/362880 + O(x**6))
f = atan(sqrt(x)) / x**2
assert fps(f, x).truncate() == \
(x**Rational(-3, 2) - x**Rational(-1, 2)/3 + x**Rational(1, 2)/5 -
x**Rational(3, 2)/7 + x**Rational(5, 2)/9 - x**Rational(7, 2)/11 +
x**Rational(9, 2)/13 - x**Rational(11, 2)/15 + O(x**6))
f = exp(sqrt(x))
assert fps(f, x).truncate().expand() == \
(1 + x/2 + x**2/24 + x**3/720 + x**4/40320 + x**5/3628800 + sqrt(x) +
x**Rational(3, 2)/6 + x**Rational(5, 2)/120 + x**Rational(7, 2)/5040 +
x**Rational(9, 2)/362880 + x**Rational(11, 2)/39916800 + O(x**6))
f = exp(sqrt(x))*x
assert fps(f, x).truncate().expand() == \
(x + x**2/2 + x**3/24 + x**4/720 + x**5/40320 + x**Rational(3, 2) +
x**Rational(5, 2)/6 + x**Rational(7, 2)/120 + x**Rational(9, 2)/5040 +
x**Rational(11, 2)/362880 + O(x**6))
def test_fps__logarithmic_singularity():
f = log(1 + 1/x)
assert fps(f, x) != \
-log(x) + x - x**2/2 + x**3/3 - x**4/4 + x**5/5 + O(x**6)
assert fps(f, x, rational=False) != \
-log(x) + x - x**2/2 + x**3/3 - x**4/4 + x**5/5 + O(x**6)
@XFAIL
def test_fps__logarithmic_singularity_fail():
f = asech(x) # Algorithms for computing limits probably needs improvemnts
assert fps(f, x) == log(2) - log(x) - x**2/4 - 3*x**4/64 + O(x**6)
@XFAIL
def test_fps__symbolic():
f = x**n*sin(x**2)
assert fps(f, x).truncate(8) == x**2*x**n - x**6*x**n/6 + O(x**(n + 8), x)
f = x**(n - 2)*cos(x)
assert fps(f, x).truncate() == \
(x**n*(-S(1)/2 + x**(-2)) + x**2*x**n/24 - x**4*x**n/720 +
O(x**(n + 6), x))
f = x**n*log(1 + x)
fp = fps(f, x)
k = fp.ak.variables[0]
assert fp.infinite == \
Sum((-(-1)**(-k)*x**k*x**n)/k, (k, 1, oo))
f = x**(n - 2)*sin(x) + x**n*exp(x)
assert fps(f, x).truncate() == \
(x**n*(1 + 1/x) + 5*x*x**n/6 + x**2*x**n/2 + 7*x**3*x**n/40 +
x**4*x**n/24 + 41*x**5*x**n/5040 + O(x**(n + 6), x))
f = (x - 2)**n*log(1 + x)
assert fps(f, x, 2).truncate() == \
((x - 2)**n*log(3) - (x - 2)**2*(x - 2)**n/18 +
(x - 2)**3*(x - 2)**n/81 - (x - 2)**4*(x - 2)**n/324 +
(x - 2)**5*(x - 2)**n/1215 + (x/3 - S(2)/3)*(x - 2)**n +
O((x - 2)**(n + 6), (x, 2)))
f = x**n*atan(x)
assert fps(f, x, oo).truncate() == \
(-x**n/(5*x**5) + x**n/(3*x**3) + x**n*(pi/2 - 1/x) +
O(x**(n - 6), (x, oo)))
def test_fps__slow():
f = x*exp(x)*sin(2*x) # TODO: rsolve needs improvement
assert fps(f, x).truncate() == 2*x**2 + 2*x**3 - x**4/3 - x**5 + O(x**6)
def test_fps__operations():
f1, f2 = fps(sin(x)), fps(cos(x))
fsum = f1 + f2
assert fsum.function == sin(x) + cos(x)
assert fsum.truncate() == \
1 + x - x**2/2 - x**3/6 + x**4/24 + x**5/120 + O(x**6)
fsum = f1 + 1
assert fsum.function == sin(x) + 1
assert fsum.truncate() == 1 + x - x**3/6 + x**5/120 + O(x**6)
fsum = 1 + f2
assert fsum.function == cos(x) + 1
assert fsum.truncate() == 2 - x**2/2 + x**4/24 + O(x**6)
assert (f1 + x) == Add(f1, x)
assert -f2.truncate() == -1 + x**2/2 - x**4/24 + O(x**6)
assert (f1 - f1) == S.Zero
fsub = f1 - f2
assert fsub.function == sin(x) - cos(x)
assert fsub.truncate() == \
-1 + x + x**2/2 - x**3/6 - x**4/24 + x**5/120 + O(x**6)
fsub = f1 - 1
assert fsub.function == sin(x) - 1
assert fsub.truncate() == -1 + x - x**3/6 + x**5/120 + O(x**6)
fsub = 1 - f2
assert fsub.function == -cos(x) + 1
assert fsub.truncate() == x**2/2 - x**4/24 + O(x**6)
raises(ValueError, lambda: f1 + fps(exp(x), dir=-1))
raises(ValueError, lambda: f1 + fps(exp(x), x0=1))
fm = f1 * 3
assert fm.function == 3*sin(x)
assert fm.truncate() == 3*x - x**3/2 + x**5/40 + O(x**6)
fm = 3 * f2
assert fm.function == 3*cos(x)
assert fm.truncate() == 3 - 3*x**2/2 + x**4/8 + O(x**6)
assert (f1 * f2) == Mul(f1, f2)
assert (f1 * x) == Mul(f1, x)
fd = f1.diff()
assert fd.function == cos(x)
assert fd.truncate() == 1 - x**2/2 + x**4/24 + O(x**6)
fd = f2.diff()
assert fd.function == -sin(x)
assert fd.truncate() == -x + x**3/6 - x**5/120 + O(x**6)
fd = f2.diff().diff()
assert fd.function == -cos(x)
assert fd.truncate() == -1 + x**2/2 - x**4/24 + O(x**6)
f3 = fps(exp(sqrt(x)))
fd = f3.diff()
assert fd.truncate().expand() == \
(1/(2*sqrt(x)) + S(1)/2 + x/12 + x**2/240 + x**3/10080 + x**4/725760 +
x**5/79833600 + sqrt(x)/4 + x**(S(3)/2)/48 + x**(S(5)/2)/1440 +
x**(S(7)/2)/80640 + x**(S(9)/2)/7257600 + x**(S(11)/2)/958003200 +
O(x**6))
assert f1.integrate((x, 0, 1)) == -cos(1) + 1
assert integrate(f1, (x, 0, 1)) == -cos(1) + 1
fi = integrate(f1, x)
assert fi.function == -cos(x)
assert fi.truncate() == -1 + x**2/2 - x**4/24 + O(x**6)
fi = f2.integrate(x)
assert fi.function == sin(x)
assert fi.truncate() == x - x**3/6 + x**5/120 + O(x**6)
|
0f9287cf8a0707c995e3c29ef313e15d37cccf1ff3fb07cbd8ac248df38055d7
|
from sympy import (symbols, pi, Piecewise, sin, cos, sinc, Rational,
oo, fourier_series, Add, log, exp, tan)
from sympy.series.fourier import FourierSeries
from sympy.utilities.pytest import raises
from sympy.core.cache import lru_cache
x, y, z = symbols('x y z')
# Don't declare these during import because they are slow
@lru_cache()
def _get_examples():
fo = fourier_series(x, (x, -pi, pi))
fe = fourier_series(x**2, (-pi, pi))
fp = fourier_series(Piecewise((0, x < 0), (pi, True)), (x, -pi, pi))
return fo, fe, fp
def test_FourierSeries():
fo, fe, fp = _get_examples()
assert fourier_series(1, (-pi, pi)) == 1
assert (Piecewise((0, x < 0), (pi, True)).
fourier_series((x, -pi, pi)).truncate()) == fp.truncate()
assert isinstance(fo, FourierSeries)
assert fo.function == x
assert fo.x == x
assert fo.period == (-pi, pi)
assert fo.term(3) == 2*sin(3*x) / 3
assert fe.term(3) == -4*cos(3*x) / 9
assert fp.term(3) == 2*sin(3*x) / 3
assert fo.as_leading_term(x) == 2*sin(x)
assert fe.as_leading_term(x) == pi**2 / 3
assert fp.as_leading_term(x) == pi / 2
assert fo.truncate() == 2*sin(x) - sin(2*x) + (2*sin(3*x) / 3)
assert fe.truncate() == -4*cos(x) + cos(2*x) + pi**2 / 3
assert fp.truncate() == 2*sin(x) + (2*sin(3*x) / 3) + pi / 2
fot = fo.truncate(n=None)
s = [0, 2*sin(x), -sin(2*x)]
for i, t in enumerate(fot):
if i == 3:
break
assert s[i] == t
def _check_iter(f, i):
for ind, t in enumerate(f):
assert t == f[ind]
if ind == i:
break
_check_iter(fo, 3)
_check_iter(fe, 3)
_check_iter(fp, 3)
assert fo.subs(x, x**2) == fo
raises(ValueError, lambda: fourier_series(x, (0, 1, 2)))
raises(ValueError, lambda: fourier_series(x, (x, 0, oo)))
raises(ValueError, lambda: fourier_series(x*y, (0, oo)))
def test_FourierSeries_2():
p = Piecewise((0, x < 0), (x, True))
f = fourier_series(p, (x, -2, 2))
assert f.term(3) == (2*sin(3*pi*x / 2) / (3*pi) -
4*cos(3*pi*x / 2) / (9*pi**2))
assert f.truncate() == (2*sin(pi*x / 2) / pi - sin(pi*x) / pi -
4*cos(pi*x / 2) / pi**2 + Rational(1, 2))
def test_fourier_series_square_wave():
"""Test if fourier_series approximates discontinuous function correctly."""
square_wave = Piecewise((1, x < pi), (-1, True))
s = fourier_series(square_wave, (x, 0, 2*pi))
assert s.truncate(3) == 4 / pi * sin(x) + 4 / (3 * pi) * sin(3 * x) + \
4 / (5 * pi) * sin(5 * x)
assert s.sigma_approximation(4) == 4 / pi * sin(x) * sinc(pi / 4) + \
4 / (3 * pi) * sin(3 * x) * sinc(3 * pi / 4)
def test_FourierSeries__operations():
fo, fe, fp = _get_examples()
fes = fe.scale(-1).shift(pi**2)
assert fes.truncate() == 4*cos(x) - cos(2*x) + 2*pi**2 / 3
assert fp.shift(-pi/2).truncate() == (2*sin(x) + (2*sin(3*x) / 3) +
(2*sin(5*x) / 5))
fos = fo.scale(3)
assert fos.truncate() == 6*sin(x) - 3*sin(2*x) + 2*sin(3*x)
fx = fe.scalex(2).shiftx(1)
assert fx.truncate() == -4*cos(2*x + 2) + cos(4*x + 4) + pi**2 / 3
fl = fe.scalex(3).shift(-pi).scalex(2).shiftx(1).scale(4)
assert fl.truncate() == (-16*cos(6*x + 6) + 4*cos(12*x + 12) -
4*pi + 4*pi**2 / 3)
raises(ValueError, lambda: fo.shift(x))
raises(ValueError, lambda: fo.shiftx(sin(x)))
raises(ValueError, lambda: fo.scale(x*y))
raises(ValueError, lambda: fo.scalex(x**2))
def test_FourierSeries__neg():
fo, fe, fp = _get_examples()
assert (-fo).truncate() == -2*sin(x) + sin(2*x) - (2*sin(3*x) / 3)
assert (-fe).truncate() == +4*cos(x) - cos(2*x) - pi**2 / 3
def test_FourierSeries__add__sub():
fo, fe, fp = _get_examples()
assert fo + fo == fo.scale(2)
assert fo - fo == 0
assert -fe - fe == fe.scale(-2)
assert (fo + fe).truncate() == 2*sin(x) - sin(2*x) - 4*cos(x) + cos(2*x) \
+ pi**2 / 3
assert (fo - fe).truncate() == 2*sin(x) - sin(2*x) + 4*cos(x) - cos(2*x) \
- pi**2 / 3
assert isinstance(fo + 1, Add)
raises(ValueError, lambda: fo + fourier_series(x, (x, 0, 2)))
def test_FourierSeries_finite():
assert fourier_series(sin(x)).truncate(1) == sin(x)
# assert type(fourier_series(sin(x)*log(x))).truncate() == FourierSeries
# assert type(fourier_series(sin(x**2+6))).truncate() == FourierSeries
assert fourier_series(sin(x)*log(y)*exp(z),(x,pi,-pi)).truncate() == sin(x)*log(y)*exp(z)
assert fourier_series(sin(x)**6).truncate(oo) == -15*cos(2*x)/32 + 3*cos(4*x)/16 - cos(6*x)/32 \
+ Rational(5, 16)
assert fourier_series(sin(x) ** 6).truncate() == -15 * cos(2 * x) / 32 + 3 * cos(4 * x) / 16 \
+ Rational(5, 16)
assert fourier_series(sin(4*x+3) + cos(3*x+4)).truncate(oo) == -sin(4)*sin(3*x) + sin(4*x)*cos(3) \
+ cos(4)*cos(3*x) + sin(3)*cos(4*x)
assert fourier_series(sin(x)+cos(x)*tan(x)).truncate(oo) == 2*sin(x)
assert fourier_series(cos(pi*x), (x, -1, 1)).truncate(oo) == cos(pi*x)
assert fourier_series(cos(3*pi*x + 4) - sin(4*pi*x)*log(pi*y) , (x, -1, 1)).truncate(oo) == -log(pi*y)*sin(4*pi*x)\
- sin(4)*sin(3*pi*x) + cos(4)*cos(3*pi*x)
|
e9f2273c052b904f333f53a5d0f216109c58de07aabc97bc33856d737618570d
|
from itertools import product as cartes
from sympy import (
limit, exp, oo, log, sqrt, Limit, sin, floor, cos, ceiling,
atan, gamma, Symbol, S, pi, Integral, Rational, I, EulerGamma,
tan, cot, integrate, Sum, sign, Function, subfactorial, symbols,
binomial, simplify, frac, Float, sec, zoo, fresnelc, fresnels)
from sympy.calculus.util import AccumBounds
from sympy.core.add import Add
from sympy.core.mul import Mul
from sympy.series.limits import heuristics
from sympy.series.order import Order
from sympy.utilities.pytest import XFAIL, raises
from sympy.core.numbers import GoldenRatio
from sympy.functions.combinatorial.numbers import fibonacci
from sympy.abc import x, y, z, k
n = Symbol('n', integer=True, positive=True)
def test_basic1():
assert limit(x, x, oo) == oo
assert limit(x, x, -oo) == -oo
assert limit(-x, x, oo) == -oo
assert limit(x**2, x, -oo) == oo
assert limit(-x**2, x, oo) == -oo
assert limit(x*log(x), x, 0, dir="+") == 0
assert limit(1/x, x, oo) == 0
assert limit(exp(x), x, oo) == oo
assert limit(-exp(x), x, oo) == -oo
assert limit(exp(x)/x, x, oo) == oo
assert limit(1/x - exp(-x), x, oo) == 0
assert limit(x + 1/x, x, oo) == oo
assert limit(x - x**2, x, oo) == -oo
assert limit((1 + x)**(1 + sqrt(2)), x, 0) == 1
assert limit((1 + x)**oo, x, 0) == oo
assert limit((1 + x)**oo, x, 0, dir='-') == 0
assert limit((1 + x + y)**oo, x, 0, dir='-') == (1 + y)**(oo)
assert limit(y/x/log(x), x, 0) == -oo*sign(y)
assert limit(cos(x + y)/x, x, 0) == sign(cos(y))*oo
assert limit(gamma(1/x + 3), x, oo) == 2
assert limit(S.NaN, x, -oo) == S.NaN
assert limit(Order(2)*x, x, S.NaN) == S.NaN
assert limit(1/(x - 1), x, 1, dir="+") == oo
assert limit(1/(x - 1), x, 1, dir="-") == -oo
assert limit(1/(5 - x)**3, x, 5, dir="+") == -oo
assert limit(1/(5 - x)**3, x, 5, dir="-") == oo
assert limit(1/sin(x), x, pi, dir="+") == -oo
assert limit(1/sin(x), x, pi, dir="-") == oo
assert limit(1/cos(x), x, pi/2, dir="+") == -oo
assert limit(1/cos(x), x, pi/2, dir="-") == oo
assert limit(1/tan(x**3), x, (2*pi)**(S(1)/3), dir="+") == oo
assert limit(1/tan(x**3), x, (2*pi)**(S(1)/3), dir="-") == -oo
assert limit(1/cot(x)**3, x, (3*pi/2), dir="+") == -oo
assert limit(1/cot(x)**3, x, (3*pi/2), dir="-") == oo
# test bi-directional limits
assert limit(sin(x)/x, x, 0, dir="+-") == 1
assert limit(x**2, x, 0, dir="+-") == 0
assert limit(1/x**2, x, 0, dir="+-") == oo
# test failing bi-directional limits
raises(ValueError, lambda: limit(1/x, x, 0, dir="+-"))
# approaching 0
# from dir="+"
assert limit(1 + 1/x, x, 0) == oo
# from dir='-'
# Add
assert limit(1 + 1/x, x, 0, dir='-') == -oo
# Pow
assert limit(x**(-2), x, 0, dir='-') == oo
assert limit(x**(-3), x, 0, dir='-') == -oo
assert limit(1/sqrt(x), x, 0, dir='-') == (-oo)*I
assert limit(x**2, x, 0, dir='-') == 0
assert limit(sqrt(x), x, 0, dir='-') == 0
assert limit(x**-pi, x, 0, dir='-') == oo*sign((-1)**(-pi))
assert limit((1 + cos(x))**oo, x, 0) == oo
def test_basic2():
assert limit(x**x, x, 0, dir="+") == 1
assert limit((exp(x) - 1)/x, x, 0) == 1
assert limit(1 + 1/x, x, oo) == 1
assert limit(-exp(1/x), x, oo) == -1
assert limit(x + exp(-x), x, oo) == oo
assert limit(x + exp(-x**2), x, oo) == oo
assert limit(x + exp(-exp(x)), x, oo) == oo
assert limit(13 + 1/x - exp(-x), x, oo) == 13
def test_basic3():
assert limit(1/x, x, 0, dir="+") == oo
assert limit(1/x, x, 0, dir="-") == -oo
def test_basic4():
assert limit(2*x + y*x, x, 0) == 0
assert limit(2*x + y*x, x, 1) == 2 + y
assert limit(2*x**8 + y*x**(-3), x, -2) == 512 - y/8
assert limit(sqrt(x + 1) - sqrt(x), x, oo) == 0
assert integrate(1/(x**3 + 1), (x, 0, oo)) == 2*pi*sqrt(3)/9
def test_basic5():
class my(Function):
@classmethod
def eval(cls, arg):
if arg is S.Infinity:
return S.NaN
assert limit(my(x), x, oo) == Limit(my(x), x, oo)
def test_issue_3885():
assert limit(x*y + x*z, z, 2) == x*y + 2*x
def test_Limit():
assert Limit(sin(x)/x, x, 0) != 1
assert Limit(sin(x)/x, x, 0).doit() == 1
assert Limit(x, x, 0, dir='+-').args == (x, x, 0, Symbol('+-'))
def test_floor():
assert limit(floor(x), x, -2, "+") == -2
assert limit(floor(x), x, -2, "-") == -3
assert limit(floor(x), x, -1, "+") == -1
assert limit(floor(x), x, -1, "-") == -2
assert limit(floor(x), x, 0, "+") == 0
assert limit(floor(x), x, 0, "-") == -1
assert limit(floor(x), x, 1, "+") == 1
assert limit(floor(x), x, 1, "-") == 0
assert limit(floor(x), x, 2, "+") == 2
assert limit(floor(x), x, 2, "-") == 1
assert limit(floor(x), x, 248, "+") == 248
assert limit(floor(x), x, 248, "-") == 247
def test_floor_requires_robust_assumptions():
assert limit(floor(sin(x)), x, 0, "+") == 0
assert limit(floor(sin(x)), x, 0, "-") == -1
assert limit(floor(cos(x)), x, 0, "+") == 0
assert limit(floor(cos(x)), x, 0, "-") == 0
assert limit(floor(5 + sin(x)), x, 0, "+") == 5
assert limit(floor(5 + sin(x)), x, 0, "-") == 4
assert limit(floor(5 + cos(x)), x, 0, "+") == 5
assert limit(floor(5 + cos(x)), x, 0, "-") == 5
def test_ceiling():
assert limit(ceiling(x), x, -2, "+") == -1
assert limit(ceiling(x), x, -2, "-") == -2
assert limit(ceiling(x), x, -1, "+") == 0
assert limit(ceiling(x), x, -1, "-") == -1
assert limit(ceiling(x), x, 0, "+") == 1
assert limit(ceiling(x), x, 0, "-") == 0
assert limit(ceiling(x), x, 1, "+") == 2
assert limit(ceiling(x), x, 1, "-") == 1
assert limit(ceiling(x), x, 2, "+") == 3
assert limit(ceiling(x), x, 2, "-") == 2
assert limit(ceiling(x), x, 248, "+") == 249
assert limit(ceiling(x), x, 248, "-") == 248
def test_ceiling_requires_robust_assumptions():
assert limit(ceiling(sin(x)), x, 0, "+") == 1
assert limit(ceiling(sin(x)), x, 0, "-") == 0
assert limit(ceiling(cos(x)), x, 0, "+") == 1
assert limit(ceiling(cos(x)), x, 0, "-") == 1
assert limit(ceiling(5 + sin(x)), x, 0, "+") == 6
assert limit(ceiling(5 + sin(x)), x, 0, "-") == 5
assert limit(ceiling(5 + cos(x)), x, 0, "+") == 6
assert limit(ceiling(5 + cos(x)), x, 0, "-") == 6
def test_atan():
x = Symbol("x", real=True)
assert limit(atan(x)*sin(1/x), x, 0) == 0
assert limit(atan(x) + sqrt(x + 1) - sqrt(x), x, oo) == pi/2
def test_abs():
assert limit(abs(x), x, 0) == 0
assert limit(abs(sin(x)), x, 0) == 0
assert limit(abs(cos(x)), x, 0) == 1
assert limit(abs(sin(x + 1)), x, 0) == sin(1)
def test_heuristic():
x = Symbol("x", real=True)
assert heuristics(sin(1/x) + atan(x), x, 0, '+') == AccumBounds(-1, 1)
assert limit(log(2 + sqrt(atan(x))*sqrt(sin(1/x))), x, 0) == log(2)
def test_issue_3871():
z = Symbol("z", positive=True)
f = -1/z*exp(-z*x)
assert limit(f, x, oo) == 0
assert f.limit(x, oo) == 0
def test_exponential():
n = Symbol('n')
x = Symbol('x', real=True)
assert limit((1 + x/n)**n, n, oo) == exp(x)
assert limit((1 + x/(2*n))**n, n, oo) == exp(x/2)
assert limit((1 + x/(2*n + 1))**n, n, oo) == exp(x/2)
assert limit(((x - 1)/(x + 1))**x, x, oo) == exp(-2)
assert limit(1 + (1 + 1/x)**x, x, oo) == 1 + S.Exp1
@XFAIL
def test_exponential2():
n = Symbol('n')
assert limit((1 + x/(n + sin(n)))**n, n, oo) == exp(x)
def test_doit():
f = Integral(2 * x, x)
l = Limit(f, x, oo)
assert l.doit() == oo
def test_AccumBounds():
assert limit(sin(k) - sin(k + 1), k, oo) == AccumBounds(-2, 2)
assert limit(cos(k) - cos(k + 1) + 1, k, oo) == AccumBounds(-1, 3)
# not the exact bound
assert limit(sin(k) - sin(k)*cos(k), k, oo) == AccumBounds(-2, 2)
# test for issue #9934
t1 = Mul(S(1)/2, 1/(-1 + cos(1)), Add(AccumBounds(-3, 1), cos(1)))
assert limit(simplify(Sum(cos(n).rewrite(exp), (n, 0, k)).doit().rewrite(sin)), k, oo) == t1
t2 = Mul(S(1)/2, Add(AccumBounds(-2, 2), sin(1)), 1/(-cos(1) + 1))
assert limit(simplify(Sum(sin(n).rewrite(exp), (n, 0, k)).doit().rewrite(sin)), k, oo) == t2
assert limit(frac(x)**x, x, oo) == AccumBounds(0, oo)
assert limit(((sin(x) + 1)/2)**x, x, oo) == AccumBounds(0, oo)
# Possible improvement: AccumBounds(0, 1)
@XFAIL
def test_doit2():
f = Integral(2 * x, x)
l = Limit(f, x, oo)
# limit() breaks on the contained Integral.
assert l.doit(deep=False) == l
def test_issue_3792():
assert limit((1 - cos(x))/x**2, x, S(1)/2) == 4 - 4*cos(S(1)/2)
assert limit(sin(sin(x + 1) + 1), x, 0) == sin(1 + sin(1))
assert limit(abs(sin(x + 1) + 1), x, 0) == 1 + sin(1)
def test_issue_4090():
assert limit(1/(x + 3), x, 2) == S(1)/5
assert limit(1/(x + pi), x, 2) == S(1)/(2 + pi)
assert limit(log(x)/(x**2 + 3), x, 2) == log(2)/7
assert limit(log(x)/(x**2 + pi), x, 2) == log(2)/(4 + pi)
def test_issue_4547():
assert limit(cot(x), x, 0, dir='+') == oo
assert limit(cot(x), x, pi/2, dir='+') == 0
def test_issue_5164():
assert limit(x**0.5, x, oo) == oo**0.5 == oo
assert limit(x**0.5, x, 16) == S(16)**0.5
assert limit(x**0.5, x, 0) == 0
assert limit(x**(-0.5), x, oo) == 0
assert limit(x**(-0.5), x, 4) == S(4)**(-0.5)
def test_issue_5183():
# using list(...) so py.test can recalculate values
tests = list(cartes([x, -x],
[-1, 1],
[2, 3, Rational(1, 2), Rational(2, 3)],
['-', '+']))
results = (oo, oo, -oo, oo, -oo*I, oo, -oo*(-1)**Rational(1, 3), oo,
0, 0, 0, 0, 0, 0, 0, 0,
oo, oo, oo, -oo, oo, -oo*I, oo, -oo*(-1)**Rational(1, 3),
0, 0, 0, 0, 0, 0, 0, 0)
assert len(tests) == len(results)
for i, (args, res) in enumerate(zip(tests, results)):
y, s, e, d = args
eq = y**(s*e)
try:
assert limit(eq, x, 0, dir=d) == res
except AssertionError:
if 0: # change to 1 if you want to see the failing tests
print()
print(i, res, eq, d, limit(eq, x, 0, dir=d))
else:
assert None
def test_issue_5184():
assert limit(sin(x)/x, x, oo) == 0
assert limit(atan(x), x, oo) == pi/2
assert limit(gamma(x), x, oo) == oo
assert limit(cos(x)/x, x, oo) == 0
assert limit(gamma(x), x, Rational(1, 2)) == sqrt(pi)
r = Symbol('r', real=True, finite=True)
assert limit(r*sin(1/r), r, 0) == 0
def test_issue_5229():
assert limit((1 + y)**(1/y) - S.Exp1, y, 0) == 0
def test_issue_4546():
# using list(...) so py.test can recalculate values
tests = list(cartes([cot, tan],
[-pi/2, 0, pi/2, pi, 3*pi/2],
['-', '+']))
results = (0, 0, -oo, oo, 0, 0, -oo, oo, 0, 0,
oo, -oo, 0, 0, oo, -oo, 0, 0, oo, -oo)
assert len(tests) == len(results)
for i, (args, res) in enumerate(zip(tests, results)):
f, l, d = args
eq = f(x)
try:
assert limit(eq, x, l, dir=d) == res
except AssertionError:
if 0: # change to 1 if you want to see the failing tests
print()
print(i, res, eq, l, d, limit(eq, x, l, dir=d))
else:
assert None
def test_issue_3934():
assert limit((1 + x**log(3))**(1/x), x, 0) == 1
assert limit((5**(1/x) + 3**(1/x))**x, x, 0) == 5
def test_calculate_series():
# needs gruntz calculate_series to go to n = 32
assert limit(x**(S(77)/3)/(1 + x**(S(77)/3)), x, oo) == 1
# needs gruntz calculate_series to go to n = 128
assert limit(x**101.1/(1 + x**101.1), x, oo) == 1
def test_issue_5955():
assert limit((x**16)/(1 + x**16), x, oo) == 1
assert limit((x**100)/(1 + x**100), x, oo) == 1
assert limit((x**1885)/(1 + x**1885), x, oo) == 1
assert limit((x**1000/((x + 1)**1000 + exp(-x))), x, oo) == 1
def test_newissue():
assert limit(exp(1/sin(x))/exp(cot(x)), x, 0) == 1
def test_extended_real_line():
assert limit(x - oo, x, oo) == -oo
assert limit(oo - x, x, -oo) == oo
assert limit(x**2/(x - 5) - oo, x, oo) == -oo
assert limit(1/(x + sin(x)) - oo, x, 0) == -oo
assert limit(oo/x, x, oo) == oo
assert limit(x - oo + 1/x, x, oo) == -oo
assert limit(x - oo + 1/x, x, 0) == -oo
@XFAIL
def test_order_oo():
x = Symbol('x', positive=True, finite=True)
assert Order(x)*oo != Order(1, x)
assert limit(oo/(x**2 - 4), x, oo) == oo
def test_issue_5436():
raises(NotImplementedError, lambda: limit(exp(x*y), x, oo))
raises(NotImplementedError, lambda: limit(exp(-x*y), x, oo))
def test_Limit_dir():
raises(TypeError, lambda: Limit(x, x, 0, dir=0))
raises(ValueError, lambda: Limit(x, x, 0, dir='0'))
def test_polynomial():
assert limit((x + 1)**1000/((x + 1)**1000 + 1), x, oo) == 1
assert limit((x + 1)**1000/((x + 1)**1000 + 1), x, -oo) == 1
def test_rational():
assert limit(1/y - (1/(y + x) + x/(y + x)/y)/z, x, oo) == (z - 1)/(y*z)
assert limit(1/y - (1/(y + x) + x/(y + x)/y)/z, x, -oo) == (z - 1)/(y*z)
def test_issue_5740():
assert limit(log(x)*z - log(2*x)*y, x, 0) == oo*sign(y - z)
def test_issue_6366():
n = Symbol('n', integer=True, positive=True)
r = (n + 1)*x**(n + 1)/(x**(n + 1) - 1) - x/(x - 1)
assert limit(r, x, 1).simplify() == n/2
def test_factorial():
from sympy import factorial, E
f = factorial(x)
assert limit(f, x, oo) == oo
assert limit(x/f, x, oo) == 0
# see Stirling's approximation:
# https://en.wikipedia.org/wiki/Stirling's_approximation
assert limit(f/(sqrt(2*pi*x)*(x/E)**x), x, oo) == 1
assert limit(f, x, -oo) == factorial(-oo)
assert limit(f, x, x**2) == factorial(x**2)
assert limit(f, x, -x**2) == factorial(-x**2)
def test_issue_6560():
e = (5*x**3/4 - 3*x/4 + (y*(3*x**2/2 - S(1)/2) +
35*x**4/8 - 15*x**2/4 + S(3)/8)/(2*(y + 1)))
assert limit(e, y, oo) == (5*x**3 + 3*x**2 - 3*x - 1)/4
def test_issue_5172():
n = Symbol('n')
r = Symbol('r', positive=True)
c = Symbol('c')
p = Symbol('p', positive=True)
m = Symbol('m', negative=True)
expr = ((2*n*(n - r + 1)/(n + r*(n - r + 1)))**c +
(r - 1)*(n*(n - r + 2)/(n + r*(n - r + 1)))**c - n)/(n**c - n)
expr = expr.subs(c, c + 1)
raises(NotImplementedError, lambda: limit(expr, n, oo))
assert limit(expr.subs(c, m), n, oo) == 1
assert limit(expr.subs(c, p), n, oo).simplify() == \
(2**(p + 1) + r - 1)/(r + 1)**(p + 1)
def test_issue_7088():
a = Symbol('a')
assert limit(sqrt(x/(x + a)), x, oo) == 1
def test_issue_6364():
a = Symbol('a')
e = z/(1 - sqrt(1 + z)*sin(a)**2 - sqrt(1 - z)*cos(a)**2)
assert limit(e, z, 0).simplify() == 2/cos(2*a)
def test_issue_4099():
a = Symbol('a')
assert limit(a/x, x, 0) == oo*sign(a)
assert limit(-a/x, x, 0) == -oo*sign(a)
assert limit(-a*x, x, oo) == -oo*sign(a)
assert limit(a*x, x, oo) == oo*sign(a)
def test_issue_4503():
dx = Symbol('dx')
assert limit((sqrt(1 + exp(x + dx)) - sqrt(1 + exp(x)))/dx, dx, 0) == \
exp(x)/(2*sqrt(exp(x) + 1))
def test_issue_8730():
assert limit(subfactorial(x), x, oo) == oo
def test_issue_10801():
# make sure limits work with binomial
assert limit(16**k / (k * binomial(2*k, k)**2), k, oo) == pi
def test_issue_9205():
x, y, a = symbols('x, y, a')
assert Limit(x, x, a).free_symbols == {a}
assert Limit(x, x, a, '-').free_symbols == {a}
assert Limit(x + y, x + y, a).free_symbols == {a}
assert Limit(-x**2 + y, x**2, a).free_symbols == {y, a}
def test_issue_11879():
assert simplify(limit(((x+y)**n-x**n)/y, y, 0)) == n*x**(n-1)
def test_limit_with_Float():
k = symbols("k")
assert limit(1.0 ** k, k, oo) == 1
assert limit(0.3*1.0**k, k, oo) == Float(0.3)
def test_issue_10610():
assert limit(3**x*3**(-x - 1)*(x + 1)**2/x**2, x, oo) == S(1)/3
def test_issue_6599():
assert limit((n + cos(n))/n, n, oo) == 1
def test_issue_12555():
assert limit((3**x + 2* x**10) / (x**10 + exp(x)), x, -oo) == 2
assert limit((3**x + 2* x**10) / (x**10 + exp(x)), x, oo) == oo
def test_issue_12564():
assert limit(x**2 + x*sin(x) + cos(x), x, -oo) == oo
assert limit(x**2 + x*sin(x) + cos(x), x, oo) == oo
assert limit(((x + cos(x))**2).expand(), x, oo) == oo
assert limit(((x + sin(x))**2).expand(), x, oo) == oo
assert limit(((x + cos(x))**2).expand(), x, -oo) == oo
assert limit(((x + sin(x))**2).expand(), x, -oo) == oo
def test_issue_14456():
raises(NotImplementedError, lambda: Limit(exp(x), x, zoo).doit())
raises(NotImplementedError, lambda: Limit(x**2/(x+1), x, zoo).doit())
def test_issue_14411():
assert limit(3*sec(4*pi*x - x/3), x, 3*pi/(24*pi - 2)) == -oo
def test_issue_14574():
assert limit(sqrt(x)*cos(x - x**2) / (x + 1), x, oo) == 0
def test_issue_10102():
assert limit(fresnels(x), x, oo) == S.Half
assert limit(3 + fresnels(x), x, oo) == 3 + S.Half
assert limit(5*fresnels(x), x, oo) == 5*S.Half
assert limit(fresnelc(x), x, oo) == S.Half
assert limit(fresnels(x), x, -oo) == -S.Half
assert limit(4*fresnelc(x), x, -oo) == -2
def test_issue_14377():
raises(NotImplementedError, lambda: limit(exp(I*x)*sin(pi*x), x, oo))
def test_issue_15984():
assert limit((-x + log(exp(x) + 1))/x, x, oo, dir='-').doit() == 0
|
6caad3eae9c7ff9a2f9572847ef15917024de5d13d75d8c035fe2bf53218aefc
|
from sympy import (
sqrt, Derivative, symbols, collect, Function, factor, Wild, S,
collect_const, log, fraction, I, cos, Add, O,sin, rcollect,
Mul, radsimp, diff, root, Symbol, Rational, exp)
from sympy.core.mul import _unevaluated_Mul as umul
from sympy.simplify.radsimp import _unevaluated_Add, collect_sqrt, fraction_expand
from sympy.utilities.pytest import XFAIL, raises
from sympy.abc import x, y, z, a, b, c, d
def test_radsimp():
r2 = sqrt(2)
r3 = sqrt(3)
r5 = sqrt(5)
r7 = sqrt(7)
assert fraction(radsimp(1/r2)) == (sqrt(2), 2)
assert radsimp(1/(1 + r2)) == \
-1 + sqrt(2)
assert radsimp(1/(r2 + r3)) == \
-sqrt(2) + sqrt(3)
assert fraction(radsimp(1/(1 + r2 + r3))) == \
(-sqrt(6) + sqrt(2) + 2, 4)
assert fraction(radsimp(1/(r2 + r3 + r5))) == \
(-sqrt(30) + 2*sqrt(3) + 3*sqrt(2), 12)
assert fraction(radsimp(1/(1 + r2 + r3 + r5))) == (
(-34*sqrt(10) - 26*sqrt(15) - 55*sqrt(3) - 61*sqrt(2) + 14*sqrt(30) +
93 + 46*sqrt(6) + 53*sqrt(5), 71))
assert fraction(radsimp(1/(r2 + r3 + r5 + r7))) == (
(-50*sqrt(42) - 133*sqrt(5) - 34*sqrt(70) - 145*sqrt(3) + 22*sqrt(105)
+ 185*sqrt(2) + 62*sqrt(30) + 135*sqrt(7), 215))
z = radsimp(1/(1 + r2/3 + r3/5 + r5 + r7))
assert len((3616791619821680643598*z).args) == 16
assert radsimp(1/z) == 1/z
assert radsimp(1/z, max_terms=20).expand() == 1 + r2/3 + r3/5 + r5 + r7
assert radsimp(1/(r2*3)) == \
sqrt(2)/6
assert radsimp(1/(r2*a + r3 + r5 + r7)) == (
(8*sqrt(2)*a**7 - 8*sqrt(7)*a**6 - 8*sqrt(5)*a**6 - 8*sqrt(3)*a**6 -
180*sqrt(2)*a**5 + 8*sqrt(30)*a**5 + 8*sqrt(42)*a**5 + 8*sqrt(70)*a**5
- 24*sqrt(105)*a**4 + 84*sqrt(3)*a**4 + 100*sqrt(5)*a**4 +
116*sqrt(7)*a**4 - 72*sqrt(70)*a**3 - 40*sqrt(42)*a**3 -
8*sqrt(30)*a**3 + 782*sqrt(2)*a**3 - 462*sqrt(3)*a**2 -
302*sqrt(7)*a**2 - 254*sqrt(5)*a**2 + 120*sqrt(105)*a**2 -
795*sqrt(2)*a - 62*sqrt(30)*a + 82*sqrt(42)*a + 98*sqrt(70)*a -
118*sqrt(105) + 59*sqrt(7) + 295*sqrt(5) + 531*sqrt(3))/(16*a**8 -
480*a**6 + 3128*a**4 - 6360*a**2 + 3481))
assert radsimp(1/(r2*a + r2*b + r3 + r7)) == (
(sqrt(2)*a*(a + b)**2 - 5*sqrt(2)*a + sqrt(42)*a + sqrt(2)*b*(a +
b)**2 - 5*sqrt(2)*b + sqrt(42)*b - sqrt(7)*(a + b)**2 - sqrt(3)*(a +
b)**2 - 2*sqrt(3) + 2*sqrt(7))/(2*a**4 + 8*a**3*b + 12*a**2*b**2 -
20*a**2 + 8*a*b**3 - 40*a*b + 2*b**4 - 20*b**2 + 8))
assert radsimp(1/(r2*a + r2*b + r2*c + r2*d)) == \
sqrt(2)/(2*a + 2*b + 2*c + 2*d)
assert radsimp(1/(1 + r2*a + r2*b + r2*c + r2*d)) == (
(sqrt(2)*a + sqrt(2)*b + sqrt(2)*c + sqrt(2)*d - 1)/(2*a**2 + 4*a*b +
4*a*c + 4*a*d + 2*b**2 + 4*b*c + 4*b*d + 2*c**2 + 4*c*d + 2*d**2 - 1))
assert radsimp((y**2 - x)/(y - sqrt(x))) == \
sqrt(x) + y
assert radsimp(-(y**2 - x)/(y - sqrt(x))) == \
-(sqrt(x) + y)
assert radsimp(1/(1 - I + a*I)) == \
(-I*a + 1 + I)/(a**2 - 2*a + 2)
assert radsimp(1/((-x + y)*(x - sqrt(y)))) == \
(-x - sqrt(y))/((x - y)*(x**2 - y))
e = (3 + 3*sqrt(2))*x*(3*x - 3*sqrt(y))
assert radsimp(e) == x*(3 + 3*sqrt(2))*(3*x - 3*sqrt(y))
assert radsimp(1/e) == (
(-9*x + 9*sqrt(2)*x - 9*sqrt(y) + 9*sqrt(2)*sqrt(y))/(9*x*(9*x**2 -
9*y)))
assert radsimp(1 + 1/(1 + sqrt(3))) == \
Mul(S.Half, -1 + sqrt(3), evaluate=False) + 1
A = symbols("A", commutative=False)
assert radsimp(x**2 + sqrt(2)*x**2 - sqrt(2)*x*A) == \
x**2 + sqrt(2)*x**2 - sqrt(2)*x*A
assert radsimp(1/sqrt(5 + 2 * sqrt(6))) == -sqrt(2) + sqrt(3)
assert radsimp(1/sqrt(5 + 2 * sqrt(6))**3) == -(-sqrt(3) + sqrt(2))**3
# issue 6532
assert fraction(radsimp(1/sqrt(x))) == (sqrt(x), x)
assert fraction(radsimp(1/sqrt(2*x + 3))) == (sqrt(2*x + 3), 2*x + 3)
assert fraction(radsimp(1/sqrt(2*(x + 3)))) == (sqrt(2*x + 6), 2*x + 6)
# issue 5994
e = S('-(2 + 2*sqrt(2) + 4*2**(1/4))/'
'(1 + 2**(3/4) + 3*2**(1/4) + 3*sqrt(2))')
assert radsimp(e).expand() == -2*2**(S(3)/4) - 2*2**(S(1)/4) + 2 + 2*sqrt(2)
# issue 5986 (modifications to radimp didn't initially recognize this so
# the test is included here)
assert radsimp(1/(-sqrt(5)/2 - S(1)/2 + (-sqrt(5)/2 - S(1)/2)**2)) == 1
# from issue 5934
eq = (
(-240*sqrt(2)*sqrt(sqrt(5) + 5)*sqrt(8*sqrt(5) + 40) -
360*sqrt(2)*sqrt(-8*sqrt(5) + 40)*sqrt(-sqrt(5) + 5) -
120*sqrt(10)*sqrt(-8*sqrt(5) + 40)*sqrt(-sqrt(5) + 5) +
120*sqrt(2)*sqrt(-sqrt(5) + 5)*sqrt(8*sqrt(5) + 40) +
120*sqrt(2)*sqrt(-8*sqrt(5) + 40)*sqrt(sqrt(5) + 5) +
120*sqrt(10)*sqrt(-sqrt(5) + 5)*sqrt(8*sqrt(5) + 40) +
120*sqrt(10)*sqrt(-8*sqrt(5) + 40)*sqrt(sqrt(5) + 5))/(-36000 -
7200*sqrt(5) + (12*sqrt(10)*sqrt(sqrt(5) + 5) +
24*sqrt(10)*sqrt(-sqrt(5) + 5))**2))
assert radsimp(eq) is S.NaN # it's 0/0
# work with normal form
e = 1/sqrt(sqrt(7)/7 + 2*sqrt(2) + 3*sqrt(3) + 5*sqrt(5)) + 3
assert radsimp(e) == (
-sqrt(sqrt(7) + 14*sqrt(2) + 21*sqrt(3) +
35*sqrt(5))*(-11654899*sqrt(35) - 1577436*sqrt(210) - 1278438*sqrt(15)
- 1346996*sqrt(10) + 1635060*sqrt(6) + 5709765 + 7539830*sqrt(14) +
8291415*sqrt(21))/1300423175 + 3)
# obey power rules
base = sqrt(3) - sqrt(2)
assert radsimp(1/base**3) == (sqrt(3) + sqrt(2))**3
assert radsimp(1/(-base)**3) == -(sqrt(2) + sqrt(3))**3
assert radsimp(1/(-base)**x) == (-base)**(-x)
assert radsimp(1/base**x) == (sqrt(2) + sqrt(3))**x
assert radsimp(root(1/(-1 - sqrt(2)), -x)) == (-1)**(-1/x)*(1 + sqrt(2))**(1/x)
# recurse
e = cos(1/(1 + sqrt(2)))
assert radsimp(e) == cos(-sqrt(2) + 1)
assert radsimp(e/2) == cos(-sqrt(2) + 1)/2
assert radsimp(1/e) == 1/cos(-sqrt(2) + 1)
assert radsimp(2/e) == 2/cos(-sqrt(2) + 1)
assert fraction(radsimp(e/sqrt(x))) == (sqrt(x)*cos(-sqrt(2)+1), x)
# test that symbolic denominators are not processed
r = 1 + sqrt(2)
assert radsimp(x/r, symbolic=False) == -x*(-sqrt(2) + 1)
assert radsimp(x/(y + r), symbolic=False) == x/(y + 1 + sqrt(2))
assert radsimp(x/(y + r)/r, symbolic=False) == \
-x*(-sqrt(2) + 1)/(y + 1 + sqrt(2))
# issue 7408
eq = sqrt(x)/sqrt(y)
assert radsimp(eq) == umul(sqrt(x), sqrt(y), 1/y)
assert radsimp(eq, symbolic=False) == eq
# issue 7498
assert radsimp(sqrt(x)/sqrt(y)**3) == umul(sqrt(x), sqrt(y**3), 1/y**3)
# for coverage
eq = sqrt(x)/y**2
assert radsimp(eq) == eq
def test_radsimp_issue_3214():
c, p = symbols('c p', positive=True)
s = sqrt(c**2 - p**2)
b = (c + I*p - s)/(c + I*p + s)
assert radsimp(b) == -I*(c + I*p - sqrt(c**2 - p**2))**2/(2*c*p)
def test_collect_1():
"""Collect with respect to a Symbol"""
x, y, z, n = symbols('x,y,z,n')
assert collect(1, x) == 1
assert collect( x + y*x, x ) == x * (1 + y)
assert collect( x + x**2, x ) == x + x**2
assert collect( x**2 + y*x**2, x ) == (x**2)*(1 + y)
assert collect( x**2 + y*x, x ) == x*y + x**2
assert collect( 2*x**2 + y*x**2 + 3*x*y, [x] ) == x**2*(2 + y) + 3*x*y
assert collect( 2*x**2 + y*x**2 + 3*x*y, [y] ) == 2*x**2 + y*(x**2 + 3*x)
assert collect( ((1 + y + x)**4).expand(), x) == ((1 + y)**4).expand() + \
x*(4*(1 + y)**3).expand() + x**2*(6*(1 + y)**2).expand() + \
x**3*(4*(1 + y)).expand() + x**4
# symbols can be given as any iterable
expr = x + y
assert collect(expr, expr.free_symbols) == expr
def test_collect_2():
"""Collect with respect to a sum"""
a, b, x = symbols('a,b,x')
assert collect(a*(cos(x) + sin(x)) + b*(cos(x) + sin(x)),
sin(x) + cos(x)) == (a + b)*(cos(x) + sin(x))
def test_collect_3():
"""Collect with respect to a product"""
a, b, c = symbols('a,b,c')
f = Function('f')
x, y, z, n = symbols('x,y,z,n')
assert collect(-x/8 + x*y, -x) == x*(y - S(1)/8)
assert collect( 1 + x*(y**2), x*y ) == 1 + x*(y**2)
assert collect( x*y + a*x*y, x*y) == x*y*(1 + a)
assert collect( 1 + x*y + a*x*y, x*y) == 1 + x*y*(1 + a)
assert collect(a*x*f(x) + b*(x*f(x)), x*f(x)) == x*(a + b)*f(x)
assert collect(a*x*log(x) + b*(x*log(x)), x*log(x)) == x*(a + b)*log(x)
assert collect(a*x**2*log(x)**2 + b*(x*log(x))**2, x*log(x)) == \
x**2*log(x)**2*(a + b)
# with respect to a product of three symbols
assert collect(y*x*z + a*x*y*z, x*y*z) == (1 + a)*x*y*z
def test_collect_4():
"""Collect with respect to a power"""
a, b, c, x = symbols('a,b,c,x')
assert collect(a*x**c + b*x**c, x**c) == x**c*(a + b)
# issue 6096: 2 stays with c (unless c is integer or x is positive0
assert collect(a*x**(2*c) + b*x**(2*c), x**c) == x**(2*c)*(a + b)
def test_collect_5():
"""Collect with respect to a tuple"""
a, x, y, z, n = symbols('a,x,y,z,n')
assert collect(x**2*y**4 + z*(x*y**2)**2 + z + a*z, [x*y**2, z]) in [
z*(1 + a + x**2*y**4) + x**2*y**4,
z*(1 + a) + x**2*y**4*(1 + z) ]
assert collect((1 + (x + y) + (x + y)**2).expand(),
[x, y]) == 1 + y + x*(1 + 2*y) + x**2 + y**2
def test_collect_D():
D = Derivative
f = Function('f')
x, a, b = symbols('x,a,b')
fx = D(f(x), x)
fxx = D(f(x), x, x)
assert collect(a*fx + b*fx, fx) == (a + b)*fx
assert collect(a*D(fx, x) + b*D(fx, x), fx) == (a + b)*D(fx, x)
assert collect(a*fxx + b*fxx, fx) == (a + b)*D(fx, x)
# issue 4784
assert collect(5*f(x) + 3*fx, fx) == 5*f(x) + 3*fx
assert collect(f(x) + f(x)*diff(f(x), x) + x*diff(f(x), x)*f(x), f(x).diff(x)) == \
(x*f(x) + f(x))*D(f(x), x) + f(x)
assert collect(f(x) + f(x)*diff(f(x), x) + x*diff(f(x), x)*f(x), f(x).diff(x), exact=True) == \
(x*f(x) + f(x))*D(f(x), x) + f(x)
assert collect(1/f(x) + 1/f(x)*diff(f(x), x) + x*diff(f(x), x)/f(x), f(x).diff(x), exact=True) == \
(1/f(x) + x/f(x))*D(f(x), x) + 1/f(x)
e = (1 + x*fx + fx)/f(x)
assert collect(e.expand(), fx) == fx*(x/f(x) + 1/f(x)) + 1/f(x)
def test_collect_func():
f = ((x + a + 1)**3).expand()
assert collect(f, x) == a**3 + 3*a**2 + 3*a + x**3 + x**2*(3*a + 3) + \
x*(3*a**2 + 6*a + 3) + 1
assert collect(f, x, factor) == x**3 + 3*x**2*(a + 1) + 3*x*(a + 1)**2 + \
(a + 1)**3
assert collect(f, x, evaluate=False) == {
S.One: a**3 + 3*a**2 + 3*a + 1,
x: 3*a**2 + 6*a + 3, x**2: 3*a + 3,
x**3: 1
}
assert collect(f, x, factor, evaluate=False) == {
S.One: (a + 1)**3, x: 3*(a + 1)**2,
x**2: umul(S(3), a + 1), x**3: 1}
def test_collect_order():
a, b, x, t = symbols('a,b,x,t')
assert collect(t + t*x + t*x**2 + O(x**3), t) == t*(1 + x + x**2 + O(x**3))
assert collect(t + t*x + x**2 + O(x**3), t) == \
t*(1 + x + O(x**3)) + x**2 + O(x**3)
f = a*x + b*x + c*x**2 + d*x**2 + O(x**3)
g = x*(a + b) + x**2*(c + d) + O(x**3)
assert collect(f, x) == g
assert collect(f, x, distribute_order_term=False) == g
f = sin(a + b).series(b, 0, 10)
assert collect(f, [sin(a), cos(a)]) == \
sin(a)*cos(b).series(b, 0, 10) + cos(a)*sin(b).series(b, 0, 10)
assert collect(f, [sin(a), cos(a)], distribute_order_term=False) == \
sin(a)*cos(b).series(b, 0, 10).removeO() + \
cos(a)*sin(b).series(b, 0, 10).removeO() + O(b**10)
def test_rcollect():
assert rcollect((x**2*y + x*y + x + y)/(x + y), y) == \
(x + y*(1 + x + x**2))/(x + y)
assert rcollect(sqrt(-((x + 1)*(y + 1))), z) == sqrt(-((x + 1)*(y + 1)))
def test_collect_D_0():
D = Derivative
f = Function('f')
x, a, b = symbols('x,a,b')
fxx = D(f(x), x, x)
assert collect(a*fxx + b*fxx, fxx) == (a + b)*fxx
def test_collect_Wild():
"""Collect with respect to functions with Wild argument"""
a, b, x, y = symbols('a b x y')
f = Function('f')
w1 = Wild('.1')
w2 = Wild('.2')
assert collect(f(x) + a*f(x), f(w1)) == (1 + a)*f(x)
assert collect(f(x, y) + a*f(x, y), f(w1)) == f(x, y) + a*f(x, y)
assert collect(f(x, y) + a*f(x, y), f(w1, w2)) == (1 + a)*f(x, y)
assert collect(f(x, y) + a*f(x, y), f(w1, w1)) == f(x, y) + a*f(x, y)
assert collect(f(x, x) + a*f(x, x), f(w1, w1)) == (1 + a)*f(x, x)
assert collect(a*(x + 1)**y + (x + 1)**y, w1**y) == (1 + a)*(x + 1)**y
assert collect(a*(x + 1)**y + (x + 1)**y, w1**b) == \
a*(x + 1)**y + (x + 1)**y
assert collect(a*(x + 1)**y + (x + 1)**y, (x + 1)**w2) == \
(1 + a)*(x + 1)**y
assert collect(a*(x + 1)**y + (x + 1)**y, w1**w2) == (1 + a)*(x + 1)**y
def test_collect_const():
# coverage not provided by above tests
assert collect_const(2*sqrt(3) + 4*a*sqrt(5)) == \
2*(2*sqrt(5)*a + sqrt(3)) # let the primitive reabsorb
assert collect_const(2*sqrt(3) + 4*a*sqrt(5), sqrt(3)) == \
2*sqrt(3) + 4*a*sqrt(5)
assert collect_const(sqrt(2)*(1 + sqrt(2)) + sqrt(3) + x*sqrt(2)) == \
sqrt(2)*(x + 1 + sqrt(2)) + sqrt(3)
# issue 5290
assert collect_const(2*x + 2*y + 1, 2) == \
collect_const(2*x + 2*y + 1) == \
Add(S(1), Mul(2, x + y, evaluate=False), evaluate=False)
assert collect_const(-y - z) == Mul(-1, y + z, evaluate=False)
assert collect_const(2*x - 2*y - 2*z, 2) == \
Mul(2, x - y - z, evaluate=False)
assert collect_const(2*x - 2*y - 2*z, -2) == \
_unevaluated_Add(2*x, Mul(-2, y + z, evaluate=False))
# this is why the content_primitive is used
eq = (sqrt(15 + 5*sqrt(2))*x + sqrt(3 + sqrt(2))*y)*2
assert collect_sqrt(eq + 2) == \
2*sqrt(sqrt(2) + 3)*(sqrt(5)*x + y) + 2
def test_issue_13143():
f = Function('f')
fx = f(x).diff(x)
e = f(x) + fx + f(x)*fx
# collect function before derivative
assert collect(e, Wild('w')) == f(x)*(fx + 1) + fx
e = f(x) + f(x)*fx + x*fx*f(x)
assert collect(e, fx) == (x*f(x) + f(x))*fx + f(x)
assert collect(e, f(x)) == (x*fx + fx + 1)*f(x)
e = f(x) + fx + f(x)*fx
assert collect(e, [f(x), fx]) == f(x)*(1 + fx) + fx
assert collect(e, [fx, f(x)]) == fx*(1 + f(x)) + f(x)
def test_issue_6097():
assert collect(a*y**(2.0*x) + b*y**(2.0*x), y**x) == y**(2.0*x)*(a + b)
assert collect(a*2**(2.0*x) + b*2**(2.0*x), 2**x) == 2**(2.0*x)*(a + b)
def test_fraction_expand():
eq = (x + y)*y/x
assert eq.expand(frac=True) == fraction_expand(eq) == (x*y + y**2)/x
assert eq.expand() == y + y**2/x
def test_fraction():
x, y, z = map(Symbol, 'xyz')
A = Symbol('A', commutative=False)
assert fraction(Rational(1, 2)) == (1, 2)
assert fraction(x) == (x, 1)
assert fraction(1/x) == (1, x)
assert fraction(x/y) == (x, y)
assert fraction(x/2) == (x, 2)
assert fraction(x*y/z) == (x*y, z)
assert fraction(x/(y*z)) == (x, y*z)
assert fraction(1/y**2) == (1, y**2)
assert fraction(x/y**2) == (x, y**2)
assert fraction((x**2 + 1)/y) == (x**2 + 1, y)
assert fraction(x*(y + 1)/y**7) == (x*(y + 1), y**7)
assert fraction(exp(-x), exact=True) == (exp(-x), 1)
assert fraction((1/(x + y))/2, exact=True) == (1, Mul(2,(x + y), evaluate=False))
assert fraction(x*A/y) == (x*A, y)
assert fraction(x*A**-1/y) == (x*A**-1, y)
n = symbols('n', negative=True)
assert fraction(exp(n)) == (1, exp(-n))
assert fraction(exp(-n)) == (exp(-n), 1)
p = symbols('p', positive=True)
assert fraction(exp(-p)*log(p), exact=True) == (exp(-p)*log(p), 1)
def test_issue_5615():
aA, Re, a, b, D = symbols('aA Re a b D')
e = ((D**3*a + b*aA**3)/Re).expand()
assert collect(e, [aA**3/Re, a]) == e
def test_issue_5933():
from sympy import Polygon, RegularPolygon, denom
x = Polygon(*RegularPolygon((0, 0), 1, 5).vertices).centroid.x
assert abs(denom(x).n()) > 1e-12
assert abs(denom(radsimp(x))) > 1e-12 # in case simplify didn't handle it
def test_issue_14608():
a, b = symbols('a b', commutative=False)
x, y = symbols('x y')
raises(AttributeError, lambda: collect(a*b + b*a, a))
assert collect(x*y + y*(x+1), a) == x*y + y*(x+1)
assert collect(x*y + y*(x+1) + a*b + b*a, y) == y*(2*x + 1) + a*b + b*a
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4aee997f92a5ed842e952334ffc48b79b4bc087a8141905393d380fef5ac86f9
|
from random import randrange
from sympy.simplify.hyperexpand import (ShiftA, ShiftB, UnShiftA, UnShiftB,
MeijerShiftA, MeijerShiftB, MeijerShiftC, MeijerShiftD,
MeijerUnShiftA, MeijerUnShiftB, MeijerUnShiftC,
MeijerUnShiftD,
ReduceOrder, reduce_order, apply_operators,
devise_plan, make_derivative_operator, Formula,
hyperexpand, Hyper_Function, G_Function,
reduce_order_meijer,
build_hypergeometric_formula)
from sympy import hyper, I, S, meijerg, Piecewise, Tuple, Sum, binomial, Expr
from sympy.abc import z, a, b, c
from sympy.utilities.pytest import XFAIL, raises, slow, ON_TRAVIS, skip
from sympy.utilities.randtest import verify_numerically as tn
from sympy.core.compatibility import range
from sympy import (cos, sin, log, exp, asin, lowergamma, atanh, besseli,
gamma, sqrt, pi, erf, exp_polar, Rational)
def test_branch_bug():
assert hyperexpand(hyper((-S(1)/3, S(1)/2), (S(2)/3, S(3)/2), -z)) == \
-z**S('1/3')*lowergamma(exp_polar(I*pi)/3, z)/5 \
+ sqrt(pi)*erf(sqrt(z))/(5*sqrt(z))
assert hyperexpand(meijerg([S(7)/6, 1], [], [S(2)/3], [S(1)/6, 0], z)) == \
2*z**S('2/3')*(2*sqrt(pi)*erf(sqrt(z))/sqrt(z) - 2*lowergamma(
S(2)/3, z)/z**S('2/3'))*gamma(S(2)/3)/gamma(S(5)/3)
def test_hyperexpand():
# Luke, Y. L. (1969), The Special Functions and Their Approximations,
# Volume 1, section 6.2
assert hyperexpand(hyper([], [], z)) == exp(z)
assert hyperexpand(hyper([1, 1], [2], -z)*z) == log(1 + z)
assert hyperexpand(hyper([], [S.Half], -z**2/4)) == cos(z)
assert hyperexpand(z*hyper([], [S('3/2')], -z**2/4)) == sin(z)
assert hyperexpand(hyper([S('1/2'), S('1/2')], [S('3/2')], z**2)*z) \
== asin(z)
assert isinstance(Sum(binomial(2, z)*z**2, (z, 0, a)).doit(), Expr)
def can_do(ap, bq, numerical=True, div=1, lowerplane=False):
from sympy import exp_polar, exp
r = hyperexpand(hyper(ap, bq, z))
if r.has(hyper):
return False
if not numerical:
return True
repl = {}
randsyms = r.free_symbols - {z}
while randsyms:
# Only randomly generated parameters are checked.
for n, a in enumerate(randsyms):
repl[a] = randcplx(n)/div
if not any([b.is_Integer and b <= 0 for b in Tuple(*bq).subs(repl)]):
break
[a, b, c, d] = [2, -1, 3, 1]
if lowerplane:
[a, b, c, d] = [2, -2, 3, -1]
return tn(
hyper(ap, bq, z).subs(repl),
r.replace(exp_polar, exp).subs(repl),
z, a=a, b=b, c=c, d=d)
def test_roach():
# Kelly B. Roach. Meijer G Function Representations.
# Section "Gallery"
assert can_do([S(1)/2], [S(9)/2])
assert can_do([], [1, S(5)/2, 4])
assert can_do([-S.Half, 1, 2], [3, 4])
assert can_do([S(1)/3], [-S(2)/3, -S(1)/2, S(1)/2, 1])
assert can_do([-S(3)/2, -S(1)/2], [-S(5)/2, 1])
assert can_do([-S(3)/2, ], [-S(1)/2, S(1)/2]) # shine-integral
assert can_do([-S(3)/2, -S(1)/2], [2]) # elliptic integrals
@XFAIL
def test_roach_fail():
assert can_do([-S(1)/2, 1], [S(1)/4, S(1)/2, S(3)/4]) # PFDD
assert can_do([S(3)/2], [S(5)/2, 5]) # struve function
assert can_do([-S(1)/2, S(1)/2, 1], [S(3)/2, S(5)/2]) # polylog, pfdd
assert can_do([1, 2, 3], [S(1)/2, 4]) # XXX ?
assert can_do([S(1)/2], [-S(1)/3, -S(1)/2, -S(2)/3]) # PFDD ?
# For the long table tests, see end of file
def test_polynomial():
from sympy import oo
assert hyperexpand(hyper([], [-1], z)) == oo
assert hyperexpand(hyper([-2], [-1], z)) == oo
assert hyperexpand(hyper([0, 0], [-1], z)) == 1
assert can_do([-5, -2, randcplx(), randcplx()], [-10, randcplx()])
assert hyperexpand(hyper((-1, 1), (-2,), z)) == 1 + z/2
def test_hyperexpand_bases():
assert hyperexpand(hyper([2], [a], z)) == \
a + z**(-a + 1)*(-a**2 + 3*a + z*(a - 1) - 2)*exp(z)* \
lowergamma(a - 1, z) - 1
# TODO [a+1, a-S.Half], [2*a]
assert hyperexpand(hyper([1, 2], [3], z)) == -2/z - 2*log(-z + 1)/z**2
assert hyperexpand(hyper([S.Half, 2], [S(3)/2], z)) == \
-1/(2*z - 2) + atanh(sqrt(z))/sqrt(z)/2
assert hyperexpand(hyper([S(1)/2, S(1)/2], [S(5)/2], z)) == \
(-3*z + 3)/4/(z*sqrt(-z + 1)) \
+ (6*z - 3)*asin(sqrt(z))/(4*z**(S(3)/2))
assert hyperexpand(hyper([1, 2], [S(3)/2], z)) == -1/(2*z - 2) \
- asin(sqrt(z))/(sqrt(z)*(2*z - 2)*sqrt(-z + 1))
assert hyperexpand(hyper([-S.Half - 1, 1, 2], [S.Half, 3], z)) == \
sqrt(z)*(6*z/7 - S(6)/5)*atanh(sqrt(z)) \
+ (-30*z**2 + 32*z - 6)/35/z - 6*log(-z + 1)/(35*z**2)
assert hyperexpand(hyper([1 + S.Half, 1, 1], [2, 2], z)) == \
-4*log(sqrt(-z + 1)/2 + S(1)/2)/z
# TODO hyperexpand(hyper([a], [2*a + 1], z))
# TODO [S.Half, a], [S(3)/2, a+1]
assert hyperexpand(hyper([2], [b, 1], z)) == \
z**(-b/2 + S(1)/2)*besseli(b - 1, 2*sqrt(z))*gamma(b) \
+ z**(-b/2 + 1)*besseli(b, 2*sqrt(z))*gamma(b)
# TODO [a], [a - S.Half, 2*a]
def test_hyperexpand_parametric():
assert hyperexpand(hyper([a, S(1)/2 + a], [S(1)/2], z)) \
== (1 + sqrt(z))**(-2*a)/2 + (1 - sqrt(z))**(-2*a)/2
assert hyperexpand(hyper([a, -S(1)/2 + a], [2*a], z)) \
== 2**(2*a - 1)*((-z + 1)**(S(1)/2) + 1)**(-2*a + 1)
def test_shifted_sum():
from sympy import simplify
assert simplify(hyperexpand(z**4*hyper([2], [3, S('3/2')], -z**2))) \
== z*sin(2*z) + (-z**2 + S.Half)*cos(2*z) - S.Half
def _randrat():
""" Steer clear of integers. """
return S(randrange(25) + 10)/50
def randcplx(offset=-1):
""" Polys is not good with real coefficients. """
return _randrat() + I*_randrat() + I*(1 + offset)
@slow
def test_formulae():
from sympy.simplify.hyperexpand import FormulaCollection
formulae = FormulaCollection().formulae
for formula in formulae:
h = formula.func(formula.z)
rep = {}
for n, sym in enumerate(formula.symbols):
rep[sym] = randcplx(n)
# NOTE hyperexpand returns truly branched functions. We know we are
# on the main sheet, but numerical evaluation can still go wrong
# (e.g. if exp_polar cannot be evalf'd).
# Just replace all exp_polar by exp, this usually works.
# first test if the closed-form is actually correct
h = h.subs(rep)
closed_form = formula.closed_form.subs(rep).rewrite('nonrepsmall')
z = formula.z
assert tn(h, closed_form.replace(exp_polar, exp), z)
# now test the computed matrix
cl = (formula.C * formula.B)[0].subs(rep).rewrite('nonrepsmall')
assert tn(closed_form.replace(
exp_polar, exp), cl.replace(exp_polar, exp), z)
deriv1 = z*formula.B.applyfunc(lambda t: t.rewrite(
'nonrepsmall')).diff(z)
deriv2 = formula.M * formula.B
for d1, d2 in zip(deriv1, deriv2):
assert tn(d1.subs(rep).replace(exp_polar, exp),
d2.subs(rep).rewrite('nonrepsmall').replace(exp_polar, exp), z)
def test_meijerg_formulae():
from sympy.simplify.hyperexpand import MeijerFormulaCollection
formulae = MeijerFormulaCollection().formulae
for sig in formulae:
for formula in formulae[sig]:
g = meijerg(formula.func.an, formula.func.ap,
formula.func.bm, formula.func.bq,
formula.z)
rep = {}
for sym in formula.symbols:
rep[sym] = randcplx()
# first test if the closed-form is actually correct
g = g.subs(rep)
closed_form = formula.closed_form.subs(rep)
z = formula.z
assert tn(g, closed_form, z)
# now test the computed matrix
cl = (formula.C * formula.B)[0].subs(rep)
assert tn(closed_form, cl, z)
deriv1 = z*formula.B.diff(z)
deriv2 = formula.M * formula.B
for d1, d2 in zip(deriv1, deriv2):
assert tn(d1.subs(rep), d2.subs(rep), z)
def op(f):
return z*f.diff(z)
def test_plan():
assert devise_plan(Hyper_Function([0], ()),
Hyper_Function([0], ()), z) == []
with raises(ValueError):
devise_plan(Hyper_Function([1], ()), Hyper_Function((), ()), z)
with raises(ValueError):
devise_plan(Hyper_Function([2], [1]), Hyper_Function([2], [2]), z)
with raises(ValueError):
devise_plan(Hyper_Function([2], []), Hyper_Function([S("1/2")], []), z)
# We cannot use pi/(10000 + n) because polys is insanely slow.
a1, a2, b1 = (randcplx(n) for n in range(3))
b1 += 2*I
h = hyper([a1, a2], [b1], z)
h2 = hyper((a1 + 1, a2), [b1], z)
assert tn(apply_operators(h,
devise_plan(Hyper_Function((a1 + 1, a2), [b1]),
Hyper_Function((a1, a2), [b1]), z), op),
h2, z)
h2 = hyper((a1 + 1, a2 - 1), [b1], z)
assert tn(apply_operators(h,
devise_plan(Hyper_Function((a1 + 1, a2 - 1), [b1]),
Hyper_Function((a1, a2), [b1]), z), op),
h2, z)
def test_plan_derivatives():
a1, a2, a3 = 1, 2, S('1/2')
b1, b2 = 3, S('5/2')
h = Hyper_Function((a1, a2, a3), (b1, b2))
h2 = Hyper_Function((a1 + 1, a2 + 1, a3 + 2), (b1 + 1, b2 + 1))
ops = devise_plan(h2, h, z)
f = Formula(h, z, h(z), [])
deriv = make_derivative_operator(f.M, z)
assert tn((apply_operators(f.C, ops, deriv)*f.B)[0], h2(z), z)
h2 = Hyper_Function((a1, a2 - 1, a3 - 2), (b1 - 1, b2 - 1))
ops = devise_plan(h2, h, z)
assert tn((apply_operators(f.C, ops, deriv)*f.B)[0], h2(z), z)
def test_reduction_operators():
a1, a2, b1 = (randcplx(n) for n in range(3))
h = hyper([a1], [b1], z)
assert ReduceOrder(2, 0) is None
assert ReduceOrder(2, -1) is None
assert ReduceOrder(1, S('1/2')) is None
h2 = hyper((a1, a2), (b1, a2), z)
assert tn(ReduceOrder(a2, a2).apply(h, op), h2, z)
h2 = hyper((a1, a2 + 1), (b1, a2), z)
assert tn(ReduceOrder(a2 + 1, a2).apply(h, op), h2, z)
h2 = hyper((a2 + 4, a1), (b1, a2), z)
assert tn(ReduceOrder(a2 + 4, a2).apply(h, op), h2, z)
# test several step order reduction
ap = (a2 + 4, a1, b1 + 1)
bq = (a2, b1, b1)
func, ops = reduce_order(Hyper_Function(ap, bq))
assert func.ap == (a1,)
assert func.bq == (b1,)
assert tn(apply_operators(h, ops, op), hyper(ap, bq, z), z)
def test_shift_operators():
a1, a2, b1, b2, b3 = (randcplx(n) for n in range(5))
h = hyper((a1, a2), (b1, b2, b3), z)
raises(ValueError, lambda: ShiftA(0))
raises(ValueError, lambda: ShiftB(1))
assert tn(ShiftA(a1).apply(h, op), hyper((a1 + 1, a2), (b1, b2, b3), z), z)
assert tn(ShiftA(a2).apply(h, op), hyper((a1, a2 + 1), (b1, b2, b3), z), z)
assert tn(ShiftB(b1).apply(h, op), hyper((a1, a2), (b1 - 1, b2, b3), z), z)
assert tn(ShiftB(b2).apply(h, op), hyper((a1, a2), (b1, b2 - 1, b3), z), z)
assert tn(ShiftB(b3).apply(h, op), hyper((a1, a2), (b1, b2, b3 - 1), z), z)
def test_ushift_operators():
a1, a2, b1, b2, b3 = (randcplx(n) for n in range(5))
h = hyper((a1, a2), (b1, b2, b3), z)
raises(ValueError, lambda: UnShiftA((1,), (), 0, z))
raises(ValueError, lambda: UnShiftB((), (-1,), 0, z))
raises(ValueError, lambda: UnShiftA((1,), (0, -1, 1), 0, z))
raises(ValueError, lambda: UnShiftB((0, 1), (1,), 0, z))
s = UnShiftA((a1, a2), (b1, b2, b3), 0, z)
assert tn(s.apply(h, op), hyper((a1 - 1, a2), (b1, b2, b3), z), z)
s = UnShiftA((a1, a2), (b1, b2, b3), 1, z)
assert tn(s.apply(h, op), hyper((a1, a2 - 1), (b1, b2, b3), z), z)
s = UnShiftB((a1, a2), (b1, b2, b3), 0, z)
assert tn(s.apply(h, op), hyper((a1, a2), (b1 + 1, b2, b3), z), z)
s = UnShiftB((a1, a2), (b1, b2, b3), 1, z)
assert tn(s.apply(h, op), hyper((a1, a2), (b1, b2 + 1, b3), z), z)
s = UnShiftB((a1, a2), (b1, b2, b3), 2, z)
assert tn(s.apply(h, op), hyper((a1, a2), (b1, b2, b3 + 1), z), z)
def can_do_meijer(a1, a2, b1, b2, numeric=True):
"""
This helper function tries to hyperexpand() the meijer g-function
corresponding to the parameters a1, a2, b1, b2.
It returns False if this expansion still contains g-functions.
If numeric is True, it also tests the so-obtained formula numerically
(at random values) and returns False if the test fails.
Else it returns True.
"""
from sympy import unpolarify, expand
r = hyperexpand(meijerg(a1, a2, b1, b2, z))
if r.has(meijerg):
return False
# NOTE hyperexpand() returns a truly branched function, whereas numerical
# evaluation only works on the main branch. Since we are evaluating on
# the main branch, this should not be a problem, but expressions like
# exp_polar(I*pi/2*x)**a are evaluated incorrectly. We thus have to get
# rid of them. The expand heuristically does this...
r = unpolarify(expand(r, force=True, power_base=True, power_exp=False,
mul=False, log=False, multinomial=False, basic=False))
if not numeric:
return True
repl = {}
for n, a in enumerate(meijerg(a1, a2, b1, b2, z).free_symbols - {z}):
repl[a] = randcplx(n)
return tn(meijerg(a1, a2, b1, b2, z).subs(repl), r.subs(repl), z)
@slow
def test_meijerg_expand():
from sympy import gammasimp, simplify
# from mpmath docs
assert hyperexpand(meijerg([[], []], [[0], []], -z)) == exp(z)
assert hyperexpand(meijerg([[1, 1], []], [[1], [0]], z)) == \
log(z + 1)
assert hyperexpand(meijerg([[1, 1], []], [[1], [1]], z)) == \
z/(z + 1)
assert hyperexpand(meijerg([[], []], [[S(1)/2], [0]], (z/2)**2)) \
== sin(z)/sqrt(pi)
assert hyperexpand(meijerg([[], []], [[0], [S(1)/2]], (z/2)**2)) \
== cos(z)/sqrt(pi)
assert can_do_meijer([], [a], [a - 1, a - S.Half], [])
assert can_do_meijer([], [], [a/2], [-a/2], False) # branches...
assert can_do_meijer([a], [b], [a], [b, a - 1])
# wikipedia
assert hyperexpand(meijerg([1], [], [], [0], z)) == \
Piecewise((0, abs(z) < 1), (1, abs(1/z) < 1),
(meijerg([1], [], [], [0], z), True))
assert hyperexpand(meijerg([], [1], [0], [], z)) == \
Piecewise((1, abs(z) < 1), (0, abs(1/z) < 1),
(meijerg([], [1], [0], [], z), True))
# The Special Functions and their Approximations
assert can_do_meijer([], [], [a + b/2], [a, a - b/2, a + S.Half])
assert can_do_meijer(
[], [], [a], [b], False) # branches only agree for small z
assert can_do_meijer([], [S.Half], [a], [-a])
assert can_do_meijer([], [], [a, b], [])
assert can_do_meijer([], [], [a, b], [])
assert can_do_meijer([], [], [a, a + S.Half], [b, b + S.Half])
assert can_do_meijer([], [], [a, -a], [0, S.Half], False) # dito
assert can_do_meijer([], [], [a, a + S.Half, b, b + S.Half], [])
assert can_do_meijer([S.Half], [], [0], [a, -a])
assert can_do_meijer([S.Half], [], [a], [0, -a], False) # dito
assert can_do_meijer([], [a - S.Half], [a, b], [a - S.Half], False)
assert can_do_meijer([], [a + S.Half], [a + b, a - b, a], [], False)
assert can_do_meijer([a + S.Half], [], [b, 2*a - b, a], [], False)
# This for example is actually zero.
assert can_do_meijer([], [], [], [a, b])
# Testing a bug:
assert hyperexpand(meijerg([0, 2], [], [], [-1, 1], z)) == \
Piecewise((0, abs(z) < 1),
(z/2 - 1/(2*z), abs(1/z) < 1),
(meijerg([0, 2], [], [], [-1, 1], z), True))
# Test that the simplest possible answer is returned:
assert gammasimp(simplify(hyperexpand(
meijerg([1], [1 - a], [-a/2, -a/2 + S(1)/2], [], 1/z)))) == \
-2*sqrt(pi)*(sqrt(z + 1) + 1)**a/a
# Test that hyper is returned
assert hyperexpand(meijerg([1], [], [a], [0, 0], z)) == hyper(
(a,), (a + 1, a + 1), z*exp_polar(I*pi))*z**a*gamma(a)/gamma(a + 1)**2
# Test place option
f = meijerg(((0, 1), ()), ((S(1)/2,), (0,)), z**2)
assert hyperexpand(f) == sqrt(pi)/sqrt(1 + z**(-2))
assert hyperexpand(f, place=0) == sqrt(pi)*z/sqrt(z**2 + 1)
def test_meijerg_lookup():
from sympy import uppergamma, Si, Ci
assert hyperexpand(meijerg([a], [], [b, a], [], z)) == \
z**b*exp(z)*gamma(-a + b + 1)*uppergamma(a - b, z)
assert hyperexpand(meijerg([0], [], [0, 0], [], z)) == \
exp(z)*uppergamma(0, z)
assert can_do_meijer([a], [], [b, a + 1], [])
assert can_do_meijer([a], [], [b + 2, a], [])
assert can_do_meijer([a], [], [b - 2, a], [])
assert hyperexpand(meijerg([a], [], [a, a, a - S(1)/2], [], z)) == \
-sqrt(pi)*z**(a - S(1)/2)*(2*cos(2*sqrt(z))*(Si(2*sqrt(z)) - pi/2)
- 2*sin(2*sqrt(z))*Ci(2*sqrt(z))) == \
hyperexpand(meijerg([a], [], [a, a - S(1)/2, a], [], z)) == \
hyperexpand(meijerg([a], [], [a - S(1)/2, a, a], [], z))
assert can_do_meijer([a - 1], [], [a + 2, a - S(3)/2, a + 1], [])
@XFAIL
def test_meijerg_expand_fail():
# These basically test hyper([], [1/2 - a, 1/2 + 1, 1/2], z),
# which is *very* messy. But since the meijer g actually yields a
# sum of bessel functions, things can sometimes be simplified a lot and
# are then put into tables...
assert can_do_meijer([], [], [a + S.Half], [a, a - b/2, a + b/2])
assert can_do_meijer([], [], [0, S.Half], [a, -a])
assert can_do_meijer([], [], [3*a - S.Half, a, -a - S.Half], [a - S.Half])
assert can_do_meijer([], [], [0, a - S.Half, -a - S.Half], [S.Half])
assert can_do_meijer([], [], [a, b + S(1)/2, b], [2*b - a])
assert can_do_meijer([], [], [a, b + S(1)/2, b, 2*b - a])
assert can_do_meijer([S.Half], [], [-a, a], [0])
@slow
def test_meijerg():
# carefully set up the parameters.
# NOTE: this used to fail sometimes. I believe it is fixed, but if you
# hit an inexplicable test failure here, please let me know the seed.
a1, a2 = (randcplx(n) - 5*I - n*I for n in range(2))
b1, b2 = (randcplx(n) + 5*I + n*I for n in range(2))
b3, b4, b5, a3, a4, a5 = (randcplx() for n in range(6))
g = meijerg([a1], [a3, a4], [b1], [b3, b4], z)
assert ReduceOrder.meijer_minus(3, 4) is None
assert ReduceOrder.meijer_plus(4, 3) is None
g2 = meijerg([a1, a2], [a3, a4], [b1], [b3, b4, a2], z)
assert tn(ReduceOrder.meijer_plus(a2, a2).apply(g, op), g2, z)
g2 = meijerg([a1, a2], [a3, a4], [b1], [b3, b4, a2 + 1], z)
assert tn(ReduceOrder.meijer_plus(a2, a2 + 1).apply(g, op), g2, z)
g2 = meijerg([a1, a2 - 1], [a3, a4], [b1], [b3, b4, a2 + 2], z)
assert tn(ReduceOrder.meijer_plus(a2 - 1, a2 + 2).apply(g, op), g2, z)
g2 = meijerg([a1], [a3, a4, b2 - 1], [b1, b2 + 2], [b3, b4], z)
assert tn(ReduceOrder.meijer_minus(
b2 + 2, b2 - 1).apply(g, op), g2, z, tol=1e-6)
# test several-step reduction
an = [a1, a2]
bq = [b3, b4, a2 + 1]
ap = [a3, a4, b2 - 1]
bm = [b1, b2 + 1]
niq, ops = reduce_order_meijer(G_Function(an, ap, bm, bq))
assert niq.an == (a1,)
assert set(niq.ap) == {a3, a4}
assert niq.bm == (b1,)
assert set(niq.bq) == {b3, b4}
assert tn(apply_operators(g, ops, op), meijerg(an, ap, bm, bq, z), z)
def test_meijerg_shift_operators():
# carefully set up the parameters. XXX this still fails sometimes
a1, a2, a3, a4, a5, b1, b2, b3, b4, b5 = (randcplx(n) for n in range(10))
g = meijerg([a1], [a3, a4], [b1], [b3, b4], z)
assert tn(MeijerShiftA(b1).apply(g, op),
meijerg([a1], [a3, a4], [b1 + 1], [b3, b4], z), z)
assert tn(MeijerShiftB(a1).apply(g, op),
meijerg([a1 - 1], [a3, a4], [b1], [b3, b4], z), z)
assert tn(MeijerShiftC(b3).apply(g, op),
meijerg([a1], [a3, a4], [b1], [b3 + 1, b4], z), z)
assert tn(MeijerShiftD(a3).apply(g, op),
meijerg([a1], [a3 - 1, a4], [b1], [b3, b4], z), z)
s = MeijerUnShiftA([a1], [a3, a4], [b1], [b3, b4], 0, z)
assert tn(
s.apply(g, op), meijerg([a1], [a3, a4], [b1 - 1], [b3, b4], z), z)
s = MeijerUnShiftC([a1], [a3, a4], [b1], [b3, b4], 0, z)
assert tn(
s.apply(g, op), meijerg([a1], [a3, a4], [b1], [b3 - 1, b4], z), z)
s = MeijerUnShiftB([a1], [a3, a4], [b1], [b3, b4], 0, z)
assert tn(
s.apply(g, op), meijerg([a1 + 1], [a3, a4], [b1], [b3, b4], z), z)
s = MeijerUnShiftD([a1], [a3, a4], [b1], [b3, b4], 0, z)
assert tn(
s.apply(g, op), meijerg([a1], [a3 + 1, a4], [b1], [b3, b4], z), z)
@slow
def test_meijerg_confluence():
def t(m, a, b):
from sympy import sympify, Piecewise
a, b = sympify([a, b])
m_ = m
m = hyperexpand(m)
if not m == Piecewise((a, abs(z) < 1), (b, abs(1/z) < 1), (m_, True)):
return False
if not (m.args[0].args[0] == a and m.args[1].args[0] == b):
return False
z0 = randcplx()/10
if abs(m.subs(z, z0).n() - a.subs(z, z0).n()).n() > 1e-10:
return False
if abs(m.subs(z, 1/z0).n() - b.subs(z, 1/z0).n()).n() > 1e-10:
return False
return True
assert t(meijerg([], [1, 1], [0, 0], [], z), -log(z), 0)
assert t(meijerg(
[], [3, 1], [0, 0], [], z), -z**2/4 + z - log(z)/2 - S(3)/4, 0)
assert t(meijerg([], [3, 1], [-1, 0], [], z),
z**2/12 - z/2 + log(z)/2 + S(1)/4 + 1/(6*z), 0)
assert t(meijerg([], [1, 1, 1, 1], [0, 0, 0, 0], [], z), -log(z)**3/6, 0)
assert t(meijerg([1, 1], [], [], [0, 0], z), 0, -log(1/z))
assert t(meijerg([1, 1], [2, 2], [1, 1], [0, 0], z),
-z*log(z) + 2*z, -log(1/z) + 2)
assert t(meijerg([S(1)/2], [1, 1], [0, 0], [S(3)/2], z), log(z)/2 - 1, 0)
def u(an, ap, bm, bq):
m = meijerg(an, ap, bm, bq, z)
m2 = hyperexpand(m, allow_hyper=True)
if m2.has(meijerg) and not (m2.is_Piecewise and len(m2.args) == 3):
return False
return tn(m, m2, z)
assert u([], [1], [0, 0], [])
assert u([1, 1], [], [], [0])
assert u([1, 1], [2, 2, 5], [1, 1, 6], [0, 0])
assert u([1, 1], [2, 2, 5], [1, 1, 6], [0])
def test_meijerg_with_Floats():
# see issue #10681
from sympy import RR
f = meijerg(((3.0, 1), ()), ((S(3)/2,), (0,)), z)
a = -2.3632718012073
g = a*z**(S(3)/2)*hyper((-0.5, S(3)/2), (S(5)/2,), z*exp_polar(I*pi))
assert RR.almosteq((hyperexpand(f)/g).n(), 1.0, 1e-12)
def test_lerchphi():
from sympy import gammasimp, exp_polar, polylog, log, lerchphi
assert hyperexpand(hyper([1, a], [a + 1], z)/a) == lerchphi(z, 1, a)
assert hyperexpand(
hyper([1, a, a], [a + 1, a + 1], z)/a**2) == lerchphi(z, 2, a)
assert hyperexpand(hyper([1, a, a, a], [a + 1, a + 1, a + 1], z)/a**3) == \
lerchphi(z, 3, a)
assert hyperexpand(hyper([1] + [a]*10, [a + 1]*10, z)/a**10) == \
lerchphi(z, 10, a)
assert gammasimp(hyperexpand(meijerg([0, 1 - a], [], [0],
[-a], exp_polar(-I*pi)*z))) == lerchphi(z, 1, a)
assert gammasimp(hyperexpand(meijerg([0, 1 - a, 1 - a], [], [0],
[-a, -a], exp_polar(-I*pi)*z))) == lerchphi(z, 2, a)
assert gammasimp(hyperexpand(meijerg([0, 1 - a, 1 - a, 1 - a], [], [0],
[-a, -a, -a], exp_polar(-I*pi)*z))) == lerchphi(z, 3, a)
assert hyperexpand(z*hyper([1, 1], [2], z)) == -log(1 + -z)
assert hyperexpand(z*hyper([1, 1, 1], [2, 2], z)) == polylog(2, z)
assert hyperexpand(z*hyper([1, 1, 1, 1], [2, 2, 2], z)) == polylog(3, z)
assert hyperexpand(hyper([1, a, 1 + S(1)/2], [a + 1, S(1)/2], z)) == \
-2*a/(z - 1) + (-2*a**2 + a)*lerchphi(z, 1, a)
# Now numerical tests. These make sure reductions etc are carried out
# correctly
# a rational function (polylog at negative integer order)
assert can_do([2, 2, 2], [1, 1])
# NOTE these contain log(1-x) etc ... better make sure we have |z| < 1
# reduction of order for polylog
assert can_do([1, 1, 1, b + 5], [2, 2, b], div=10)
# reduction of order for lerchphi
# XXX lerchphi in mpmath is flaky
assert can_do(
[1, a, a, a, b + 5], [a + 1, a + 1, a + 1, b], numerical=False)
# test a bug
from sympy import Abs
assert hyperexpand(hyper([S(1)/2, S(1)/2, S(1)/2, 1],
[S(3)/2, S(3)/2, S(3)/2], S(1)/4)) == \
Abs(-polylog(3, exp_polar(I*pi)/2) + polylog(3, S(1)/2))
def test_partial_simp():
# First test that hypergeometric function formulae work.
a, b, c, d, e = (randcplx() for _ in range(5))
for func in [Hyper_Function([a, b, c], [d, e]),
Hyper_Function([], [a, b, c, d, e])]:
f = build_hypergeometric_formula(func)
z = f.z
assert f.closed_form == func(z)
deriv1 = f.B.diff(z)*z
deriv2 = f.M*f.B
for func1, func2 in zip(deriv1, deriv2):
assert tn(func1, func2, z)
# Now test that formulae are partially simplified.
from sympy.abc import a, b, z
assert hyperexpand(hyper([3, a], [1, b], z)) == \
(-a*b/2 + a*z/2 + 2*a)*hyper([a + 1], [b], z) \
+ (a*b/2 - 2*a + 1)*hyper([a], [b], z)
assert tn(
hyperexpand(hyper([3, d], [1, e], z)), hyper([3, d], [1, e], z), z)
assert hyperexpand(hyper([3], [1, a, b], z)) == \
hyper((), (a, b), z) \
+ z*hyper((), (a + 1, b), z)/(2*a) \
- z*(b - 4)*hyper((), (a + 1, b + 1), z)/(2*a*b)
assert tn(
hyperexpand(hyper([3], [1, d, e], z)), hyper([3], [1, d, e], z), z)
def test_hyperexpand_special():
assert hyperexpand(hyper([a, b], [c], 1)) == \
gamma(c)*gamma(c - a - b)/gamma(c - a)/gamma(c - b)
assert hyperexpand(hyper([a, b], [1 + a - b], -1)) == \
gamma(1 + a/2)*gamma(1 + a - b)/gamma(1 + a)/gamma(1 + a/2 - b)
assert hyperexpand(hyper([a, b], [1 + b - a], -1)) == \
gamma(1 + b/2)*gamma(1 + b - a)/gamma(1 + b)/gamma(1 + b/2 - a)
assert hyperexpand(meijerg([1 - z - a/2], [1 - z + a/2], [b/2], [-b/2], 1)) == \
gamma(1 - 2*z)*gamma(z + a/2 + b/2)/gamma(1 - z + a/2 - b/2) \
/gamma(1 - z - a/2 + b/2)/gamma(1 - z + a/2 + b/2)
assert hyperexpand(hyper([a], [b], 0)) == 1
assert hyper([a], [b], 0) != 0
def test_Mod1_behavior():
from sympy import Symbol, simplify, lowergamma
n = Symbol('n', integer=True)
# Note: this should not hang.
assert simplify(hyperexpand(meijerg([1], [], [n + 1], [0], z))) == \
lowergamma(n + 1, z)
@slow
def test_prudnikov_misc():
assert can_do([1, (3 + I)/2, (3 - I)/2], [S(3)/2, 2])
assert can_do([S.Half, a - 1], [S(3)/2, a + 1], lowerplane=True)
assert can_do([], [b + 1])
assert can_do([a], [a - 1, b + 1])
assert can_do([a], [a - S.Half, 2*a])
assert can_do([a], [a - S.Half, 2*a + 1])
assert can_do([a], [a - S.Half, 2*a - 1])
assert can_do([a], [a + S.Half, 2*a])
assert can_do([a], [a + S.Half, 2*a + 1])
assert can_do([a], [a + S.Half, 2*a - 1])
assert can_do([S.Half], [b, 2 - b])
assert can_do([S.Half], [b, 3 - b])
assert can_do([1], [2, b])
assert can_do([a, a + S.Half], [2*a, b, 2*a - b + 1])
assert can_do([a, a + S.Half], [S.Half, 2*a, 2*a + S.Half])
assert can_do([a], [a + 1], lowerplane=True) # lowergamma
def test_prudnikov_1():
# A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev (1990).
# Integrals and Series: More Special Functions, Vol. 3,.
# Gordon and Breach Science Publisher
# 7.3.1
assert can_do([a, -a], [S.Half])
assert can_do([a, 1 - a], [S.Half])
assert can_do([a, 1 - a], [S(3)/2])
assert can_do([a, 2 - a], [S.Half])
assert can_do([a, 2 - a], [S(3)/2])
assert can_do([a, 2 - a], [S(3)/2])
assert can_do([a, a + S(1)/2], [2*a - 1])
assert can_do([a, a + S(1)/2], [2*a])
assert can_do([a, a + S(1)/2], [2*a + 1])
assert can_do([a, a + S(1)/2], [S(1)/2])
assert can_do([a, a + S(1)/2], [S(3)/2])
assert can_do([a, a/2 + 1], [a/2])
assert can_do([1, b], [2])
assert can_do([1, b], [b + 1], numerical=False) # Lerch Phi
# NOTE: branches are complicated for |z| > 1
assert can_do([a], [2*a])
assert can_do([a], [2*a + 1])
assert can_do([a], [2*a - 1])
@slow
def test_prudnikov_2():
h = S.Half
assert can_do([-h, -h], [h])
assert can_do([-h, h], [3*h])
assert can_do([-h, h], [5*h])
assert can_do([-h, h], [7*h])
assert can_do([-h, 1], [h])
for p in [-h, h]:
for n in [-h, h, 1, 3*h, 2, 5*h, 3, 7*h, 4]:
for m in [-h, h, 3*h, 5*h, 7*h]:
assert can_do([p, n], [m])
for n in [1, 2, 3, 4]:
for m in [1, 2, 3, 4]:
assert can_do([p, n], [m])
@slow
def test_prudnikov_3():
if ON_TRAVIS:
# See https://github.com/sympy/sympy/pull/12795
skip("Too slow for travis.")
h = S.Half
assert can_do([S(1)/4, S(3)/4], [h])
assert can_do([S(1)/4, S(3)/4], [3*h])
assert can_do([S(1)/3, S(2)/3], [3*h])
assert can_do([S(3)/4, S(5)/4], [h])
assert can_do([S(3)/4, S(5)/4], [3*h])
for p in [1, 2, 3, 4]:
for n in [-h, h, 1, 3*h, 2, 5*h, 3, 7*h, 4, 9*h]:
for m in [1, 3*h, 2, 5*h, 3, 7*h, 4]:
assert can_do([p, m], [n])
@slow
def test_prudnikov_4():
h = S.Half
for p in [3*h, 5*h, 7*h]:
for n in [-h, h, 3*h, 5*h, 7*h]:
for m in [3*h, 2, 5*h, 3, 7*h, 4]:
assert can_do([p, m], [n])
for n in [1, 2, 3, 4]:
for m in [2, 3, 4]:
assert can_do([p, m], [n])
@slow
def test_prudnikov_5():
h = S.Half
for p in [1, 2, 3]:
for q in range(p, 4):
for r in [1, 2, 3]:
for s in range(r, 4):
assert can_do([-h, p, q], [r, s])
for p in [h, 1, 3*h, 2, 5*h, 3]:
for q in [h, 3*h, 5*h]:
for r in [h, 3*h, 5*h]:
for s in [h, 3*h, 5*h]:
if s <= q and s <= r:
assert can_do([-h, p, q], [r, s])
for p in [h, 1, 3*h, 2, 5*h, 3]:
for q in [1, 2, 3]:
for r in [h, 3*h, 5*h]:
for s in [1, 2, 3]:
assert can_do([-h, p, q], [r, s])
@slow
def test_prudnikov_6():
h = S.Half
for m in [3*h, 5*h]:
for n in [1, 2, 3]:
for q in [h, 1, 2]:
for p in [1, 2, 3]:
assert can_do([h, q, p], [m, n])
for q in [1, 2, 3]:
for p in [3*h, 5*h]:
assert can_do([h, q, p], [m, n])
for q in [1, 2]:
for p in [1, 2, 3]:
for m in [1, 2, 3]:
for n in [1, 2, 3]:
assert can_do([h, q, p], [m, n])
assert can_do([h, h, 5*h], [3*h, 3*h])
assert can_do([h, 1, 5*h], [3*h, 3*h])
assert can_do([h, 2, 2], [1, 3])
# pages 435 to 457 contain more PFDD and stuff like this
@slow
def test_prudnikov_7():
assert can_do([3], [6])
h = S.Half
for n in [h, 3*h, 5*h, 7*h]:
assert can_do([-h], [n])
for m in [-h, h, 1, 3*h, 2, 5*h, 3, 7*h, 4]: # HERE
for n in [-h, h, 3*h, 5*h, 7*h, 1, 2, 3, 4]:
assert can_do([m], [n])
@slow
def test_prudnikov_8():
h = S.Half
# 7.12.2
for a in [1, 2, 3]:
for b in [1, 2, 3]:
for c in range(1, a + 1):
for d in [h, 1, 3*h, 2, 5*h, 3]:
assert can_do([a, b], [c, d])
for b in [3*h, 5*h]:
for c in [h, 1, 3*h, 2, 5*h, 3]:
for d in [1, 2, 3]:
assert can_do([a, b], [c, d])
for a in [-h, h, 3*h, 5*h]:
for b in [1, 2, 3]:
for c in [h, 1, 3*h, 2, 5*h, 3]:
for d in [1, 2, 3]:
assert can_do([a, b], [c, d])
for b in [h, 3*h, 5*h]:
for c in [h, 3*h, 5*h, 3]:
for d in [h, 1, 3*h, 2, 5*h, 3]:
if c <= b:
assert can_do([a, b], [c, d])
def test_prudnikov_9():
# 7.13.1 [we have a general formula ... so this is a bit pointless]
for i in range(9):
assert can_do([], [(S(i) + 1)/2])
for i in range(5):
assert can_do([], [-(2*S(i) + 1)/2])
@slow
def test_prudnikov_10():
# 7.14.2
h = S.Half
for p in [-h, h, 1, 3*h, 2, 5*h, 3, 7*h, 4]:
for m in [1, 2, 3, 4]:
for n in range(m, 5):
assert can_do([p], [m, n])
for p in [1, 2, 3, 4]:
for n in [h, 3*h, 5*h, 7*h]:
for m in [1, 2, 3, 4]:
assert can_do([p], [n, m])
for p in [3*h, 5*h, 7*h]:
for m in [h, 1, 2, 5*h, 3, 7*h, 4]:
assert can_do([p], [h, m])
assert can_do([p], [3*h, m])
for m in [h, 1, 2, 5*h, 3, 7*h, 4]:
assert can_do([7*h], [5*h, m])
assert can_do([-S(1)/2], [S(1)/2, S(1)/2]) # shine-integral shi
def test_prudnikov_11():
# 7.15
assert can_do([a, a + S.Half], [2*a, b, 2*a - b])
assert can_do([a, a + S.Half], [S(3)/2, 2*a, 2*a - S(1)/2])
assert can_do([S(1)/4, S(3)/4], [S(1)/2, S(1)/2, 1])
assert can_do([S(5)/4, S(3)/4], [S(3)/2, S(1)/2, 2])
assert can_do([S(5)/4, S(3)/4], [S(3)/2, S(3)/2, 1])
assert can_do([S(5)/4, S(7)/4], [S(3)/2, S(5)/2, 2])
assert can_do([1, 1], [S(3)/2, 2, 2]) # cosh-integral chi
def test_prudnikov_12():
# 7.16
assert can_do(
[], [a, a + S.Half, 2*a], False) # branches only agree for some z!
assert can_do([], [a, a + S.Half, 2*a + 1], False) # dito
assert can_do([], [S.Half, a, a + S.Half])
assert can_do([], [S(3)/2, a, a + S.Half])
assert can_do([], [S(1)/4, S(1)/2, S(3)/4])
assert can_do([], [S(1)/2, S(1)/2, 1])
assert can_do([], [S(1)/2, S(3)/2, 1])
assert can_do([], [S(3)/4, S(3)/2, S(5)/4])
assert can_do([], [1, 1, S(3)/2])
assert can_do([], [1, 2, S(3)/2])
assert can_do([], [1, S(3)/2, S(3)/2])
assert can_do([], [S(5)/4, S(3)/2, S(7)/4])
assert can_do([], [2, S(3)/2, S(3)/2])
@slow
def test_prudnikov_2F1():
h = S.Half
# Elliptic integrals
for p in [-h, h]:
for m in [h, 3*h, 5*h, 7*h]:
for n in [1, 2, 3, 4]:
assert can_do([p, m], [n])
@XFAIL
def test_prudnikov_fail_2F1():
assert can_do([a, b], [b + 1]) # incomplete beta function
assert can_do([-1, b], [c]) # Poly. also -2, -3 etc
# TODO polys
# Legendre functions:
assert can_do([a, b], [a + b + S.Half])
assert can_do([a, b], [a + b - S.Half])
assert can_do([a, b], [a + b + S(3)/2])
assert can_do([a, b], [(a + b + 1)/2])
assert can_do([a, b], [(a + b)/2 + 1])
assert can_do([a, b], [a - b + 1])
assert can_do([a, b], [a - b + 2])
assert can_do([a, b], [2*b])
assert can_do([a, b], [S.Half])
assert can_do([a, b], [S(3)/2])
assert can_do([a, 1 - a], [c])
assert can_do([a, 2 - a], [c])
assert can_do([a, 3 - a], [c])
assert can_do([a, a + S(1)/2], [c])
assert can_do([1, b], [c])
assert can_do([1, b], [S(3)/2])
assert can_do([S(1)/4, S(3)/4], [1])
# PFDD
o = S(1)
assert can_do([o/8, 1], [o/8*9])
assert can_do([o/6, 1], [o/6*7])
assert can_do([o/6, 1], [o/6*13])
assert can_do([o/5, 1], [o/5*6])
assert can_do([o/5, 1], [o/5*11])
assert can_do([o/4, 1], [o/4*5])
assert can_do([o/4, 1], [o/4*9])
assert can_do([o/3, 1], [o/3*4])
assert can_do([o/3, 1], [o/3*7])
assert can_do([o/8*3, 1], [o/8*11])
assert can_do([o/5*2, 1], [o/5*7])
assert can_do([o/5*2, 1], [o/5*12])
assert can_do([o/5*3, 1], [o/5*8])
assert can_do([o/5*3, 1], [o/5*13])
assert can_do([o/8*5, 1], [o/8*13])
assert can_do([o/4*3, 1], [o/4*7])
assert can_do([o/4*3, 1], [o/4*11])
assert can_do([o/3*2, 1], [o/3*5])
assert can_do([o/3*2, 1], [o/3*8])
assert can_do([o/5*4, 1], [o/5*9])
assert can_do([o/5*4, 1], [o/5*14])
assert can_do([o/6*5, 1], [o/6*11])
assert can_do([o/6*5, 1], [o/6*17])
assert can_do([o/8*7, 1], [o/8*15])
@XFAIL
def test_prudnikov_fail_3F2():
assert can_do([a, a + S(1)/3, a + S(2)/3], [S(1)/3, S(2)/3])
assert can_do([a, a + S(1)/3, a + S(2)/3], [S(2)/3, S(4)/3])
assert can_do([a, a + S(1)/3, a + S(2)/3], [S(4)/3, S(5)/3])
# page 421
assert can_do([a, a + S(1)/3, a + S(2)/3], [3*a/2, (3*a + 1)/2])
# pages 422 ...
assert can_do([-S.Half, S.Half, S.Half], [1, 1]) # elliptic integrals
assert can_do([-S.Half, S.Half, 1], [S(3)/2, S(3)/2])
# TODO LOTS more
# PFDD
assert can_do([S(1)/8, S(3)/8, 1], [S(9)/8, S(11)/8])
assert can_do([S(1)/8, S(5)/8, 1], [S(9)/8, S(13)/8])
assert can_do([S(1)/8, S(7)/8, 1], [S(9)/8, S(15)/8])
assert can_do([S(1)/6, S(1)/3, 1], [S(7)/6, S(4)/3])
assert can_do([S(1)/6, S(2)/3, 1], [S(7)/6, S(5)/3])
assert can_do([S(1)/6, S(2)/3, 1], [S(5)/3, S(13)/6])
assert can_do([S.Half, 1, 1], [S(1)/4, S(3)/4])
# LOTS more
@XFAIL
def test_prudnikov_fail_other():
# 7.11.2
# 7.12.1
assert can_do([1, a], [b, 1 - 2*a + b]) # ???
# 7.14.2
assert can_do([-S(1)/2], [S(1)/2, 1]) # struve
assert can_do([1], [S(1)/2, S(1)/2]) # struve
assert can_do([S(1)/4], [S(1)/2, S(5)/4]) # PFDD
assert can_do([S(3)/4], [S(3)/2, S(7)/4]) # PFDD
assert can_do([1], [S(1)/4, S(3)/4]) # PFDD
assert can_do([1], [S(3)/4, S(5)/4]) # PFDD
assert can_do([1], [S(5)/4, S(7)/4]) # PFDD
# TODO LOTS more
# 7.15.2
assert can_do([S(1)/2, 1], [S(3)/4, S(5)/4, S(3)/2]) # PFDD
assert can_do([S(1)/2, 1], [S(7)/4, S(5)/4, S(3)/2]) # PFDD
# 7.16.1
assert can_do([], [S(1)/3, S(2/3)]) # PFDD
assert can_do([], [S(2)/3, S(4/3)]) # PFDD
assert can_do([], [S(5)/3, S(4/3)]) # PFDD
# XXX this does not *evaluate* right??
assert can_do([], [a, a + S.Half, 2*a - 1])
def test_bug():
h = hyper([-1, 1], [z], -1)
assert hyperexpand(h) == (z + 1)/z
def test_omgissue_203():
h = hyper((-5, -3, -4), (-6, -6), 1)
assert hyperexpand(h) == Rational(1, 30)
h = hyper((-6, -7, -5), (-6, -6), 1)
assert hyperexpand(h) == -Rational(1, 6)
|
b53c9e34e7b22ac74cceb0eae7ddf26d5556f0600a3e4bed049bd211f30b7dac
|
from sympy import (
Abs, acos, Add, asin, atan, Basic, binomial, besselsimp,
collect,cos, cosh, cot, coth, count_ops, csch, Derivative, diff, E,
Eq, erf, exp, exp_polar, expand, expand_multinomial, factor,
factorial, Float, fraction, Function, gamma, GoldenRatio, hyper,
hypersimp, I, Integral, integrate, log, logcombine, Lt, Matrix,
MatrixSymbol, Mul, nsimplify, O, oo, pi, Piecewise, posify, rad,
Rational, root, S, separatevars, signsimp, simplify, sign, sin,
sinc, sinh, solve, sqrt, Sum, Symbol, symbols, sympify, tan, tanh,
zoo)
from sympy.core.mul import _keep_coeff
from sympy.simplify.simplify import nthroot, inversecombine
from sympy.utilities.pytest import XFAIL, slow
from sympy.core.compatibility import range
from sympy.abc import x, y, z, t, a, b, c, d, e, f, g, h, i, k
def test_issue_7263():
assert abs((simplify(30.8**2 - 82.5**2 * sin(rad(11.6))**2)).evalf() - \
673.447451402970) < 1e-12
@XFAIL
def test_factorial_simplify():
# There are more tests in test_factorials.py. These are just to
# ensure that simplify() calls factorial_simplify correctly
from sympy.specfun.factorials import factorial
x = Symbol('x')
assert simplify(factorial(x)/x) == factorial(x - 1)
assert simplify(factorial(factorial(x))) == factorial(factorial(x))
def test_simplify_expr():
x, y, z, k, n, m, w, s, A = symbols('x,y,z,k,n,m,w,s,A')
f = Function('f')
assert all(simplify(tmp) == tmp for tmp in [I, E, oo, x, -x, -oo, -E, -I])
e = 1/x + 1/y
assert e != (x + y)/(x*y)
assert simplify(e) == (x + y)/(x*y)
e = A**2*s**4/(4*pi*k*m**3)
assert simplify(e) == e
e = (4 + 4*x - 2*(2 + 2*x))/(2 + 2*x)
assert simplify(e) == 0
e = (-4*x*y**2 - 2*y**3 - 2*x**2*y)/(x + y)**2
assert simplify(e) == -2*y
e = -x - y - (x + y)**(-1)*y**2 + (x + y)**(-1)*x**2
assert simplify(e) == -2*y
e = (x + x*y)/x
assert simplify(e) == 1 + y
e = (f(x) + y*f(x))/f(x)
assert simplify(e) == 1 + y
e = (2 * (1/n - cos(n * pi)/n))/pi
assert simplify(e) == (-cos(pi*n) + 1)/(pi*n)*2
e = integrate(1/(x**3 + 1), x).diff(x)
assert simplify(e) == 1/(x**3 + 1)
e = integrate(x/(x**2 + 3*x + 1), x).diff(x)
assert simplify(e) == x/(x**2 + 3*x + 1)
f = Symbol('f')
A = Matrix([[2*k - m*w**2, -k], [-k, k - m*w**2]]).inv()
assert simplify((A*Matrix([0, f]))[1]) == \
-f*(2*k - m*w**2)/(k**2 - (k - m*w**2)*(2*k - m*w**2))
f = -x + y/(z + t) + z*x/(z + t) + z*a/(z + t) + t*x/(z + t)
assert simplify(f) == (y + a*z)/(z + t)
# issue 10347
expr = -x*(y**2 - 1)*(2*y**2*(x**2 - 1)/(a*(x**2 - y**2)**2) + (x**2 - 1)
/(a*(x**2 - y**2)))/(a*(x**2 - y**2)) + x*(-2*x**2*sqrt(-x**2*y**2 + x**2
+ y**2 - 1)*sin(z)/(a*(x**2 - y**2)**2) - x**2*sqrt(-x**2*y**2 + x**2 +
y**2 - 1)*sin(z)/(a*(x**2 - 1)*(x**2 - y**2)) + (x**2*sqrt((-x**2 + 1)*
(y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*sin(z)/(x**2 - 1) + sqrt(
(-x**2 + 1)*(y**2 - 1))*(x*(-x*y**2 + x)/sqrt(-x**2*y**2 + x**2 + y**2 -
1) + sqrt(-x**2*y**2 + x**2 + y**2 - 1))*sin(z))/(a*sqrt((-x**2 + 1)*(
y**2 - 1))*(x**2 - y**2)))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*sin(z)/(a*
(x**2 - y**2)) + x*(-2*x**2*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z)/(a*
(x**2 - y**2)**2) - x**2*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z)/(a*
(x**2 - 1)*(x**2 - y**2)) + (x**2*sqrt((-x**2 + 1)*(y**2 - 1))*sqrt(-x**2
*y**2 + x**2 + y**2 - 1)*cos(z)/(x**2 - 1) + x*sqrt((-x**2 + 1)*(y**2 -
1))*(-x*y**2 + x)*cos(z)/sqrt(-x**2*y**2 + x**2 + y**2 - 1) + sqrt((-x**2
+ 1)*(y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z))/(a*sqrt((-x**2
+ 1)*(y**2 - 1))*(x**2 - y**2)))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(
z)/(a*(x**2 - y**2)) - y*sqrt((-x**2 + 1)*(y**2 - 1))*(-x*y*sqrt(-x**2*
y**2 + x**2 + y**2 - 1)*sin(z)/(a*(x**2 - y**2)*(y**2 - 1)) + 2*x*y*sqrt(
-x**2*y**2 + x**2 + y**2 - 1)*sin(z)/(a*(x**2 - y**2)**2) + (x*y*sqrt((
-x**2 + 1)*(y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*sin(z)/(y**2 -
1) + x*sqrt((-x**2 + 1)*(y**2 - 1))*(-x**2*y + y)*sin(z)/sqrt(-x**2*y**2
+ x**2 + y**2 - 1))/(a*sqrt((-x**2 + 1)*(y**2 - 1))*(x**2 - y**2)))*sin(
z)/(a*(x**2 - y**2)) + y*(x**2 - 1)*(-2*x*y*(x**2 - 1)/(a*(x**2 - y**2)
**2) + 2*x*y/(a*(x**2 - y**2)))/(a*(x**2 - y**2)) + y*(x**2 - 1)*(y**2 -
1)*(-x*y*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z)/(a*(x**2 - y**2)*(y**2
- 1)) + 2*x*y*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z)/(a*(x**2 - y**2)
**2) + (x*y*sqrt((-x**2 + 1)*(y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 -
1)*cos(z)/(y**2 - 1) + x*sqrt((-x**2 + 1)*(y**2 - 1))*(-x**2*y + y)*cos(
z)/sqrt(-x**2*y**2 + x**2 + y**2 - 1))/(a*sqrt((-x**2 + 1)*(y**2 - 1)
)*(x**2 - y**2)))*cos(z)/(a*sqrt((-x**2 + 1)*(y**2 - 1))*(x**2 - y**2)
) - x*sqrt((-x**2 + 1)*(y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*sin(
z)**2/(a**2*(x**2 - 1)*(x**2 - y**2)*(y**2 - 1)) - x*sqrt((-x**2 + 1)*(
y**2 - 1))*sqrt(-x**2*y**2 + x**2 + y**2 - 1)*cos(z)**2/(a**2*(x**2 - 1)*(
x**2 - y**2)*(y**2 - 1))
assert simplify(expr) == 2*x/(a**2*(x**2 - y**2))
A, B = symbols('A,B', commutative=False)
assert simplify(A*B - B*A) == A*B - B*A
assert simplify(A/(1 + y/x)) == x*A/(x + y)
assert simplify(A*(1/x + 1/y)) == A/x + A/y #(x + y)*A/(x*y)
assert simplify(log(2) + log(3)) == log(6)
assert simplify(log(2*x) - log(2)) == log(x)
assert simplify(hyper([], [], x)) == exp(x)
def test_issue_3557():
f_1 = x*a + y*b + z*c - 1
f_2 = x*d + y*e + z*f - 1
f_3 = x*g + y*h + z*i - 1
solutions = solve([f_1, f_2, f_3], x, y, z, simplify=False)
assert simplify(solutions[y]) == \
(a*i + c*d + f*g - a*f - c*g - d*i)/ \
(a*e*i + b*f*g + c*d*h - a*f*h - b*d*i - c*e*g)
def test_simplify_other():
assert simplify(sin(x)**2 + cos(x)**2) == 1
assert simplify(gamma(x + 1)/gamma(x)) == x
assert simplify(sin(x)**2 + cos(x)**2 + factorial(x)/gamma(x)) == 1 + x
assert simplify(
Eq(sin(x)**2 + cos(x)**2, factorial(x)/gamma(x))) == Eq(x, 1)
nc = symbols('nc', commutative=False)
assert simplify(x + x*nc) == x*(1 + nc)
# issue 6123
# f = exp(-I*(k*sqrt(t) + x/(2*sqrt(t)))**2)
# ans = integrate(f, (k, -oo, oo), conds='none')
ans = I*(-pi*x*exp(-3*I*pi/4 + I*x**2/(4*t))*erf(x*exp(-3*I*pi/4)/
(2*sqrt(t)))/(2*sqrt(t)) + pi*x*exp(-3*I*pi/4 + I*x**2/(4*t))/
(2*sqrt(t)))*exp(-I*x**2/(4*t))/(sqrt(pi)*x) - I*sqrt(pi) * \
(-erf(x*exp(I*pi/4)/(2*sqrt(t))) + 1)*exp(I*pi/4)/(2*sqrt(t))
assert simplify(ans) == -(-1)**(S(3)/4)*sqrt(pi)/sqrt(t)
# issue 6370
assert simplify(2**(2 + x)/4) == 2**x
def test_simplify_complex():
cosAsExp = cos(x)._eval_rewrite_as_exp(x)
tanAsExp = tan(x)._eval_rewrite_as_exp(x)
assert simplify(cosAsExp*tanAsExp) == sin(x) # issue 4341
# issue 10124
assert simplify(exp(Matrix([[0, -1], [1, 0]]))) == Matrix([[cos(1),
-sin(1)], [sin(1), cos(1)]])
def test_simplify_ratio():
# roots of x**3-3*x+5
roots = ['(1/2 - sqrt(3)*I/2)*(sqrt(21)/2 + 5/2)**(1/3) + 1/((1/2 - '
'sqrt(3)*I/2)*(sqrt(21)/2 + 5/2)**(1/3))',
'1/((1/2 + sqrt(3)*I/2)*(sqrt(21)/2 + 5/2)**(1/3)) + '
'(1/2 + sqrt(3)*I/2)*(sqrt(21)/2 + 5/2)**(1/3)',
'-(sqrt(21)/2 + 5/2)**(1/3) - 1/(sqrt(21)/2 + 5/2)**(1/3)']
for r in roots:
r = S(r)
assert count_ops(simplify(r, ratio=1)) <= count_ops(r)
# If ratio=oo, simplify() is always applied:
assert simplify(r, ratio=oo) is not r
def test_simplify_measure():
measure1 = lambda expr: len(str(expr))
measure2 = lambda expr: -count_ops(expr)
# Return the most complicated result
expr = (x + 1)/(x + sin(x)**2 + cos(x)**2)
assert measure1(simplify(expr, measure=measure1)) <= measure1(expr)
assert measure2(simplify(expr, measure=measure2)) <= measure2(expr)
expr2 = Eq(sin(x)**2 + cos(x)**2, 1)
assert measure1(simplify(expr2, measure=measure1)) <= measure1(expr2)
assert measure2(simplify(expr2, measure=measure2)) <= measure2(expr2)
def test_simplify_rational():
expr = 2**x*2.**y
assert simplify(expr, rational = True) == 2**(x+y)
assert simplify(expr, rational = None) == 2.0**(x+y)
assert simplify(expr, rational = False) == expr
def test_simplify_issue_1308():
assert simplify(exp(-Rational(1, 2)) + exp(-Rational(3, 2))) == \
(1 + E)*exp(-Rational(3, 2))
def test_issue_5652():
assert simplify(E + exp(-E)) == exp(-E) + E
n = symbols('n', commutative=False)
assert simplify(n + n**(-n)) == n + n**(-n)
def test_simplify_fail1():
x = Symbol('x')
y = Symbol('y')
e = (x + y)**2/(-4*x*y**2 - 2*y**3 - 2*x**2*y)
assert simplify(e) == 1 / (-2*y)
def test_nthroot():
assert nthroot(90 + 34*sqrt(7), 3) == sqrt(7) + 3
q = 1 + sqrt(2) - 2*sqrt(3) + sqrt(6) + sqrt(7)
assert nthroot(expand_multinomial(q**3), 3) == q
assert nthroot(41 + 29*sqrt(2), 5) == 1 + sqrt(2)
assert nthroot(-41 - 29*sqrt(2), 5) == -1 - sqrt(2)
expr = 1320*sqrt(10) + 4216 + 2576*sqrt(6) + 1640*sqrt(15)
assert nthroot(expr, 5) == 1 + sqrt(6) + sqrt(15)
q = 1 + sqrt(2) + sqrt(3) + sqrt(5)
assert expand_multinomial(nthroot(expand_multinomial(q**5), 5)) == q
q = 1 + sqrt(2) + 7*sqrt(6) + 2*sqrt(10)
assert nthroot(expand_multinomial(q**5), 5, 8) == q
q = 1 + sqrt(2) - 2*sqrt(3) + 1171*sqrt(6)
assert nthroot(expand_multinomial(q**3), 3) == q
assert nthroot(expand_multinomial(q**6), 6) == q
def test_nthroot1():
q = 1 + sqrt(2) + sqrt(3) + S(1)/10**20
p = expand_multinomial(q**5)
assert nthroot(p, 5) == q
q = 1 + sqrt(2) + sqrt(3) + S(1)/10**30
p = expand_multinomial(q**5)
assert nthroot(p, 5) == q
def test_separatevars():
x, y, z, n = symbols('x,y,z,n')
assert separatevars(2*n*x*z + 2*x*y*z) == 2*x*z*(n + y)
assert separatevars(x*z + x*y*z) == x*z*(1 + y)
assert separatevars(pi*x*z + pi*x*y*z) == pi*x*z*(1 + y)
assert separatevars(x*y**2*sin(x) + x*sin(x)*sin(y)) == \
x*(sin(y) + y**2)*sin(x)
assert separatevars(x*exp(x + y) + x*exp(x)) == x*(1 + exp(y))*exp(x)
assert separatevars((x*(y + 1))**z).is_Pow # != x**z*(1 + y)**z
assert separatevars(1 + x + y + x*y) == (x + 1)*(y + 1)
assert separatevars(y/pi*exp(-(z - x)/cos(n))) == \
y*exp(x/cos(n))*exp(-z/cos(n))/pi
assert separatevars((x + y)*(x - y) + y**2 + 2*x + 1) == (x + 1)**2
# issue 4858
p = Symbol('p', positive=True)
assert separatevars(sqrt(p**2 + x*p**2)) == p*sqrt(1 + x)
assert separatevars(sqrt(y*(p**2 + x*p**2))) == p*sqrt(y*(1 + x))
assert separatevars(sqrt(y*(p**2 + x*p**2)), force=True) == \
p*sqrt(y)*sqrt(1 + x)
# issue 4865
assert separatevars(sqrt(x*y)).is_Pow
assert separatevars(sqrt(x*y), force=True) == sqrt(x)*sqrt(y)
# issue 4957
# any type sequence for symbols is fine
assert separatevars(((2*x + 2)*y), dict=True, symbols=()) == \
{'coeff': 1, x: 2*x + 2, y: y}
# separable
assert separatevars(((2*x + 2)*y), dict=True, symbols=[x]) == \
{'coeff': y, x: 2*x + 2}
assert separatevars(((2*x + 2)*y), dict=True, symbols=[]) == \
{'coeff': 1, x: 2*x + 2, y: y}
assert separatevars(((2*x + 2)*y), dict=True) == \
{'coeff': 1, x: 2*x + 2, y: y}
assert separatevars(((2*x + 2)*y), dict=True, symbols=None) == \
{'coeff': y*(2*x + 2)}
# not separable
assert separatevars(3, dict=True) is None
assert separatevars(2*x + y, dict=True, symbols=()) is None
assert separatevars(2*x + y, dict=True) is None
assert separatevars(2*x + y, dict=True, symbols=None) == {'coeff': 2*x + y}
# issue 4808
n, m = symbols('n,m', commutative=False)
assert separatevars(m + n*m) == (1 + n)*m
assert separatevars(x + x*n) == x*(1 + n)
# issue 4910
f = Function('f')
assert separatevars(f(x) + x*f(x)) == f(x) + x*f(x)
# a noncommutable object present
eq = x*(1 + hyper((), (), y*z))
assert separatevars(eq) == eq
def test_separatevars_advanced_factor():
x, y, z = symbols('x,y,z')
assert separatevars(1 + log(x)*log(y) + log(x) + log(y)) == \
(log(x) + 1)*(log(y) + 1)
assert separatevars(1 + x - log(z) - x*log(z) - exp(y)*log(z) -
x*exp(y)*log(z) + x*exp(y) + exp(y)) == \
-((x + 1)*(log(z) - 1)*(exp(y) + 1))
x, y = symbols('x,y', positive=True)
assert separatevars(1 + log(x**log(y)) + log(x*y)) == \
(log(x) + 1)*(log(y) + 1)
def test_hypersimp():
n, k = symbols('n,k', integer=True)
assert hypersimp(factorial(k), k) == k + 1
assert hypersimp(factorial(k**2), k) is None
assert hypersimp(1/factorial(k), k) == 1/(k + 1)
assert hypersimp(2**k/factorial(k)**2, k) == 2/(k + 1)**2
assert hypersimp(binomial(n, k), k) == (n - k)/(k + 1)
assert hypersimp(binomial(n + 1, k), k) == (n - k + 1)/(k + 1)
term = (4*k + 1)*factorial(k)/factorial(2*k + 1)
assert hypersimp(term, k) == (S(1)/2)*((4*k + 5)/(3 + 14*k + 8*k**2))
term = 1/((2*k - 1)*factorial(2*k + 1))
assert hypersimp(term, k) == (k - S(1)/2)/((k + 1)*(2*k + 1)*(2*k + 3))
term = binomial(n, k)*(-1)**k/factorial(k)
assert hypersimp(term, k) == (k - n)/(k + 1)**2
def test_nsimplify():
x = Symbol("x")
assert nsimplify(0) == 0
assert nsimplify(-1) == -1
assert nsimplify(1) == 1
assert nsimplify(1 + x) == 1 + x
assert nsimplify(2.7) == Rational(27, 10)
assert nsimplify(1 - GoldenRatio) == (1 - sqrt(5))/2
assert nsimplify((1 + sqrt(5))/4, [GoldenRatio]) == GoldenRatio/2
assert nsimplify(2/GoldenRatio, [GoldenRatio]) == 2*GoldenRatio - 2
assert nsimplify(exp(5*pi*I/3, evaluate=False)) == \
sympify('1/2 - sqrt(3)*I/2')
assert nsimplify(sin(3*pi/5, evaluate=False)) == \
sympify('sqrt(sqrt(5)/8 + 5/8)')
assert nsimplify(sqrt(atan('1', evaluate=False))*(2 + I), [pi]) == \
sqrt(pi) + sqrt(pi)/2*I
assert nsimplify(2 + exp(2*atan('1/4')*I)) == sympify('49/17 + 8*I/17')
assert nsimplify(pi, tolerance=0.01) == Rational(22, 7)
assert nsimplify(pi, tolerance=0.001) == Rational(355, 113)
assert nsimplify(0.33333, tolerance=1e-4) == Rational(1, 3)
assert nsimplify(2.0**(1/3.), tolerance=0.001) == Rational(635, 504)
assert nsimplify(2.0**(1/3.), tolerance=0.001, full=True) == \
2**Rational(1, 3)
assert nsimplify(x + .5, rational=True) == Rational(1, 2) + x
assert nsimplify(1/.3 + x, rational=True) == Rational(10, 3) + x
assert nsimplify(log(3).n(), rational=True) == \
sympify('109861228866811/100000000000000')
assert nsimplify(Float(0.272198261287950), [pi, log(2)]) == pi*log(2)/8
assert nsimplify(Float(0.272198261287950).n(3), [pi, log(2)]) == \
-pi/4 - log(2) + S(7)/4
assert nsimplify(x/7.0) == x/7
assert nsimplify(pi/1e2) == pi/100
assert nsimplify(pi/1e2, rational=False) == pi/100.0
assert nsimplify(pi/1e-7) == 10000000*pi
assert not nsimplify(
factor(-3.0*z**2*(z**2)**(-2.5) + 3*(z**2)**(-1.5))).atoms(Float)
e = x**0.0
assert e.is_Pow and nsimplify(x**0.0) == 1
assert nsimplify(3.333333, tolerance=0.1, rational=True) == Rational(10, 3)
assert nsimplify(3.333333, tolerance=0.01, rational=True) == Rational(10, 3)
assert nsimplify(3.666666, tolerance=0.1, rational=True) == Rational(11, 3)
assert nsimplify(3.666666, tolerance=0.01, rational=True) == Rational(11, 3)
assert nsimplify(33, tolerance=10, rational=True) == Rational(33)
assert nsimplify(33.33, tolerance=10, rational=True) == Rational(30)
assert nsimplify(37.76, tolerance=10, rational=True) == Rational(40)
assert nsimplify(-203.1) == -S(2031)/10
assert nsimplify(.2, tolerance=0) == S.One/5
assert nsimplify(-.2, tolerance=0) == -S.One/5
assert nsimplify(.2222, tolerance=0) == S(1111)/5000
assert nsimplify(-.2222, tolerance=0) == -S(1111)/5000
# issue 7211, PR 4112
assert nsimplify(S(2e-8)) == S(1)/50000000
# issue 7322 direct test
assert nsimplify(1e-42, rational=True) != 0
# issue 10336
inf = Float('inf')
infs = (-oo, oo, inf, -inf)
for i in infs:
ans = sign(i)*oo
assert nsimplify(i) == ans
assert nsimplify(i + x) == x + ans
assert nsimplify(0.33333333, rational=True, rational_conversion='exact') == Rational(0.33333333)
# Make sure nsimplify on expressions uses full precision
assert nsimplify(pi.evalf(100)*x, rational_conversion='exact').evalf(100) == pi.evalf(100)*x
def test_issue_9448():
tmp = sympify("1/(1 - (-1)**(2/3) - (-1)**(1/3)) + 1/(1 + (-1)**(2/3) + (-1)**(1/3))")
assert nsimplify(tmp) == S(1)/2
def test_extract_minus_sign():
x = Symbol("x")
y = Symbol("y")
a = Symbol("a")
b = Symbol("b")
assert simplify(-x/-y) == x/y
assert simplify(-x/y) == -x/y
assert simplify(x/y) == x/y
assert simplify(x/-y) == -x/y
assert simplify(-x/0) == zoo*x
assert simplify(S(-5)/0) == zoo
assert simplify(-a*x/(-y - b)) == a*x/(b + y)
def test_diff():
x = Symbol("x")
y = Symbol("y")
f = Function("f")
g = Function("g")
assert simplify(g(x).diff(x)*f(x).diff(x) - f(x).diff(x)*g(x).diff(x)) == 0
assert simplify(2*f(x)*f(x).diff(x) - diff(f(x)**2, x)) == 0
assert simplify(diff(1/f(x), x) + f(x).diff(x)/f(x)**2) == 0
assert simplify(f(x).diff(x, y) - f(x).diff(y, x)) == 0
def test_logcombine_1():
x, y = symbols("x,y")
a = Symbol("a")
z, w = symbols("z,w", positive=True)
b = Symbol("b", real=True)
assert logcombine(log(x) + 2*log(y)) == log(x) + 2*log(y)
assert logcombine(log(x) + 2*log(y), force=True) == log(x*y**2)
assert logcombine(a*log(w) + log(z)) == a*log(w) + log(z)
assert logcombine(b*log(z) + b*log(x)) == log(z**b) + b*log(x)
assert logcombine(b*log(z) - log(w)) == log(z**b/w)
assert logcombine(log(x)*log(z)) == log(x)*log(z)
assert logcombine(log(w)*log(x)) == log(w)*log(x)
assert logcombine(cos(-2*log(z) + b*log(w))) in [cos(log(w**b/z**2)),
cos(log(z**2/w**b))]
assert logcombine(log(log(x) - log(y)) - log(z), force=True) == \
log(log(x/y)/z)
assert logcombine((2 + I)*log(x), force=True) == (2 + I)*log(x)
assert logcombine((x**2 + log(x) - log(y))/(x*y), force=True) == \
(x**2 + log(x/y))/(x*y)
# the following could also give log(z*x**log(y**2)), what we
# are testing is that a canonical result is obtained
assert logcombine(log(x)*2*log(y) + log(z), force=True) == \
log(z*y**log(x**2))
assert logcombine((x*y + sqrt(x**4 + y**4) + log(x) - log(y))/(pi*x**Rational(2, 3)*
sqrt(y)**3), force=True) == (
x*y + sqrt(x**4 + y**4) + log(x/y))/(pi*x**(S(2)/3)*y**(S(3)/2))
assert logcombine(gamma(-log(x/y))*acos(-log(x/y)), force=True) == \
acos(-log(x/y))*gamma(-log(x/y))
assert logcombine(2*log(z)*log(w)*log(x) + log(z) + log(w)) == \
log(z**log(w**2))*log(x) + log(w*z)
assert logcombine(3*log(w) + 3*log(z)) == log(w**3*z**3)
assert logcombine(x*(y + 1) + log(2) + log(3)) == x*(y + 1) + log(6)
assert logcombine((x + y)*log(w) + (-x - y)*log(3)) == (x + y)*log(w/3)
# a single unknown can combine
assert logcombine(log(x) + log(2)) == log(2*x)
eq = log(abs(x)) + log(abs(y))
assert logcombine(eq) == eq
reps = {x: 0, y: 0}
assert log(abs(x)*abs(y)).subs(reps) != eq.subs(reps)
def test_logcombine_complex_coeff():
i = Integral((sin(x**2) + cos(x**3))/x, x)
assert logcombine(i, force=True) == i
assert logcombine(i + 2*log(x), force=True) == \
i + log(x**2)
def test_issue_5950():
x, y = symbols("x,y", positive=True)
assert logcombine(log(3) - log(2)) == log(Rational(3,2), evaluate=False)
assert logcombine(log(x) - log(y)) == log(x/y)
assert logcombine(log(Rational(3,2), evaluate=False) - log(2)) == \
log(Rational(3,4), evaluate=False)
def test_posify():
from sympy.abc import x
assert str(posify(
x +
Symbol('p', positive=True) +
Symbol('n', negative=True))) == '(_x + n + p, {_x: x})'
eq, rep = posify(1/x)
assert log(eq).expand().subs(rep) == -log(x)
assert str(posify([x, 1 + x])) == '([_x, _x + 1], {_x: x})'
x = symbols('x')
p = symbols('p', positive=True)
n = symbols('n', negative=True)
orig = [x, n, p]
modified, reps = posify(orig)
assert str(modified) == '[_x, n, p]'
assert [w.subs(reps) for w in modified] == orig
assert str(Integral(posify(1/x + y)[0], (y, 1, 3)).expand()) == \
'Integral(1/_x, (y, 1, 3)) + Integral(_y, (y, 1, 3))'
assert str(Sum(posify(1/x**n)[0], (n,1,3)).expand()) == \
'Sum(_x**(-n), (n, 1, 3))'
def test_issue_4194():
# simplify should call cancel
from sympy.abc import x, y
f = Function('f')
assert simplify((4*x + 6*f(y))/(2*x + 3*f(y))) == 2
@XFAIL
def test_simplify_float_vs_integer():
# Test for issue 4473:
# https://github.com/sympy/sympy/issues/4473
assert simplify(x**2.0 - x**2) == 0
assert simplify(x**2 - x**2.0) == 0
def test_as_content_primitive():
assert (x/2 + y).as_content_primitive() == (S.Half, x + 2*y)
assert (x/2 + y).as_content_primitive(clear=False) == (S.One, x/2 + y)
assert (y*(x/2 + y)).as_content_primitive() == (S.Half, y*(x + 2*y))
assert (y*(x/2 + y)).as_content_primitive(clear=False) == (S.One, y*(x/2 + y))
# although the _as_content_primitive methods do not alter the underlying structure,
# the as_content_primitive function will touch up the expression and join
# bases that would otherwise have not been joined.
assert ((x*(2 + 2*x)*(3*x + 3)**2)).as_content_primitive() == \
(18, x*(x + 1)**3)
assert (2 + 2*x + 2*y*(3 + 3*y)).as_content_primitive() == \
(2, x + 3*y*(y + 1) + 1)
assert ((2 + 6*x)**2).as_content_primitive() == \
(4, (3*x + 1)**2)
assert ((2 + 6*x)**(2*y)).as_content_primitive() == \
(1, (_keep_coeff(S(2), (3*x + 1)))**(2*y))
assert (5 + 10*x + 2*y*(3 + 3*y)).as_content_primitive() == \
(1, 10*x + 6*y*(y + 1) + 5)
assert ((5*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive() == \
(11, x*(y + 1))
assert ((5*(x*(1 + y)) + 2*x*(3 + 3*y))**2).as_content_primitive() == \
(121, x**2*(y + 1)**2)
assert (y**2).as_content_primitive() == \
(1, y**2)
assert (S.Infinity).as_content_primitive() == (1, oo)
eq = x**(2 + y)
assert (eq).as_content_primitive() == (1, eq)
assert (S.Half**(2 + x)).as_content_primitive() == (S(1)/4, 2**-x)
assert ((-S.Half)**(2 + x)).as_content_primitive() == \
(S(1)/4, (-S.Half)**x)
assert ((-S.Half)**(2 + x)).as_content_primitive() == \
(S(1)/4, (-S.Half)**x)
assert (4**((1 + y)/2)).as_content_primitive() == (2, 4**(y/2))
assert (3**((1 + y)/2)).as_content_primitive() == \
(1, 3**(Mul(S(1)/2, 1 + y, evaluate=False)))
assert (5**(S(3)/4)).as_content_primitive() == (1, 5**(S(3)/4))
assert (5**(S(7)/4)).as_content_primitive() == (5, 5**(S(3)/4))
assert Add(5*z/7, 0.5*x, 3*y/2, evaluate=False).as_content_primitive() == \
(S(1)/14, 7.0*x + 21*y + 10*z)
assert (2**(S(3)/4) + 2**(S(1)/4)*sqrt(3)).as_content_primitive(radical=True) == \
(1, 2**(S(1)/4)*(sqrt(2) + sqrt(3)))
def test_signsimp():
e = x*(-x + 1) + x*(x - 1)
assert signsimp(Eq(e, 0)) is S.true
assert Abs(x - 1) == Abs(1 - x)
assert signsimp(y - x) == y - x
assert signsimp(y - x, evaluate=False) == Mul(-1, x - y, evaluate=False)
def test_besselsimp():
from sympy import besselj, besseli, exp_polar, cosh, cosine_transform
assert besselsimp(exp(-I*pi*y/2)*besseli(y, z*exp_polar(I*pi/2))) == \
besselj(y, z)
assert besselsimp(exp(-I*pi*a/2)*besseli(a, 2*sqrt(x)*exp_polar(I*pi/2))) == \
besselj(a, 2*sqrt(x))
assert besselsimp(sqrt(2)*sqrt(pi)*x**(S(1)/4)*exp(I*pi/4)*exp(-I*pi*a/2) *
besseli(-S(1)/2, sqrt(x)*exp_polar(I*pi/2)) *
besseli(a, sqrt(x)*exp_polar(I*pi/2))/2) == \
besselj(a, sqrt(x)) * cos(sqrt(x))
assert besselsimp(besseli(S(-1)/2, z)) == \
sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besseli(a, z*exp_polar(-I*pi/2))) == \
exp(-I*pi*a/2)*besselj(a, z)
assert cosine_transform(1/t*sin(a/t), t, y) == \
sqrt(2)*sqrt(pi)*besselj(0, 2*sqrt(a)*sqrt(y))/2
def test_Piecewise():
e1 = x*(x + y) - y*(x + y)
e2 = sin(x)**2 + cos(x)**2
e3 = expand((x + y)*y/x)
s1 = simplify(e1)
s2 = simplify(e2)
s3 = simplify(e3)
assert simplify(Piecewise((e1, x < e2), (e3, True))) == \
Piecewise((s1, x < s2), (s3, True))
def test_polymorphism():
class A(Basic):
def _eval_simplify(x, **kwargs):
return 1
a = A(5, 2)
assert simplify(a) == 1
def test_issue_from_PR1599():
n1, n2, n3, n4 = symbols('n1 n2 n3 n4', negative=True)
assert simplify(I*sqrt(n1)) == -sqrt(-n1)
def test_issue_6811():
eq = (x + 2*y)*(2*x + 2)
assert simplify(eq) == (x + 1)*(x + 2*y)*2
# reject the 2-arg Mul -- these are a headache for test writing
assert simplify(eq.expand()) == \
2*x**2 + 4*x*y + 2*x + 4*y
def test_issue_6920():
e = [cos(x) + I*sin(x), cos(x) - I*sin(x),
cosh(x) - sinh(x), cosh(x) + sinh(x)]
ok = [exp(I*x), exp(-I*x), exp(-x), exp(x)]
# wrap in f to show that the change happens wherever ei occurs
f = Function('f')
assert [simplify(f(ei)).args[0] for ei in e] == ok
def test_issue_7001():
from sympy.abc import r, R
assert simplify(-(r*Piecewise((4*pi/3, r <= R),
(-8*pi*R**3/(3*r**3), True)) + 2*Piecewise((4*pi*r/3, r <= R),
(4*pi*R**3/(3*r**2), True)))/(4*pi*r)) == \
Piecewise((-1, r <= R), (0, True))
def test_inequality_no_auto_simplify():
# no simplify on creation but can be simplified
lhs = cos(x)**2 + sin(x)**2
rhs = 2
e = Lt(lhs, rhs, evaluate=False)
assert e is not S.true
assert simplify(e)
def test_issue_9398():
from sympy import Number, cancel
assert cancel(1e-14) != 0
assert cancel(1e-14*I) != 0
assert simplify(1e-14) != 0
assert simplify(1e-14*I) != 0
assert (I*Number(1.)*Number(10)**Number(-14)).simplify() != 0
assert cancel(1e-20) != 0
assert cancel(1e-20*I) != 0
assert simplify(1e-20) != 0
assert simplify(1e-20*I) != 0
assert cancel(1e-100) != 0
assert cancel(1e-100*I) != 0
assert simplify(1e-100) != 0
assert simplify(1e-100*I) != 0
f = Float("1e-1000")
assert cancel(f) != 0
assert cancel(f*I) != 0
assert simplify(f) != 0
assert simplify(f*I) != 0
def test_issue_9324_simplify():
M = MatrixSymbol('M', 10, 10)
e = M[0, 0] + M[5, 4] + 1304
assert simplify(e) == e
def test_issue_13474():
x = Symbol('x')
assert simplify(x + csch(sinc(1))) == x + csch(sinc(1))
def test_simplify_function_inverse():
# "inverse" attribute does not guarantee that f(g(x)) is x
# so this simplification should not happen automatically.
# See issue #12140
x, y = symbols('x, y')
g = Function('g')
class f(Function):
def inverse(self, argindex=1):
return g
assert simplify(f(g(x))) == f(g(x))
assert inversecombine(f(g(x))) == x
assert simplify(f(g(x)), inverse=True) == x
assert simplify(f(g(sin(x)**2 + cos(x)**2)), inverse=True) == 1
assert simplify(f(g(x, y)), inverse=True) == f(g(x, y))
assert simplify(2*asin(sin(3*x)), inverse=True) == 6*x
assert simplify(log(exp(x))) == log(exp(x))
assert simplify(log(exp(x)), inverse=True) == x
assert simplify(log(exp(x), 2), inverse=True) == x/log(2)
assert simplify(log(exp(x), 2, evaluate=False), inverse=True) == x/log(2)
def test_clear_coefficients():
from sympy.simplify.simplify import clear_coefficients
assert clear_coefficients(4*y*(6*x + 3)) == (y*(2*x + 1), 0)
assert clear_coefficients(4*y*(6*x + 3) - 2) == (y*(2*x + 1), S(1)/6)
assert clear_coefficients(4*y*(6*x + 3) - 2, x) == (y*(2*x + 1), x/12 + S(1)/6)
assert clear_coefficients(sqrt(2) - 2) == (sqrt(2), 2)
assert clear_coefficients(4*sqrt(2) - 2) == (sqrt(2), S.Half)
assert clear_coefficients(S(3), x) == (0, x - 3)
assert clear_coefficients(S.Infinity, x) == (S.Infinity, x)
assert clear_coefficients(-S.Pi, x) == (S.Pi, -x)
assert clear_coefficients(2 - S.Pi/3, x) == (pi, -3*x + 6)
def test_nc_simplify():
from sympy.simplify.simplify import nc_simplify
from sympy.matrices.expressions import (MatrixExpr, MatAdd, MatMul,
MatPow, Identity)
from sympy.core import Pow
from functools import reduce
a, b, c, d = symbols('a b c d', commutative = False)
x = Symbol('x')
A = MatrixSymbol("A", x, x)
B = MatrixSymbol("B", x, x)
C = MatrixSymbol("C", x, x)
D = MatrixSymbol("D", x, x)
subst = {a: A, b: B, c: C, d:D}
funcs = {Add: lambda x,y: x+y, Mul: lambda x,y: x*y }
def _to_matrix(expr):
if expr in subst:
return subst[expr]
if isinstance(expr, Pow):
return MatPow(_to_matrix(expr.args[0]), expr.args[1])
elif isinstance(expr, (Add, Mul)):
return reduce(funcs[expr.func],[_to_matrix(a) for a in expr.args])
else:
return expr*Identity(x)
def _check(expr, simplified, deep=True, matrix=True):
assert nc_simplify(expr, deep=deep) == simplified
assert expand(expr) == expand(simplified)
if matrix:
m_simp = _to_matrix(simplified).doit(inv_expand=False)
assert nc_simplify(_to_matrix(expr), deep=deep) == m_simp
_check(a*b*a*b*a*b*c*(a*b)**3*c, ((a*b)**3*c)**2)
_check(a*b*(a*b)**-2*a*b, 1)
_check(a**2*b*a*b*a*b*(a*b)**-1, a*(a*b)**2, matrix=False)
_check(b*a*b**2*a*b**2*a*b**2, b*(a*b**2)**3)
_check(a*b*a**2*b*a**2*b*a**3, (a*b*a)**3*a**2)
_check(a**2*b*a**4*b*a**4*b*a**2, (a**2*b*a**2)**3)
_check(a**3*b*a**4*b*a**4*b*a, a**3*(b*a**4)**3*a**-3)
_check(a*b*a*b + a*b*c*x*a*b*c, (a*b)**2 + x*(a*b*c)**2)
_check(a*b*a*b*c*a*b*a*b*c, ((a*b)**2*c)**2)
_check(b**-1*a**-1*(a*b)**2, a*b)
_check(a**-1*b*c**-1, (c*b**-1*a)**-1)
expr = a**3*b*a**4*b*a**4*b*a**2*b*a**2*(b*a**2)**2*b*a**2*b*a**2
for i in range(10):
expr *= a*b
_check(expr, a**3*(b*a**4)**2*(b*a**2)**6*(a*b)**10)
_check((a*b*a*b)**2, (a*b*a*b)**2, deep=False)
_check(a*b*(c*d)**2, a*b*(c*d)**2)
expr = b**-1*(a**-1*b**-1 - a**-1*c*b**-1)**-1*a**-1
assert nc_simplify(expr) == (1-c)**-1
# commutative expressions should be returned without an error
assert nc_simplify(2*x**2) == 2*x**2
|
62ccc66b91df49264e98666b103b854011b6a96680cdfdfee6010d9b79b3810e
|
from __future__ import (absolute_import, print_function)
import math
from sympy import symbols, exp, S, Poly
from sympy.codegen.rewriting import optimize
from sympy.codegen.approximations import SumApprox, SeriesApprox
def test_SumApprox_trivial():
x = symbols('x')
expr1 = 1 + x
sum_approx = SumApprox(bounds={x: (-1e-20, 1e-20)}, reltol=1e-16)
apx1 = optimize(expr1, [sum_approx])
assert apx1 - 1 == 0
def test_SumApprox_monotone_terms():
x, y, z = symbols('x y z')
expr1 = exp(z)*(x**2 + y**2 + 1)
bnds1 = {x: (0, 1e-3), y: (100, 1000)}
sum_approx_m2 = SumApprox(bounds=bnds1, reltol=1e-2)
sum_approx_m5 = SumApprox(bounds=bnds1, reltol=1e-5)
sum_approx_m11 = SumApprox(bounds=bnds1, reltol=1e-11)
assert (optimize(expr1, [sum_approx_m2])/exp(z) - (y**2)).simplify() == 0
assert (optimize(expr1, [sum_approx_m5])/exp(z) - (y**2 + 1)).simplify() == 0
assert (optimize(expr1, [sum_approx_m11])/exp(z) - (y**2 + 1 + x**2)).simplify() == 0
def test_SeriesApprox_trivial():
x, z = symbols('x z')
for factor in [1, exp(z)]:
x = symbols('x')
expr1 = exp(x)*factor
bnds1 = {x: (-1, 1)}
series_approx_50 = SeriesApprox(bounds=bnds1, reltol=0.50)
series_approx_10 = SeriesApprox(bounds=bnds1, reltol=0.10)
series_approx_05 = SeriesApprox(bounds=bnds1, reltol=0.05)
c = (bnds1[x][1] + bnds1[x][0])/2 # 0.0
f0 = math.exp(c) # 1.0
ref_50 = f0 + x + x**2/2
ref_10 = f0 + x + x**2/2 + x**3/6
ref_05 = f0 + x + x**2/2 + x**3/6 + x**4/24
res_50 = optimize(expr1, [series_approx_50])
res_10 = optimize(expr1, [series_approx_10])
res_05 = optimize(expr1, [series_approx_05])
assert (res_50/factor - ref_50).simplify() == 0
assert (res_10/factor - ref_10).simplify() == 0
assert (res_05/factor - ref_05).simplify() == 0
max_ord3 = SeriesApprox(bounds=bnds1, reltol=0.05, max_order=3)
assert optimize(expr1, [max_ord3]) == expr1
|
641fb9122ea4aa6a77aa464283cd7ac1fad95c9db5ef43e7abc18f0122d2b7d0
|
from __future__ import (absolute_import, print_function)
from sympy import log, exp, Symbol, Pow, sin
from sympy.printing.ccode import ccode
from sympy.codegen.cfunctions import log2, exp2, expm1, log1p
from sympy.codegen.rewriting import (
optimize, log2_opt, exp2_opt, expm1_opt, log1p_opt, optims_c99,
create_expand_pow_optimization
)
from sympy.utilities.pytest import XFAIL
def test_log2_opt():
x = Symbol('x')
expr1 = 7*log(3*x + 5)/(log(2))
opt1 = optimize(expr1, [log2_opt])
assert opt1 == 7*log2(3*x + 5)
assert opt1.rewrite(log) == expr1
expr2 = 3*log(5*x + 7)/(13*log(2))
opt2 = optimize(expr2, [log2_opt])
assert opt2 == 3*log2(5*x + 7)/13
assert opt2.rewrite(log) == expr2
expr3 = log(x)/log(2)
opt3 = optimize(expr3, [log2_opt])
assert opt3 == log2(x)
assert opt3.rewrite(log) == expr3
expr4 = log(x)/log(2) + log(x+1)
opt4 = optimize(expr4, [log2_opt])
assert opt4 == log2(x) + log(2)*log2(x+1)
assert opt4.rewrite(log) == expr4
expr5 = log(17)
opt5 = optimize(expr5, [log2_opt])
assert opt5 == expr5
expr6 = log(x + 3)/log(2)
opt6 = optimize(expr6, [log2_opt])
assert str(opt6) == 'log2(x + 3)'
assert opt6.rewrite(log) == expr6
def test_exp2_opt():
x = Symbol('x')
expr1 = 1 + 2**x
opt1 = optimize(expr1, [exp2_opt])
assert opt1 == 1 + exp2(x)
assert opt1.rewrite(Pow) == expr1
expr2 = 1 + 3**x
assert expr2 == optimize(expr2, [exp2_opt])
def test_expm1_opt():
x = Symbol('x')
expr1 = exp(x) - 1
opt1 = optimize(expr1, [expm1_opt])
assert expm1(x) - opt1 == 0
assert opt1.rewrite(exp) == expr1
expr2 = 3*exp(x) - 3
opt2 = optimize(expr2, [expm1_opt])
assert 3*expm1(x) == opt2
assert opt2.rewrite(exp) == expr2
expr3 = 3*exp(x) - 5
assert expr3 == optimize(expr3, [expm1_opt])
expr4 = 3*exp(x) + log(x) - 3
opt4 = optimize(expr4, [expm1_opt])
assert 3*expm1(x) + log(x) == opt4
assert opt4.rewrite(exp) == expr4
expr5 = 3*exp(2*x) - 3
opt5 = optimize(expr5, [expm1_opt])
assert 3*expm1(2*x) == opt5
assert opt5.rewrite(exp) == expr5
@XFAIL
def test_expm1_two_exp_terms():
x, y = map(Symbol, 'x y'.split())
expr1 = exp(x) + exp(y) - 2
opt1 = optimize(expr1, [expm1_opt])
assert opt1 == expm1(x) + expm1(y)
def test_log1p_opt():
x = Symbol('x')
expr1 = log(x + 1)
opt1 = optimize(expr1, [log1p_opt])
assert log1p(x) - opt1 == 0
assert opt1.rewrite(log) == expr1
expr2 = log(3*x + 3)
opt2 = optimize(expr2, [log1p_opt])
assert log1p(x) + log(3) == opt2
assert (opt2.rewrite(log) - expr2).simplify() == 0
expr3 = log(2*x + 1)
opt3 = optimize(expr3, [log1p_opt])
assert log1p(2*x) - opt3 == 0
assert opt3.rewrite(log) == expr3
expr4 = log(x+3)
opt4 = optimize(expr4, [log1p_opt])
assert str(opt4) == 'log(x + 3)'
def test_optims_c99():
x = Symbol('x')
expr1 = 2**x + log(x)/log(2) + log(x + 1) + exp(x) - 1
opt1 = optimize(expr1, optims_c99).simplify()
assert opt1 == exp2(x) + log2(x) + log1p(x) + expm1(x)
assert opt1.rewrite(exp).rewrite(log).rewrite(Pow) == expr1
expr2 = log(x)/log(2) + log(x + 1)
opt2 = optimize(expr2, optims_c99)
assert opt2 == log2(x) + log1p(x)
assert opt2.rewrite(log) == expr2
expr3 = log(x)/log(2) + log(17*x + 17)
opt3 = optimize(expr3, optims_c99)
delta3 = opt3 - (log2(x) + log(17) + log1p(x))
assert delta3 == 0
assert (opt3.rewrite(log) - expr3).simplify() == 0
expr4 = 2**x + 3*log(5*x + 7)/(13*log(2)) + 11*exp(x) - 11 + log(17*x + 17)
opt4 = optimize(expr4, optims_c99).simplify()
delta4 = opt4 - (exp2(x) + 3*log2(5*x + 7)/13 + 11*expm1(x) + log(17) + log1p(x))
assert delta4 == 0
assert (opt4.rewrite(exp).rewrite(log).rewrite(Pow) - expr4).simplify() == 0
expr5 = 3*exp(2*x) - 3
opt5 = optimize(expr5, optims_c99)
delta5 = opt5 - 3*expm1(2*x)
assert delta5 == 0
assert opt5.rewrite(exp) == expr5
expr6 = exp(2*x) - 3
opt6 = optimize(expr6, optims_c99)
delta6 = opt6 - (exp(2*x) - 3)
assert delta6 == 0
expr7 = log(3*x + 3)
opt7 = optimize(expr7, optims_c99)
delta7 = opt7 - (log(3) + log1p(x))
assert delta7 == 0
assert (opt7.rewrite(log) - expr7).simplify() == 0
expr8 = log(2*x + 3)
opt8 = optimize(expr8, optims_c99)
assert opt8 == expr8
def test_create_expand_pow_optimization():
my_opt = create_expand_pow_optimization(4)
x = Symbol('x')
assert ccode(optimize(x**4, [my_opt])) == 'x*x*x*x'
x5x4 = x**5 + x**4
assert ccode(optimize(x5x4, [my_opt])) == 'pow(x, 5) + x*x*x*x'
sin4x = sin(x)**4
assert ccode(optimize(sin4x, [my_opt])) == 'pow(sin(x), 4)'
assert ccode(optimize((x**(-4)), [my_opt])) == 'pow(x, -4)'
|
fab8d1b6c950333bb43659d14b54cd8afb6c9a9be3a693a5a5c32dc2bbaacb1c
|
from sympy.core import symbols
from sympy.core.compatibility import range
from sympy.crypto.crypto import (cycle_list,
encipher_shift, encipher_affine, encipher_substitution,
check_and_join, encipher_vigenere, decipher_vigenere,
encipher_hill, decipher_hill, encipher_bifid5, encipher_bifid6,
bifid5_square, bifid6_square, bifid5, bifid6, bifid10,
decipher_bifid5, decipher_bifid6, encipher_kid_rsa,
decipher_kid_rsa, kid_rsa_private_key, kid_rsa_public_key,
decipher_rsa, rsa_private_key, rsa_public_key, encipher_rsa,
lfsr_connection_polynomial, lfsr_autocorrelation, lfsr_sequence,
encode_morse, decode_morse, elgamal_private_key, elgamal_public_key,
encipher_elgamal, decipher_elgamal, dh_private_key, dh_public_key,
dh_shared_key, decipher_shift, decipher_affine, encipher_bifid,
decipher_bifid, bifid_square, padded_key, uniq, decipher_gm,
encipher_gm, gm_public_key, gm_private_key, encipher_bg, decipher_bg,
bg_private_key, bg_public_key)
from sympy.matrices import Matrix
from sympy.ntheory import isprime, is_primitive_root
from sympy.polys.domains import FF
from sympy.utilities.pytest import raises, slow, warns_deprecated_sympy
from sympy.utilities.exceptions import SymPyDeprecationWarning
from random import randrange
def test_cycle_list():
assert cycle_list(3, 4) == [3, 0, 1, 2]
assert cycle_list(-1, 4) == [3, 0, 1, 2]
assert cycle_list(1, 4) == [1, 2, 3, 0]
def test_encipher_shift():
assert encipher_shift("ABC", 0) == "ABC"
assert encipher_shift("ABC", 1) == "BCD"
assert encipher_shift("ABC", -1) == "ZAB"
assert decipher_shift("ZAB", -1) == "ABC"
def test_encipher_affine():
assert encipher_affine("ABC", (1, 0)) == "ABC"
assert encipher_affine("ABC", (1, 1)) == "BCD"
assert encipher_affine("ABC", (-1, 0)) == "AZY"
assert encipher_affine("ABC", (-1, 1), symbols="ABCD") == "BAD"
assert encipher_affine("123", (-1, 1), symbols="1234") == "214"
assert encipher_affine("ABC", (3, 16)) == "QTW"
assert decipher_affine("QTW", (3, 16)) == "ABC"
def test_encipher_substitution():
assert encipher_substitution("ABC", "BAC", "ABC") == "BAC"
assert encipher_substitution("123", "1243", "1234") == "124"
def test_check_and_join():
assert check_and_join("abc") == "abc"
assert check_and_join(uniq("aaabc")) == "abc"
assert check_and_join("ab c".split()) == "abc"
assert check_and_join("abc", "a", filter=True) == "a"
raises(ValueError, lambda: check_and_join('ab', 'a'))
def test_encipher_vigenere():
assert encipher_vigenere("ABC", "ABC") == "ACE"
assert encipher_vigenere("ABC", "ABC", symbols="ABCD") == "ACA"
assert encipher_vigenere("ABC", "AB", symbols="ABCD") == "ACC"
assert encipher_vigenere("AB", "ABC", symbols="ABCD") == "AC"
assert encipher_vigenere("A", "ABC", symbols="ABCD") == "A"
def test_decipher_vigenere():
assert decipher_vigenere("ABC", "ABC") == "AAA"
assert decipher_vigenere("ABC", "ABC", symbols="ABCD") == "AAA"
assert decipher_vigenere("ABC", "AB", symbols="ABCD") == "AAC"
assert decipher_vigenere("AB", "ABC", symbols="ABCD") == "AA"
assert decipher_vigenere("A", "ABC", symbols="ABCD") == "A"
def test_encipher_hill():
A = Matrix(2, 2, [1, 2, 3, 5])
assert encipher_hill("ABCD", A) == "CFIV"
A = Matrix(2, 2, [1, 0, 0, 1])
assert encipher_hill("ABCD", A) == "ABCD"
assert encipher_hill("ABCD", A, symbols="ABCD") == "ABCD"
A = Matrix(2, 2, [1, 2, 3, 5])
assert encipher_hill("ABCD", A, symbols="ABCD") == "CBAB"
assert encipher_hill("AB", A, symbols="ABCD") == "CB"
# message length, n, does not need to be a multiple of k;
# it is padded
assert encipher_hill("ABA", A) == "CFGC"
assert encipher_hill("ABA", A, pad="Z") == "CFYV"
def test_decipher_hill():
A = Matrix(2, 2, [1, 2, 3, 5])
assert decipher_hill("CFIV", A) == "ABCD"
A = Matrix(2, 2, [1, 0, 0, 1])
assert decipher_hill("ABCD", A) == "ABCD"
assert decipher_hill("ABCD", A, symbols="ABCD") == "ABCD"
A = Matrix(2, 2, [1, 2, 3, 5])
assert decipher_hill("CBAB", A, symbols="ABCD") == "ABCD"
assert decipher_hill("CB", A, symbols="ABCD") == "AB"
# n does not need to be a multiple of k
assert decipher_hill("CFA", A) == "ABAA"
def test_encipher_bifid5():
assert encipher_bifid5("AB", "AB") == "AB"
assert encipher_bifid5("AB", "CD") == "CO"
assert encipher_bifid5("ab", "c") == "CH"
assert encipher_bifid5("a bc", "b") == "BAC"
def test_bifid5_square():
A = bifid5
f = lambda i, j: symbols(A[5*i + j])
M = Matrix(5, 5, f)
assert bifid5_square("") == M
def test_decipher_bifid5():
assert decipher_bifid5("AB", "AB") == "AB"
assert decipher_bifid5("CO", "CD") == "AB"
assert decipher_bifid5("ch", "c") == "AB"
assert decipher_bifid5("b ac", "b") == "ABC"
def test_encipher_bifid6():
assert encipher_bifid6("AB", "AB") == "AB"
assert encipher_bifid6("AB", "CD") == "CP"
assert encipher_bifid6("ab", "c") == "CI"
assert encipher_bifid6("a bc", "b") == "BAC"
def test_decipher_bifid6():
assert decipher_bifid6("AB", "AB") == "AB"
assert decipher_bifid6("CP", "CD") == "AB"
assert decipher_bifid6("ci", "c") == "AB"
assert decipher_bifid6("b ac", "b") == "ABC"
def test_bifid6_square():
A = bifid6
f = lambda i, j: symbols(A[6*i + j])
M = Matrix(6, 6, f)
assert bifid6_square("") == M
def test_rsa_public_key():
assert rsa_public_key(2, 3, 1) == (6, 1)
assert rsa_public_key(5, 3, 3) == (15, 3)
assert rsa_public_key(8, 8, 8) is False
raises(SymPyDeprecationWarning, lambda: rsa_public_key(2, 2, 1))
with warns_deprecated_sympy():
assert rsa_public_key(2, 2, 1) == (4, 1)
def test_rsa_private_key():
assert rsa_private_key(2, 3, 1) == (6, 1)
assert rsa_private_key(5, 3, 3) == (15, 3)
assert rsa_private_key(23,29,5) == (667,493)
assert rsa_private_key(8, 8, 8) is False
raises(SymPyDeprecationWarning, lambda: rsa_private_key(2, 2, 1))
with warns_deprecated_sympy():
assert rsa_private_key(2, 2, 1) == (4, 1)
def test_rsa_large_key():
# Sample from
# http://www.herongyang.com/Cryptography/JCE-Public-Key-RSA-Private-Public-Key-Pair-Sample.html
p = int('101565610013301240713207239558950144682174355406589305284428666'\
'903702505233009')
q = int('894687191887545488935455605955948413812376003053143521429242133'\
'12069293984003')
e = int('65537')
d = int('893650581832704239530398858744759129594796235440844479456143566'\
'6999402846577625762582824202269399672579058991442587406384754958587'\
'400493169361356902030209')
assert rsa_public_key(p, q, e) == (p*q, e)
assert rsa_private_key(p, q, e) == (p*q, d)
def test_encipher_rsa():
puk = rsa_public_key(2, 3, 1)
assert encipher_rsa(2, puk) == 2
puk = rsa_public_key(5, 3, 3)
assert encipher_rsa(2, puk) == 8
with warns_deprecated_sympy():
puk = rsa_public_key(2, 2, 1)
assert encipher_rsa(2, puk) == 2
def test_decipher_rsa():
prk = rsa_private_key(2, 3, 1)
assert decipher_rsa(2, prk) == 2
prk = rsa_private_key(5, 3, 3)
assert decipher_rsa(8, prk) == 2
with warns_deprecated_sympy():
prk = rsa_private_key(2, 2, 1)
assert decipher_rsa(2, prk) == 2
def test_kid_rsa_public_key():
assert kid_rsa_public_key(1, 2, 1, 1) == (5, 2)
assert kid_rsa_public_key(1, 2, 2, 1) == (8, 3)
assert kid_rsa_public_key(1, 2, 1, 2) == (7, 2)
def test_kid_rsa_private_key():
assert kid_rsa_private_key(1, 2, 1, 1) == (5, 3)
assert kid_rsa_private_key(1, 2, 2, 1) == (8, 3)
assert kid_rsa_private_key(1, 2, 1, 2) == (7, 4)
def test_encipher_kid_rsa():
assert encipher_kid_rsa(1, (5, 2)) == 2
assert encipher_kid_rsa(1, (8, 3)) == 3
assert encipher_kid_rsa(1, (7, 2)) == 2
def test_decipher_kid_rsa():
assert decipher_kid_rsa(2, (5, 3)) == 1
assert decipher_kid_rsa(3, (8, 3)) == 1
assert decipher_kid_rsa(2, (7, 4)) == 1
def test_encode_morse():
assert encode_morse('ABC') == '.-|-...|-.-.'
assert encode_morse('SMS ') == '...|--|...||'
assert encode_morse('SMS\n') == '...|--|...||'
assert encode_morse('') == ''
assert encode_morse(' ') == '||'
assert encode_morse(' ', sep='`') == '``'
assert encode_morse(' ', sep='``') == '````'
assert encode_morse('!@#$%^&*()_+') == '-.-.--|.--.-.|...-..-|-.--.|-.--.-|..--.-|.-.-.'
def test_decode_morse():
assert decode_morse('-.-|.|-.--') == 'KEY'
assert decode_morse('.-.|..-|-.||') == 'RUN'
raises(KeyError, lambda: decode_morse('.....----'))
def test_lfsr_sequence():
raises(TypeError, lambda: lfsr_sequence(1, [1], 1))
raises(TypeError, lambda: lfsr_sequence([1], 1, 1))
F = FF(2)
assert lfsr_sequence([F(1)], [F(1)], 2) == [F(1), F(1)]
assert lfsr_sequence([F(0)], [F(1)], 2) == [F(1), F(0)]
F = FF(3)
assert lfsr_sequence([F(1)], [F(1)], 2) == [F(1), F(1)]
assert lfsr_sequence([F(0)], [F(2)], 2) == [F(2), F(0)]
assert lfsr_sequence([F(1)], [F(2)], 2) == [F(2), F(2)]
def test_lfsr_autocorrelation():
raises(TypeError, lambda: lfsr_autocorrelation(1, 2, 3))
F = FF(2)
s = lfsr_sequence([F(1), F(0)], [F(0), F(1)], 5)
assert lfsr_autocorrelation(s, 2, 0) == 1
assert lfsr_autocorrelation(s, 2, 1) == -1
def test_lfsr_connection_polynomial():
F = FF(2)
x = symbols("x")
s = lfsr_sequence([F(1), F(0)], [F(0), F(1)], 5)
assert lfsr_connection_polynomial(s) == x**2 + 1
s = lfsr_sequence([F(1), F(1)], [F(0), F(1)], 5)
assert lfsr_connection_polynomial(s) == x**2 + x + 1
def test_elgamal_private_key():
a, b, _ = elgamal_private_key(digit=100)
assert isprime(a)
assert is_primitive_root(b, a)
assert len(bin(a)) >= 102
def test_elgamal():
dk = elgamal_private_key(5)
ek = elgamal_public_key(dk)
P = ek[0]
assert P - 1 == decipher_elgamal(encipher_elgamal(P - 1, ek), dk)
raises(ValueError, lambda: encipher_elgamal(P, dk))
raises(ValueError, lambda: encipher_elgamal(-1, dk))
def test_dh_private_key():
p, g, _ = dh_private_key(digit = 100)
assert isprime(p)
assert is_primitive_root(g, p)
assert len(bin(p)) >= 102
def test_dh_public_key():
p1, g1, a = dh_private_key(digit = 100)
p2, g2, ga = dh_public_key((p1, g1, a))
assert p1 == p2
assert g1 == g2
assert ga == pow(g1, a, p1)
def test_dh_shared_key():
prk = dh_private_key(digit = 100)
p, _, ga = dh_public_key(prk)
b = randrange(2, p)
sk = dh_shared_key((p, _, ga), b)
assert sk == pow(ga, b, p)
raises(ValueError, lambda: dh_shared_key((1031, 14, 565), 2000))
def test_padded_key():
assert padded_key('b', 'ab') == 'ba'
raises(ValueError, lambda: padded_key('ab', 'ace'))
raises(ValueError, lambda: padded_key('ab', 'abba'))
def test_bifid():
raises(ValueError, lambda: encipher_bifid('abc', 'b', 'abcde'))
assert encipher_bifid('abc', 'b', 'abcd') == 'bdb'
raises(ValueError, lambda: decipher_bifid('bdb', 'b', 'abcde'))
assert encipher_bifid('bdb', 'b', 'abcd') == 'abc'
raises(ValueError, lambda: bifid_square('abcde'))
assert bifid5_square("B") == \
bifid5_square('BACDEFGHIKLMNOPQRSTUVWXYZ')
assert bifid6_square('B0') == \
bifid6_square('B0ACDEFGHIJKLMNOPQRSTUVWXYZ123456789')
def test_encipher_decipher_gm():
ps = [131, 137, 139, 149, 151, 157, 163, 167,
173, 179, 181, 191, 193, 197, 199]
qs = [89, 97, 101, 103, 107, 109, 113, 127,
131, 137, 139, 149, 151, 157, 47]
messages = [
0, 32855, 34303, 14805, 1280, 75859, 38368,
724, 60356, 51675, 76697, 61854, 18661,
]
for p, q in zip(ps, qs):
pri = gm_private_key(p, q)
for msg in messages:
pub = gm_public_key(p, q)
enc = encipher_gm(msg, pub)
dec = decipher_gm(enc, pri)
assert dec == msg
def test_gm_private_key():
raises(ValueError, lambda: gm_public_key(13, 15))
raises(ValueError, lambda: gm_public_key(0, 0))
raises(ValueError, lambda: gm_public_key(0, 5))
assert 17, 19 == gm_public_key(17, 19)
def test_gm_public_key():
assert 323 == gm_public_key(17, 19)[1]
assert 15 == gm_public_key(3, 5)[1]
raises(ValueError, lambda: gm_public_key(15, 19))
def test_encipher_decipher_bg():
ps = [67, 7, 71, 103, 11, 43, 107, 47,
79, 19, 83, 23, 59, 127, 31]
qs = qs = [7, 71, 103, 11, 43, 107, 47,
79, 19, 83, 23, 59, 127, 31, 67]
messages = [
0, 328, 343, 148, 1280, 758, 383,
724, 603, 516, 766, 618, 186,
]
for p, q in zip(ps, qs):
pri = bg_private_key(p, q)
for msg in messages:
pub = bg_public_key(p, q)
enc = encipher_bg(msg, pub)
dec = decipher_bg(enc, pri)
assert dec == msg
def test_bg_private_key():
raises(ValueError, lambda: bg_private_key(8, 16))
raises(ValueError, lambda: bg_private_key(8, 8))
raises(ValueError, lambda: bg_private_key(13, 17))
assert 23, 31 == bg_private_key(23, 31)
def test_bg_public_key():
assert 5293 == bg_public_key(67, 79)
assert 713 == bg_public_key(23, 31)
raises(ValueError, lambda: bg_private_key(13, 17))
|
521f0165b68898ca2ccc98079180dc147e8eb827f0083f176f0145f419ac1c80
|
"""
Handlers for keys related to number theory: prime, even, odd, etc.
"""
from __future__ import print_function, division
from sympy.assumptions import Q, ask
from sympy.assumptions.handlers import CommonHandler
from sympy.ntheory import isprime
from sympy.core import S, Float
class AskPrimeHandler(CommonHandler):
"""
Handler for key 'prime'
Test that an expression represents a prime number. When the
expression is an exact number, the result (when True) is subject to
the limitations of isprime() which is used to return the result.
"""
@staticmethod
def Expr(expr, assumptions):
return expr.is_prime
@staticmethod
def _number(expr, assumptions):
# helper method
exact = not expr.atoms(Float)
try:
i = int(expr.round())
if (expr - i).equals(0) is False:
raise TypeError
except TypeError:
return False
if exact:
return isprime(i)
# when not exact, we won't give a True or False
# since the number represents an approximate value
@staticmethod
def Basic(expr, assumptions):
if expr.is_number:
return AskPrimeHandler._number(expr, assumptions)
@staticmethod
def Mul(expr, assumptions):
if expr.is_number:
return AskPrimeHandler._number(expr, assumptions)
for arg in expr.args:
if not ask(Q.integer(arg), assumptions):
return None
for arg in expr.args:
if arg.is_number and arg.is_composite:
return False
@staticmethod
def Pow(expr, assumptions):
"""
Integer**Integer -> !Prime
"""
if expr.is_number:
return AskPrimeHandler._number(expr, assumptions)
if ask(Q.integer(expr.exp), assumptions) and \
ask(Q.integer(expr.base), assumptions):
return False
@staticmethod
def Integer(expr, assumptions):
return isprime(expr)
Rational, Infinity, NegativeInfinity, ImaginaryUnit = [staticmethod(CommonHandler.AlwaysFalse)]*4
@staticmethod
def Float(expr, assumptions):
return AskPrimeHandler._number(expr, assumptions)
@staticmethod
def NumberSymbol(expr, assumptions):
return AskPrimeHandler._number(expr, assumptions)
class AskCompositeHandler(CommonHandler):
@staticmethod
def Expr(expr, assumptions):
return expr.is_composite
@staticmethod
def Basic(expr, assumptions):
_positive = ask(Q.positive(expr), assumptions)
if _positive:
_integer = ask(Q.integer(expr), assumptions)
if _integer:
_prime = ask(Q.prime(expr), assumptions)
if _prime is None:
return
# Positive integer which is not prime is not
# necessarily composite
if expr.equals(1):
return False
return not _prime
else:
return _integer
else:
return _positive
class AskEvenHandler(CommonHandler):
@staticmethod
def Expr(expr, assumptions):
return expr.is_even
@staticmethod
def _number(expr, assumptions):
# helper method
try:
i = int(expr.round())
if not (expr - i).equals(0):
raise TypeError
except TypeError:
return False
if isinstance(expr, (float, Float)):
return False
return i % 2 == 0
@staticmethod
def Basic(expr, assumptions):
if expr.is_number:
return AskEvenHandler._number(expr, assumptions)
@staticmethod
def Mul(expr, assumptions):
"""
Even * Integer -> Even
Even * Odd -> Even
Integer * Odd -> ?
Odd * Odd -> Odd
Even * Even -> Even
Integer * Integer -> Even if Integer + Integer = Odd
otherwise -> ?
"""
if expr.is_number:
return AskEvenHandler._number(expr, assumptions)
even, odd, irrational, acc = False, 0, False, 1
for arg in expr.args:
# check for all integers and at least one even
if ask(Q.integer(arg), assumptions):
if ask(Q.even(arg), assumptions):
even = True
elif ask(Q.odd(arg), assumptions):
odd += 1
elif not even and acc != 1:
if ask(Q.odd(acc + arg), assumptions):
even = True
elif ask(Q.irrational(arg), assumptions):
# one irrational makes the result False
# two makes it undefined
if irrational:
break
irrational = True
else:
break
acc = arg
else:
if irrational:
return False
if even:
return True
if odd == len(expr.args):
return False
@staticmethod
def Add(expr, assumptions):
"""
Even + Odd -> Odd
Even + Even -> Even
Odd + Odd -> Even
"""
if expr.is_number:
return AskEvenHandler._number(expr, assumptions)
_result = True
for arg in expr.args:
if ask(Q.even(arg), assumptions):
pass
elif ask(Q.odd(arg), assumptions):
_result = not _result
else:
break
else:
return _result
@staticmethod
def Pow(expr, assumptions):
if expr.is_number:
return AskEvenHandler._number(expr, assumptions)
if ask(Q.integer(expr.exp), assumptions):
if ask(Q.positive(expr.exp), assumptions):
return ask(Q.even(expr.base), assumptions)
elif ask(~Q.negative(expr.exp) & Q.odd(expr.base), assumptions):
return False
elif expr.base is S.NegativeOne:
return False
@staticmethod
def Integer(expr, assumptions):
return not bool(expr.p & 1)
Rational, Infinity, NegativeInfinity, ImaginaryUnit = [staticmethod(CommonHandler.AlwaysFalse)]*4
@staticmethod
def NumberSymbol(expr, assumptions):
return AskEvenHandler._number(expr, assumptions)
@staticmethod
def Abs(expr, assumptions):
if ask(Q.real(expr.args[0]), assumptions):
return ask(Q.even(expr.args[0]), assumptions)
@staticmethod
def re(expr, assumptions):
if ask(Q.real(expr.args[0]), assumptions):
return ask(Q.even(expr.args[0]), assumptions)
@staticmethod
def im(expr, assumptions):
if ask(Q.real(expr.args[0]), assumptions):
return True
class AskOddHandler(CommonHandler):
"""
Handler for key 'odd'
Test that an expression represents an odd number
"""
@staticmethod
def Expr(expr, assumptions):
return expr.is_odd
@staticmethod
def Basic(expr, assumptions):
_integer = ask(Q.integer(expr), assumptions)
if _integer:
_even = ask(Q.even(expr), assumptions)
if _even is None:
return None
return not _even
return _integer
|
3c02b44a25c9eb560ea8198027bfde31b436ab8599b58c59dabfbf4164d2d828
|
from sympy import (Abs, exp, Expr, I, pi, Q, Rational, refine, S, sqrt,
atan, atan2, nan, Symbol)
from sympy.abc import x, y, z
from sympy.core.relational import Eq, Ne
from sympy.functions.elementary.piecewise import Piecewise
from sympy.utilities.pytest import slow
def test_Abs():
assert refine(Abs(x), Q.positive(x)) == x
assert refine(1 + Abs(x), Q.positive(x)) == 1 + x
assert refine(Abs(x), Q.negative(x)) == -x
assert refine(1 + Abs(x), Q.negative(x)) == 1 - x
assert refine(Abs(x**2)) != x**2
assert refine(Abs(x**2), Q.real(x)) == x**2
def test_pow1():
assert refine((-1)**x, Q.even(x)) == 1
assert refine((-1)**x, Q.odd(x)) == -1
assert refine((-2)**x, Q.even(x)) == 2**x
# nested powers
assert refine(sqrt(x**2)) != Abs(x)
assert refine(sqrt(x**2), Q.complex(x)) != Abs(x)
assert refine(sqrt(x**2), Q.real(x)) == Abs(x)
assert refine(sqrt(x**2), Q.positive(x)) == x
assert refine((x**3)**(S(1)/3)) != x
assert refine((x**3)**(S(1)/3), Q.real(x)) != x
assert refine((x**3)**(S(1)/3), Q.positive(x)) == x
assert refine(sqrt(1/x), Q.real(x)) != 1/sqrt(x)
assert refine(sqrt(1/x), Q.positive(x)) == 1/sqrt(x)
@slow
def test_pow2():
# powers of (-1)
assert refine((-1)**(x + y), Q.even(x)) == (-1)**y
assert refine((-1)**(x + y + z), Q.odd(x) & Q.odd(z)) == (-1)**y
assert refine((-1)**(x + y + 1), Q.odd(x)) == (-1)**y
assert refine((-1)**(x + y + 2), Q.odd(x)) == (-1)**(y + 1)
assert refine((-1)**(x + 3)) == (-1)**(x + 1)
@slow
def test_pow3():
# continuation
assert refine((-1)**((-1)**x/2 - S.Half), Q.integer(x)) == (-1)**x
assert refine((-1)**((-1)**x/2 + S.Half), Q.integer(x)) == (-1)**(x + 1)
assert refine((-1)**((-1)**x/2 + 5*S.Half), Q.integer(x)) == (-1)**(x + 1)
@slow
def test_pow4():
assert refine((-1)**((-1)**x/2 - 7*S.Half), Q.integer(x)) == (-1)**(x + 1)
assert refine((-1)**((-1)**x/2 - 9*S.Half), Q.integer(x)) == (-1)**x
# powers of Abs
assert refine(Abs(x)**2, Q.real(x)) == x**2
assert refine(Abs(x)**3, Q.real(x)) == Abs(x)**3
assert refine(Abs(x)**2) == Abs(x)**2
def test_exp():
x = Symbol('x', integer=True)
assert refine(exp(pi*I*2*x)) == 1
assert refine(exp(pi*I*2*(x + Rational(1, 2)))) == -1
assert refine(exp(pi*I*2*(x + Rational(1, 4)))) == I
assert refine(exp(pi*I*2*(x + Rational(3, 4)))) == -I
def test_Relational():
assert not refine(x < 0, ~Q.is_true(x < 0))
assert refine(x < 0, Q.is_true(x < 0))
assert refine(x < 0, Q.is_true(0 > x)) == True
assert refine(x < 0, Q.is_true(y < 0)) == (x < 0)
assert not refine(x <= 0, ~Q.is_true(x <= 0))
assert refine(x <= 0, Q.is_true(x <= 0))
assert refine(x <= 0, Q.is_true(0 >= x)) == True
assert refine(x <= 0, Q.is_true(y <= 0)) == (x <= 0)
assert not refine(x > 0, ~Q.is_true(x > 0))
assert refine(x > 0, Q.is_true(x > 0))
assert refine(x > 0, Q.is_true(0 < x)) == True
assert refine(x > 0, Q.is_true(y > 0)) == (x > 0)
assert not refine(x >= 0, ~Q.is_true(x >= 0))
assert refine(x >= 0, Q.is_true(x >= 0))
assert refine(x >= 0, Q.is_true(0 <= x)) == True
assert refine(x >= 0, Q.is_true(y >= 0)) == (x >= 0)
assert not refine(Eq(x, 0), ~Q.is_true(Eq(x, 0)))
assert refine(Eq(x, 0), Q.is_true(Eq(x, 0)))
assert refine(Eq(x, 0), Q.is_true(Eq(0, x))) == True
assert refine(Eq(x, 0), Q.is_true(Eq(y, 0))) == Eq(x, 0)
assert not refine(Ne(x, 0), ~Q.is_true(Ne(x, 0)))
assert refine(Ne(x, 0), Q.is_true(Ne(0, x))) == True
assert refine(Ne(x, 0), Q.is_true(Ne(x, 0)))
assert refine(Ne(x, 0), Q.is_true(Ne(y, 0))) == (Ne(x, 0))
def test_Piecewise():
assert refine(Piecewise((1, x < 0), (3, True)), Q.is_true(x < 0)) == 1
assert refine(Piecewise((1, x < 0), (3, True)), ~Q.is_true(x < 0)) == 3
assert refine(Piecewise((1, x < 0), (3, True)), Q.is_true(y < 0)) == \
Piecewise((1, x < 0), (3, True))
assert refine(Piecewise((1, x > 0), (3, True)), Q.is_true(x > 0)) == 1
assert refine(Piecewise((1, x > 0), (3, True)), ~Q.is_true(x > 0)) == 3
assert refine(Piecewise((1, x > 0), (3, True)), Q.is_true(y > 0)) == \
Piecewise((1, x > 0), (3, True))
assert refine(Piecewise((1, x <= 0), (3, True)), Q.is_true(x <= 0)) == 1
assert refine(Piecewise((1, x <= 0), (3, True)), ~Q.is_true(x <= 0)) == 3
assert refine(Piecewise((1, x <= 0), (3, True)), Q.is_true(y <= 0)) == \
Piecewise((1, x <= 0), (3, True))
assert refine(Piecewise((1, x >= 0), (3, True)), Q.is_true(x >= 0)) == 1
assert refine(Piecewise((1, x >= 0), (3, True)), ~Q.is_true(x >= 0)) == 3
assert refine(Piecewise((1, x >= 0), (3, True)), Q.is_true(y >= 0)) == \
Piecewise((1, x >= 0), (3, True))
assert refine(Piecewise((1, Eq(x, 0)), (3, True)), Q.is_true(Eq(x, 0)))\
== 1
assert refine(Piecewise((1, Eq(x, 0)), (3, True)), Q.is_true(Eq(0, x)))\
== 1
assert refine(Piecewise((1, Eq(x, 0)), (3, True)), ~Q.is_true(Eq(x, 0)))\
== 3
assert refine(Piecewise((1, Eq(x, 0)), (3, True)), ~Q.is_true(Eq(0, x)))\
== 3
assert refine(Piecewise((1, Eq(x, 0)), (3, True)), Q.is_true(Eq(y, 0)))\
== Piecewise((1, Eq(x, 0)), (3, True))
assert refine(Piecewise((1, Ne(x, 0)), (3, True)), Q.is_true(Ne(x, 0)))\
== 1
assert refine(Piecewise((1, Ne(x, 0)), (3, True)), ~Q.is_true(Ne(x, 0)))\
== 3
assert refine(Piecewise((1, Ne(x, 0)), (3, True)), Q.is_true(Ne(y, 0)))\
== Piecewise((1, Ne(x, 0)), (3, True))
def test_atan2():
assert refine(atan2(y, x), Q.real(y) & Q.positive(x)) == atan(y/x)
assert refine(atan2(y, x), Q.negative(y) & Q.positive(x)) == atan(y/x)
assert refine(atan2(y, x), Q.negative(y) & Q.negative(x)) == atan(y/x) - pi
assert refine(atan2(y, x), Q.positive(y) & Q.negative(x)) == atan(y/x) + pi
assert refine(atan2(y, x), Q.zero(y) & Q.negative(x)) == pi
assert refine(atan2(y, x), Q.positive(y) & Q.zero(x)) == pi/2
assert refine(atan2(y, x), Q.negative(y) & Q.zero(x)) == -pi/2
assert refine(atan2(y, x), Q.zero(y) & Q.zero(x)) == nan
def test_func_args():
class MyClass(Expr):
# A class with nontrivial .func
def __init__(self, *args):
self.my_member = ""
@property
def func(self):
def my_func(*args):
obj = MyClass(*args)
obj.my_member = self.my_member
return obj
return my_func
x = MyClass()
x.my_member = "A very important value"
assert x.my_member == refine(x).my_member
def test_eval_refine():
from sympy.core.expr import Expr
class MockExpr(Expr):
def _eval_refine(self, assumptions):
return True
mock_obj = MockExpr()
assert refine(mock_obj)
def test_refine_issue_12724():
expr1 = refine(Abs(x * y), Q.positive(x))
expr2 = refine(Abs(x * y * z), Q.positive(x))
assert expr1 == x * Abs(y)
assert expr2 == x * Abs(y * z)
y1 = Symbol('y1', real = True)
expr3 = refine(Abs(x * y1**2 * z), Q.positive(x))
assert expr3 == x * y1**2 * Abs(z)
|
da6809e05c25e5c44cf93c013de2737b6ec34dfc5665cbc51d09fa6993396506
|
from sympy.assumptions.satask import satask
from sympy import symbols, Q, assuming, Implies, MatrixSymbol, I, pi, Rational
from sympy.utilities.pytest import raises, XFAIL, slow
x, y, z = symbols('x y z')
def test_satask():
# No relevant facts
assert satask(Q.real(x), Q.real(x)) is True
assert satask(Q.real(x), ~Q.real(x)) is False
assert satask(Q.real(x)) is None
assert satask(Q.real(x), Q.positive(x)) is True
assert satask(Q.positive(x), Q.real(x)) is None
assert satask(Q.real(x), ~Q.positive(x)) is None
assert satask(Q.positive(x), ~Q.real(x)) is False
raises(ValueError, lambda: satask(Q.real(x), Q.real(x) & ~Q.real(x)))
with assuming(Q.positive(x)):
assert satask(Q.real(x)) is True
assert satask(~Q.positive(x)) is False
raises(ValueError, lambda: satask(Q.real(x), ~Q.positive(x)))
assert satask(Q.zero(x), Q.nonzero(x)) is False
assert satask(Q.positive(x), Q.zero(x)) is False
assert satask(Q.real(x), Q.zero(x)) is True
assert satask(Q.zero(x), Q.zero(x*y)) is None
assert satask(Q.zero(x*y), Q.zero(x))
def test_zero():
"""
Everything in this test doesn't work with the ask handlers, and most
things would be very difficult or impossible to make work under that
model.
"""
assert satask(Q.zero(x) | Q.zero(y), Q.zero(x*y)) is True
assert satask(Q.zero(x*y), Q.zero(x) | Q.zero(y)) is True
assert satask(Implies(Q.zero(x), Q.zero(x*y))) is True
# This one in particular requires computing the fixed-point of the
# relevant facts, because going from Q.nonzero(x*y) -> ~Q.zero(x*y) and
# Q.zero(x*y) -> Equivalent(Q.zero(x*y), Q.zero(x) | Q.zero(y)) takes two
# steps.
assert satask(Q.zero(x) | Q.zero(y), Q.nonzero(x*y)) is False
assert satask(Q.zero(x), Q.zero(x**2)) is True
def test_zero_positive():
assert satask(Q.zero(x + y), Q.positive(x) & Q.positive(y)) is False
assert satask(Q.positive(x) & Q.positive(y), Q.zero(x + y)) is False
assert satask(Q.nonzero(x + y), Q.positive(x) & Q.positive(y)) is True
assert satask(Q.positive(x) & Q.positive(y), Q.nonzero(x + y)) is None
# This one requires several levels of forward chaining
assert satask(Q.zero(x*(x + y)), Q.positive(x) & Q.positive(y)) is False
assert satask(Q.positive(pi*x*y + 1), Q.positive(x) & Q.positive(y)) is True
assert satask(Q.positive(pi*x*y - 5), Q.positive(x) & Q.positive(y)) is None
def test_zero_pow():
assert satask(Q.zero(x**y), Q.zero(x) & Q.positive(y)) is True
assert satask(Q.zero(x**y), Q.nonzero(x) & Q.zero(y)) is False
assert satask(Q.zero(x), Q.zero(x**y)) is True
assert satask(Q.zero(x**y), Q.zero(x)) is None
@XFAIL
# Requires correct Q.square calculation first
def test_invertible():
A = MatrixSymbol('A', 5, 5)
B = MatrixSymbol('B', 5, 5)
assert satask(Q.invertible(A*B), Q.invertible(A) & Q.invertible(B)) is True
assert satask(Q.invertible(A), Q.invertible(A*B))
assert satask(Q.invertible(A) & Q.invertible(B), Q.invertible(A*B))
def test_prime():
assert satask(Q.prime(5)) is True
assert satask(Q.prime(6)) is False
assert satask(Q.prime(-5)) is False
assert satask(Q.prime(x*y), Q.integer(x) & Q.integer(y)) is None
assert satask(Q.prime(x*y), Q.prime(x) & Q.prime(y)) is False
def test_old_assump():
assert satask(Q.positive(1)) is True
assert satask(Q.positive(-1)) is False
assert satask(Q.positive(0)) is False
assert satask(Q.positive(I)) is False
assert satask(Q.positive(pi)) is True
assert satask(Q.negative(1)) is False
assert satask(Q.negative(-1)) is True
assert satask(Q.negative(0)) is False
assert satask(Q.negative(I)) is False
assert satask(Q.negative(pi)) is False
assert satask(Q.zero(1)) is False
assert satask(Q.zero(-1)) is False
assert satask(Q.zero(0)) is True
assert satask(Q.zero(I)) is False
assert satask(Q.zero(pi)) is False
assert satask(Q.nonzero(1)) is True
assert satask(Q.nonzero(-1)) is True
assert satask(Q.nonzero(0)) is False
assert satask(Q.nonzero(I)) is False
assert satask(Q.nonzero(pi)) is True
assert satask(Q.nonpositive(1)) is False
assert satask(Q.nonpositive(-1)) is True
assert satask(Q.nonpositive(0)) is True
assert satask(Q.nonpositive(I)) is False
assert satask(Q.nonpositive(pi)) is False
assert satask(Q.nonnegative(1)) is True
assert satask(Q.nonnegative(-1)) is False
assert satask(Q.nonnegative(0)) is True
assert satask(Q.nonnegative(I)) is False
assert satask(Q.nonnegative(pi)) is True
def test_rational_irrational():
assert satask(Q.irrational(2)) is False
assert satask(Q.rational(2)) is True
assert satask(Q.irrational(pi)) is True
assert satask(Q.rational(pi)) is False
assert satask(Q.irrational(I)) is False
assert satask(Q.rational(I)) is False
assert satask(Q.irrational(x*y*z), Q.irrational(x) & Q.irrational(y) &
Q.rational(z)) is None
assert satask(Q.irrational(x*y*z), Q.irrational(x) & Q.rational(y) &
Q.rational(z)) is True
assert satask(Q.irrational(pi*x*y), Q.rational(x) & Q.rational(y)) is True
assert satask(Q.irrational(x + y + z), Q.irrational(x) & Q.irrational(y) &
Q.rational(z)) is None
assert satask(Q.irrational(x + y + z), Q.irrational(x) & Q.rational(y) &
Q.rational(z)) is True
assert satask(Q.irrational(pi + x + y), Q.rational(x) & Q.rational(y)) is True
assert satask(Q.irrational(x*y*z), Q.rational(x) & Q.rational(y) &
Q.rational(z)) is False
assert satask(Q.rational(x*y*z), Q.rational(x) & Q.rational(y) &
Q.rational(z)) is True
assert satask(Q.irrational(x + y + z), Q.rational(x) & Q.rational(y) &
Q.rational(z)) is False
assert satask(Q.rational(x + y + z), Q.rational(x) & Q.rational(y) &
Q.rational(z)) is True
def test_even_satask():
assert satask(Q.even(2)) is True
assert satask(Q.even(3)) is False
assert satask(Q.even(x*y), Q.even(x) & Q.odd(y)) is True
assert satask(Q.even(x*y), Q.even(x) & Q.integer(y)) is True
assert satask(Q.even(x*y), Q.even(x) & Q.even(y)) is True
assert satask(Q.even(x*y), Q.odd(x) & Q.odd(y)) is False
assert satask(Q.even(x*y), Q.even(x)) is None
assert satask(Q.even(x*y), Q.odd(x) & Q.integer(y)) is None
assert satask(Q.even(x*y), Q.odd(x) & Q.odd(y)) is False
assert satask(Q.even(abs(x)), Q.even(x)) is True
assert satask(Q.even(abs(x)), Q.odd(x)) is False
assert satask(Q.even(x), Q.even(abs(x))) is None # x could be complex
def test_odd_satask():
assert satask(Q.odd(2)) is False
assert satask(Q.odd(3)) is True
assert satask(Q.odd(x*y), Q.even(x) & Q.odd(y)) is False
assert satask(Q.odd(x*y), Q.even(x) & Q.integer(y)) is False
assert satask(Q.odd(x*y), Q.even(x) & Q.even(y)) is False
assert satask(Q.odd(x*y), Q.odd(x) & Q.odd(y)) is True
assert satask(Q.odd(x*y), Q.even(x)) is None
assert satask(Q.odd(x*y), Q.odd(x) & Q.integer(y)) is None
assert satask(Q.odd(x*y), Q.odd(x) & Q.odd(y)) is True
assert satask(Q.odd(abs(x)), Q.even(x)) is False
assert satask(Q.odd(abs(x)), Q.odd(x)) is True
assert satask(Q.odd(x), Q.odd(abs(x))) is None # x could be complex
def test_integer():
assert satask(Q.integer(1)) is True
assert satask(Q.integer(Rational(1, 2))) is False
assert satask(Q.integer(x + y), Q.integer(x) & Q.integer(y)) is True
assert satask(Q.integer(x + y), Q.integer(x)) is None
assert satask(Q.integer(x + y), Q.integer(x) & ~Q.integer(y)) is False
assert satask(Q.integer(x + y + z), Q.integer(x) & Q.integer(y) &
~Q.integer(z)) is False
assert satask(Q.integer(x + y + z), Q.integer(x) & ~Q.integer(y) &
~Q.integer(z)) is None
assert satask(Q.integer(x + y + z), Q.integer(x) & ~Q.integer(y)) is None
assert satask(Q.integer(x + y), Q.integer(x) & Q.irrational(y)) is False
assert satask(Q.integer(x*y), Q.integer(x) & Q.integer(y)) is True
assert satask(Q.integer(x*y), Q.integer(x)) is None
assert satask(Q.integer(x*y), Q.integer(x) & ~Q.integer(y)) is None
assert satask(Q.integer(x*y), Q.integer(x) & ~Q.rational(y)) is False
assert satask(Q.integer(x*y*z), Q.integer(x) & Q.integer(y) &
~Q.rational(z)) is False
assert satask(Q.integer(x*y*z), Q.integer(x) & ~Q.rational(y) &
~Q.rational(z)) is None
assert satask(Q.integer(x*y*z), Q.integer(x) & ~Q.rational(y)) is None
assert satask(Q.integer(x*y), Q.integer(x) & Q.irrational(y)) is False
def test_abs():
assert satask(Q.nonnegative(abs(x))) is True
assert satask(Q.positive(abs(x)), ~Q.zero(x)) is True
assert satask(Q.zero(x), ~Q.zero(abs(x))) is False
assert satask(Q.zero(x), Q.zero(abs(x))) is True
assert satask(Q.nonzero(x), ~Q.zero(abs(x))) is None # x could be complex
assert satask(Q.zero(abs(x)), Q.zero(x)) is True
def test_imaginary():
assert satask(Q.imaginary(2*I)) is True
assert satask(Q.imaginary(x*y), Q.imaginary(x)) is None
assert satask(Q.imaginary(x*y), Q.imaginary(x) & Q.real(y)) is True
assert satask(Q.imaginary(x), Q.real(x)) is False
assert satask(Q.imaginary(1)) is False
assert satask(Q.imaginary(x*y), Q.real(x) & Q.real(y)) is False
assert satask(Q.imaginary(x + y), Q.real(x) & Q.real(y)) is False
def test_real():
assert satask(Q.real(x*y), Q.real(x) & Q.real(y)) is True
assert satask(Q.real(x + y), Q.real(x) & Q.real(y)) is True
assert satask(Q.real(x*y*z), Q.real(x) & Q.real(y) & Q.real(z)) is True
assert satask(Q.real(x*y*z), Q.real(x) & Q.real(y)) is None
assert satask(Q.real(x*y*z), Q.real(x) & Q.real(y) & Q.imaginary(z)) is False
assert satask(Q.real(x + y + z), Q.real(x) & Q.real(y) & Q.real(z)) is True
assert satask(Q.real(x + y + z), Q.real(x) & Q.real(y)) is None
def test_pos_neg():
assert satask(~Q.positive(x), Q.negative(x)) is True
assert satask(~Q.negative(x), Q.positive(x)) is True
assert satask(Q.positive(x + y), Q.positive(x) & Q.positive(y)) is True
assert satask(Q.negative(x + y), Q.negative(x) & Q.negative(y)) is True
assert satask(Q.positive(x + y), Q.negative(x) & Q.negative(y)) is False
assert satask(Q.negative(x + y), Q.positive(x) & Q.positive(y)) is False
@slow
def test_pow_pos_neg():
assert satask(Q.nonnegative(x**2), Q.positive(x)) is True
assert satask(Q.nonpositive(x**2), Q.positive(x)) is False
assert satask(Q.positive(x**2), Q.positive(x)) is True
assert satask(Q.negative(x**2), Q.positive(x)) is False
assert satask(Q.real(x**2), Q.positive(x)) is True
assert satask(Q.nonnegative(x**2), Q.negative(x)) is True
assert satask(Q.nonpositive(x**2), Q.negative(x)) is False
assert satask(Q.positive(x**2), Q.negative(x)) is True
assert satask(Q.negative(x**2), Q.negative(x)) is False
assert satask(Q.real(x**2), Q.negative(x)) is True
assert satask(Q.nonnegative(x**2), Q.nonnegative(x)) is True
assert satask(Q.nonpositive(x**2), Q.nonnegative(x)) is None
assert satask(Q.positive(x**2), Q.nonnegative(x)) is None
assert satask(Q.negative(x**2), Q.nonnegative(x)) is False
assert satask(Q.real(x**2), Q.nonnegative(x)) is True
assert satask(Q.nonnegative(x**2), Q.nonpositive(x)) is True
assert satask(Q.nonpositive(x**2), Q.nonpositive(x)) is None
assert satask(Q.positive(x**2), Q.nonpositive(x)) is None
assert satask(Q.negative(x**2), Q.nonpositive(x)) is False
assert satask(Q.real(x**2), Q.nonpositive(x)) is True
assert satask(Q.nonnegative(x**3), Q.positive(x)) is True
assert satask(Q.nonpositive(x**3), Q.positive(x)) is False
assert satask(Q.positive(x**3), Q.positive(x)) is True
assert satask(Q.negative(x**3), Q.positive(x)) is False
assert satask(Q.real(x**3), Q.positive(x)) is True
assert satask(Q.nonnegative(x**3), Q.negative(x)) is False
assert satask(Q.nonpositive(x**3), Q.negative(x)) is True
assert satask(Q.positive(x**3), Q.negative(x)) is False
assert satask(Q.negative(x**3), Q.negative(x)) is True
assert satask(Q.real(x**3), Q.negative(x)) is True
assert satask(Q.nonnegative(x**3), Q.nonnegative(x)) is True
assert satask(Q.nonpositive(x**3), Q.nonnegative(x)) is None
assert satask(Q.positive(x**3), Q.nonnegative(x)) is None
assert satask(Q.negative(x**3), Q.nonnegative(x)) is False
assert satask(Q.real(x**3), Q.nonnegative(x)) is True
assert satask(Q.nonnegative(x**3), Q.nonpositive(x)) is None
assert satask(Q.nonpositive(x**3), Q.nonpositive(x)) is True
assert satask(Q.positive(x**3), Q.nonpositive(x)) is False
assert satask(Q.negative(x**3), Q.nonpositive(x)) is None
assert satask(Q.real(x**3), Q.nonpositive(x)) is True
# If x is zero, x**negative is not real.
assert satask(Q.nonnegative(x**-2), Q.nonpositive(x)) is None
assert satask(Q.nonpositive(x**-2), Q.nonpositive(x)) is None
assert satask(Q.positive(x**-2), Q.nonpositive(x)) is None
assert satask(Q.negative(x**-2), Q.nonpositive(x)) is None
assert satask(Q.real(x**-2), Q.nonpositive(x)) is None
# We could deduce things for negative powers if x is nonzero, but it
# isn't implemented yet.
|
18f5d3221727bd16c91a48489c4ea58e4dce9cd2384b59d74ae769471e88ce9d
|
from sympy.abc import t, w, x, y, z, n, k, m, p, i
from sympy.assumptions import (ask, AssumptionsContext, Q, register_handler,
remove_handler)
from sympy.assumptions.assume import global_assumptions
from sympy.assumptions.ask import compute_known_facts, single_fact_lookup
from sympy.assumptions.handlers import AskHandler
from sympy.core.add import Add
from sympy.core.numbers import (I, Integer, Rational, oo, pi)
from sympy.core.singleton import S
from sympy.core.power import Pow
from sympy.core.symbol import symbols
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.elementary.complexes import (Abs, im, re, sign)
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (
acos, acot, asin, atan, cos, cot, sin, tan)
from sympy.logic.boolalg import Equivalent, Implies, Xor, And, to_cnf
from sympy.utilities.pytest import XFAIL, slow, raises, warns_deprecated_sympy
from sympy.assumptions.assume import assuming
import math
def test_int_1():
z = 1
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is True
assert ask(Q.rational(z)) is True
assert ask(Q.real(z)) is True
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is False
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is True
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is False
assert ask(Q.odd(z)) is True
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is True
assert ask(Q.antihermitian(z)) is False
def test_int_11():
z = 11
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is True
assert ask(Q.rational(z)) is True
assert ask(Q.real(z)) is True
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is False
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is True
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is False
assert ask(Q.odd(z)) is True
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is True
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is True
assert ask(Q.antihermitian(z)) is False
def test_int_12():
z = 12
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is True
assert ask(Q.rational(z)) is True
assert ask(Q.real(z)) is True
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is False
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is True
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is True
assert ask(Q.odd(z)) is False
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is True
assert ask(Q.hermitian(z)) is True
assert ask(Q.antihermitian(z)) is False
def test_float_1():
z = 1.0
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is False
assert ask(Q.rational(z)) is None
assert ask(Q.real(z)) is True
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is None
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is True
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is False
assert ask(Q.odd(z)) is False
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is True
assert ask(Q.antihermitian(z)) is False
z = 7.2123
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is False
assert ask(Q.rational(z)) is None
assert ask(Q.real(z)) is True
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is None
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is True
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is False
assert ask(Q.odd(z)) is False
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is True
assert ask(Q.antihermitian(z)) is False
# test for issue #12168
assert ask(Q.rational(math.pi)) is None
def test_zero_0():
z = Integer(0)
assert ask(Q.nonzero(z)) is False
assert ask(Q.zero(z)) is True
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is True
assert ask(Q.rational(z)) is True
assert ask(Q.real(z)) is True
assert ask(Q.complex(z)) is True
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is False
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is True
assert ask(Q.odd(z)) is False
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is True
assert ask(Q.antihermitian(z)) is False
def test_negativeone():
z = Integer(-1)
assert ask(Q.nonzero(z)) is True
assert ask(Q.zero(z)) is False
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is True
assert ask(Q.rational(z)) is True
assert ask(Q.real(z)) is True
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is False
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is False
assert ask(Q.negative(z)) is True
assert ask(Q.even(z)) is False
assert ask(Q.odd(z)) is True
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is True
assert ask(Q.antihermitian(z)) is False
def test_infinity():
assert ask(Q.commutative(oo)) is True
assert ask(Q.integer(oo)) is False
assert ask(Q.rational(oo)) is False
assert ask(Q.algebraic(oo)) is False
assert ask(Q.real(oo)) is False
assert ask(Q.extended_real(oo)) is True
assert ask(Q.complex(oo)) is False
assert ask(Q.irrational(oo)) is False
assert ask(Q.imaginary(oo)) is False
assert ask(Q.positive(oo)) is True
assert ask(Q.negative(oo)) is False
assert ask(Q.even(oo)) is False
assert ask(Q.odd(oo)) is False
assert ask(Q.finite(oo)) is False
assert ask(Q.prime(oo)) is False
assert ask(Q.composite(oo)) is False
assert ask(Q.hermitian(oo)) is False
assert ask(Q.antihermitian(oo)) is False
def test_neg_infinity():
mm = S.NegativeInfinity
assert ask(Q.commutative(mm)) is True
assert ask(Q.integer(mm)) is False
assert ask(Q.rational(mm)) is False
assert ask(Q.algebraic(mm)) is False
assert ask(Q.real(mm)) is False
assert ask(Q.extended_real(mm)) is True
assert ask(Q.complex(mm)) is False
assert ask(Q.irrational(mm)) is False
assert ask(Q.imaginary(mm)) is False
assert ask(Q.positive(mm)) is False
assert ask(Q.negative(mm)) is True
assert ask(Q.even(mm)) is False
assert ask(Q.odd(mm)) is False
assert ask(Q.finite(mm)) is False
assert ask(Q.prime(mm)) is False
assert ask(Q.composite(mm)) is False
assert ask(Q.hermitian(mm)) is False
assert ask(Q.antihermitian(mm)) is False
def test_nan():
nan = S.NaN
assert ask(Q.commutative(nan)) is True
assert ask(Q.integer(nan)) is False
assert ask(Q.rational(nan)) is False
assert ask(Q.algebraic(nan)) is False
assert ask(Q.real(nan)) is False
assert ask(Q.extended_real(nan)) is False
assert ask(Q.complex(nan)) is False
assert ask(Q.irrational(nan)) is False
assert ask(Q.imaginary(nan)) is False
assert ask(Q.positive(nan)) is False
assert ask(Q.nonzero(nan)) is True
assert ask(Q.zero(nan)) is False
assert ask(Q.even(nan)) is False
assert ask(Q.odd(nan)) is False
assert ask(Q.finite(nan)) is False
assert ask(Q.prime(nan)) is False
assert ask(Q.composite(nan)) is False
assert ask(Q.hermitian(nan)) is False
assert ask(Q.antihermitian(nan)) is False
def test_Rational_number():
r = Rational(3, 4)
assert ask(Q.commutative(r)) is True
assert ask(Q.integer(r)) is False
assert ask(Q.rational(r)) is True
assert ask(Q.real(r)) is True
assert ask(Q.complex(r)) is True
assert ask(Q.irrational(r)) is False
assert ask(Q.imaginary(r)) is False
assert ask(Q.positive(r)) is True
assert ask(Q.negative(r)) is False
assert ask(Q.even(r)) is False
assert ask(Q.odd(r)) is False
assert ask(Q.finite(r)) is True
assert ask(Q.prime(r)) is False
assert ask(Q.composite(r)) is False
assert ask(Q.hermitian(r)) is True
assert ask(Q.antihermitian(r)) is False
r = Rational(1, 4)
assert ask(Q.positive(r)) is True
assert ask(Q.negative(r)) is False
r = Rational(5, 4)
assert ask(Q.negative(r)) is False
assert ask(Q.positive(r)) is True
r = Rational(5, 3)
assert ask(Q.positive(r)) is True
assert ask(Q.negative(r)) is False
r = Rational(-3, 4)
assert ask(Q.positive(r)) is False
assert ask(Q.negative(r)) is True
r = Rational(-1, 4)
assert ask(Q.positive(r)) is False
assert ask(Q.negative(r)) is True
r = Rational(-5, 4)
assert ask(Q.negative(r)) is True
assert ask(Q.positive(r)) is False
r = Rational(-5, 3)
assert ask(Q.positive(r)) is False
assert ask(Q.negative(r)) is True
def test_sqrt_2():
z = sqrt(2)
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is False
assert ask(Q.rational(z)) is False
assert ask(Q.real(z)) is True
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is True
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is True
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is False
assert ask(Q.odd(z)) is False
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is True
assert ask(Q.antihermitian(z)) is False
def test_pi():
z = S.Pi
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is False
assert ask(Q.rational(z)) is False
assert ask(Q.algebraic(z)) is False
assert ask(Q.real(z)) is True
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is True
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is True
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is False
assert ask(Q.odd(z)) is False
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is True
assert ask(Q.antihermitian(z)) is False
z = S.Pi + 1
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is False
assert ask(Q.rational(z)) is False
assert ask(Q.algebraic(z)) is False
assert ask(Q.real(z)) is True
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is True
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is True
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is False
assert ask(Q.odd(z)) is False
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is True
assert ask(Q.antihermitian(z)) is False
z = 2*S.Pi
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is False
assert ask(Q.rational(z)) is False
assert ask(Q.algebraic(z)) is False
assert ask(Q.real(z)) is True
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is True
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is True
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is False
assert ask(Q.odd(z)) is False
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is True
assert ask(Q.antihermitian(z)) is False
z = S.Pi ** 2
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is False
assert ask(Q.rational(z)) is False
assert ask(Q.algebraic(z)) is False
assert ask(Q.real(z)) is True
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is True
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is True
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is False
assert ask(Q.odd(z)) is False
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is True
assert ask(Q.antihermitian(z)) is False
z = (1 + S.Pi) ** 2
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is False
assert ask(Q.rational(z)) is False
assert ask(Q.algebraic(z)) is False
assert ask(Q.real(z)) is True
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is True
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is True
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is False
assert ask(Q.odd(z)) is False
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is True
assert ask(Q.antihermitian(z)) is False
def test_E():
z = S.Exp1
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is False
assert ask(Q.rational(z)) is False
assert ask(Q.algebraic(z)) is False
assert ask(Q.real(z)) is True
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is True
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is True
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is False
assert ask(Q.odd(z)) is False
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is True
assert ask(Q.antihermitian(z)) is False
def test_GoldenRatio():
z = S.GoldenRatio
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is False
assert ask(Q.rational(z)) is False
assert ask(Q.algebraic(z)) is True
assert ask(Q.real(z)) is True
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is True
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is True
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is False
assert ask(Q.odd(z)) is False
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is True
assert ask(Q.antihermitian(z)) is False
def test_TribonacciConstant():
z = S.TribonacciConstant
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is False
assert ask(Q.rational(z)) is False
assert ask(Q.algebraic(z)) is True
assert ask(Q.real(z)) is True
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is True
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is True
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is False
assert ask(Q.odd(z)) is False
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is True
assert ask(Q.antihermitian(z)) is False
def test_I():
z = I
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is False
assert ask(Q.rational(z)) is False
assert ask(Q.algebraic(z)) is True
assert ask(Q.real(z)) is False
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is False
assert ask(Q.imaginary(z)) is True
assert ask(Q.positive(z)) is False
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is False
assert ask(Q.odd(z)) is False
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is False
assert ask(Q.antihermitian(z)) is True
z = 1 + I
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is False
assert ask(Q.rational(z)) is False
assert ask(Q.algebraic(z)) is True
assert ask(Q.real(z)) is False
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is False
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is False
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is False
assert ask(Q.odd(z)) is False
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is False
assert ask(Q.antihermitian(z)) is False
z = I*(1 + I)
assert ask(Q.commutative(z)) is True
assert ask(Q.integer(z)) is False
assert ask(Q.rational(z)) is False
assert ask(Q.algebraic(z)) is True
assert ask(Q.real(z)) is False
assert ask(Q.complex(z)) is True
assert ask(Q.irrational(z)) is False
assert ask(Q.imaginary(z)) is False
assert ask(Q.positive(z)) is False
assert ask(Q.negative(z)) is False
assert ask(Q.even(z)) is False
assert ask(Q.odd(z)) is False
assert ask(Q.finite(z)) is True
assert ask(Q.prime(z)) is False
assert ask(Q.composite(z)) is False
assert ask(Q.hermitian(z)) is False
assert ask(Q.antihermitian(z)) is False
z = I**(I)
assert ask(Q.imaginary(z)) is False
assert ask(Q.real(z)) is True
z = (-I)**(I)
assert ask(Q.imaginary(z)) is False
assert ask(Q.real(z)) is True
z = (3*I)**(I)
assert ask(Q.imaginary(z)) is False
assert ask(Q.real(z)) is False
z = (1)**(I)
assert ask(Q.imaginary(z)) is False
assert ask(Q.real(z)) is True
z = (-1)**(I)
assert ask(Q.imaginary(z)) is False
assert ask(Q.real(z)) is True
z = (1+I)**(I)
assert ask(Q.imaginary(z)) is False
assert ask(Q.real(z)) is False
z = (I)**(I+3)
assert ask(Q.imaginary(z)) is True
assert ask(Q.real(z)) is False
z = (I)**(I+2)
assert ask(Q.imaginary(z)) is False
assert ask(Q.real(z)) is True
z = (I)**(2)
assert ask(Q.imaginary(z)) is False
assert ask(Q.real(z)) is True
z = (I)**(3)
assert ask(Q.imaginary(z)) is True
assert ask(Q.real(z)) is False
z = (3)**(I)
assert ask(Q.imaginary(z)) is False
assert ask(Q.real(z)) is False
z = (I)**(0)
assert ask(Q.imaginary(z)) is False
assert ask(Q.real(z)) is True
@slow
def test_bounded1():
x, y, z = symbols('x,y,z')
assert ask(Q.finite(x)) is None
assert ask(Q.finite(x), Q.finite(x)) is True
assert ask(Q.finite(x), Q.finite(y)) is None
assert ask(Q.finite(x), Q.complex(x)) is None
assert ask(Q.finite(x + 1)) is None
assert ask(Q.finite(x + 1), Q.finite(x)) is True
a = x + y
x, y = a.args
# B + B
assert ask(Q.finite(a), Q.finite(x) & Q.finite(y)) is True
assert ask(
Q.finite(a), Q.finite(x) & Q.finite(y) & Q.positive(x)) is True
assert ask(
Q.finite(a), Q.finite(x) & Q.finite(y) & Q.positive(y)) is True
assert ask(Q.finite(a),
Q.finite(x) & Q.finite(y) & Q.positive(x) & Q.positive(y)) is True
assert ask(Q.finite(a),
Q.finite(x) & Q.finite(y) & Q.positive(x) & ~Q.positive(y)) is True
assert ask(Q.finite(a),
Q.finite(x) & Q.finite(y) & ~Q.positive(x) & Q.positive(y)) is True
assert ask(Q.finite(a),
Q.finite(x) & Q.finite(y) & ~Q.positive(x) & ~Q.positive(y)) is True
# B + U
assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(y)) is False
assert ask(
Q.finite(a), Q.finite(x) & ~Q.finite(y) & Q.positive(x)) is False
assert ask(
Q.finite(a), Q.finite(x) & ~Q.finite(y) & Q.positive(y)) is False
assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(y) & Q.positive(x) &
Q.positive(y)) is False
assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(y) & Q.positive(x) &
~Q.positive(y)) is False
assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(y) & ~Q.positive(x) &
Q.positive(y)) is False
assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(y) & ~Q.positive(x) &
~Q.positive(y)) is False
# B + ?
assert ask(Q.finite(a), Q.finite(x)) is None
assert ask(Q.finite(a), Q.finite(x) & Q.positive(x)) is None
assert ask(Q.finite(a), Q.finite(x) & Q.positive(y)) is None
assert ask(
Q.finite(a), Q.finite(x) & Q.positive(x) & Q.positive(y)) is None
assert ask(
Q.finite(a), Q.finite(x) & Q.positive(x) & ~Q.positive(y)) is None
assert ask(
Q.finite(a), Q.finite(x) & ~Q.positive(x) & Q.positive(y)) is None
assert ask(
Q.finite(a), Q.finite(x) & ~Q.positive(x) & ~Q.positive(y)) is None
# U + U
assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y)) is None
assert ask(
Q.finite(a), ~Q.finite(x) & ~Q.finite(y) & Q.positive(x)) is None
assert ask(
Q.finite(a), ~Q.finite(x) & ~Q.finite(y) & Q.positive(y)) is None
assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y) & Q.positive(x) &
Q.positive(y)) is False
assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y) & Q.positive(x) &
~Q.positive(y)) is None
assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y) & ~Q.positive(x) &
Q.positive(y)) is None
assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y) & ~Q.positive(x) &
~Q.positive(y)) is False
# U + ?
assert ask(Q.finite(a), ~Q.finite(y)) is None
assert ask(Q.finite(a), ~Q.finite(y) & Q.positive(x)) is None
assert ask(Q.finite(a), ~Q.finite(y) & Q.positive(y)) is None
assert ask(
Q.finite(a), ~Q.finite(y) & Q.positive(x) & Q.positive(y)) is False
assert ask(
Q.finite(a), ~Q.finite(y) & Q.positive(x) & ~Q.positive(y)) is None
assert ask(
Q.finite(a), ~Q.finite(y) & ~Q.positive(x) & Q.positive(y)) is None
assert ask(
Q.finite(a), ~Q.finite(y) & ~Q.positive(x) & ~Q.positive(y)) is False
# ? + ?
assert ask(Q.finite(a),) is None
assert ask(Q.finite(a), Q.positive(x)) is None
assert ask(Q.finite(a), Q.positive(y)) is None
assert ask(Q.finite(a), Q.positive(x) & Q.positive(y)) is None
assert ask(Q.finite(a), Q.positive(x) & ~Q.positive(y)) is None
assert ask(Q.finite(a), ~Q.positive(x) & Q.positive(y)) is None
assert ask(Q.finite(a), ~Q.positive(x) & ~Q.positive(y)) is None
@slow
def test_bounded2a():
x, y, z = symbols('x,y,z')
a = x + y + z
x, y, z = a.args
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) & Q.negative(y) &
Q.finite(y) & Q.negative(z) & Q.finite(z)) is True
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) &
Q.negative(y) & Q.finite(y) & Q.finite(z)) is True
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) & Q.negative(y) &
Q.finite(y) & Q.positive(z) & Q.finite(z)) is True
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) & Q.negative(y) &
Q.finite(y) & Q.negative(z) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) &
Q.negative(y) & Q.finite(y) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) & Q.negative(y) &
Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) &
Q.negative(y) & Q.finite(y) & Q.negative(z)) is None
assert ask(Q.finite(a), Q.negative(x) &
Q.finite(x) & Q.negative(y) & Q.finite(y)) is None
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) &
Q.negative(y) & Q.finite(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.negative(x) &
Q.finite(x) & Q.finite(y) & Q.finite(z)) is True
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) &
Q.finite(y) & Q.positive(z) & Q.finite(z)) is True
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) &
Q.finite(y) & Q.negative(z) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.negative(x) &
Q.finite(x) & Q.finite(y) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) &
Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.negative(x) &
Q.finite(x) & Q.finite(y) & Q.negative(z)) is None
assert ask(
Q.finite(a), Q.negative(x) & Q.finite(x) & Q.finite(y)) is None
assert ask(Q.finite(a), Q.negative(x) &
Q.finite(x) & Q.finite(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) & Q.positive(y) &
Q.finite(y) & Q.positive(z) & Q.finite(z)) is True
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) & Q.positive(y) &
Q.finite(y) & Q.negative(z) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) &
Q.positive(y) & Q.finite(y) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) & Q.positive(y) &
Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) &
Q.positive(y) & Q.finite(y) & Q.negative(z)) is None
assert ask(Q.finite(a), Q.negative(x) &
Q.finite(x) & Q.positive(y) & Q.finite(y)) is None
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) &
Q.positive(y) & Q.finite(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) & Q.negative(y) &
~Q.finite(y) & Q.negative(z) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) &
Q.negative(y) & ~Q.finite(y) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) & Q.negative(y) &
~Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) &
Q.negative(y) & ~Q.finite(y) & Q.negative(z)) is False
assert ask(Q.finite(a), Q.negative(x) &
Q.finite(x) & Q.negative(y) & ~Q.finite(y)) is None
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) &
Q.negative(y) & ~Q.finite(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.negative(x) &
Q.finite(x) & ~Q.finite(y) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) &
~Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.negative(x) &
Q.finite(x) & ~Q.finite(y) & Q.negative(z)) is None
assert ask(
Q.finite(a), Q.negative(x) & Q.finite(x) & ~Q.finite(y)) is None
assert ask(Q.finite(a), Q.negative(x) &
Q.finite(x) & ~Q.finite(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) & Q.positive(y) &
~Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) &
Q.positive(y) & ~Q.finite(y) & Q.negative(z)) is None
assert ask(Q.finite(a), Q.negative(x) &
Q.finite(x) & Q.positive(y) & ~Q.finite(y)) is None
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x) &
Q.positive(y) & ~Q.finite(y) & Q.positive(z)) is False
assert ask(Q.finite(a), Q.negative(x) &
Q.finite(x) & Q.negative(y) & Q.negative(z)) is None
assert ask(
Q.finite(a), Q.negative(x) & Q.finite(x) & Q.negative(y)) is None
assert ask(Q.finite(a), Q.negative(x) &
Q.finite(x) & Q.negative(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.negative(x) & Q.finite(x)) is None
assert ask(
Q.finite(a), Q.negative(x) & Q.finite(x) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.negative(x) &
Q.finite(x) & Q.positive(y) & Q.positive(z)) is None
assert ask(
Q.finite(a), Q.finite(x) & Q.finite(y) & Q.finite(z)) is True
assert ask(Q.finite(a),
Q.finite(x) & Q.finite(y) & Q.positive(z) & Q.finite(z)) is True
assert ask(Q.finite(a), Q.finite(x) &
Q.finite(y) & Q.negative(z) & ~Q.finite(z)) is False
assert ask(
Q.finite(a), Q.finite(x) & Q.finite(y) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.finite(x) &
Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is False
assert ask(
Q.finite(a), Q.finite(x) & Q.finite(y) & Q.negative(z)) is None
assert ask(Q.finite(a), Q.finite(x) & Q.finite(y)) is None
assert ask(
Q.finite(a), Q.finite(x) & Q.finite(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.finite(x) & Q.positive(y) &
Q.finite(y) & Q.positive(z) & Q.finite(z)) is True
assert ask(Q.finite(a), Q.finite(x) & Q.positive(y) &
Q.finite(y) & Q.negative(z) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.finite(x) &
Q.positive(y) & Q.finite(y) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.finite(x) & Q.positive(y) &
Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.finite(x) &
Q.positive(y) & Q.finite(y) & Q.negative(z)) is None
assert ask(
Q.finite(a), Q.finite(x) & Q.positive(y) & Q.finite(y)) is None
assert ask(Q.finite(a), Q.finite(x) &
Q.positive(y) & Q.finite(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.finite(x) & Q.negative(y) &
~Q.finite(y) & Q.negative(z) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.finite(x) &
Q.negative(y) & ~Q.finite(y) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.finite(x) & Q.negative(y) &
~Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.finite(x) &
Q.negative(y) & ~Q.finite(y) & Q.negative(z)) is False
assert ask(
Q.finite(a), Q.finite(x) & Q.negative(y) & ~Q.finite(y)) is None
assert ask(Q.finite(a), Q.finite(x) &
Q.negative(y) & ~Q.finite(y) & Q.positive(z)) is None
assert ask(
Q.finite(a), Q.finite(x) & ~Q.finite(y) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.finite(x) &
~Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is None
assert ask(
Q.finite(a), Q.finite(x) & ~Q.finite(y) & Q.negative(z)) is None
assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(y)) is None
assert ask(
Q.finite(a), Q.finite(x) & ~Q.finite(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.finite(x) & Q.positive(y) &
~Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is False
@slow
def test_bounded2b():
x, y, z = symbols('x,y,z')
a = x + y + z
x, y, z = a.args
assert ask(Q.finite(a), Q.finite(x) &
Q.positive(y) & ~Q.finite(y) & Q.negative(z)) is None
assert ask(
Q.finite(a), Q.finite(x) & Q.positive(y) & ~Q.finite(y)) is None
assert ask(Q.finite(a), Q.finite(x) &
Q.positive(y) & ~Q.finite(y) & Q.positive(z)) is False
assert ask(
Q.finite(a), Q.finite(x) & Q.negative(y) & Q.negative(z)) is None
assert ask(Q.finite(a), Q.finite(x) & Q.negative(y)) is None
assert ask(
Q.finite(a), Q.finite(x) & Q.negative(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.finite(x)) is None
assert ask(Q.finite(a), Q.finite(x) & Q.positive(z)) is None
assert ask(
Q.finite(a), Q.finite(x) & Q.positive(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.positive(x) & Q.finite(x) & Q.positive(y) &
Q.finite(y) & Q.positive(z) & Q.finite(z)) is True
assert ask(Q.finite(a), Q.positive(x) & Q.finite(x) & Q.positive(y) &
Q.finite(y) & Q.negative(z) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.positive(x) & Q.finite(x) &
Q.positive(y) & Q.finite(y) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.positive(x) & Q.finite(x) & Q.positive(y) &
Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.positive(x) & Q.finite(x) &
Q.positive(y) & Q.finite(y) & Q.negative(z)) is None
assert ask(Q.finite(a), Q.positive(x) &
Q.finite(x) & Q.positive(y) & Q.finite(y)) is None
assert ask(Q.finite(a), Q.positive(x) & Q.finite(x) &
Q.positive(y) & Q.finite(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.positive(x) & Q.finite(x) & Q.negative(y) &
~Q.finite(y) & Q.negative(z) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.positive(x) & Q.finite(x) &
Q.negative(y) & ~Q.finite(y) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.positive(x) & Q.finite(x) & Q.negative(y) &
~Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.positive(x) & Q.finite(x) &
Q.negative(y) & ~Q.finite(y) & Q.negative(z)) is False
assert ask(Q.finite(a), Q.positive(x) &
Q.finite(x) & Q.negative(y) & ~Q.finite(y)) is None
assert ask(Q.finite(a), Q.positive(x) & Q.finite(x) &
Q.negative(y) & ~Q.finite(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.positive(x) &
Q.finite(x) & ~Q.finite(y) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.positive(x) & Q.finite(x) &
~Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.positive(x) &
Q.finite(x) & ~Q.finite(y) & Q.negative(z)) is None
assert ask(
Q.finite(a), Q.positive(x) & Q.finite(x) & ~Q.finite(y)) is None
assert ask(Q.finite(a), Q.positive(x) &
Q.finite(x) & ~Q.finite(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.positive(x) & Q.finite(x) & Q.positive(y) &
~Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.positive(x) & Q.finite(x) &
Q.positive(y) & ~Q.finite(y) & Q.negative(z)) is None
assert ask(Q.finite(a), Q.positive(x) &
Q.finite(x) & Q.positive(y) & ~Q.finite(y)) is None
assert ask(Q.finite(a), Q.positive(x) & Q.finite(x) &
Q.positive(y) & ~Q.finite(y) & Q.positive(z)) is False
assert ask(Q.finite(a), Q.positive(x) &
Q.finite(x) & Q.negative(y) & Q.negative(z)) is None
assert ask(
Q.finite(a), Q.positive(x) & Q.finite(x) & Q.negative(y)) is None
assert ask(Q.finite(a), Q.positive(x) &
Q.finite(x) & Q.negative(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.positive(x) & Q.finite(x)) is None
assert ask(
Q.finite(a), Q.positive(x) & Q.finite(x) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.positive(x) &
Q.finite(x) & Q.positive(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.negative(x) & ~Q.finite(x) & Q.negative(y) &
~Q.finite(y) & Q.negative(z) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.negative(x) & ~Q.finite(x) &
Q.negative(y) & ~Q.finite(y) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.negative(x) & ~Q.finite(x) & Q.negative(y) &
~Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.negative(x) & ~Q.finite(x) &
Q.negative(y) & ~Q.finite(y) & Q.negative(z)) is False
assert ask(Q.finite(a), Q.negative(x) &
~Q.finite(x) & Q.negative(y) & ~Q.finite(y)) is None
assert ask(Q.finite(a), Q.negative(x) & ~Q.finite(x) &
Q.negative(y) & ~Q.finite(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.negative(x) &
~Q.finite(x) & ~Q.finite(y) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.negative(x) & ~Q.finite(x) &
~Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.negative(x) &
~Q.finite(x) & ~Q.finite(y) & Q.negative(z)) is None
assert ask(
Q.finite(a), Q.negative(x) & ~Q.finite(x) & ~Q.finite(y)) is None
assert ask(Q.finite(a), Q.negative(x) &
~Q.finite(x) & ~Q.finite(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.negative(x) & ~Q.finite(x) & Q.positive(y) &
~Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.negative(x) & ~Q.finite(x) &
Q.positive(y) & ~Q.finite(y) & Q.negative(z)) is None
assert ask(Q.finite(a), Q.negative(x) &
~Q.finite(x) & Q.positive(y) & ~Q.finite(y)) is None
assert ask(Q.finite(a), Q.negative(x) & ~Q.finite(x) &
Q.positive(y) & ~Q.finite(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.negative(x) &
~Q.finite(x) & Q.negative(y) & Q.negative(z)) is False
assert ask(
Q.finite(a), Q.negative(x) & ~Q.finite(x) & Q.negative(y)) is None
assert ask(Q.finite(a), Q.negative(x) &
~Q.finite(x) & Q.negative(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.negative(x) & ~Q.finite(x)) is None
assert ask(
Q.finite(a), Q.negative(x) & ~Q.finite(x) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.negative(x) &
~Q.finite(x) & Q.positive(y) & Q.positive(z)) is None
assert ask(
Q.finite(a), ~Q.finite(x) & ~Q.finite(y) & ~Q.finite(z)) is None
assert ask(Q.finite(a), ~Q.finite(x) &
~Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is None
assert ask(
Q.finite(a), ~Q.finite(x) & ~Q.finite(y) & Q.negative(z)) is None
assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y)) is None
assert ask(
Q.finite(a), ~Q.finite(x) & ~Q.finite(y) & Q.positive(z)) is None
assert ask(Q.finite(a), ~Q.finite(x) & Q.positive(y) &
~Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is None
assert ask(Q.finite(a), ~Q.finite(x) &
Q.positive(y) & ~Q.finite(y) & Q.negative(z)) is None
assert ask(
Q.finite(a), ~Q.finite(x) & Q.positive(y) & ~Q.finite(y)) is None
assert ask(Q.finite(a), ~Q.finite(x) &
Q.positive(y) & ~Q.finite(y) & Q.positive(z)) is None
assert ask(
Q.finite(a), ~Q.finite(x) & Q.negative(y) & Q.negative(z)) is None
assert ask(Q.finite(a), ~Q.finite(x) & Q.negative(y)) is None
assert ask(
Q.finite(a), ~Q.finite(x) & Q.negative(y) & Q.positive(z)) is None
assert ask(Q.finite(a), ~Q.finite(x)) is None
assert ask(Q.finite(a), ~Q.finite(x) & Q.positive(z)) is None
assert ask(
Q.finite(a), ~Q.finite(x) & Q.positive(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.positive(x) & ~Q.finite(x) & Q.positive(y) &
~Q.finite(y) & Q.positive(z) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.positive(x) & ~Q.finite(x) &
Q.positive(y) & ~Q.finite(y) & Q.negative(z)) is None
assert ask(Q.finite(a), Q.positive(x) &
~Q.finite(x) & Q.positive(y) & ~Q.finite(y)) is None
assert ask(Q.finite(a), Q.positive(x) & ~Q.finite(x) &
Q.positive(y) & ~Q.finite(y) & Q.positive(z)) is False
assert ask(Q.finite(a), Q.positive(x) &
~Q.finite(x) & Q.negative(y) & Q.negative(z)) is None
assert ask(
Q.finite(a), Q.positive(x) & ~Q.finite(x) & Q.negative(y)) is None
assert ask(Q.finite(a), Q.positive(x) &
~Q.finite(x) & Q.negative(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.positive(x) & ~Q.finite(x)) is None
assert ask(
Q.finite(a), Q.positive(x) & ~Q.finite(x) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.positive(x) &
~Q.finite(x) & Q.positive(y) & Q.positive(z)) is False
assert ask(
Q.finite(a), Q.negative(x) & Q.negative(y) & Q.negative(z)) is None
assert ask(Q.finite(a), Q.negative(x) & Q.negative(y)) is None
assert ask(
Q.finite(a), Q.negative(x) & Q.negative(y) & Q.positive(z)) is None
assert ask(Q.finite(a), Q.negative(x)) is None
assert ask(Q.finite(a), Q.negative(x) & Q.positive(z)) is None
assert ask(
Q.finite(a), Q.negative(x) & Q.positive(y) & Q.positive(z)) is None
assert ask(Q.finite(a)) is None
assert ask(Q.finite(a), Q.positive(z)) is None
assert ask(Q.finite(a), Q.positive(y) & Q.positive(z)) is None
assert ask(
Q.finite(a), Q.positive(x) & Q.positive(y) & Q.positive(z)) is None
assert ask(Q.finite(2*x)) is None
assert ask(Q.finite(2*x), Q.finite(x)) is True
@slow
def test_bounded3():
x, y, z = symbols('x,y,z')
a = x*y
x, y = a.args
assert ask(Q.finite(a), Q.finite(x) & Q.finite(y)) is True
assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(y)) is False
assert ask(Q.finite(a), Q.finite(x)) is None
assert ask(Q.finite(a), ~Q.finite(x) & Q.finite(y)) is False
assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y)) is False
assert ask(Q.finite(a), ~Q.finite(x)) is None
assert ask(Q.finite(a), Q.finite(y)) is None
assert ask(Q.finite(a), ~Q.finite(y)) is None
assert ask(Q.finite(a)) is None
a = x*y*z
x, y, z = a.args
assert ask(
Q.finite(a), Q.finite(x) & Q.finite(y) & Q.finite(z)) is True
assert ask(
Q.finite(a), Q.finite(x) & Q.finite(y) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.finite(x) & Q.finite(y)) is None
assert ask(
Q.finite(a), Q.finite(x) & ~Q.finite(y) & Q.finite(z)) is False
assert ask(
Q.finite(a), Q.finite(x) & ~Q.finite(y) & ~Q.finite(z)) is False
assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(y)) is None
assert ask(Q.finite(a), Q.finite(x) & Q.finite(z)) is None
assert ask(Q.finite(a), Q.finite(x) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.finite(x)) is None
assert ask(
Q.finite(a), ~Q.finite(x) & Q.finite(y) & Q.finite(z)) is False
assert ask(
Q.finite(a), ~Q.finite(x) & Q.finite(y) & ~Q.finite(z)) is False
assert ask(Q.finite(a), ~Q.finite(x) & Q.finite(y)) is None
assert ask(
Q.finite(a), ~Q.finite(x) & ~Q.finite(y) & Q.finite(z)) is False
assert ask(
Q.finite(a), ~Q.finite(x) & ~Q.finite(y) & ~Q.finite(z)) is False
assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(y)) is None
assert ask(Q.finite(a), ~Q.finite(x) & Q.finite(z)) is None
assert ask(Q.finite(a), ~Q.finite(x) & ~Q.finite(z)) is None
assert ask(Q.finite(a), ~Q.finite(x)) is None
assert ask(Q.finite(a), Q.finite(y) & Q.finite(z)) is None
assert ask(Q.finite(a), Q.finite(y) & ~Q.finite(z)) is None
assert ask(Q.finite(a), Q.finite(y)) is None
assert ask(Q.finite(a), ~Q.finite(y) & Q.finite(z)) is None
assert ask(Q.finite(a), ~Q.finite(y) & ~Q.finite(z)) is None
assert ask(Q.finite(a), ~Q.finite(y)) is None
assert ask(Q.finite(a), Q.finite(z)) is None
assert ask(Q.finite(a), ~Q.finite(z)) is None
assert ask(Q.finite(a), ~Q.finite(z) &
Q.nonzero(x) & Q.nonzero(y) & Q.nonzero(z)) is None
assert ask(Q.finite(a), ~Q.finite(y) & ~Q.finite(z) &
Q.nonzero(x) & Q.nonzero(y) & Q.nonzero(z)) is False
x, y, z = symbols('x,y,z')
assert ask(Q.finite(x**2)) is None
assert ask(Q.finite(2**x)) is None
assert ask(Q.finite(2**x), Q.finite(x)) is True
assert ask(Q.finite(x**x)) is None
assert ask(Q.finite(Rational(1, 2) ** x)) is None
assert ask(Q.finite(Rational(1, 2) ** x), Q.positive(x)) is True
assert ask(Q.finite(Rational(1, 2) ** x), Q.negative(x)) is None
assert ask(Q.finite(2**x), Q.negative(x)) is True
assert ask(Q.finite(sqrt(x))) is None
assert ask(Q.finite(2**x), ~Q.finite(x)) is False
assert ask(Q.finite(x**2), ~Q.finite(x)) is False
# sign function
assert ask(Q.finite(sign(x))) is True
assert ask(Q.finite(sign(x)), ~Q.finite(x)) is True
# exponential functions
assert ask(Q.finite(log(x))) is None
assert ask(Q.finite(log(x)), Q.finite(x)) is True
assert ask(Q.finite(exp(x))) is None
assert ask(Q.finite(exp(x)), Q.finite(x)) is True
assert ask(Q.finite(exp(2))) is True
# trigonometric functions
assert ask(Q.finite(sin(x))) is True
assert ask(Q.finite(sin(x)), ~Q.finite(x)) is True
assert ask(Q.finite(cos(x))) is True
assert ask(Q.finite(cos(x)), ~Q.finite(x)) is True
assert ask(Q.finite(2*sin(x))) is True
assert ask(Q.finite(sin(x)**2)) is True
assert ask(Q.finite(cos(x)**2)) is True
assert ask(Q.finite(cos(x) + sin(x))) is True
@XFAIL
def test_bounded_xfail():
"""We need to support relations in ask for this to work"""
assert ask(Q.finite(sin(x)**x)) is True
assert ask(Q.finite(cos(x)**x)) is True
def test_commutative():
"""By default objects are Q.commutative that is why it returns True
for both key=True and key=False"""
assert ask(Q.commutative(x)) is True
assert ask(Q.commutative(x), ~Q.commutative(x)) is False
assert ask(Q.commutative(x), Q.complex(x)) is True
assert ask(Q.commutative(x), Q.imaginary(x)) is True
assert ask(Q.commutative(x), Q.real(x)) is True
assert ask(Q.commutative(x), Q.positive(x)) is True
assert ask(Q.commutative(x), ~Q.commutative(y)) is True
assert ask(Q.commutative(2*x)) is True
assert ask(Q.commutative(2*x), ~Q.commutative(x)) is False
assert ask(Q.commutative(x + 1)) is True
assert ask(Q.commutative(x + 1), ~Q.commutative(x)) is False
assert ask(Q.commutative(x**2)) is True
assert ask(Q.commutative(x**2), ~Q.commutative(x)) is False
assert ask(Q.commutative(log(x))) is True
def test_complex():
assert ask(Q.complex(x)) is None
assert ask(Q.complex(x), Q.complex(x)) is True
assert ask(Q.complex(x), Q.complex(y)) is None
assert ask(Q.complex(x), ~Q.complex(x)) is False
assert ask(Q.complex(x), Q.real(x)) is True
assert ask(Q.complex(x), ~Q.real(x)) is None
assert ask(Q.complex(x), Q.rational(x)) is True
assert ask(Q.complex(x), Q.irrational(x)) is True
assert ask(Q.complex(x), Q.positive(x)) is True
assert ask(Q.complex(x), Q.imaginary(x)) is True
assert ask(Q.complex(x), Q.algebraic(x)) is True
# a+b
assert ask(Q.complex(x + 1), Q.complex(x)) is True
assert ask(Q.complex(x + 1), Q.real(x)) is True
assert ask(Q.complex(x + 1), Q.rational(x)) is True
assert ask(Q.complex(x + 1), Q.irrational(x)) is True
assert ask(Q.complex(x + 1), Q.imaginary(x)) is True
assert ask(Q.complex(x + 1), Q.integer(x)) is True
assert ask(Q.complex(x + 1), Q.even(x)) is True
assert ask(Q.complex(x + 1), Q.odd(x)) is True
assert ask(Q.complex(x + y), Q.complex(x) & Q.complex(y)) is True
assert ask(Q.complex(x + y), Q.real(x) & Q.imaginary(y)) is True
# a*x +b
assert ask(Q.complex(2*x + 1), Q.complex(x)) is True
assert ask(Q.complex(2*x + 1), Q.real(x)) is True
assert ask(Q.complex(2*x + 1), Q.positive(x)) is True
assert ask(Q.complex(2*x + 1), Q.rational(x)) is True
assert ask(Q.complex(2*x + 1), Q.irrational(x)) is True
assert ask(Q.complex(2*x + 1), Q.imaginary(x)) is True
assert ask(Q.complex(2*x + 1), Q.integer(x)) is True
assert ask(Q.complex(2*x + 1), Q.even(x)) is True
assert ask(Q.complex(2*x + 1), Q.odd(x)) is True
# x**2
assert ask(Q.complex(x**2), Q.complex(x)) is True
assert ask(Q.complex(x**2), Q.real(x)) is True
assert ask(Q.complex(x**2), Q.positive(x)) is True
assert ask(Q.complex(x**2), Q.rational(x)) is True
assert ask(Q.complex(x**2), Q.irrational(x)) is True
assert ask(Q.complex(x**2), Q.imaginary(x)) is True
assert ask(Q.complex(x**2), Q.integer(x)) is True
assert ask(Q.complex(x**2), Q.even(x)) is True
assert ask(Q.complex(x**2), Q.odd(x)) is True
# 2**x
assert ask(Q.complex(2**x), Q.complex(x)) is True
assert ask(Q.complex(2**x), Q.real(x)) is True
assert ask(Q.complex(2**x), Q.positive(x)) is True
assert ask(Q.complex(2**x), Q.rational(x)) is True
assert ask(Q.complex(2**x), Q.irrational(x)) is True
assert ask(Q.complex(2**x), Q.imaginary(x)) is True
assert ask(Q.complex(2**x), Q.integer(x)) is True
assert ask(Q.complex(2**x), Q.even(x)) is True
assert ask(Q.complex(2**x), Q.odd(x)) is True
assert ask(Q.complex(x**y), Q.complex(x) & Q.complex(y)) is True
# trigonometric expressions
assert ask(Q.complex(sin(x))) is True
assert ask(Q.complex(sin(2*x + 1))) is True
assert ask(Q.complex(cos(x))) is True
assert ask(Q.complex(cos(2*x + 1))) is True
# exponential
assert ask(Q.complex(exp(x))) is True
assert ask(Q.complex(exp(x))) is True
# Q.complexes
assert ask(Q.complex(Abs(x))) is True
assert ask(Q.complex(re(x))) is True
assert ask(Q.complex(im(x))) is True
@slow
def test_even_query():
assert ask(Q.even(x)) is None
assert ask(Q.even(x), Q.integer(x)) is None
assert ask(Q.even(x), ~Q.integer(x)) is False
assert ask(Q.even(x), Q.rational(x)) is None
assert ask(Q.even(x), Q.positive(x)) is None
assert ask(Q.even(2*x)) is None
assert ask(Q.even(2*x), Q.integer(x)) is True
assert ask(Q.even(2*x), Q.even(x)) is True
assert ask(Q.even(2*x), Q.irrational(x)) is False
assert ask(Q.even(2*x), Q.odd(x)) is True
assert ask(Q.even(2*x), ~Q.integer(x)) is None
assert ask(Q.even(3*x), Q.integer(x)) is None
assert ask(Q.even(3*x), Q.even(x)) is True
assert ask(Q.even(3*x), Q.odd(x)) is False
assert ask(Q.even(x + 1), Q.odd(x)) is True
assert ask(Q.even(x + 1), Q.even(x)) is False
assert ask(Q.even(x + 2), Q.odd(x)) is False
assert ask(Q.even(x + 2), Q.even(x)) is True
assert ask(Q.even(7 - x), Q.odd(x)) is True
assert ask(Q.even(7 + x), Q.odd(x)) is True
assert ask(Q.even(x + y), Q.odd(x) & Q.odd(y)) is True
assert ask(Q.even(x + y), Q.odd(x) & Q.even(y)) is False
assert ask(Q.even(x + y), Q.even(x) & Q.even(y)) is True
assert ask(Q.even(2*x + 1), Q.integer(x)) is False
assert ask(Q.even(2*x*y), Q.rational(x) & Q.rational(x)) is None
assert ask(Q.even(2*x*y), Q.irrational(x) & Q.irrational(x)) is None
assert ask(Q.even(x + y + z), Q.odd(x) & Q.odd(y) & Q.even(z)) is True
assert ask(Q.even(x + y + z + t),
Q.odd(x) & Q.odd(y) & Q.even(z) & Q.integer(t)) is None
assert ask(Q.even(Abs(x)), Q.even(x)) is True
assert ask(Q.even(Abs(x)), ~Q.even(x)) is None
assert ask(Q.even(re(x)), Q.even(x)) is True
assert ask(Q.even(re(x)), ~Q.even(x)) is None
assert ask(Q.even(im(x)), Q.even(x)) is True
assert ask(Q.even(im(x)), Q.real(x)) is True
assert ask(Q.even((-1)**n), Q.integer(n)) is False
assert ask(Q.even(k**2), Q.even(k)) is True
assert ask(Q.even(n**2), Q.odd(n)) is False
assert ask(Q.even(2**k), Q.even(k)) is None
assert ask(Q.even(x**2)) is None
assert ask(Q.even(k**m), Q.even(k) & Q.integer(m) & ~Q.negative(m)) is None
assert ask(Q.even(n**m), Q.odd(n) & Q.integer(m) & ~Q.negative(m)) is False
assert ask(Q.even(k**p), Q.even(k) & Q.integer(p) & Q.positive(p)) is True
assert ask(Q.even(n**p), Q.odd(n) & Q.integer(p) & Q.positive(p)) is False
assert ask(Q.even(m**k), Q.even(k) & Q.integer(m) & ~Q.negative(m)) is None
assert ask(Q.even(p**k), Q.even(k) & Q.integer(p) & Q.positive(p)) is None
assert ask(Q.even(m**n), Q.odd(n) & Q.integer(m) & ~Q.negative(m)) is None
assert ask(Q.even(p**n), Q.odd(n) & Q.integer(p) & Q.positive(p)) is None
assert ask(Q.even(k**x), Q.even(k)) is None
assert ask(Q.even(n**x), Q.odd(n)) is None
assert ask(Q.even(x*y), Q.integer(x) & Q.integer(y)) is None
assert ask(Q.even(x*x), Q.integer(x)) is None
assert ask(Q.even(x*(x + y)), Q.integer(x) & Q.odd(y)) is True
assert ask(Q.even(x*(x + y)), Q.integer(x) & Q.even(y)) is None
@XFAIL
def test_evenness_in_ternary_integer_product_with_odd():
# Tests that oddness inference is independent of term ordering.
# Term ordering at the point of testing depends on SymPy's symbol order, so
# we try to force a different order by modifying symbol names.
assert ask(Q.even(x*y*(y + z)), Q.integer(x) & Q.integer(y) & Q.odd(z)) is True
assert ask(Q.even(y*x*(x + z)), Q.integer(x) & Q.integer(y) & Q.odd(z)) is True
def test_evenness_in_ternary_integer_product_with_even():
assert ask(Q.even(x*y*(y + z)), Q.integer(x) & Q.integer(y) & Q.even(z)) is None
def test_extended_real():
assert ask(Q.extended_real(x), Q.positive(x)) is True
assert ask(Q.extended_real(-x), Q.positive(x)) is True
assert ask(Q.extended_real(-x), Q.negative(x)) is True
assert ask(Q.extended_real(x + S.Infinity), Q.real(x)) is True
def test_rational():
assert ask(Q.rational(x), Q.integer(x)) is True
assert ask(Q.rational(x), Q.irrational(x)) is False
assert ask(Q.rational(x), Q.real(x)) is None
assert ask(Q.rational(x), Q.positive(x)) is None
assert ask(Q.rational(x), Q.negative(x)) is None
assert ask(Q.rational(x), Q.nonzero(x)) is None
assert ask(Q.rational(x), ~Q.algebraic(x)) is False
assert ask(Q.rational(2*x), Q.rational(x)) is True
assert ask(Q.rational(2*x), Q.integer(x)) is True
assert ask(Q.rational(2*x), Q.even(x)) is True
assert ask(Q.rational(2*x), Q.odd(x)) is True
assert ask(Q.rational(2*x), Q.irrational(x)) is False
assert ask(Q.rational(x/2), Q.rational(x)) is True
assert ask(Q.rational(x/2), Q.integer(x)) is True
assert ask(Q.rational(x/2), Q.even(x)) is True
assert ask(Q.rational(x/2), Q.odd(x)) is True
assert ask(Q.rational(x/2), Q.irrational(x)) is False
assert ask(Q.rational(1/x), Q.rational(x)) is True
assert ask(Q.rational(1/x), Q.integer(x)) is True
assert ask(Q.rational(1/x), Q.even(x)) is True
assert ask(Q.rational(1/x), Q.odd(x)) is True
assert ask(Q.rational(1/x), Q.irrational(x)) is False
assert ask(Q.rational(2/x), Q.rational(x)) is True
assert ask(Q.rational(2/x), Q.integer(x)) is True
assert ask(Q.rational(2/x), Q.even(x)) is True
assert ask(Q.rational(2/x), Q.odd(x)) is True
assert ask(Q.rational(2/x), Q.irrational(x)) is False
assert ask(Q.rational(x), ~Q.algebraic(x)) is False
# with multiple symbols
assert ask(Q.rational(x*y), Q.irrational(x) & Q.irrational(y)) is None
assert ask(Q.rational(y/x), Q.rational(x) & Q.rational(y)) is True
assert ask(Q.rational(y/x), Q.integer(x) & Q.rational(y)) is True
assert ask(Q.rational(y/x), Q.even(x) & Q.rational(y)) is True
assert ask(Q.rational(y/x), Q.odd(x) & Q.rational(y)) is True
assert ask(Q.rational(y/x), Q.irrational(x) & Q.rational(y)) is False
for f in [exp, sin, tan, asin, atan, cos]:
assert ask(Q.rational(f(7))) is False
assert ask(Q.rational(f(7, evaluate=False))) is False
assert ask(Q.rational(f(0, evaluate=False))) is True
assert ask(Q.rational(f(x)), Q.rational(x)) is None
assert ask(Q.rational(f(x)), Q.rational(x) & Q.nonzero(x)) is False
for g in [log, acos]:
assert ask(Q.rational(g(7))) is False
assert ask(Q.rational(g(7, evaluate=False))) is False
assert ask(Q.rational(g(1, evaluate=False))) is True
assert ask(Q.rational(g(x)), Q.rational(x)) is None
assert ask(Q.rational(g(x)), Q.rational(x) & Q.nonzero(x - 1)) is False
for h in [cot, acot]:
assert ask(Q.rational(h(7))) is False
assert ask(Q.rational(h(7, evaluate=False))) is False
assert ask(Q.rational(h(x)), Q.rational(x)) is False
@slow
def test_hermitian():
assert ask(Q.hermitian(x)) is None
assert ask(Q.hermitian(x), Q.antihermitian(x)) is False
assert ask(Q.hermitian(x), Q.imaginary(x)) is False
assert ask(Q.hermitian(x), Q.prime(x)) is True
assert ask(Q.hermitian(x), Q.real(x)) is True
assert ask(Q.hermitian(x + 1), Q.antihermitian(x)) is False
assert ask(Q.hermitian(x + 1), Q.complex(x)) is None
assert ask(Q.hermitian(x + 1), Q.hermitian(x)) is True
assert ask(Q.hermitian(x + 1), Q.imaginary(x)) is False
assert ask(Q.hermitian(x + 1), Q.real(x)) is True
assert ask(Q.hermitian(x + I), Q.antihermitian(x)) is None
assert ask(Q.hermitian(x + I), Q.complex(x)) is None
assert ask(Q.hermitian(x + I), Q.hermitian(x)) is False
assert ask(Q.hermitian(x + I), Q.imaginary(x)) is None
assert ask(Q.hermitian(x + I), Q.real(x)) is False
assert ask(
Q.hermitian(x + y), Q.antihermitian(x) & Q.antihermitian(y)) is None
assert ask(Q.hermitian(x + y), Q.antihermitian(x) & Q.complex(y)) is None
assert ask(
Q.hermitian(x + y), Q.antihermitian(x) & Q.hermitian(y)) is False
assert ask(Q.hermitian(x + y), Q.antihermitian(x) & Q.imaginary(y)) is None
assert ask(Q.hermitian(x + y), Q.antihermitian(x) & Q.real(y)) is False
assert ask(Q.hermitian(x + y), Q.hermitian(x) & Q.complex(y)) is None
assert ask(Q.hermitian(x + y), Q.hermitian(x) & Q.hermitian(y)) is True
assert ask(Q.hermitian(x + y), Q.hermitian(x) & Q.imaginary(y)) is False
assert ask(Q.hermitian(x + y), Q.hermitian(x) & Q.real(y)) is True
assert ask(Q.hermitian(x + y), Q.imaginary(x) & Q.complex(y)) is None
assert ask(Q.hermitian(x + y), Q.imaginary(x) & Q.imaginary(y)) is None
assert ask(Q.hermitian(x + y), Q.imaginary(x) & Q.real(y)) is False
assert ask(Q.hermitian(x + y), Q.real(x) & Q.complex(y)) is None
assert ask(Q.hermitian(x + y), Q.real(x) & Q.real(y)) is True
assert ask(Q.hermitian(I*x), Q.antihermitian(x)) is True
assert ask(Q.hermitian(I*x), Q.complex(x)) is None
assert ask(Q.hermitian(I*x), Q.hermitian(x)) is False
assert ask(Q.hermitian(I*x), Q.imaginary(x)) is True
assert ask(Q.hermitian(I*x), Q.real(x)) is False
assert ask(Q.hermitian(x*y), Q.hermitian(x) & Q.real(y)) is True
assert ask(
Q.hermitian(x + y + z), Q.real(x) & Q.real(y) & Q.real(z)) is True
assert ask(Q.hermitian(x + y + z),
Q.real(x) & Q.real(y) & Q.imaginary(z)) is False
assert ask(Q.hermitian(x + y + z),
Q.real(x) & Q.imaginary(y) & Q.imaginary(z)) is None
assert ask(Q.hermitian(x + y + z),
Q.imaginary(x) & Q.imaginary(y) & Q.imaginary(z)) is None
assert ask(Q.antihermitian(x)) is None
assert ask(Q.antihermitian(x), Q.real(x)) is False
assert ask(Q.antihermitian(x), Q.prime(x)) is False
assert ask(Q.antihermitian(x + 1), Q.antihermitian(x)) is False
assert ask(Q.antihermitian(x + 1), Q.complex(x)) is None
assert ask(Q.antihermitian(x + 1), Q.hermitian(x)) is None
assert ask(Q.antihermitian(x + 1), Q.imaginary(x)) is False
assert ask(Q.antihermitian(x + 1), Q.real(x)) is False
assert ask(Q.antihermitian(x + I), Q.antihermitian(x)) is True
assert ask(Q.antihermitian(x + I), Q.complex(x)) is None
assert ask(Q.antihermitian(x + I), Q.hermitian(x)) is False
assert ask(Q.antihermitian(x + I), Q.imaginary(x)) is True
assert ask(Q.antihermitian(x + I), Q.real(x)) is False
assert ask(
Q.antihermitian(x + y), Q.antihermitian(x) & Q.antihermitian(y)
) is True
assert ask(
Q.antihermitian(x + y), Q.antihermitian(x) & Q.complex(y)) is None
assert ask(
Q.antihermitian(x + y), Q.antihermitian(x) & Q.hermitian(y)) is False
assert ask(
Q.antihermitian(x + y), Q.antihermitian(x) & Q.imaginary(y)) is True
assert ask(Q.antihermitian(x + y), Q.antihermitian(x) & Q.real(y)
) is False
assert ask(Q.antihermitian(x + y), Q.hermitian(x) & Q.complex(y)) is None
assert ask(Q.antihermitian(x + y), Q.hermitian(x) & Q.hermitian(y)
) is None
assert ask(
Q.antihermitian(x + y), Q.hermitian(x) & Q.imaginary(y)) is False
assert ask(Q.antihermitian(x + y), Q.hermitian(x) & Q.real(y)) is None
assert ask(Q.antihermitian(x + y), Q.imaginary(x) & Q.complex(y)) is None
assert ask(Q.antihermitian(x + y), Q.imaginary(x) & Q.imaginary(y)) is True
assert ask(Q.antihermitian(x + y), Q.imaginary(x) & Q.real(y)) is False
assert ask(Q.antihermitian(x + y), Q.real(x) & Q.complex(y)) is None
assert ask(Q.antihermitian(x + y), Q.real(x) & Q.real(y)) is False
assert ask(Q.antihermitian(I*x), Q.real(x)) is True
assert ask(Q.antihermitian(I*x), Q.antihermitian(x)) is False
assert ask(Q.antihermitian(I*x), Q.complex(x)) is None
assert ask(Q.antihermitian(x*y), Q.antihermitian(x) & Q.real(y)) is True
assert ask(Q.antihermitian(x + y + z),
Q.real(x) & Q.real(y) & Q.real(z)) is False
assert ask(Q.antihermitian(x + y + z),
Q.real(x) & Q.real(y) & Q.imaginary(z)) is None
assert ask(Q.antihermitian(x + y + z),
Q.real(x) & Q.imaginary(y) & Q.imaginary(z)) is False
assert ask(Q.antihermitian(x + y + z),
Q.imaginary(x) & Q.imaginary(y) & Q.imaginary(z)) is True
@slow
def test_imaginary():
assert ask(Q.imaginary(x)) is None
assert ask(Q.imaginary(x), Q.real(x)) is False
assert ask(Q.imaginary(x), Q.prime(x)) is False
assert ask(Q.imaginary(x + 1), Q.real(x)) is False
assert ask(Q.imaginary(x + 1), Q.imaginary(x)) is False
assert ask(Q.imaginary(x + I), Q.real(x)) is False
assert ask(Q.imaginary(x + I), Q.imaginary(x)) is True
assert ask(Q.imaginary(x + y), Q.imaginary(x) & Q.imaginary(y)) is True
assert ask(Q.imaginary(x + y), Q.real(x) & Q.real(y)) is False
assert ask(Q.imaginary(x + y), Q.imaginary(x) & Q.real(y)) is False
assert ask(Q.imaginary(x + y), Q.complex(x) & Q.real(y)) is None
assert ask(
Q.imaginary(x + y + z), Q.real(x) & Q.real(y) & Q.real(z)) is False
assert ask(Q.imaginary(x + y + z),
Q.real(x) & Q.real(y) & Q.imaginary(z)) is None
assert ask(Q.imaginary(x + y + z),
Q.real(x) & Q.imaginary(y) & Q.imaginary(z)) is False
assert ask(Q.imaginary(I*x), Q.real(x)) is True
assert ask(Q.imaginary(I*x), Q.imaginary(x)) is False
assert ask(Q.imaginary(I*x), Q.complex(x)) is None
assert ask(Q.imaginary(x*y), Q.imaginary(x) & Q.real(y)) is True
assert ask(Q.imaginary(x*y), Q.real(x) & Q.real(y)) is False
assert ask(Q.imaginary(I**x), Q.negative(x)) is None
assert ask(Q.imaginary(I**x), Q.positive(x)) is None
assert ask(Q.imaginary(I**x), Q.even(x)) is False
assert ask(Q.imaginary(I**x), Q.odd(x)) is True
assert ask(Q.imaginary(I**x), Q.imaginary(x)) is False
assert ask(Q.imaginary((2*I)**x), Q.imaginary(x)) is False
assert ask(Q.imaginary(x**0), Q.imaginary(x)) is False
assert ask(Q.imaginary(x**y), Q.imaginary(x) & Q.imaginary(y)) is None
assert ask(Q.imaginary(x**y), Q.imaginary(x) & Q.real(y)) is None
assert ask(Q.imaginary(x**y), Q.real(x) & Q.imaginary(y)) is None
assert ask(Q.imaginary(x**y), Q.real(x) & Q.real(y)) is None
assert ask(Q.imaginary(x**y), Q.imaginary(x) & Q.integer(y)) is None
assert ask(Q.imaginary(x**y), Q.imaginary(y) & Q.integer(x)) is None
assert ask(Q.imaginary(x**y), Q.imaginary(x) & Q.odd(y)) is True
assert ask(Q.imaginary(x**y), Q.imaginary(x) & Q.rational(y)) is None
assert ask(Q.imaginary(x**y), Q.imaginary(x) & Q.even(y)) is False
assert ask(Q.imaginary(x**y), Q.real(x) & Q.integer(y)) is False
assert ask(Q.imaginary(x**y), Q.positive(x) & Q.real(y)) is False
assert ask(Q.imaginary(x**y), Q.negative(x) & Q.real(y)) is None
assert ask(Q.imaginary(x**y), Q.negative(x) & Q.real(y) & ~Q.rational(y)) is False
assert ask(Q.imaginary(x**y), Q.integer(x) & Q.imaginary(y)) is None
assert ask(Q.imaginary(x**y), Q.negative(x) & Q.rational(y) & Q.integer(2*y)) is True
assert ask(Q.imaginary(x**y), Q.negative(x) & Q.rational(y) & ~Q.integer(2*y)) is False
assert ask(Q.imaginary(x**y), Q.negative(x) & Q.rational(y)) is None
assert ask(Q.imaginary(x**y), Q.real(x) & Q.rational(y) & ~Q.integer(2*y)) is False
assert ask(Q.imaginary(x**y), Q.real(x) & Q.rational(y) & Q.integer(2*y)) is None
# logarithm
assert ask(Q.imaginary(log(I))) is True
assert ask(Q.imaginary(log(2*I))) is False
assert ask(Q.imaginary(log(I + 1))) is False
assert ask(Q.imaginary(log(x)), Q.complex(x)) is None
assert ask(Q.imaginary(log(x)), Q.imaginary(x)) is None
assert ask(Q.imaginary(log(x)), Q.positive(x)) is False
assert ask(Q.imaginary(log(exp(x))), Q.complex(x)) is None
assert ask(Q.imaginary(log(exp(x))), Q.imaginary(x)) is None # zoo/I/a+I*b
assert ask(Q.imaginary(log(exp(I)))) is True
# exponential
assert ask(Q.imaginary(exp(x)**x), Q.imaginary(x)) is False
eq = Pow(exp(pi*I*x, evaluate=False), x, evaluate=False)
assert ask(Q.imaginary(eq), Q.even(x)) is False
eq = Pow(exp(pi*I*x/2, evaluate=False), x, evaluate=False)
assert ask(Q.imaginary(eq), Q.odd(x)) is True
assert ask(Q.imaginary(exp(3*I*pi*x)**x), Q.integer(x)) is False
assert ask(Q.imaginary(exp(2*pi*I, evaluate=False))) is False
assert ask(Q.imaginary(exp(pi*I/2, evaluate=False))) is True
# issue 7886
assert ask(Q.imaginary(Pow(x, S.One/4)), Q.real(x) & Q.negative(x)) is False
def test_integer():
assert ask(Q.integer(x)) is None
assert ask(Q.integer(x), Q.integer(x)) is True
assert ask(Q.integer(x), ~Q.integer(x)) is False
assert ask(Q.integer(x), ~Q.real(x)) is False
assert ask(Q.integer(x), ~Q.positive(x)) is None
assert ask(Q.integer(x), Q.even(x) | Q.odd(x)) is True
assert ask(Q.integer(2*x), Q.integer(x)) is True
assert ask(Q.integer(2*x), Q.even(x)) is True
assert ask(Q.integer(2*x), Q.prime(x)) is True
assert ask(Q.integer(2*x), Q.rational(x)) is None
assert ask(Q.integer(2*x), Q.real(x)) is None
assert ask(Q.integer(sqrt(2)*x), Q.integer(x)) is False
assert ask(Q.integer(sqrt(2)*x), Q.irrational(x)) is None
assert ask(Q.integer(x/2), Q.odd(x)) is False
assert ask(Q.integer(x/2), Q.even(x)) is True
assert ask(Q.integer(x/3), Q.odd(x)) is None
assert ask(Q.integer(x/3), Q.even(x)) is None
@slow
def test_negative():
assert ask(Q.negative(x), Q.negative(x)) is True
assert ask(Q.negative(x), Q.positive(x)) is False
assert ask(Q.negative(x), ~Q.real(x)) is False
assert ask(Q.negative(x), Q.prime(x)) is False
assert ask(Q.negative(x), ~Q.prime(x)) is None
assert ask(Q.negative(-x), Q.positive(x)) is True
assert ask(Q.negative(-x), ~Q.positive(x)) is None
assert ask(Q.negative(-x), Q.negative(x)) is False
assert ask(Q.negative(-x), Q.positive(x)) is True
assert ask(Q.negative(x - 1), Q.negative(x)) is True
assert ask(Q.negative(x + y)) is None
assert ask(Q.negative(x + y), Q.negative(x)) is None
assert ask(Q.negative(x + y), Q.negative(x) & Q.negative(y)) is True
assert ask(Q.negative(x + y), Q.negative(x) & Q.nonpositive(y)) is True
assert ask(Q.negative(2 + I)) is False
# although this could be False, it is representative of expressions
# that don't evaluate to a zero with precision
assert ask(Q.negative(cos(I)**2 + sin(I)**2 - 1)) is None
assert ask(Q.negative(-I + I*(cos(2)**2 + sin(2)**2))) is None
assert ask(Q.negative(x**2)) is None
assert ask(Q.negative(x**2), Q.real(x)) is False
assert ask(Q.negative(x**1.4), Q.real(x)) is None
assert ask(Q.negative(x**I), Q.positive(x)) is None
assert ask(Q.negative(x*y)) is None
assert ask(Q.negative(x*y), Q.positive(x) & Q.positive(y)) is False
assert ask(Q.negative(x*y), Q.positive(x) & Q.negative(y)) is True
assert ask(Q.negative(x*y), Q.complex(x) & Q.complex(y)) is None
assert ask(Q.negative(x**y)) is None
assert ask(Q.negative(x**y), Q.negative(x) & Q.even(y)) is False
assert ask(Q.negative(x**y), Q.negative(x) & Q.odd(y)) is True
assert ask(Q.negative(x**y), Q.positive(x) & Q.integer(y)) is False
assert ask(Q.negative(Abs(x))) is False
def test_nonzero():
assert ask(Q.nonzero(x)) is None
assert ask(Q.nonzero(x), Q.real(x)) is None
assert ask(Q.nonzero(x), Q.positive(x)) is True
assert ask(Q.nonzero(x), Q.negative(x)) is True
assert ask(Q.nonzero(x), Q.negative(x) | Q.positive(x)) is True
assert ask(Q.nonzero(x + y)) is None
assert ask(Q.nonzero(x + y), Q.positive(x) & Q.positive(y)) is True
assert ask(Q.nonzero(x + y), Q.positive(x) & Q.negative(y)) is None
assert ask(Q.nonzero(x + y), Q.negative(x) & Q.negative(y)) is True
assert ask(Q.nonzero(2*x)) is None
assert ask(Q.nonzero(2*x), Q.positive(x)) is True
assert ask(Q.nonzero(2*x), Q.negative(x)) is True
assert ask(Q.nonzero(x*y), Q.nonzero(x)) is None
assert ask(Q.nonzero(x*y), Q.nonzero(x) & Q.nonzero(y)) is True
assert ask(Q.nonzero(x**y), Q.nonzero(x)) is True
assert ask(Q.nonzero(Abs(x))) is None
assert ask(Q.nonzero(Abs(x)), Q.nonzero(x)) is True
assert ask(Q.nonzero(log(exp(2*I)))) is False
# although this could be False, it is representative of expressions
# that don't evaluate to a zero with precision
assert ask(Q.nonzero(cos(1)**2 + sin(1)**2 - 1)) is None
@slow
def test_zero():
assert ask(Q.zero(x)) is None
assert ask(Q.zero(x), Q.real(x)) is None
assert ask(Q.zero(x), Q.positive(x)) is False
assert ask(Q.zero(x), Q.negative(x)) is False
assert ask(Q.zero(x), Q.negative(x) | Q.positive(x)) is False
assert ask(Q.zero(x), Q.nonnegative(x) & Q.nonpositive(x)) is True
assert ask(Q.zero(x + y)) is None
assert ask(Q.zero(x + y), Q.positive(x) & Q.positive(y)) is False
assert ask(Q.zero(x + y), Q.positive(x) & Q.negative(y)) is None
assert ask(Q.zero(x + y), Q.negative(x) & Q.negative(y)) is False
assert ask(Q.zero(2*x)) is None
assert ask(Q.zero(2*x), Q.positive(x)) is False
assert ask(Q.zero(2*x), Q.negative(x)) is False
assert ask(Q.zero(x*y), Q.nonzero(x)) is None
assert ask(Q.zero(Abs(x))) is None
assert ask(Q.zero(Abs(x)), Q.zero(x)) is True
assert ask(Q.integer(x), Q.zero(x)) is True
assert ask(Q.even(x), Q.zero(x)) is True
assert ask(Q.odd(x), Q.zero(x)) is False
assert ask(Q.zero(x), Q.even(x)) is None
assert ask(Q.zero(x), Q.odd(x)) is False
assert ask(Q.zero(x) | Q.zero(y), Q.zero(x*y)) is True
@slow
def test_odd_query():
assert ask(Q.odd(x)) is None
assert ask(Q.odd(x), Q.odd(x)) is True
assert ask(Q.odd(x), Q.integer(x)) is None
assert ask(Q.odd(x), ~Q.integer(x)) is False
assert ask(Q.odd(x), Q.rational(x)) is None
assert ask(Q.odd(x), Q.positive(x)) is None
assert ask(Q.odd(-x), Q.odd(x)) is True
assert ask(Q.odd(2*x)) is None
assert ask(Q.odd(2*x), Q.integer(x)) is False
assert ask(Q.odd(2*x), Q.odd(x)) is False
assert ask(Q.odd(2*x), Q.irrational(x)) is False
assert ask(Q.odd(2*x), ~Q.integer(x)) is None
assert ask(Q.odd(3*x), Q.integer(x)) is None
assert ask(Q.odd(x/3), Q.odd(x)) is None
assert ask(Q.odd(x/3), Q.even(x)) is None
assert ask(Q.odd(x + 1), Q.even(x)) is True
assert ask(Q.odd(x + 2), Q.even(x)) is False
assert ask(Q.odd(x + 2), Q.odd(x)) is True
assert ask(Q.odd(3 - x), Q.odd(x)) is False
assert ask(Q.odd(3 - x), Q.even(x)) is True
assert ask(Q.odd(3 + x), Q.odd(x)) is False
assert ask(Q.odd(3 + x), Q.even(x)) is True
assert ask(Q.odd(x + y), Q.odd(x) & Q.odd(y)) is False
assert ask(Q.odd(x + y), Q.odd(x) & Q.even(y)) is True
assert ask(Q.odd(x - y), Q.even(x) & Q.odd(y)) is True
assert ask(Q.odd(x - y), Q.odd(x) & Q.odd(y)) is False
assert ask(Q.odd(x + y + z), Q.odd(x) & Q.odd(y) & Q.even(z)) is False
assert ask(Q.odd(x + y + z + t),
Q.odd(x) & Q.odd(y) & Q.even(z) & Q.integer(t)) is None
assert ask(Q.odd(2*x + 1), Q.integer(x)) is True
assert ask(Q.odd(2*x + y), Q.integer(x) & Q.odd(y)) is True
assert ask(Q.odd(2*x + y), Q.integer(x) & Q.even(y)) is False
assert ask(Q.odd(2*x + y), Q.integer(x) & Q.integer(y)) is None
assert ask(Q.odd(x*y), Q.odd(x) & Q.even(y)) is False
assert ask(Q.odd(x*y), Q.odd(x) & Q.odd(y)) is True
assert ask(Q.odd(2*x*y), Q.rational(x) & Q.rational(x)) is None
assert ask(Q.odd(2*x*y), Q.irrational(x) & Q.irrational(x)) is None
assert ask(Q.odd(Abs(x)), Q.odd(x)) is True
assert ask(Q.odd((-1)**n), Q.integer(n)) is True
assert ask(Q.odd(k**2), Q.even(k)) is False
assert ask(Q.odd(n**2), Q.odd(n)) is True
assert ask(Q.odd(3**k), Q.even(k)) is None
assert ask(Q.odd(k**m), Q.even(k) & Q.integer(m) & ~Q.negative(m)) is None
assert ask(Q.odd(n**m), Q.odd(n) & Q.integer(m) & ~Q.negative(m)) is True
assert ask(Q.odd(k**p), Q.even(k) & Q.integer(p) & Q.positive(p)) is False
assert ask(Q.odd(n**p), Q.odd(n) & Q.integer(p) & Q.positive(p)) is True
assert ask(Q.odd(m**k), Q.even(k) & Q.integer(m) & ~Q.negative(m)) is None
assert ask(Q.odd(p**k), Q.even(k) & Q.integer(p) & Q.positive(p)) is None
assert ask(Q.odd(m**n), Q.odd(n) & Q.integer(m) & ~Q.negative(m)) is None
assert ask(Q.odd(p**n), Q.odd(n) & Q.integer(p) & Q.positive(p)) is None
assert ask(Q.odd(k**x), Q.even(k)) is None
assert ask(Q.odd(n**x), Q.odd(n)) is None
assert ask(Q.odd(x*y), Q.integer(x) & Q.integer(y)) is None
assert ask(Q.odd(x*x), Q.integer(x)) is None
assert ask(Q.odd(x*(x + y)), Q.integer(x) & Q.odd(y)) is False
assert ask(Q.odd(x*(x + y)), Q.integer(x) & Q.even(y)) is None
@XFAIL
def test_oddness_in_ternary_integer_product_with_odd():
# Tests that oddness inference is independent of term ordering.
# Term ordering at the point of testing depends on SymPy's symbol order, so
# we try to force a different order by modifying symbol names.
assert ask(Q.odd(x*y*(y + z)), Q.integer(x) & Q.integer(y) & Q.odd(z)) is False
assert ask(Q.odd(y*x*(x + z)), Q.integer(x) & Q.integer(y) & Q.odd(z)) is False
def test_oddness_in_ternary_integer_product_with_even():
assert ask(Q.odd(x*y*(y + z)), Q.integer(x) & Q.integer(y) & Q.even(z)) is None
def test_prime():
assert ask(Q.prime(x), Q.prime(x)) is True
assert ask(Q.prime(x), ~Q.prime(x)) is False
assert ask(Q.prime(x), Q.integer(x)) is None
assert ask(Q.prime(x), ~Q.integer(x)) is False
assert ask(Q.prime(2*x), Q.integer(x)) is None
assert ask(Q.prime(x*y)) is None
assert ask(Q.prime(x*y), Q.prime(x)) is None
assert ask(Q.prime(x*y), Q.integer(x) & Q.integer(y)) is None
assert ask(Q.prime(4*x), Q.integer(x)) is False
assert ask(Q.prime(4*x)) is None
assert ask(Q.prime(x**2), Q.integer(x)) is False
assert ask(Q.prime(x**2), Q.prime(x)) is False
assert ask(Q.prime(x**y), Q.integer(x) & Q.integer(y)) is False
@slow
def test_positive():
assert ask(Q.positive(x), Q.positive(x)) is True
assert ask(Q.positive(x), Q.negative(x)) is False
assert ask(Q.positive(x), Q.nonzero(x)) is None
assert ask(Q.positive(-x), Q.positive(x)) is False
assert ask(Q.positive(-x), Q.negative(x)) is True
assert ask(Q.positive(x + y), Q.positive(x) & Q.positive(y)) is True
assert ask(Q.positive(x + y), Q.positive(x) & Q.nonnegative(y)) is True
assert ask(Q.positive(x + y), Q.positive(x) & Q.negative(y)) is None
assert ask(Q.positive(x + y), Q.positive(x) & Q.imaginary(y)) is False
assert ask(Q.positive(2*x), Q.positive(x)) is True
assumptions = Q.positive(x) & Q.negative(y) & Q.negative(z) & Q.positive(w)
assert ask(Q.positive(x*y*z)) is None
assert ask(Q.positive(x*y*z), assumptions) is True
assert ask(Q.positive(-x*y*z), assumptions) is False
assert ask(Q.positive(x**I), Q.positive(x)) is None
assert ask(Q.positive(x**2), Q.positive(x)) is True
assert ask(Q.positive(x**2), Q.negative(x)) is True
assert ask(Q.positive(x**3), Q.negative(x)) is False
assert ask(Q.positive(1/(1 + x**2)), Q.real(x)) is True
assert ask(Q.positive(2**I)) is False
assert ask(Q.positive(2 + I)) is False
# although this could be False, it is representative of expressions
# that don't evaluate to a zero with precision
assert ask(Q.positive(cos(I)**2 + sin(I)**2 - 1)) is None
assert ask(Q.positive(-I + I*(cos(2)**2 + sin(2)**2))) is None
#exponential
assert ask(Q.positive(exp(x)), Q.real(x)) is True
assert ask(~Q.negative(exp(x)), Q.real(x)) is True
assert ask(Q.positive(x + exp(x)), Q.real(x)) is None
assert ask(Q.positive(exp(x)), Q.imaginary(x)) is None
assert ask(Q.positive(exp(2*pi*I, evaluate=False)), Q.imaginary(x)) is True
assert ask(Q.negative(exp(pi*I, evaluate=False)), Q.imaginary(x)) is True
assert ask(Q.positive(exp(x*pi*I)), Q.even(x)) is True
assert ask(Q.positive(exp(x*pi*I)), Q.odd(x)) is False
assert ask(Q.positive(exp(x*pi*I)), Q.real(x)) is None
# logarithm
assert ask(Q.positive(log(x)), Q.imaginary(x)) is False
assert ask(Q.positive(log(x)), Q.negative(x)) is False
assert ask(Q.positive(log(x)), Q.positive(x)) is None
assert ask(Q.positive(log(x + 2)), Q.positive(x)) is True
# factorial
assert ask(Q.positive(factorial(x)), Q.integer(x) & Q.positive(x))
assert ask(Q.positive(factorial(x)), Q.integer(x)) is None
#absolute value
assert ask(Q.positive(Abs(x))) is None # Abs(0) = 0
assert ask(Q.positive(Abs(x)), Q.positive(x)) is True
def test_nonpositive():
assert ask(Q.nonpositive(-1))
assert ask(Q.nonpositive(0))
assert ask(Q.nonpositive(1)) is False
assert ask(~Q.positive(x), Q.nonpositive(x))
assert ask(Q.nonpositive(x), Q.positive(x)) is False
assert ask(Q.nonpositive(sqrt(-1))) is False
assert ask(Q.nonpositive(x), Q.imaginary(x)) is False
def test_nonnegative():
assert ask(Q.nonnegative(-1)) is False
assert ask(Q.nonnegative(0))
assert ask(Q.nonnegative(1))
assert ask(~Q.negative(x), Q.nonnegative(x))
assert ask(Q.nonnegative(x), Q.negative(x)) is False
assert ask(Q.nonnegative(sqrt(-1))) is False
assert ask(Q.nonnegative(x), Q.imaginary(x)) is False
def test_real_basic():
assert ask(Q.real(x)) is None
assert ask(Q.real(x), Q.real(x)) is True
assert ask(Q.real(x), Q.nonzero(x)) is True
assert ask(Q.real(x), Q.positive(x)) is True
assert ask(Q.real(x), Q.negative(x)) is True
assert ask(Q.real(x), Q.integer(x)) is True
assert ask(Q.real(x), Q.even(x)) is True
assert ask(Q.real(x), Q.prime(x)) is True
assert ask(Q.real(x/sqrt(2)), Q.real(x)) is True
assert ask(Q.real(x/sqrt(-2)), Q.real(x)) is False
assert ask(Q.real(x + 1), Q.real(x)) is True
assert ask(Q.real(x + I), Q.real(x)) is False
assert ask(Q.real(x + I), Q.complex(x)) is None
assert ask(Q.real(2*x), Q.real(x)) is True
assert ask(Q.real(I*x), Q.real(x)) is False
assert ask(Q.real(I*x), Q.imaginary(x)) is True
assert ask(Q.real(I*x), Q.complex(x)) is None
@slow
def test_real_pow():
assert ask(Q.real(x**2), Q.real(x)) is True
assert ask(Q.real(sqrt(x)), Q.negative(x)) is False
assert ask(Q.real(x**y), Q.real(x) & Q.integer(y)) is True
assert ask(Q.real(x**y), Q.real(x) & Q.real(y)) is None
assert ask(Q.real(x**y), Q.positive(x) & Q.real(y)) is True
assert ask(Q.real(x**y), Q.imaginary(x) & Q.imaginary(y)) is None # I**I or (2*I)**I
assert ask(Q.real(x**y), Q.imaginary(x) & Q.real(y)) is None # I**1 or I**0
assert ask(Q.real(x**y), Q.real(x) & Q.imaginary(y)) is None # could be exp(2*pi*I) or 2**I
assert ask(Q.real(x**0), Q.imaginary(x)) is True
assert ask(Q.real(x**y), Q.real(x) & Q.integer(y)) is True
assert ask(Q.real(x**y), Q.positive(x) & Q.real(y)) is True
assert ask(Q.real(x**y), Q.real(x) & Q.rational(y)) is None
assert ask(Q.real(x**y), Q.imaginary(x) & Q.integer(y)) is None
assert ask(Q.real(x**y), Q.imaginary(x) & Q.odd(y)) is False
assert ask(Q.real(x**y), Q.imaginary(x) & Q.even(y)) is True
assert ask(Q.real(x**(y/z)), Q.real(x) & Q.real(y/z) & Q.rational(y/z) & Q.even(z) & Q.positive(x)) is True
assert ask(Q.real(x**(y/z)), Q.real(x) & Q.rational(y/z) & Q.even(z) & Q.negative(x)) is False
assert ask(Q.real(x**(y/z)), Q.real(x) & Q.integer(y/z)) is True
assert ask(Q.real(x**(y/z)), Q.real(x) & Q.real(y/z) & Q.positive(x)) is True
assert ask(Q.real(x**(y/z)), Q.real(x) & Q.real(y/z) & Q.negative(x)) is False
assert ask(Q.real((-I)**i), Q.imaginary(i)) is True
assert ask(Q.real(I**i), Q.imaginary(i)) is True
assert ask(Q.real(i**i), Q.imaginary(i)) is None # i might be 2*I
assert ask(Q.real(x**i), Q.imaginary(i)) is None # x could be 0
assert ask(Q.real(x**(I*pi/log(x))), Q.real(x)) is True
def test_real_functions():
# trigonometric functions
assert ask(Q.real(sin(x))) is None
assert ask(Q.real(cos(x))) is None
assert ask(Q.real(sin(x)), Q.real(x)) is True
assert ask(Q.real(cos(x)), Q.real(x)) is True
# exponential function
assert ask(Q.real(exp(x))) is None
assert ask(Q.real(exp(x)), Q.real(x)) is True
assert ask(Q.real(x + exp(x)), Q.real(x)) is True
assert ask(Q.real(exp(2*pi*I, evaluate=False))) is True
assert ask(Q.real(exp(pi*I, evaluate=False))) is True
assert ask(Q.real(exp(pi*I/2, evaluate=False))) is False
# logarithm
assert ask(Q.real(log(I))) is False
assert ask(Q.real(log(2*I))) is False
assert ask(Q.real(log(I + 1))) is False
assert ask(Q.real(log(x)), Q.complex(x)) is None
assert ask(Q.real(log(x)), Q.imaginary(x)) is False
assert ask(Q.real(log(exp(x))), Q.imaginary(x)) is None # exp(2*pi*I) is 1, log(exp(pi*I)) is pi*I (disregarding periodicity)
assert ask(Q.real(log(exp(x))), Q.complex(x)) is None
eq = Pow(exp(2*pi*I*x, evaluate=False), x, evaluate=False)
assert ask(Q.real(eq), Q.integer(x)) is True
assert ask(Q.real(exp(x)**x), Q.imaginary(x)) is True
assert ask(Q.real(exp(x)**x), Q.complex(x)) is None
# Q.complexes
assert ask(Q.real(re(x))) is True
assert ask(Q.real(im(x))) is True
def test_algebraic():
assert ask(Q.algebraic(x)) is None
assert ask(Q.algebraic(I)) is True
assert ask(Q.algebraic(2*I)) is True
assert ask(Q.algebraic(I/3)) is True
assert ask(Q.algebraic(sqrt(7))) is True
assert ask(Q.algebraic(2*sqrt(7))) is True
assert ask(Q.algebraic(sqrt(7)/3)) is True
assert ask(Q.algebraic(I*sqrt(3))) is True
assert ask(Q.algebraic(sqrt(1 + I*sqrt(3)))) is True
assert ask(Q.algebraic((1 + I*sqrt(3)**(S(17)/31)))) is True
assert ask(Q.algebraic((1 + I*sqrt(3)**(S(17)/pi)))) is False
for f in [exp, sin, tan, asin, atan, cos]:
assert ask(Q.algebraic(f(7))) is False
assert ask(Q.algebraic(f(7, evaluate=False))) is False
assert ask(Q.algebraic(f(0, evaluate=False))) is True
assert ask(Q.algebraic(f(x)), Q.algebraic(x)) is None
assert ask(Q.algebraic(f(x)), Q.algebraic(x) & Q.nonzero(x)) is False
for g in [log, acos]:
assert ask(Q.algebraic(g(7))) is False
assert ask(Q.algebraic(g(7, evaluate=False))) is False
assert ask(Q.algebraic(g(1, evaluate=False))) is True
assert ask(Q.algebraic(g(x)), Q.algebraic(x)) is None
assert ask(Q.algebraic(g(x)), Q.algebraic(x) & Q.nonzero(x - 1)) is False
for h in [cot, acot]:
assert ask(Q.algebraic(h(7))) is False
assert ask(Q.algebraic(h(7, evaluate=False))) is False
assert ask(Q.algebraic(h(x)), Q.algebraic(x)) is False
assert ask(Q.algebraic(sqrt(sin(7)))) is False
assert ask(Q.algebraic(sqrt(y + I*sqrt(7)))) is None
assert ask(Q.algebraic(2.47)) is True
assert ask(Q.algebraic(x), Q.transcendental(x)) is False
assert ask(Q.transcendental(x), Q.algebraic(x)) is False
def test_global():
"""Test ask with global assumptions"""
assert ask(Q.integer(x)) is None
global_assumptions.add(Q.integer(x))
assert ask(Q.integer(x)) is True
global_assumptions.clear()
assert ask(Q.integer(x)) is None
def test_custom_context():
"""Test ask with custom assumptions context"""
assert ask(Q.integer(x)) is None
local_context = AssumptionsContext()
local_context.add(Q.integer(x))
assert ask(Q.integer(x), context=local_context) is True
assert ask(Q.integer(x)) is None
def test_functions_in_assumptions():
assert ask(Q.negative(x), Q.real(x) >> Q.positive(x)) is False
assert ask(Q.negative(x), Equivalent(Q.real(x), Q.positive(x))) is False
assert ask(Q.negative(x), Xor(Q.real(x), Q.negative(x))) is False
def test_composite_ask():
assert ask(Q.negative(x) & Q.integer(x),
assumptions=Q.real(x) >> Q.positive(x)) is False
def test_composite_proposition():
assert ask(True) is True
assert ask(False) is False
assert ask(~Q.negative(x), Q.positive(x)) is True
assert ask(~Q.real(x), Q.commutative(x)) is None
assert ask(Q.negative(x) & Q.integer(x), Q.positive(x)) is False
assert ask(Q.negative(x) & Q.integer(x)) is None
assert ask(Q.real(x) | Q.integer(x), Q.positive(x)) is True
assert ask(Q.real(x) | Q.integer(x)) is None
assert ask(Q.real(x) >> Q.positive(x), Q.negative(x)) is False
assert ask(Implies(
Q.real(x), Q.positive(x), evaluate=False), Q.negative(x)) is False
assert ask(Implies(Q.real(x), Q.positive(x), evaluate=False)) is None
assert ask(Equivalent(Q.integer(x), Q.even(x)), Q.even(x)) is True
assert ask(Equivalent(Q.integer(x), Q.even(x))) is None
assert ask(Equivalent(Q.positive(x), Q.integer(x)), Q.integer(x)) is None
assert ask(Q.real(x) | Q.integer(x), Q.real(x) | Q.integer(x)) is True
def test_tautology():
assert ask(Q.real(x) | ~Q.real(x)) is True
assert ask(Q.real(x) & ~Q.real(x)) is False
def test_composite_assumptions():
assert ask(Q.real(x), Q.real(x) & Q.real(y)) is True
assert ask(Q.positive(x), Q.positive(x) | Q.positive(y)) is None
assert ask(Q.positive(x), Q.real(x) >> Q.positive(y)) is None
assert ask(Q.real(x), ~(Q.real(x) >> Q.real(y))) is True
def test_incompatible_resolutors():
class Prime2AskHandler(AskHandler):
@staticmethod
def Number(expr, assumptions):
return True
register_handler('prime', Prime2AskHandler)
raises(ValueError, lambda: ask(Q.prime(4)))
remove_handler('prime', Prime2AskHandler)
class InconclusiveHandler(AskHandler):
@staticmethod
def Number(expr, assumptions):
return None
register_handler('prime', InconclusiveHandler)
assert ask(Q.prime(3)) is True
remove_handler('prime', InconclusiveHandler)
def test_key_extensibility():
"""test that you can add keys to the ask system at runtime"""
# make sure the key is not defined
raises(AttributeError, lambda: ask(Q.my_key(x)))
class MyAskHandler(AskHandler):
@staticmethod
def Symbol(expr, assumptions):
return True
register_handler('my_key', MyAskHandler)
assert ask(Q.my_key(x)) is True
assert ask(Q.my_key(x + 1)) is None
remove_handler('my_key', MyAskHandler)
del Q.my_key
raises(AttributeError, lambda: ask(Q.my_key(x)))
def test_type_extensibility():
"""test that new types can be added to the ask system at runtime
We create a custom type MyType, and override ask Q.prime=True with handler
MyAskHandler for this type
TODO: test incompatible resolutors
"""
from sympy.core import Basic
class MyType(Basic):
pass
class MyAskHandler(AskHandler):
@staticmethod
def MyType(expr, assumptions):
return True
a = MyType()
register_handler(Q.prime, MyAskHandler)
assert ask(Q.prime(a)) is True
def test_single_fact_lookup():
known_facts = And(Implies(Q.integer, Q.rational),
Implies(Q.rational, Q.real),
Implies(Q.real, Q.complex))
known_facts_keys = {Q.integer, Q.rational, Q.real, Q.complex}
known_facts_cnf = to_cnf(known_facts)
mapping = single_fact_lookup(known_facts_keys, known_facts_cnf)
assert mapping[Q.rational] == {Q.real, Q.rational, Q.complex}
def test_compute_known_facts():
known_facts = And(Implies(Q.integer, Q.rational),
Implies(Q.rational, Q.real),
Implies(Q.real, Q.complex))
known_facts_keys = {Q.integer, Q.rational, Q.real, Q.complex}
s = compute_known_facts(known_facts, known_facts_keys)
@slow
def test_known_facts_consistent():
""""Test that ask_generated.py is up-to-date"""
from sympy.assumptions.ask import get_known_facts, get_known_facts_keys
from os.path import abspath, dirname, join
filename = join(dirname(dirname(abspath(__file__))), 'ask_generated.py')
with open(filename, 'r') as f:
assert f.read() == \
compute_known_facts(get_known_facts(), get_known_facts_keys())
def test_Add_queries():
assert ask(Q.prime(12345678901234567890 + (cos(1)**2 + sin(1)**2))) is True
assert ask(Q.even(Add(S(2), S(2), evaluate=0))) is True
assert ask(Q.prime(Add(S(2), S(2), evaluate=0))) is False
assert ask(Q.integer(Add(S(2), S(2), evaluate=0))) is True
def test_positive_assuming():
with assuming(Q.positive(x + 1)):
assert not ask(Q.positive(x))
def test_issue_5421():
raises(TypeError, lambda: ask(pi/log(x), Q.real))
def test_issue_3906():
raises(TypeError, lambda: ask(Q.positive))
def test_issue_5833():
assert ask(Q.positive(log(x)**2), Q.positive(x)) is None
assert ask(~Q.negative(log(x)**2), Q.positive(x)) is True
def test_issue_6732():
raises(ValueError, lambda: ask(Q.positive(x), Q.positive(x) & Q.negative(x)))
raises(ValueError, lambda: ask(Q.negative(x), Q.positive(x) & Q.negative(x)))
def test_issue_7246():
assert ask(Q.positive(atan(p)), Q.positive(p)) is True
assert ask(Q.positive(atan(p)), Q.negative(p)) is False
assert ask(Q.positive(atan(p)), Q.zero(p)) is False
assert ask(Q.positive(atan(x))) is None
assert ask(Q.positive(asin(p)), Q.positive(p)) is None
assert ask(Q.positive(asin(p)), Q.zero(p)) is None
assert ask(Q.positive(asin(Rational(1, 7)))) is True
assert ask(Q.positive(asin(x)), Q.positive(x) & Q.nonpositive(x - 1)) is True
assert ask(Q.positive(asin(x)), Q.negative(x) & Q.nonnegative(x + 1)) is False
assert ask(Q.positive(acos(p)), Q.positive(p)) is None
assert ask(Q.positive(acos(Rational(1, 7)))) is True
assert ask(Q.positive(acos(x)), Q.nonnegative(x + 1) & Q.nonpositive(x - 1)) is True
assert ask(Q.positive(acos(x)), Q.nonnegative(x - 1)) is None
assert ask(Q.positive(acot(x)), Q.positive(x)) is True
assert ask(Q.positive(acot(x)), Q.real(x)) is True
assert ask(Q.positive(acot(x)), Q.imaginary(x)) is False
assert ask(Q.positive(acot(x))) is None
@XFAIL
def test_issue_7246_failing():
#Move this test to test_issue_7246 once
#the new assumptions module is improved.
assert ask(Q.positive(acos(x)), Q.zero(x)) is True
def test_deprecated_Q_bounded():
with warns_deprecated_sympy():
Q.bounded
def test_deprecated_Q_infinity():
with warns_deprecated_sympy():
Q.infinity
def test_check_old_assumption():
x = symbols('x', real=True)
assert ask(Q.real(x)) is True
assert ask(Q.imaginary(x)) is False
assert ask(Q.complex(x)) is True
x = symbols('x', imaginary=True)
assert ask(Q.real(x)) is False
assert ask(Q.imaginary(x)) is True
assert ask(Q.complex(x)) is True
x = symbols('x', complex=True)
assert ask(Q.real(x)) is None
assert ask(Q.complex(x)) is True
x = symbols('x', positive=True)
assert ask(Q.positive(x)) is True
assert ask(Q.negative(x)) is False
assert ask(Q.real(x)) is True
x = symbols('x', commutative=False)
assert ask(Q.commutative(x)) is False
x = symbols('x', negative=True)
assert ask(Q.positive(x)) is False
assert ask(Q.negative(x)) is True
x = symbols('x', nonnegative=True)
assert ask(Q.negative(x)) is False
assert ask(Q.positive(x)) is None
assert ask(Q.zero(x)) is None
x = symbols('x', finite=True)
assert ask(Q.finite(x)) is True
x = symbols('x', prime=True)
assert ask(Q.prime(x)) is True
assert ask(Q.composite(x)) is False
x = symbols('x', composite=True)
assert ask(Q.prime(x)) is False
assert ask(Q.composite(x)) is True
x = symbols('x', even=True)
assert ask(Q.even(x)) is True
assert ask(Q.odd(x)) is False
x = symbols('x', odd=True)
assert ask(Q.even(x)) is False
assert ask(Q.odd(x)) is True
x = symbols('x', nonzero=True)
assert ask(Q.nonzero(x)) is True
assert ask(Q.zero(x)) is False
x = symbols('x', zero=True)
assert ask(Q.zero(x)) is True
x = symbols('x', integer=True)
assert ask(Q.integer(x)) is True
x = symbols('x', rational=True)
assert ask(Q.rational(x)) is True
assert ask(Q.irrational(x)) is False
x = symbols('x', irrational=True)
assert ask(Q.irrational(x)) is True
assert ask(Q.rational(x)) is False
def test_issue_9636():
assert ask(Q.integer(1.0)) is False
assert ask(Q.prime(3.0)) is False
assert ask(Q.composite(4.0)) is False
assert ask(Q.even(2.0)) is False
assert ask(Q.odd(3.0)) is False
def test_autosimp_used_to_fail():
# See issue #9807
assert ask(Q.imaginary(0**I)) is False
assert ask(Q.imaginary(0**(-I))) is False
assert ask(Q.real(0**I)) is False
assert ask(Q.real(0**(-I))) is False
|
9151e5637dddbea9df0a23161b4555c63d2d5ff4562295b9ba713c603a61f6bd
|
"""
This module implements some special functions that commonly appear in
combinatorial contexts (e.g. in power series); in particular,
sequences of rational numbers such as Bernoulli and Fibonacci numbers.
Factorials, binomial coefficients and related functions are located in
the separate 'factorials' module.
"""
from __future__ import print_function, division
from sympy.core import S, Symbol, Rational, Integer, Add, Dummy
from sympy.core.cache import cacheit
from sympy.core.compatibility import as_int, SYMPY_INTS, range
from sympy.core.function import Function, expand_mul
from sympy.core.logic import fuzzy_not
from sympy.core.numbers import E, pi
from sympy.core.relational import LessThan, StrictGreaterThan
from sympy.functions.combinatorial.factorials import binomial, factorial
from sympy.functions.elementary.exponential import log
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt, cbrt
from sympy.functions.elementary.trigonometric import sin, cos, cot
from sympy.ntheory import isprime
from sympy.ntheory.primetest import is_square
from sympy.utilities.memoization import recurrence_memo
from mpmath import bernfrac, workprec
from mpmath.libmp import ifib as _ifib
def _product(a, b):
p = 1
for k in range(a, b + 1):
p *= k
return p
# Dummy symbol used for computing polynomial sequences
_sym = Symbol('x')
_symbols = Function('x')
#----------------------------------------------------------------------------#
# #
# Carmichael numbers #
# #
#----------------------------------------------------------------------------#
class carmichael(Function):
"""
Carmichael Numbers:
Certain cryptographic algorithms make use of big prime numbers.
However, checking whether a big number is prime is not so easy.
Randomized prime number checking tests exist that offer a high degree of confidence of
accurate determination at low cost, such as the Fermat test.
Let 'a' be a random number between 2 and n - 1, where n is the number whose primality we are testing.
Then, n is probably prime if it satisfies the modular arithmetic congruence relation :
a^(n-1) = 1(mod n).
(where mod refers to the modulo operation)
If a number passes the Fermat test several times, then it is prime with a
high probability.
Unfortunately, certain composite numbers (non-primes) still pass the Fermat test
with every number smaller than themselves.
These numbers are called Carmichael numbers.
A Carmichael number will pass a Fermat primality test to every base b relatively prime to the number,
even though it is not actually prime. This makes tests based on Fermat's Little Theorem less effective than
strong probable prime tests such as the Baillie-PSW primality test and the Miller-Rabin primality test.
mr functions given in sympy/sympy/ntheory/primetest.py will produce wrong results for each and every
carmichael number.
Examples
========
>>> from sympy import carmichael
>>> carmichael.find_first_n_carmichaels(5)
[561, 1105, 1729, 2465, 2821]
>>> carmichael.is_prime(2465)
False
>>> carmichael.is_prime(1729)
False
>>> carmichael.find_carmichael_numbers_in_range(0, 562)
[561]
>>> carmichael.find_carmichael_numbers_in_range(0,1000)
[561]
>>> carmichael.find_carmichael_numbers_in_range(0,2000)
[561, 1105, 1729]
References
==========
.. [1] https://en.wikipedia.org/wiki/Carmichael_number
.. [2] https://en.wikipedia.org/wiki/Fermat_primality_test
.. [3] https://www.jstor.org/stable/23248683?seq=1#metadata_info_tab_contents
"""
@staticmethod
def is_perfect_square(n):
return is_square(n)
@staticmethod
def divides(p, n):
return n % p == 0
@staticmethod
def is_prime(n):
return isprime(n)
@staticmethod
def is_carmichael(n):
if n >= 0:
if (n == 1) or (carmichael.is_prime(n)) or (n % 2 == 0):
return False
divisors = list([1, n])
# get divisors
for i in range(3, n // 2 + 1, 2):
if n % i == 0:
divisors.append(i)
for i in divisors:
if carmichael.is_perfect_square(i) and i != 1:
return False
if carmichael.is_prime(i):
if not carmichael.divides(i - 1, n - 1):
return False
return True
else:
raise ValueError('The provided number must be greater than or equal to 0')
@staticmethod
def find_carmichael_numbers_in_range(x, y):
if 0 <= x <= y:
if x % 2 == 0:
return list([i for i in range(x + 1, y, 2) if carmichael.is_carmichael(i)])
else:
return list([i for i in range(x, y, 2) if carmichael.is_carmichael(i)])
else:
raise ValueError('The provided range is not valid. x and y must be non-negative integers and x <= y')
@staticmethod
def find_first_n_carmichaels(n):
i = 1
carmichaels = list()
while len(carmichaels) < n:
if carmichael.is_carmichael(i):
carmichaels.append(i)
i += 2
return carmichaels
#----------------------------------------------------------------------------#
# #
# Fibonacci numbers #
# #
#----------------------------------------------------------------------------#
class fibonacci(Function):
r"""
Fibonacci numbers / Fibonacci polynomials
The Fibonacci numbers are the integer sequence defined by the
initial terms `F_0 = 0`, `F_1 = 1` and the two-term recurrence
relation `F_n = F_{n-1} + F_{n-2}`. This definition
extended to arbitrary real and complex arguments using
the formula
.. math :: F_z = \frac{\phi^z - \cos(\pi z) \phi^{-z}}{\sqrt 5}
The Fibonacci polynomials are defined by `F_1(x) = 1`,
`F_2(x) = x`, and `F_n(x) = x*F_{n-1}(x) + F_{n-2}(x)` for `n > 2`.
For all positive integers `n`, `F_n(1) = F_n`.
* ``fibonacci(n)`` gives the `n^{th}` Fibonacci number, `F_n`
* ``fibonacci(n, x)`` gives the `n^{th}` Fibonacci polynomial in `x`, `F_n(x)`
Examples
========
>>> from sympy import fibonacci, Symbol
>>> [fibonacci(x) for x in range(11)]
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
>>> fibonacci(5, Symbol('t'))
t**4 + 3*t**2 + 1
See Also
========
bell, bernoulli, catalan, euler, harmonic, lucas, genocchi, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Fibonacci_number
.. [2] http://mathworld.wolfram.com/FibonacciNumber.html
"""
@staticmethod
def _fib(n):
return _ifib(n)
@staticmethod
@recurrence_memo([None, S.One, _sym])
def _fibpoly(n, prev):
return (prev[-2] + _sym*prev[-1]).expand()
@classmethod
def eval(cls, n, sym=None):
if n is S.Infinity:
return S.Infinity
if n.is_Integer:
n = int(n)
if n < 0:
return S.NegativeOne**(n + 1) * fibonacci(-n)
if sym is None:
return Integer(cls._fib(n))
else:
if n < 1:
raise ValueError("Fibonacci polynomials are defined "
"only for positive integer indices.")
return cls._fibpoly(n).subs(_sym, sym)
def _eval_rewrite_as_sqrt(self, n, **kwargs):
return 2**(-n)*sqrt(5)*((1 + sqrt(5))**n - (-sqrt(5) + 1)**n) / 5
def _eval_rewrite_as_GoldenRatio(self,n, **kwargs):
return (S.GoldenRatio**n - 1/(-S.GoldenRatio)**n)/(2*S.GoldenRatio-1)
#----------------------------------------------------------------------------#
# #
# Lucas numbers #
# #
#----------------------------------------------------------------------------#
class lucas(Function):
"""
Lucas numbers
Lucas numbers satisfy a recurrence relation similar to that of
the Fibonacci sequence, in which each term is the sum of the
preceding two. They are generated by choosing the initial
values `L_0 = 2` and `L_1 = 1`.
* ``lucas(n)`` gives the `n^{th}` Lucas number
Examples
========
>>> from sympy import lucas
>>> [lucas(x) for x in range(11)]
[2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123]
See Also
========
bell, bernoulli, catalan, euler, fibonacci, harmonic, genocchi, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Lucas_number
.. [2] http://mathworld.wolfram.com/LucasNumber.html
"""
@classmethod
def eval(cls, n):
if n is S.Infinity:
return S.Infinity
if n.is_Integer:
return fibonacci(n + 1) + fibonacci(n - 1)
def _eval_rewrite_as_sqrt(self, n, **kwargs):
return 2**(-n)*((1 + sqrt(5))**n + (-sqrt(5) + 1)**n)
#----------------------------------------------------------------------------#
# #
# Tribonacci numbers #
# #
#----------------------------------------------------------------------------#
class tribonacci(Function):
r"""
Tribonacci numbers / Tribonacci polynomials
The Tribonacci numbers are the integer sequence defined by the
initial terms `T_0 = 0`, `T_1 = 1`, `T_2 = 1` and the three-term
recurrence relation `T_n = T_{n-1} + T_{n-2} + T_{n-3}`.
The Tribonacci polynomials are defined by `T_0(x) = 0`, `T_1(x) = 1`,
`T_2(x) = x^2`, and `T_n(x) = x^2 T_{n-1}(x) + x T_{n-2}(x) + T_{n-3}(x)`
for `n > 2`. For all positive integers `n`, `T_n(1) = T_n`.
* ``tribonacci(n)`` gives the `n^{th}` Tribonacci number, `T_n`
* ``tribonacci(n, x)`` gives the `n^{th}` Tribonacci polynomial in `x`, `T_n(x)`
Examples
========
>>> from sympy import tribonacci, Symbol
>>> [tribonacci(x) for x in range(11)]
[0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149]
>>> tribonacci(5, Symbol('t'))
t**8 + 3*t**5 + 3*t**2
See Also
========
bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition
References
==========
.. [1] https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers#Tribonacci_numbers
.. [2] http://mathworld.wolfram.com/TribonacciNumber.html
.. [3] https://oeis.org/A000073
"""
@staticmethod
@recurrence_memo([S.Zero, S.One, S.One])
def _trib(n, prev):
return (prev[-3] + prev[-2] + prev[-1])
@staticmethod
@recurrence_memo([S.Zero, S.One, _sym**2])
def _tribpoly(n, prev):
return (prev[-3] + _sym*prev[-2] + _sym**2*prev[-1]).expand()
@classmethod
def eval(cls, n, sym=None):
if n is S.Infinity:
return S.Infinity
if n.is_Integer:
n = int(n)
if n < 0:
raise ValueError("Tribonacci polynomials are defined "
"only for non-negative integer indices.")
if sym is None:
return Integer(cls._trib(n))
else:
return cls._tribpoly(n).subs(_sym, sym)
def _eval_rewrite_as_sqrt(self, n, **kwargs):
w = (-1 + S.ImaginaryUnit * sqrt(3)) / 2
a = (1 + cbrt(19 + 3*sqrt(33)) + cbrt(19 - 3*sqrt(33))) / 3
b = (1 + w*cbrt(19 + 3*sqrt(33)) + w**2*cbrt(19 - 3*sqrt(33))) / 3
c = (1 + w**2*cbrt(19 + 3*sqrt(33)) + w*cbrt(19 - 3*sqrt(33))) / 3
Tn = (a**(n + 1)/((a - b)*(a - c))
+ b**(n + 1)/((b - a)*(b - c))
+ c**(n + 1)/((c - a)*(c - b)))
return Tn
def _eval_rewrite_as_TribonacciConstant(self, n, **kwargs):
b = cbrt(586 + 102*sqrt(33))
Tn = 3 * b * S.TribonacciConstant**n / (b**2 - 2*b + 4)
return floor(Tn + S.Half)
#----------------------------------------------------------------------------#
# #
# Bernoulli numbers #
# #
#----------------------------------------------------------------------------#
class bernoulli(Function):
r"""
Bernoulli numbers / Bernoulli polynomials
The Bernoulli numbers are a sequence of rational numbers
defined by `B_0 = 1` and the recursive relation (`n > 0`):
.. math :: 0 = \sum_{k=0}^n \binom{n+1}{k} B_k
They are also commonly defined by their exponential generating
function, which is `\frac{x}{e^x - 1}`. For odd indices > 1, the
Bernoulli numbers are zero.
The Bernoulli polynomials satisfy the analogous formula:
.. math :: B_n(x) = \sum_{k=0}^n \binom{n}{k} B_k x^{n-k}
Bernoulli numbers and Bernoulli polynomials are related as
`B_n(0) = B_n`.
We compute Bernoulli numbers using Ramanujan's formula:
.. math :: B_n = \frac{A(n) - S(n)}{\binom{n+3}{n}}
where:
.. math :: A(n) = \begin{cases} \frac{n+3}{3} &
n \equiv 0\ \text{or}\ 2 \pmod{6} \\
-\frac{n+3}{6} & n \equiv 4 \pmod{6} \end{cases}
and:
.. math :: S(n) = \sum_{k=1}^{[n/6]} \binom{n+3}{n-6k} B_{n-6k}
This formula is similar to the sum given in the definition, but
cuts 2/3 of the terms. For Bernoulli polynomials, we use the
formula in the definition.
* ``bernoulli(n)`` gives the nth Bernoulli number, `B_n`
* ``bernoulli(n, x)`` gives the nth Bernoulli polynomial in `x`, `B_n(x)`
Examples
========
>>> from sympy import bernoulli
>>> [bernoulli(n) for n in range(11)]
[1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66]
>>> bernoulli(1000001)
0
See Also
========
bell, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Bernoulli_number
.. [2] https://en.wikipedia.org/wiki/Bernoulli_polynomial
.. [3] http://mathworld.wolfram.com/BernoulliNumber.html
.. [4] http://mathworld.wolfram.com/BernoulliPolynomial.html
"""
# Calculates B_n for positive even n
@staticmethod
def _calc_bernoulli(n):
s = 0
a = int(binomial(n + 3, n - 6))
for j in range(1, n//6 + 1):
s += a * bernoulli(n - 6*j)
# Avoid computing each binomial coefficient from scratch
a *= _product(n - 6 - 6*j + 1, n - 6*j)
a //= _product(6*j + 4, 6*j + 9)
if n % 6 == 4:
s = -Rational(n + 3, 6) - s
else:
s = Rational(n + 3, 3) - s
return s / binomial(n + 3, n)
# We implement a specialized memoization scheme to handle each
# case modulo 6 separately
_cache = {0: S.One, 2: Rational(1, 6), 4: Rational(-1, 30)}
_highest = {0: 0, 2: 2, 4: 4}
@classmethod
def eval(cls, n, sym=None):
if n.is_Number:
if n.is_Integer and n.is_nonnegative:
if n is S.Zero:
return S.One
elif n is S.One:
if sym is None:
return -S.Half
else:
return sym - S.Half
# Bernoulli numbers
elif sym is None:
if n.is_odd:
return S.Zero
n = int(n)
# Use mpmath for enormous Bernoulli numbers
if n > 500:
p, q = bernfrac(n)
return Rational(int(p), int(q))
case = n % 6
highest_cached = cls._highest[case]
if n <= highest_cached:
return cls._cache[n]
# To avoid excessive recursion when, say, bernoulli(1000) is
# requested, calculate and cache the entire sequence ... B_988,
# B_994, B_1000 in increasing order
for i in range(highest_cached + 6, n + 6, 6):
b = cls._calc_bernoulli(i)
cls._cache[i] = b
cls._highest[case] = i
return b
# Bernoulli polynomials
else:
n, result = int(n), []
for k in range(n + 1):
result.append(binomial(n, k)*cls(k)*sym**(n - k))
return Add(*result)
else:
raise ValueError("Bernoulli numbers are defined only"
" for nonnegative integer indices.")
if sym is None:
if n.is_odd and (n - 1).is_positive:
return S.Zero
#----------------------------------------------------------------------------#
# #
# Bell numbers #
# #
#----------------------------------------------------------------------------#
class bell(Function):
r"""
Bell numbers / Bell polynomials
The Bell numbers satisfy `B_0 = 1` and
.. math:: B_n = \sum_{k=0}^{n-1} \binom{n-1}{k} B_k.
They are also given by:
.. math:: B_n = \frac{1}{e} \sum_{k=0}^{\infty} \frac{k^n}{k!}.
The Bell polynomials are given by `B_0(x) = 1` and
.. math:: B_n(x) = x \sum_{k=1}^{n-1} \binom{n-1}{k-1} B_{k-1}(x).
The second kind of Bell polynomials (are sometimes called "partial" Bell
polynomials or incomplete Bell polynomials) are defined as
.. math:: B_{n,k}(x_1, x_2,\dotsc x_{n-k+1}) =
\sum_{j_1+j_2+j_2+\dotsb=k \atop j_1+2j_2+3j_2+\dotsb=n}
\frac{n!}{j_1!j_2!\dotsb j_{n-k+1}!}
\left(\frac{x_1}{1!} \right)^{j_1}
\left(\frac{x_2}{2!} \right)^{j_2} \dotsb
\left(\frac{x_{n-k+1}}{(n-k+1)!} \right) ^{j_{n-k+1}}.
* ``bell(n)`` gives the `n^{th}` Bell number, `B_n`.
* ``bell(n, x)`` gives the `n^{th}` Bell polynomial, `B_n(x)`.
* ``bell(n, k, (x1, x2, ...))`` gives Bell polynomials of the second kind,
`B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1})`.
Notes
=====
Not to be confused with Bernoulli numbers and Bernoulli polynomials,
which use the same notation.
Examples
========
>>> from sympy import bell, Symbol, symbols
>>> [bell(n) for n in range(11)]
[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975]
>>> bell(30)
846749014511809332450147
>>> bell(4, Symbol('t'))
t**4 + 6*t**3 + 7*t**2 + t
>>> bell(6, 2, symbols('x:6')[1:])
6*x1*x5 + 15*x2*x4 + 10*x3**2
See Also
========
bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Bell_number
.. [2] http://mathworld.wolfram.com/BellNumber.html
.. [3] http://mathworld.wolfram.com/BellPolynomial.html
"""
@staticmethod
@recurrence_memo([1, 1])
def _bell(n, prev):
s = 1
a = 1
for k in range(1, n):
a = a * (n - k) // k
s += a * prev[k]
return s
@staticmethod
@recurrence_memo([S.One, _sym])
def _bell_poly(n, prev):
s = 1
a = 1
for k in range(2, n + 1):
a = a * (n - k + 1) // (k - 1)
s += a * prev[k - 1]
return expand_mul(_sym * s)
@staticmethod
def _bell_incomplete_poly(n, k, symbols):
r"""
The second kind of Bell polynomials (incomplete Bell polynomials).
Calculated by recurrence formula:
.. math:: B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1}) =
\sum_{m=1}^{n-k+1}
\x_m \binom{n-1}{m-1} B_{n-m,k-1}(x_1, x_2, \dotsc, x_{n-m-k})
where
`B_{0,0} = 1;`
`B_{n,0} = 0; for n \ge 1`
`B_{0,k} = 0; for k \ge 1`
"""
if (n == 0) and (k == 0):
return S.One
elif (n == 0) or (k == 0):
return S.Zero
s = S.Zero
a = S.One
for m in range(1, n - k + 2):
s += a * bell._bell_incomplete_poly(
n - m, k - 1, symbols) * symbols[m - 1]
a = a * (n - m) / m
return expand_mul(s)
@classmethod
def eval(cls, n, k_sym=None, symbols=None):
if n is S.Infinity:
if k_sym is None:
return S.Infinity
else:
raise ValueError("Bell polynomial is not defined")
if n.is_negative or n.is_integer is False:
raise ValueError("a non-negative integer expected")
if n.is_Integer and n.is_nonnegative:
if k_sym is None:
return Integer(cls._bell(int(n)))
elif symbols is None:
return cls._bell_poly(int(n)).subs(_sym, k_sym)
else:
r = cls._bell_incomplete_poly(int(n), int(k_sym), symbols)
return r
def _eval_rewrite_as_Sum(self, n, k_sym=None, symbols=None, **kwargs):
from sympy import Sum
if (k_sym is not None) or (symbols is not None):
return self
# Dobinski's formula
if not n.is_nonnegative:
return self
k = Dummy('k', integer=True, nonnegative=True)
return 1 / E * Sum(k**n / factorial(k), (k, 0, S.Infinity))
#----------------------------------------------------------------------------#
# #
# Harmonic numbers #
# #
#----------------------------------------------------------------------------#
class harmonic(Function):
r"""
Harmonic numbers
The nth harmonic number is given by `\operatorname{H}_{n} =
1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}`.
More generally:
.. math:: \operatorname{H}_{n,m} = \sum_{k=1}^{n} \frac{1}{k^m}
As `n \rightarrow \infty`, `\operatorname{H}_{n,m} \rightarrow \zeta(m)`,
the Riemann zeta function.
* ``harmonic(n)`` gives the nth harmonic number, `\operatorname{H}_n`
* ``harmonic(n, m)`` gives the nth generalized harmonic number
of order `m`, `\operatorname{H}_{n,m}`, where
``harmonic(n) == harmonic(n, 1)``
Examples
========
>>> from sympy import harmonic, oo
>>> [harmonic(n) for n in range(6)]
[0, 1, 3/2, 11/6, 25/12, 137/60]
>>> [harmonic(n, 2) for n in range(6)]
[0, 1, 5/4, 49/36, 205/144, 5269/3600]
>>> harmonic(oo, 2)
pi**2/6
>>> from sympy import Symbol, Sum
>>> n = Symbol("n")
>>> harmonic(n).rewrite(Sum)
Sum(1/_k, (_k, 1, n))
We can evaluate harmonic numbers for all integral and positive
rational arguments:
>>> from sympy import S, expand_func, simplify
>>> harmonic(8)
761/280
>>> harmonic(11)
83711/27720
>>> H = harmonic(1/S(3))
>>> H
harmonic(1/3)
>>> He = expand_func(H)
>>> He
-log(6) - sqrt(3)*pi/6 + 2*Sum(log(sin(_k*pi/3))*cos(2*_k*pi/3), (_k, 1, 1))
+ 3*Sum(1/(3*_k + 1), (_k, 0, 0))
>>> He.doit()
-log(6) - sqrt(3)*pi/6 - log(sqrt(3)/2) + 3
>>> H = harmonic(25/S(7))
>>> He = simplify(expand_func(H).doit())
>>> He
log(sin(pi/7)**(-2*cos(pi/7))*sin(2*pi/7)**(2*cos(16*pi/7))*cos(pi/14)**(-2*sin(pi/14))/14)
+ pi*tan(pi/14)/2 + 30247/9900
>>> He.n(40)
1.983697455232980674869851942390639915940
>>> harmonic(25/S(7)).n(40)
1.983697455232980674869851942390639915940
We can rewrite harmonic numbers in terms of polygamma functions:
>>> from sympy import digamma, polygamma
>>> m = Symbol("m")
>>> harmonic(n).rewrite(digamma)
polygamma(0, n + 1) + EulerGamma
>>> harmonic(n).rewrite(polygamma)
polygamma(0, n + 1) + EulerGamma
>>> harmonic(n,3).rewrite(polygamma)
polygamma(2, n + 1)/2 - polygamma(2, 1)/2
>>> harmonic(n,m).rewrite(polygamma)
(-1)**m*(polygamma(m - 1, 1) - polygamma(m - 1, n + 1))/factorial(m - 1)
Integer offsets in the argument can be pulled out:
>>> from sympy import expand_func
>>> expand_func(harmonic(n+4))
harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1)
>>> expand_func(harmonic(n-4))
harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n
Some limits can be computed as well:
>>> from sympy import limit, oo
>>> limit(harmonic(n), n, oo)
oo
>>> limit(harmonic(n, 2), n, oo)
pi**2/6
>>> limit(harmonic(n, 3), n, oo)
-polygamma(2, 1)/2
However we can not compute the general relation yet:
>>> limit(harmonic(n, m), n, oo)
harmonic(oo, m)
which equals ``zeta(m)`` for ``m > 1``.
See Also
========
bell, bernoulli, catalan, euler, fibonacci, lucas, genocchi, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Harmonic_number
.. [2] http://functions.wolfram.com/GammaBetaErf/HarmonicNumber/
.. [3] http://functions.wolfram.com/GammaBetaErf/HarmonicNumber2/
"""
# Generate one memoized Harmonic number-generating function for each
# order and store it in a dictionary
_functions = {}
@classmethod
def eval(cls, n, m=None):
from sympy import zeta
if m is S.One:
return cls(n)
if m is None:
m = S.One
if m.is_zero:
return n
if n is S.Infinity and m.is_Number:
# TODO: Fix for symbolic values of m
if m.is_negative:
return S.NaN
elif LessThan(m, S.One):
return S.Infinity
elif StrictGreaterThan(m, S.One):
return zeta(m)
else:
return cls
if n == 0:
return S.Zero
if n.is_Integer and n.is_nonnegative and m.is_Integer:
if not m in cls._functions:
@recurrence_memo([0])
def f(n, prev):
return prev[-1] + S.One / n**m
cls._functions[m] = f
return cls._functions[m](int(n))
def _eval_rewrite_as_polygamma(self, n, m=1, **kwargs):
from sympy.functions.special.gamma_functions import polygamma
return S.NegativeOne**m/factorial(m - 1) * (polygamma(m - 1, 1) - polygamma(m - 1, n + 1))
def _eval_rewrite_as_digamma(self, n, m=1, **kwargs):
from sympy.functions.special.gamma_functions import polygamma
return self.rewrite(polygamma)
def _eval_rewrite_as_trigamma(self, n, m=1, **kwargs):
from sympy.functions.special.gamma_functions import polygamma
return self.rewrite(polygamma)
def _eval_rewrite_as_Sum(self, n, m=None, **kwargs):
from sympy import Sum
k = Dummy("k", integer=True)
if m is None:
m = S.One
return Sum(k**(-m), (k, 1, n))
def _eval_expand_func(self, **hints):
from sympy import Sum
n = self.args[0]
m = self.args[1] if len(self.args) == 2 else 1
if m == S.One:
if n.is_Add:
off = n.args[0]
nnew = n - off
if off.is_Integer and off.is_positive:
result = [S.One/(nnew + i) for i in range(off, 0, -1)] + [harmonic(nnew)]
return Add(*result)
elif off.is_Integer and off.is_negative:
result = [-S.One/(nnew + i) for i in range(0, off, -1)] + [harmonic(nnew)]
return Add(*result)
if n.is_Rational:
# Expansions for harmonic numbers at general rational arguments (u + p/q)
# Split n as u + p/q with p < q
p, q = n.as_numer_denom()
u = p // q
p = p - u * q
if u.is_nonnegative and p.is_positive and q.is_positive and p < q:
k = Dummy("k")
t1 = q * Sum(1 / (q * k + p), (k, 0, u))
t2 = 2 * Sum(cos((2 * pi * p * k) / S(q)) *
log(sin((pi * k) / S(q))),
(k, 1, floor((q - 1) / S(2))))
t3 = (pi / 2) * cot((pi * p) / q) + log(2 * q)
return t1 + t2 - t3
return self
def _eval_rewrite_as_tractable(self, n, m=1, **kwargs):
from sympy import polygamma
return self.rewrite(polygamma).rewrite("tractable", deep=True)
def _eval_evalf(self, prec):
from sympy import polygamma
if all(i.is_number for i in self.args):
return self.rewrite(polygamma)._eval_evalf(prec)
#----------------------------------------------------------------------------#
# #
# Euler numbers #
# #
#----------------------------------------------------------------------------#
class euler(Function):
r"""
Euler numbers / Euler polynomials
The Euler numbers are given by:
.. math:: E_{2n} = I \sum_{k=1}^{2n+1} \sum_{j=0}^k \binom{k}{j}
\frac{(-1)^j (k-2j)^{2n+1}}{2^k I^k k}
.. math:: E_{2n+1} = 0
Euler numbers and Euler polynomials are related by
.. math:: E_n = 2^n E_n\left(\frac{1}{2}\right).
We compute symbolic Euler polynomials using [5]_
.. math:: E_n(x) = \sum_{k=0}^n \binom{n}{k} \frac{E_k}{2^k}
\left(x - \frac{1}{2}\right)^{n-k}.
However, numerical evaluation of the Euler polynomial is computed
more efficiently (and more accurately) using the mpmath library.
* ``euler(n)`` gives the `n^{th}` Euler number, `E_n`.
* ``euler(n, x)`` gives the `n^{th}` Euler polynomial, `E_n(x)`.
Examples
========
>>> from sympy import Symbol, S
>>> from sympy.functions import euler
>>> [euler(n) for n in range(10)]
[1, 0, -1, 0, 5, 0, -61, 0, 1385, 0]
>>> n = Symbol("n")
>>> euler(n + 2*n)
euler(3*n)
>>> x = Symbol("x")
>>> euler(n, x)
euler(n, x)
>>> euler(0, x)
1
>>> euler(1, x)
x - 1/2
>>> euler(2, x)
x**2 - x
>>> euler(3, x)
x**3 - 3*x**2/2 + 1/4
>>> euler(4, x)
x**4 - 2*x**3 + x
>>> euler(12, S.Half)
2702765/4096
>>> euler(12)
2702765
See Also
========
bell, bernoulli, catalan, fibonacci, harmonic, lucas, genocchi, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Euler_numbers
.. [2] http://mathworld.wolfram.com/EulerNumber.html
.. [3] https://en.wikipedia.org/wiki/Alternating_permutation
.. [4] http://mathworld.wolfram.com/AlternatingPermutation.html
.. [5] http://dlmf.nist.gov/24.2#ii
"""
@classmethod
def eval(cls, m, sym=None):
if m.is_Number:
if m.is_Integer and m.is_nonnegative:
# Euler numbers
if sym is None:
if m.is_odd:
return S.Zero
from mpmath import mp
m = m._to_mpmath(mp.prec)
res = mp.eulernum(m, exact=True)
return Integer(res)
# Euler polynomial
else:
from sympy.core.evalf import pure_complex
reim = pure_complex(sym, or_real=True)
# Evaluate polynomial numerically using mpmath
if reim and all(a.is_Float or a.is_Integer for a in reim) \
and any(a.is_Float for a in reim):
from mpmath import mp
from sympy import Expr
m = int(m)
# XXX ComplexFloat (#12192) would be nice here, above
prec = min([a._prec for a in reim if a.is_Float])
with workprec(prec):
res = mp.eulerpoly(m, sym)
return Expr._from_mpmath(res, prec)
# Construct polynomial symbolically from definition
m, result = int(m), []
for k in range(m + 1):
result.append(binomial(m, k)*cls(k)/(2**k)*(sym - S.Half)**(m - k))
return Add(*result).expand()
else:
raise ValueError("Euler numbers are defined only"
" for nonnegative integer indices.")
if sym is None:
if m.is_odd and m.is_positive:
return S.Zero
def _eval_rewrite_as_Sum(self, n, x=None, **kwargs):
from sympy import Sum
if x is None and n.is_even:
k = Dummy("k", integer=True)
j = Dummy("j", integer=True)
n = n / 2
Em = (S.ImaginaryUnit * Sum(Sum(binomial(k, j) * ((-1)**j * (k - 2*j)**(2*n + 1)) /
(2**k*S.ImaginaryUnit**k * k), (j, 0, k)), (k, 1, 2*n + 1)))
return Em
if x:
k = Dummy("k", integer=True)
return Sum(binomial(n, k)*euler(k)/2**k*(x-S.Half)**(n-k), (k, 0, n))
def _eval_evalf(self, prec):
m, x = (self.args[0], None) if len(self.args) == 1 else self.args
if x is None and m.is_Integer and m.is_nonnegative:
from mpmath import mp
from sympy import Expr
m = m._to_mpmath(prec)
with workprec(prec):
res = mp.eulernum(m)
return Expr._from_mpmath(res, prec)
if x and x.is_number and m.is_Integer and m.is_nonnegative:
from mpmath import mp
from sympy import Expr
m = int(m)
x = x._to_mpmath(prec)
with workprec(prec):
res = mp.eulerpoly(m, x)
return Expr._from_mpmath(res, prec)
#----------------------------------------------------------------------------#
# #
# Catalan numbers #
# #
#----------------------------------------------------------------------------#
class catalan(Function):
r"""
Catalan numbers
The `n^{th}` catalan number is given by:
.. math :: C_n = \frac{1}{n+1} \binom{2n}{n}
* ``catalan(n)`` gives the `n^{th}` Catalan number, `C_n`
Examples
========
>>> from sympy import (Symbol, binomial, gamma, hyper, polygamma,
... catalan, diff, combsimp, Rational, I)
>>> [catalan(i) for i in range(1,10)]
[1, 2, 5, 14, 42, 132, 429, 1430, 4862]
>>> n = Symbol("n", integer=True)
>>> catalan(n)
catalan(n)
Catalan numbers can be transformed into several other, identical
expressions involving other mathematical functions
>>> catalan(n).rewrite(binomial)
binomial(2*n, n)/(n + 1)
>>> catalan(n).rewrite(gamma)
4**n*gamma(n + 1/2)/(sqrt(pi)*gamma(n + 2))
>>> catalan(n).rewrite(hyper)
hyper((1 - n, -n), (2,), 1)
For some non-integer values of n we can get closed form
expressions by rewriting in terms of gamma functions:
>>> catalan(Rational(1,2)).rewrite(gamma)
8/(3*pi)
We can differentiate the Catalan numbers C(n) interpreted as a
continuous real function in n:
>>> diff(catalan(n), n)
(polygamma(0, n + 1/2) - polygamma(0, n + 2) + log(4))*catalan(n)
As a more advanced example consider the following ratio
between consecutive numbers:
>>> combsimp((catalan(n + 1)/catalan(n)).rewrite(binomial))
2*(2*n + 1)/(n + 2)
The Catalan numbers can be generalized to complex numbers:
>>> catalan(I).rewrite(gamma)
4**I*gamma(1/2 + I)/(sqrt(pi)*gamma(2 + I))
and evaluated with arbitrary precision:
>>> catalan(I).evalf(20)
0.39764993382373624267 - 0.020884341620842555705*I
See Also
========
bell, bernoulli, euler, fibonacci, harmonic, lucas, genocchi, partition, tribonacci
sympy.functions.combinatorial.factorials.binomial
References
==========
.. [1] https://en.wikipedia.org/wiki/Catalan_number
.. [2] http://mathworld.wolfram.com/CatalanNumber.html
.. [3] http://functions.wolfram.com/GammaBetaErf/CatalanNumber/
.. [4] http://geometer.org/mathcircles/catalan.pdf
"""
@classmethod
def eval(cls, n):
from sympy import gamma
if (n.is_Integer and n.is_nonnegative) or \
(n.is_noninteger and n.is_negative):
return 4**n*gamma(n + S.Half)/(gamma(S.Half)*gamma(n + 2))
if (n.is_integer and n.is_negative):
if (n + 1).is_negative:
return S.Zero
if (n + 1).is_zero:
return -S.Half
def fdiff(self, argindex=1):
from sympy import polygamma, log
n = self.args[0]
return catalan(n)*(polygamma(0, n + Rational(1, 2)) - polygamma(0, n + 2) + log(4))
def _eval_rewrite_as_binomial(self, n, **kwargs):
return binomial(2*n, n)/(n + 1)
def _eval_rewrite_as_factorial(self, n, **kwargs):
return factorial(2*n) / (factorial(n+1) * factorial(n))
def _eval_rewrite_as_gamma(self, n, **kwargs):
from sympy import gamma
# The gamma function allows to generalize Catalan numbers to complex n
return 4**n*gamma(n + S.Half)/(gamma(S.Half)*gamma(n + 2))
def _eval_rewrite_as_hyper(self, n, **kwargs):
from sympy import hyper
return hyper([1 - n, -n], [2], 1)
def _eval_rewrite_as_Product(self, n, **kwargs):
from sympy import Product
if not (n.is_integer and n.is_nonnegative):
return self
k = Dummy('k', integer=True, positive=True)
return Product((n + k) / k, (k, 2, n))
def _eval_is_integer(self):
if self.args[0].is_integer and self.args[0].is_nonnegative:
return True
def _eval_is_positive(self):
if self.args[0].is_nonnegative:
return True
def _eval_is_composite(self):
if self.args[0].is_integer and (self.args[0] - 3).is_positive:
return True
def _eval_evalf(self, prec):
from sympy import gamma
if self.args[0].is_number:
return self.rewrite(gamma)._eval_evalf(prec)
#----------------------------------------------------------------------------#
# #
# Genocchi numbers #
# #
#----------------------------------------------------------------------------#
class genocchi(Function):
r"""
Genocchi numbers
The Genocchi numbers are a sequence of integers `G_n` that satisfy the
relation:
.. math:: \frac{2t}{e^t + 1} = \sum_{n=1}^\infty \frac{G_n t^n}{n!}
Examples
========
>>> from sympy import Symbol
>>> from sympy.functions import genocchi
>>> [genocchi(n) for n in range(1, 9)]
[1, -1, 0, 1, 0, -3, 0, 17]
>>> n = Symbol('n', integer=True, positive=True)
>>> genocchi(2*n + 1)
0
See Also
========
bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, partition, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Genocchi_number
.. [2] http://mathworld.wolfram.com/GenocchiNumber.html
"""
@classmethod
def eval(cls, n):
if n.is_Number:
if (not n.is_Integer) or n.is_nonpositive:
raise ValueError("Genocchi numbers are defined only for " +
"positive integers")
return 2 * (1 - S(2) ** n) * bernoulli(n)
if n.is_odd and (n - 1).is_positive:
return S.Zero
if (n - 1).is_zero:
return S.One
def _eval_rewrite_as_bernoulli(self, n, **kwargs):
if n.is_integer and n.is_nonnegative:
return (1 - S(2) ** n) * bernoulli(n) * 2
def _eval_is_integer(self):
if self.args[0].is_integer and self.args[0].is_positive:
return True
def _eval_is_negative(self):
n = self.args[0]
if n.is_integer and n.is_positive:
if n.is_odd:
return False
return (n / 2).is_odd
def _eval_is_positive(self):
n = self.args[0]
if n.is_integer and n.is_positive:
if n.is_odd:
return fuzzy_not((n - 1).is_positive)
return (n / 2).is_even
def _eval_is_even(self):
n = self.args[0]
if n.is_integer and n.is_positive:
if n.is_even:
return False
return (n - 1).is_positive
def _eval_is_odd(self):
n = self.args[0]
if n.is_integer and n.is_positive:
if n.is_even:
return True
return fuzzy_not((n - 1).is_positive)
def _eval_is_prime(self):
n = self.args[0]
# only G_6 = -3 and G_8 = 17 are prime,
# but SymPy does not consider negatives as prime
# so only n=8 is tested
return (n - 8).is_zero
#----------------------------------------------------------------------------#
# #
# Partition numbers #
# #
#----------------------------------------------------------------------------#
class partition(Function):
r"""
Partition numbers
The Partition numbers are a sequence of integers `p_n` that represent the
number of distinct ways of representing `n` as a sum of natural numbers
(with order irrelevant). The generating function for `p_n` is given by:
.. math:: \sum_{n=0}^\infty p_n x^n = \prod_{k=1}^\infty (1 - x^k)^{-1}
Examples
========
>>> from sympy import Symbol
>>> from sympy.functions import partition
>>> [partition(n) for n in range(9)]
[1, 1, 2, 3, 5, 7, 11, 15, 22]
>>> n = Symbol('n', integer=True, negative=True)
>>> partition(n)
0
See Also
========
bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, tribonacci
References
==========
.. [1] https://en.wikipedia.org/wiki/Partition_(number_theory%29
.. [2] https://en.wikipedia.org/wiki/Pentagonal_number_theorem
"""
@staticmethod
@recurrence_memo([1, 1])
def _partition(n, prev):
v, g, i = 0, 0, 0
while 1:
s = 0
i += 1
g = i * (3*i - 1) // 2
if n >= g:
s += prev[n - g]
g = i * (3*i + 1) // 2
if n >= g:
s += prev[n - g]
if s == 0:
break
else:
v += s if i%2 == 1 else -s
return v
@classmethod
def eval(cls, n):
is_int = n.is_integer
if is_int == False:
raise ValueError("Partition numbers are defined only for "
"integers")
elif is_int:
if n.is_negative:
return S.Zero
if n.is_zero or (n - 1).is_zero:
return S.One
if n.is_Integer:
return Integer(cls._partition(n))
def _eval_is_integer(self):
if self.args[0].is_integer:
return True
def _eval_is_negative(self):
if self.args[0].is_integer:
return False
def _eval_is_positive(self):
n = self.args[0]
if n.is_nonnegative and n.is_integer:
return True
#######################################################################
###
### Functions for enumerating partitions, permutations and combinations
###
#######################################################################
class _MultisetHistogram(tuple):
pass
_N = -1
_ITEMS = -2
_M = slice(None, _ITEMS)
def _multiset_histogram(n):
"""Return tuple used in permutation and combination counting. Input
is a dictionary giving items with counts as values or a sequence of
items (which need not be sorted).
The data is stored in a class deriving from tuple so it is easily
recognized and so it can be converted easily to a list.
"""
if isinstance(n, dict): # item: count
if not all(isinstance(v, int) and v >= 0 for v in n.values()):
raise ValueError
tot = sum(n.values())
items = sum(1 for k in n if n[k] > 0)
return _MultisetHistogram([n[k] for k in n if n[k] > 0] + [items, tot])
else:
n = list(n)
s = set(n)
if len(s) == len(n):
n = [1]*len(n)
n.extend([len(n), len(n)])
return _MultisetHistogram(n)
m = dict(zip(s, range(len(s))))
d = dict(zip(range(len(s)), [0]*len(s)))
for i in n:
d[m[i]] += 1
return _multiset_histogram(d)
def nP(n, k=None, replacement=False):
"""Return the number of permutations of ``n`` items taken ``k`` at a time.
Possible values for ``n``::
integer - set of length ``n``
sequence - converted to a multiset internally
multiset - {element: multiplicity}
If ``k`` is None then the total of all permutations of length 0
through the number of items represented by ``n`` will be returned.
If ``replacement`` is True then a given item can appear more than once
in the ``k`` items. (For example, for 'ab' permutations of 2 would
include 'aa', 'ab', 'ba' and 'bb'.) The multiplicity of elements in
``n`` is ignored when ``replacement`` is True but the total number
of elements is considered since no element can appear more times than
the number of elements in ``n``.
Examples
========
>>> from sympy.functions.combinatorial.numbers import nP
>>> from sympy.utilities.iterables import multiset_permutations, multiset
>>> nP(3, 2)
6
>>> nP('abc', 2) == nP(multiset('abc'), 2) == 6
True
>>> nP('aab', 2)
3
>>> nP([1, 2, 2], 2)
3
>>> [nP(3, i) for i in range(4)]
[1, 3, 6, 6]
>>> nP(3) == sum(_)
True
When ``replacement`` is True, each item can have multiplicity
equal to the length represented by ``n``:
>>> nP('aabc', replacement=True)
121
>>> [len(list(multiset_permutations('aaaabbbbcccc', i))) for i in range(5)]
[1, 3, 9, 27, 81]
>>> sum(_)
121
See Also
========
sympy.utilities.iterables.multiset_permutations
References
==========
.. [1] https://en.wikipedia.org/wiki/Permutation
"""
try:
n = as_int(n)
except ValueError:
return Integer(_nP(_multiset_histogram(n), k, replacement))
return Integer(_nP(n, k, replacement))
@cacheit
def _nP(n, k=None, replacement=False):
from sympy.functions.combinatorial.factorials import factorial
from sympy.core.mul import prod
if k == 0:
return 1
if isinstance(n, SYMPY_INTS): # n different items
# assert n >= 0
if k is None:
return sum(_nP(n, i, replacement) for i in range(n + 1))
elif replacement:
return n**k
elif k > n:
return 0
elif k == n:
return factorial(k)
elif k == 1:
return n
else:
# assert k >= 0
return _product(n - k + 1, n)
elif isinstance(n, _MultisetHistogram):
if k is None:
return sum(_nP(n, i, replacement) for i in range(n[_N] + 1))
elif replacement:
return n[_ITEMS]**k
elif k == n[_N]:
return factorial(k)/prod([factorial(i) for i in n[_M] if i > 1])
elif k > n[_N]:
return 0
elif k == 1:
return n[_ITEMS]
else:
# assert k >= 0
tot = 0
n = list(n)
for i in range(len(n[_M])):
if not n[i]:
continue
n[_N] -= 1
if n[i] == 1:
n[i] = 0
n[_ITEMS] -= 1
tot += _nP(_MultisetHistogram(n), k - 1)
n[_ITEMS] += 1
n[i] = 1
else:
n[i] -= 1
tot += _nP(_MultisetHistogram(n), k - 1)
n[i] += 1
n[_N] += 1
return tot
@cacheit
def _AOP_product(n):
"""for n = (m1, m2, .., mk) return the coefficients of the polynomial,
prod(sum(x**i for i in range(nj + 1)) for nj in n); i.e. the coefficients
of the product of AOPs (all-one polynomials) or order given in n. The
resulting coefficient corresponding to x**r is the number of r-length
combinations of sum(n) elements with multiplicities given in n.
The coefficients are given as a default dictionary (so if a query is made
for a key that is not present, 0 will be returned).
Examples
========
>>> from sympy.functions.combinatorial.numbers import _AOP_product
>>> from sympy.abc import x
>>> n = (2, 2, 3) # e.g. aabbccc
>>> prod = ((x**2 + x + 1)*(x**2 + x + 1)*(x**3 + x**2 + x + 1)).expand()
>>> c = _AOP_product(n); dict(c)
{0: 1, 1: 3, 2: 6, 3: 8, 4: 8, 5: 6, 6: 3, 7: 1}
>>> [c[i] for i in range(8)] == [prod.coeff(x, i) for i in range(8)]
True
The generating poly used here is the same as that listed in
http://tinyurl.com/cep849r, but in a refactored form.
"""
from collections import defaultdict
n = list(n)
ord = sum(n)
need = (ord + 2)//2
rv = [1]*(n.pop() + 1)
rv.extend([0]*(need - len(rv)))
rv = rv[:need]
while n:
ni = n.pop()
N = ni + 1
was = rv[:]
for i in range(1, min(N, len(rv))):
rv[i] += rv[i - 1]
for i in range(N, need):
rv[i] += rv[i - 1] - was[i - N]
rev = list(reversed(rv))
if ord % 2:
rv = rv + rev
else:
rv[-1:] = rev
d = defaultdict(int)
for i in range(len(rv)):
d[i] = rv[i]
return d
def nC(n, k=None, replacement=False):
"""Return the number of combinations of ``n`` items taken ``k`` at a time.
Possible values for ``n``::
integer - set of length ``n``
sequence - converted to a multiset internally
multiset - {element: multiplicity}
If ``k`` is None then the total of all combinations of length 0
through the number of items represented in ``n`` will be returned.
If ``replacement`` is True then a given item can appear more than once
in the ``k`` items. (For example, for 'ab' sets of 2 would include 'aa',
'ab', and 'bb'.) The multiplicity of elements in ``n`` is ignored when
``replacement`` is True but the total number of elements is considered
since no element can appear more times than the number of elements in
``n``.
Examples
========
>>> from sympy.functions.combinatorial.numbers import nC
>>> from sympy.utilities.iterables import multiset_combinations
>>> nC(3, 2)
3
>>> nC('abc', 2)
3
>>> nC('aab', 2)
2
When ``replacement`` is True, each item can have multiplicity
equal to the length represented by ``n``:
>>> nC('aabc', replacement=True)
35
>>> [len(list(multiset_combinations('aaaabbbbcccc', i))) for i in range(5)]
[1, 3, 6, 10, 15]
>>> sum(_)
35
If there are ``k`` items with multiplicities ``m_1, m_2, ..., m_k``
then the total of all combinations of length 0 through ``k`` is the
product, ``(m_1 + 1)*(m_2 + 1)*...*(m_k + 1)``. When the multiplicity
of each item is 1 (i.e., k unique items) then there are 2**k
combinations. For example, if there are 4 unique items, the total number
of combinations is 16:
>>> sum(nC(4, i) for i in range(5))
16
See Also
========
sympy.utilities.iterables.multiset_combinations
References
==========
.. [1] https://en.wikipedia.org/wiki/Combination
.. [2] http://tinyurl.com/cep849r
"""
from sympy.functions.combinatorial.factorials import binomial
from sympy.core.mul import prod
if isinstance(n, SYMPY_INTS):
if k is None:
if not replacement:
return 2**n
return sum(nC(n, i, replacement) for i in range(n + 1))
if k < 0:
raise ValueError("k cannot be negative")
if replacement:
return binomial(n + k - 1, k)
return binomial(n, k)
if isinstance(n, _MultisetHistogram):
N = n[_N]
if k is None:
if not replacement:
return prod(m + 1 for m in n[_M])
return sum(nC(n, i, replacement) for i in range(N + 1))
elif replacement:
return nC(n[_ITEMS], k, replacement)
# assert k >= 0
elif k in (1, N - 1):
return n[_ITEMS]
elif k in (0, N):
return 1
return _AOP_product(tuple(n[_M]))[k]
else:
return nC(_multiset_histogram(n), k, replacement)
@cacheit
def _stirling1(n, k):
if n == k == 0:
return S.One
if 0 in (n, k):
return S.Zero
n1 = n - 1
# some special values
if n == k:
return S.One
elif k == 1:
return factorial(n1)
elif k == n1:
return binomial(n, 2)
elif k == n - 2:
return (3*n - 1)*binomial(n, 3)/4
elif k == n - 3:
return binomial(n, 2)*binomial(n, 4)
# general recurrence
return n1*_stirling1(n1, k) + _stirling1(n1, k - 1)
@cacheit
def _stirling2(n, k):
if n == k == 0:
return S.One
if 0 in (n, k):
return S.Zero
n1 = n - 1
# some special values
if k == n1:
return binomial(n, 2)
elif k == 2:
return 2**n1 - 1
# general recurrence
return k*_stirling2(n1, k) + _stirling2(n1, k - 1)
def stirling(n, k, d=None, kind=2, signed=False):
r"""Return Stirling number `S(n, k)` of the first or second (default) kind.
The sum of all Stirling numbers of the second kind for `k = 1`
through `n` is ``bell(n)``. The recurrence relationship for these numbers
is:
.. math :: {0 \brace 0} = 1; {n \brace 0} = {0 \brace k} = 0;
.. math :: {{n+1} \brace k} = j {n \brace k} + {n \brace {k-1}}
where `j` is:
`n` for Stirling numbers of the first kind
`-n` for signed Stirling numbers of the first kind
`k` for Stirling numbers of the second kind
The first kind of Stirling number counts the number of permutations of
``n`` distinct items that have ``k`` cycles; the second kind counts the
ways in which ``n`` distinct items can be partitioned into ``k`` parts.
If ``d`` is given, the "reduced Stirling number of the second kind" is
returned: ``S^{d}(n, k) = S(n - d + 1, k - d + 1)`` with ``n >= k >= d``.
(This counts the ways to partition ``n`` consecutive integers into
``k`` groups with no pairwise difference less than ``d``. See example
below.)
To obtain the signed Stirling numbers of the first kind, use keyword
``signed=True``. Using this keyword automatically sets ``kind`` to 1.
Examples
========
>>> from sympy.functions.combinatorial.numbers import stirling, bell
>>> from sympy.combinatorics import Permutation
>>> from sympy.utilities.iterables import multiset_partitions, permutations
First kind (unsigned by default):
>>> [stirling(6, i, kind=1) for i in range(7)]
[0, 120, 274, 225, 85, 15, 1]
>>> perms = list(permutations(range(4)))
>>> [sum(Permutation(p).cycles == i for p in perms) for i in range(5)]
[0, 6, 11, 6, 1]
>>> [stirling(4, i, kind=1) for i in range(5)]
[0, 6, 11, 6, 1]
First kind (signed):
>>> [stirling(4, i, signed=True) for i in range(5)]
[0, -6, 11, -6, 1]
Second kind:
>>> [stirling(10, i) for i in range(12)]
[0, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1, 0]
>>> sum(_) == bell(10)
True
>>> len(list(multiset_partitions(range(4), 2))) == stirling(4, 2)
True
Reduced second kind:
>>> from sympy import subsets, oo
>>> def delta(p):
... if len(p) == 1:
... return oo
... return min(abs(i[0] - i[1]) for i in subsets(p, 2))
>>> parts = multiset_partitions(range(5), 3)
>>> d = 2
>>> sum(1 for p in parts if all(delta(i) >= d for i in p))
7
>>> stirling(5, 3, 2)
7
See Also
========
sympy.utilities.iterables.multiset_partitions
References
==========
.. [1] https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind
.. [2] https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
"""
# TODO: make this a class like bell()
n = as_int(n)
k = as_int(k)
if n < 0:
raise ValueError('n must be nonnegative')
if k > n:
return S.Zero
if d:
# assert k >= d
# kind is ignored -- only kind=2 is supported
return _stirling2(n - d + 1, k - d + 1)
elif signed:
# kind is ignored -- only kind=1 is supported
return (-1)**(n - k)*_stirling1(n, k)
if kind == 1:
return _stirling1(n, k)
elif kind == 2:
return _stirling2(n, k)
else:
raise ValueError('kind must be 1 or 2, not %s' % k)
@cacheit
def _nT(n, k):
"""Return the partitions of ``n`` items into ``k`` parts. This
is used by ``nT`` for the case when ``n`` is an integer."""
if k == 0:
return 1 if k == n else 0
return sum(_nT(n - k, j) for j in range(min(k, n - k) + 1))
def nT(n, k=None):
"""Return the number of ``k``-sized partitions of ``n`` items.
Possible values for ``n``::
integer - ``n`` identical items
sequence - converted to a multiset internally
multiset - {element: multiplicity}
Note: the convention for ``nT`` is different than that of ``nC`` and
``nP`` in that
here an integer indicates ``n`` *identical* items instead of a set of
length ``n``; this is in keeping with the ``partitions`` function which
treats its integer-``n`` input like a list of ``n`` 1s. One can use
``range(n)`` for ``n`` to indicate ``n`` distinct items.
If ``k`` is None then the total number of ways to partition the elements
represented in ``n`` will be returned.
Examples
========
>>> from sympy.functions.combinatorial.numbers import nT
Partitions of the given multiset:
>>> [nT('aabbc', i) for i in range(1, 7)]
[1, 8, 11, 5, 1, 0]
>>> nT('aabbc') == sum(_)
True
>>> [nT("mississippi", i) for i in range(1, 12)]
[1, 74, 609, 1521, 1768, 1224, 579, 197, 50, 9, 1]
Partitions when all items are identical:
>>> [nT(5, i) for i in range(1, 6)]
[1, 2, 2, 1, 1]
>>> nT('1'*5) == sum(_)
True
When all items are different:
>>> [nT(range(5), i) for i in range(1, 6)]
[1, 15, 25, 10, 1]
>>> nT(range(5)) == sum(_)
True
Partitions of an integer expressed as a sum of positive integers:
>>> from sympy.functions.combinatorial.numbers import partition
>>> partition(4)
5
>>> sum([nT(4, i) for i in range(4 + 1)])
5
>>> nT('1'*4)
5
See Also
========
sympy.utilities.iterables.partitions
sympy.utilities.iterables.multiset_partitions
sympy.functions.combinatorial.numbers.partition
References
==========
.. [1] http://undergraduate.csse.uwa.edu.au/units/CITS7209/partition.pdf
"""
from sympy.utilities.enumerative import MultisetPartitionTraverser
if isinstance(n, SYMPY_INTS):
# assert n >= 0
# all the same
if k is None:
return partition(n)
elif n == 0:
return S.One if k == 0 else S.Zero
return _nT(n, k)
if not isinstance(n, _MultisetHistogram):
try:
# if n contains hashable items there is some
# quick handling that can be done
u = len(set(n))
if u <= 1:
return nT(len(n), k)
elif u == len(n):
n = range(u)
raise TypeError
except TypeError:
n = _multiset_histogram(n)
N = n[_N]
if k is None and N == 1:
return 1
if k in (1, N):
return 1
if k == 2 or N == 2 and k is None:
m, r = divmod(N, 2)
rv = sum(nC(n, i) for i in range(1, m + 1))
if not r:
rv -= nC(n, m)//2
if k is None:
rv += 1 # for k == 1
return rv
if N == n[_ITEMS]:
# all distinct
if k is None:
return bell(N)
return stirling(N, k)
m = MultisetPartitionTraverser()
if k is None:
return m.count_partitions(n[_M])
# MultisetPartitionTraverser does not have a range-limited count
# method, so need to enumerate and count
tot = 0
for discard in m.enum_range(n[_M], k-1, k):
tot += 1
return tot
|
25d48cb78b5ce073a17adffcca82e5d1b2a7bcfdb7a1ccd113b7819daee095c0
|
from __future__ import print_function, division
from sympy.core.add import Add
from sympy.core.basic import sympify, cacheit
from sympy.core.compatibility import range
from sympy.core.function import Function, ArgumentIndexError
from sympy.core.logic import fuzzy_not, fuzzy_or
from sympy.core.numbers import igcdex, Rational, pi
from sympy.core.relational import Ne
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.functions.combinatorial.factorials import factorial, RisingFactorial
from sympy.functions.elementary.exponential import log, exp
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.hyperbolic import (acoth, asinh, atanh, cosh,
coth, HyperbolicFunction, sinh, tanh)
from sympy.functions.elementary.miscellaneous import sqrt, Min, Max
from sympy.functions.elementary.piecewise import Piecewise
from sympy.sets.sets import FiniteSet
from sympy.utilities.iterables import numbered_symbols
###############################################################################
########################## TRIGONOMETRIC FUNCTIONS ############################
###############################################################################
class TrigonometricFunction(Function):
"""Base class for trigonometric functions. """
unbranched = True
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational and fuzzy_not(s.args[0].is_zero):
return False
else:
return s.is_rational
def _eval_is_algebraic(self):
s = self.func(*self.args)
if s.func == self.func:
if fuzzy_not(self.args[0].is_zero) and self.args[0].is_algebraic:
return False
pi_coeff = _pi_coeff(self.args[0])
if pi_coeff is not None and pi_coeff.is_rational:
return True
else:
return s.is_algebraic
def _eval_expand_complex(self, deep=True, **hints):
re_part, im_part = self.as_real_imag(deep=deep, **hints)
return re_part + im_part*S.ImaginaryUnit
def _as_real_imag(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.args[0].expand(deep, **hints), S.Zero)
else:
return (self.args[0], S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
return (re, im)
def _period(self, general_period, symbol=None):
f = self.args[0]
if symbol is None:
symbol = tuple(f.free_symbols)[0]
if not f.has(symbol):
return S.Zero
if f == symbol:
return general_period
if symbol in f.free_symbols:
if f.is_Mul:
g, h = f.as_independent(symbol)
if h == symbol:
return general_period/abs(g)
if f.is_Add:
a, h = f.as_independent(symbol)
g, h = h.as_independent(symbol, as_Add=False)
if h == symbol:
return general_period/abs(g)
raise NotImplementedError("Use the periodicity function instead.")
def _peeloff_pi(arg):
"""
Split ARG into two parts, a "rest" and a multiple of pi/2.
This assumes ARG to be an Add.
The multiple of pi returned in the second position is always a Rational.
Examples
========
>>> from sympy.functions.elementary.trigonometric import _peeloff_pi as peel
>>> from sympy import pi
>>> from sympy.abc import x, y
>>> peel(x + pi/2)
(x, pi/2)
>>> peel(x + 2*pi/3 + pi*y)
(x + pi*y + pi/6, pi/2)
"""
for a in Add.make_args(arg):
if a is S.Pi:
K = S.One
break
elif a.is_Mul:
K, p = a.as_two_terms()
if p is S.Pi and K.is_Rational:
break
else:
return arg, S.Zero
m1 = (K % S.Half) * S.Pi
m2 = K*S.Pi - m1
return arg - m2, m2
def _pi_coeff(arg, cycles=1):
"""
When arg is a Number times pi (e.g. 3*pi/2) then return the Number
normalized to be in the range [0, 2], else None.
When an even multiple of pi is encountered, if it is multiplying
something with known parity then the multiple is returned as 0 otherwise
as 2.
Examples
========
>>> from sympy.functions.elementary.trigonometric import _pi_coeff as coeff
>>> from sympy import pi, Dummy
>>> from sympy.abc import x, y
>>> coeff(3*x*pi)
3*x
>>> coeff(11*pi/7)
11/7
>>> coeff(-11*pi/7)
3/7
>>> coeff(4*pi)
0
>>> coeff(5*pi)
1
>>> coeff(5.0*pi)
1
>>> coeff(5.5*pi)
3/2
>>> coeff(2 + pi)
>>> coeff(2*Dummy(integer=True)*pi)
2
>>> coeff(2*Dummy(even=True)*pi)
0
"""
arg = sympify(arg)
if arg is S.Pi:
return S.One
elif not arg:
return S.Zero
elif arg.is_Mul:
cx = arg.coeff(S.Pi)
if cx:
c, x = cx.as_coeff_Mul() # pi is not included as coeff
if c.is_Float:
# recast exact binary fractions to Rationals
f = abs(c) % 1
if f != 0:
p = -int(round(log(f, 2).evalf()))
m = 2**p
cm = c*m
i = int(cm)
if i == cm:
c = Rational(i, m)
cx = c*x
else:
c = Rational(int(c))
cx = c*x
if x.is_integer:
c2 = c % 2
if c2 == 1:
return x
elif not c2:
if x.is_even is not None: # known parity
return S.Zero
return S(2)
else:
return c2*x
return cx
class sin(TrigonometricFunction):
"""
The sine function.
Returns the sine of x (measured in radians).
Notes
=====
This function will evaluate automatically in the
case x/pi is some rational number [4]_. For example,
if x is a multiple of pi, pi/2, pi/3, pi/4 and pi/6.
Examples
========
>>> from sympy import sin, pi
>>> from sympy.abc import x
>>> sin(x**2).diff(x)
2*x*cos(x**2)
>>> sin(1).diff(x)
0
>>> sin(pi)
0
>>> sin(pi/2)
1
>>> sin(pi/6)
1/2
>>> sin(pi/12)
-sqrt(2)/4 + sqrt(6)/4
See Also
========
csc, cos, sec, tan, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Sin
.. [4] http://mathworld.wolfram.com/TrigonometryAngles.html
"""
def period(self, symbol=None):
return self._period(2*pi, symbol)
def fdiff(self, argindex=1):
if argindex == 1:
return cos(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
from sympy.calculus import AccumBounds
from sympy.sets.setexpr import SetExpr
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Zero:
return S.Zero
elif arg is S.Infinity or arg is S.NegativeInfinity:
return AccumBounds(-1, 1)
if arg is S.ComplexInfinity:
return S.NaN
if isinstance(arg, AccumBounds):
min, max = arg.min, arg.max
d = floor(min/(2*S.Pi))
if min is not S.NegativeInfinity:
min = min - d*2*S.Pi
if max is not S.Infinity:
max = max - d*2*S.Pi
if AccumBounds(min, max).intersection(FiniteSet(S.Pi/2, 5*S.Pi/2)) \
is not S.EmptySet and \
AccumBounds(min, max).intersection(FiniteSet(3*S.Pi/2,
7*S.Pi/2)) is not S.EmptySet:
return AccumBounds(-1, 1)
elif AccumBounds(min, max).intersection(FiniteSet(S.Pi/2, 5*S.Pi/2)) \
is not S.EmptySet:
return AccumBounds(Min(sin(min), sin(max)), 1)
elif AccumBounds(min, max).intersection(FiniteSet(3*S.Pi/2, 8*S.Pi/2)) \
is not S.EmptySet:
return AccumBounds(-1, Max(sin(min), sin(max)))
else:
return AccumBounds(Min(sin(min), sin(max)),
Max(sin(min), sin(max)))
elif isinstance(arg, SetExpr):
return arg._eval_func(cls)
if arg.could_extract_minus_sign():
return -cls(-arg)
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * sinh(i_coeff)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_integer:
return S.Zero
if (2*pi_coeff).is_integer:
if pi_coeff.is_even:
return S.Zero
elif pi_coeff.is_even is False:
return S.NegativeOne**(pi_coeff - S.Half)
if not pi_coeff.is_Rational:
narg = pi_coeff*S.Pi
if narg != arg:
return cls(narg)
return None
# https://github.com/sympy/sympy/issues/6048
# transform a sine to a cosine, to avoid redundant code
if pi_coeff.is_Rational:
x = pi_coeff % 2
if x > 1:
return -cls((x % 1)*S.Pi)
if 2*x > 1:
return cls((1 - x)*S.Pi)
narg = ((pi_coeff + Rational(3, 2)) % 2)*S.Pi
result = cos(narg)
if not isinstance(result, cos):
return result
if pi_coeff*S.Pi != arg:
return cls(pi_coeff*S.Pi)
return None
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
return sin(m)*cos(x) + cos(m)*sin(x)
if isinstance(arg, asin):
return arg.args[0]
if isinstance(arg, atan):
x = arg.args[0]
return x / sqrt(1 + x**2)
if isinstance(arg, atan2):
y, x = arg.args
return y / sqrt(x**2 + y**2)
if isinstance(arg, acos):
x = arg.args[0]
return sqrt(1 - x**2)
if isinstance(arg, acot):
x = arg.args[0]
return 1 / (sqrt(1 + 1 / x**2) * x)
if isinstance(arg, acsc):
x = arg.args[0]
return 1 / x
if isinstance(arg, asec):
x = arg.args[0]
return sqrt(1 - 1 / x**2)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2:
p = previous_terms[-2]
return -p * x**2 / (n*(n - 1))
else:
return (-1)**(n//2) * x**(n)/factorial(n)
def _eval_rewrite_as_exp(self, arg, **kwargs):
I = S.ImaginaryUnit
if isinstance(arg, TrigonometricFunction) or isinstance(arg, HyperbolicFunction):
arg = arg.func(arg.args[0]).rewrite(exp)
return (exp(arg*I) - exp(-arg*I)) / (2*I)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if isinstance(arg, log):
I = S.ImaginaryUnit
x = arg.args[0]
return I*x**-I / 2 - I*x**I /2
def _eval_rewrite_as_cos(self, arg, **kwargs):
return cos(arg - S.Pi / 2, evaluate=False)
def _eval_rewrite_as_tan(self, arg, **kwargs):
tan_half = tan(S.Half*arg)
return 2*tan_half/(1 + tan_half**2)
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return sin(arg)*cos(arg)/cos(arg)
def _eval_rewrite_as_cot(self, arg, **kwargs):
cot_half = cot(S.Half*arg)
return 2*cot_half/(1 + cot_half**2)
def _eval_rewrite_as_pow(self, arg, **kwargs):
return self.rewrite(cos).rewrite(pow)
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
return self.rewrite(cos).rewrite(sqrt)
def _eval_rewrite_as_csc(self, arg, **kwargs):
return 1/csc(arg)
def _eval_rewrite_as_sec(self, arg, **kwargs):
return 1 / sec(arg - S.Pi / 2, evaluate=False)
def _eval_rewrite_as_sinc(self, arg, **kwargs):
return arg*sinc(arg)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
re, im = self._as_real_imag(deep=deep, **hints)
return (sin(re)*cosh(im), cos(re)*sinh(im))
def _eval_expand_trig(self, **hints):
from sympy import expand_mul
from sympy.functions.special.polynomials import chebyshevt, chebyshevu
arg = self.args[0]
x = None
if arg.is_Add: # TODO, implement more if deep stuff here
# TODO: Do this more efficiently for more than two terms
x, y = arg.as_two_terms()
sx = sin(x, evaluate=False)._eval_expand_trig()
sy = sin(y, evaluate=False)._eval_expand_trig()
cx = cos(x, evaluate=False)._eval_expand_trig()
cy = cos(y, evaluate=False)._eval_expand_trig()
return sx*cy + sy*cx
else:
n, x = arg.as_coeff_Mul(rational=True)
if n.is_Integer: # n will be positive because of .eval
# canonicalization
# See http://mathworld.wolfram.com/Multiple-AngleFormulas.html
if n.is_odd:
return (-1)**((n - 1)/2)*chebyshevt(n, sin(x))
else:
return expand_mul((-1)**(n/2 - 1)*cos(x)*chebyshevu(n -
1, sin(x)), deep=False)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_Rational:
return self.rewrite(sqrt)
return sin(arg)
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
def _eval_is_real(self):
if self.args[0].is_real:
return True
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_real:
return True
class cos(TrigonometricFunction):
"""
The cosine function.
Returns the cosine of x (measured in radians).
Notes
=====
See :func:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import cos, pi
>>> from sympy.abc import x
>>> cos(x**2).diff(x)
-2*x*sin(x**2)
>>> cos(1).diff(x)
0
>>> cos(pi)
-1
>>> cos(pi/2)
0
>>> cos(2*pi/3)
-1/2
>>> cos(pi/12)
sqrt(2)/4 + sqrt(6)/4
See Also
========
sin, csc, sec, tan, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Cos
"""
def period(self, symbol=None):
return self._period(2*pi, symbol)
def fdiff(self, argindex=1):
if argindex == 1:
return -sin(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
from sympy.functions.special.polynomials import chebyshevt
from sympy.calculus.util import AccumBounds
from sympy.sets.setexpr import SetExpr
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Zero:
return S.One
elif arg is S.Infinity or arg is S.NegativeInfinity:
# In this case it is better to return AccumBounds(-1, 1)
# rather than returning S.NaN, since AccumBounds(-1, 1)
# preserves the information that sin(oo) is between
# -1 and 1, where S.NaN does not do that.
return AccumBounds(-1, 1)
if arg is S.ComplexInfinity:
return S.NaN
if isinstance(arg, AccumBounds):
return sin(arg + S.Pi/2)
elif isinstance(arg, SetExpr):
return arg._eval_func(cls)
if arg.could_extract_minus_sign():
return cls(-arg)
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return cosh(i_coeff)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_integer:
return (S.NegativeOne)**pi_coeff
if (2*pi_coeff).is_integer:
if pi_coeff.is_even:
return (S.NegativeOne)**(pi_coeff/2)
elif pi_coeff.is_even is False:
return S.Zero
if not pi_coeff.is_Rational:
narg = pi_coeff*S.Pi
if narg != arg:
return cls(narg)
return None
# cosine formula #####################
# https://github.com/sympy/sympy/issues/6048
# explicit calculations are preformed for
# cos(k pi/n) for n = 8,10,12,15,20,24,30,40,60,120
# Some other exact values like cos(k pi/240) can be
# calculated using a partial-fraction decomposition
# by calling cos( X ).rewrite(sqrt)
cst_table_some = {
3: S.Half,
5: (sqrt(5) + 1)/4,
}
if pi_coeff.is_Rational:
q = pi_coeff.q
p = pi_coeff.p % (2*q)
if p > q:
narg = (pi_coeff - 1)*S.Pi
return -cls(narg)
if 2*p > q:
narg = (1 - pi_coeff)*S.Pi
return -cls(narg)
# If nested sqrt's are worse than un-evaluation
# you can require q to be in (1, 2, 3, 4, 6, 12)
# q <= 12, q=15, q=20, q=24, q=30, q=40, q=60, q=120 return
# expressions with 2 or fewer sqrt nestings.
table2 = {
12: (3, 4),
20: (4, 5),
30: (5, 6),
15: (6, 10),
24: (6, 8),
40: (8, 10),
60: (20, 30),
120: (40, 60)
}
if q in table2:
a, b = p*S.Pi/table2[q][0], p*S.Pi/table2[q][1]
nvala, nvalb = cls(a), cls(b)
if None == nvala or None == nvalb:
return None
return nvala*nvalb + cls(S.Pi/2 - a)*cls(S.Pi/2 - b)
if q > 12:
return None
if q in cst_table_some:
cts = cst_table_some[pi_coeff.q]
return chebyshevt(pi_coeff.p, cts).expand()
if 0 == q % 2:
narg = (pi_coeff*2)*S.Pi
nval = cls(narg)
if None == nval:
return None
x = (2*pi_coeff + 1)/2
sign_cos = (-1)**((-1 if x < 0 else 1)*int(abs(x)))
return sign_cos*sqrt( (1 + nval)/2 )
return None
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
return cos(m)*cos(x) - sin(m)*sin(x)
if isinstance(arg, acos):
return arg.args[0]
if isinstance(arg, atan):
x = arg.args[0]
return 1 / sqrt(1 + x**2)
if isinstance(arg, atan2):
y, x = arg.args
return x / sqrt(x**2 + y**2)
if isinstance(arg, asin):
x = arg.args[0]
return sqrt(1 - x ** 2)
if isinstance(arg, acot):
x = arg.args[0]
return 1 / sqrt(1 + 1 / x**2)
if isinstance(arg, acsc):
x = arg.args[0]
return sqrt(1 - 1 / x**2)
if isinstance(arg, asec):
x = arg.args[0]
return 1 / x
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2:
p = previous_terms[-2]
return -p * x**2 / (n*(n - 1))
else:
return (-1)**(n//2)*x**(n)/factorial(n)
def _eval_rewrite_as_exp(self, arg, **kwargs):
I = S.ImaginaryUnit
if isinstance(arg, TrigonometricFunction) or isinstance(arg, HyperbolicFunction):
arg = arg.func(arg.args[0]).rewrite(exp)
return (exp(arg*I) + exp(-arg*I)) / 2
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if isinstance(arg, log):
I = S.ImaginaryUnit
x = arg.args[0]
return x**I/2 + x**-I/2
def _eval_rewrite_as_sin(self, arg, **kwargs):
return sin(arg + S.Pi / 2, evaluate=False)
def _eval_rewrite_as_tan(self, arg, **kwargs):
tan_half = tan(S.Half*arg)**2
return (1 - tan_half)/(1 + tan_half)
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return sin(arg)*cos(arg)/sin(arg)
def _eval_rewrite_as_cot(self, arg, **kwargs):
cot_half = cot(S.Half*arg)**2
return (cot_half - 1)/(cot_half + 1)
def _eval_rewrite_as_pow(self, arg, **kwargs):
return self._eval_rewrite_as_sqrt(arg)
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
from sympy.functions.special.polynomials import chebyshevt
def migcdex(x):
# recursive calcuation of gcd and linear combination
# for a sequence of integers.
# Given (x1, x2, x3)
# Returns (y1, y1, y3, g)
# such that g is the gcd and x1*y1+x2*y2+x3*y3 - g = 0
# Note, that this is only one such linear combination.
if len(x) == 1:
return (1, x[0])
if len(x) == 2:
return igcdex(x[0], x[-1])
g = migcdex(x[1:])
u, v, h = igcdex(x[0], g[-1])
return tuple([u] + [v*i for i in g[0:-1] ] + [h])
def ipartfrac(r, factors=None):
from sympy.ntheory import factorint
if isinstance(r, int):
return r
if not isinstance(r, Rational):
raise TypeError("r is not rational")
n = r.q
if 2 > r.q*r.q:
return r.q
if None == factors:
a = [n//x**y for x, y in factorint(r.q).items()]
else:
a = [n//x for x in factors]
if len(a) == 1:
return [ r ]
h = migcdex(a)
ans = [ r.p*Rational(i*j, r.q) for i, j in zip(h[:-1], a) ]
assert r == sum(ans)
return ans
pi_coeff = _pi_coeff(arg)
if pi_coeff is None:
return None
if pi_coeff.is_integer:
# it was unevaluated
return self.func(pi_coeff*S.Pi)
if not pi_coeff.is_Rational:
return None
def _cospi257():
""" Express cos(pi/257) explicitly as a function of radicals
Based upon the equations in
http://math.stackexchange.com/questions/516142/how-does-cos2-pi-257-look-like-in-real-radicals
See also http://www.susqu.edu/brakke/constructions/257-gon.m.txt
"""
def f1(a, b):
return (a + sqrt(a**2 + b))/2, (a - sqrt(a**2 + b))/2
def f2(a, b):
return (a - sqrt(a**2 + b))/2
t1, t2 = f1(-1, 256)
z1, z3 = f1(t1, 64)
z2, z4 = f1(t2, 64)
y1, y5 = f1(z1, 4*(5 + t1 + 2*z1))
y6, y2 = f1(z2, 4*(5 + t2 + 2*z2))
y3, y7 = f1(z3, 4*(5 + t1 + 2*z3))
y8, y4 = f1(z4, 4*(5 + t2 + 2*z4))
x1, x9 = f1(y1, -4*(t1 + y1 + y3 + 2*y6))
x2, x10 = f1(y2, -4*(t2 + y2 + y4 + 2*y7))
x3, x11 = f1(y3, -4*(t1 + y3 + y5 + 2*y8))
x4, x12 = f1(y4, -4*(t2 + y4 + y6 + 2*y1))
x5, x13 = f1(y5, -4*(t1 + y5 + y7 + 2*y2))
x6, x14 = f1(y6, -4*(t2 + y6 + y8 + 2*y3))
x15, x7 = f1(y7, -4*(t1 + y7 + y1 + 2*y4))
x8, x16 = f1(y8, -4*(t2 + y8 + y2 + 2*y5))
v1 = f2(x1, -4*(x1 + x2 + x3 + x6))
v2 = f2(x2, -4*(x2 + x3 + x4 + x7))
v3 = f2(x8, -4*(x8 + x9 + x10 + x13))
v4 = f2(x9, -4*(x9 + x10 + x11 + x14))
v5 = f2(x10, -4*(x10 + x11 + x12 + x15))
v6 = f2(x16, -4*(x16 + x1 + x2 + x5))
u1 = -f2(-v1, -4*(v2 + v3))
u2 = -f2(-v4, -4*(v5 + v6))
w1 = -2*f2(-u1, -4*u2)
return sqrt(sqrt(2)*sqrt(w1 + 4)/8 + S.Half)
cst_table_some = {
3: S.Half,
5: (sqrt(5) + 1)/4,
17: sqrt((15 + sqrt(17))/32 + sqrt(2)*(sqrt(17 - sqrt(17)) +
sqrt(sqrt(2)*(-8*sqrt(17 + sqrt(17)) - (1 - sqrt(17))
*sqrt(17 - sqrt(17))) + 6*sqrt(17) + 34))/32),
257: _cospi257()
# 65537 is the only other known Fermat prime and the very
# large expression is intentionally omitted from SymPy; see
# http://www.susqu.edu/brakke/constructions/65537-gon.m.txt
}
def _fermatCoords(n):
# if n can be factored in terms of Fermat primes with
# multiplicity of each being 1, return those primes, else
# False
primes = []
for p_i in cst_table_some:
quotient, remainder = divmod(n, p_i)
if remainder == 0:
n = quotient
primes.append(p_i)
if n == 1:
return tuple(primes)
return False
if pi_coeff.q in cst_table_some:
rv = chebyshevt(pi_coeff.p, cst_table_some[pi_coeff.q])
if pi_coeff.q < 257:
rv = rv.expand()
return rv
if not pi_coeff.q % 2: # recursively remove factors of 2
pico2 = pi_coeff*2
nval = cos(pico2*S.Pi).rewrite(sqrt)
x = (pico2 + 1)/2
sign_cos = -1 if int(x) % 2 else 1
return sign_cos*sqrt( (1 + nval)/2 )
FC = _fermatCoords(pi_coeff.q)
if FC:
decomp = ipartfrac(pi_coeff, FC)
X = [(x[1], x[0]*S.Pi) for x in zip(decomp, numbered_symbols('z'))]
pcls = cos(sum([x[0] for x in X]))._eval_expand_trig().subs(X)
return pcls.rewrite(sqrt)
else:
decomp = ipartfrac(pi_coeff)
X = [(x[1], x[0]*S.Pi) for x in zip(decomp, numbered_symbols('z'))]
pcls = cos(sum([x[0] for x in X]))._eval_expand_trig().subs(X)
return pcls
def _eval_rewrite_as_sec(self, arg, **kwargs):
return 1/sec(arg)
def _eval_rewrite_as_csc(self, arg, **kwargs):
return 1 / sec(arg)._eval_rewrite_as_csc(arg)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
re, im = self._as_real_imag(deep=deep, **hints)
return (cos(re)*cosh(im), -sin(re)*sinh(im))
def _eval_expand_trig(self, **hints):
from sympy.functions.special.polynomials import chebyshevt
arg = self.args[0]
x = None
if arg.is_Add: # TODO: Do this more efficiently for more than two terms
x, y = arg.as_two_terms()
sx = sin(x, evaluate=False)._eval_expand_trig()
sy = sin(y, evaluate=False)._eval_expand_trig()
cx = cos(x, evaluate=False)._eval_expand_trig()
cy = cos(y, evaluate=False)._eval_expand_trig()
return cx*cy - sx*sy
else:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff.is_Integer:
return chebyshevt(coeff, cos(terms))
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_Rational:
return self.rewrite(sqrt)
return cos(arg)
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return S.One
else:
return self.func(arg)
def _eval_is_real(self):
if self.args[0].is_real:
return True
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_real:
return True
class tan(TrigonometricFunction):
"""
The tangent function.
Returns the tangent of x (measured in radians).
Notes
=====
See :func:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import tan, pi
>>> from sympy.abc import x
>>> tan(x**2).diff(x)
2*x*(tan(x**2)**2 + 1)
>>> tan(1).diff(x)
0
>>> tan(pi/8).expand()
-1 + sqrt(2)
See Also
========
sin, csc, cos, sec, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Tan
"""
def period(self, symbol=None):
return self._period(pi, symbol)
def fdiff(self, argindex=1):
if argindex == 1:
return S.One + self**2
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return atan
@classmethod
def eval(cls, arg):
from sympy.calculus.util import AccumBounds
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Zero:
return S.Zero
elif arg is S.Infinity or arg is S.NegativeInfinity:
return AccumBounds(S.NegativeInfinity, S.Infinity)
if arg is S.ComplexInfinity:
return S.NaN
if isinstance(arg, AccumBounds):
min, max = arg.min, arg.max
d = floor(min/S.Pi)
if min is not S.NegativeInfinity:
min = min - d*S.Pi
if max is not S.Infinity:
max = max - d*S.Pi
if AccumBounds(min, max).intersection(FiniteSet(S.Pi/2, 3*S.Pi/2)):
return AccumBounds(S.NegativeInfinity, S.Infinity)
else:
return AccumBounds(tan(min), tan(max))
if arg.could_extract_minus_sign():
return -cls(-arg)
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * tanh(i_coeff)
pi_coeff = _pi_coeff(arg, 2)
if pi_coeff is not None:
if pi_coeff.is_integer:
return S.Zero
if not pi_coeff.is_Rational:
narg = pi_coeff*S.Pi
if narg != arg:
return cls(narg)
return None
if pi_coeff.is_Rational:
if not pi_coeff.q % 2:
narg = pi_coeff*S.Pi*2
cresult, sresult = cos(narg), cos(narg - S.Pi/2)
if not isinstance(cresult, cos) \
and not isinstance(sresult, cos):
if sresult == 0:
return S.ComplexInfinity
return 1/sresult - cresult/sresult
table2 = {
12: (3, 4),
20: (4, 5),
30: (5, 6),
15: (6, 10),
24: (6, 8),
40: (8, 10),
60: (20, 30),
120: (40, 60)
}
q = pi_coeff.q
p = pi_coeff.p % q
if q in table2:
nvala, nvalb = cls(p*S.Pi/table2[q][0]), cls(p*S.Pi/table2[q][1])
if None == nvala or None == nvalb:
return None
return (nvala - nvalb)/(1 + nvala*nvalb)
narg = ((pi_coeff + S.Half) % 1 - S.Half)*S.Pi
# see cos() to specify which expressions should be
# expanded automatically in terms of radicals
cresult, sresult = cos(narg), cos(narg - S.Pi/2)
if not isinstance(cresult, cos) \
and not isinstance(sresult, cos):
if cresult == 0:
return S.ComplexInfinity
return (sresult/cresult)
if narg != arg:
return cls(narg)
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
tanm = tan(m)
if tanm is S.ComplexInfinity:
return -cot(x)
else: # tanm == 0
return tan(x)
if isinstance(arg, atan):
return arg.args[0]
if isinstance(arg, atan2):
y, x = arg.args
return y/x
if isinstance(arg, asin):
x = arg.args[0]
return x / sqrt(1 - x**2)
if isinstance(arg, acos):
x = arg.args[0]
return sqrt(1 - x**2) / x
if isinstance(arg, acot):
x = arg.args[0]
return 1 / x
if isinstance(arg, acsc):
x = arg.args[0]
return 1 / (sqrt(1 - 1 / x**2) * x)
if isinstance(arg, asec):
x = arg.args[0]
return sqrt(1 - 1 / x**2) * x
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
from sympy import bernoulli
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
a, b = ((n - 1)//2), 2**(n + 1)
B = bernoulli(n + 1)
F = factorial(n + 1)
return (-1)**a * b*(b - 1) * B/F * x**n
def _eval_nseries(self, x, n, logx):
i = self.args[0].limit(x, 0)*2/S.Pi
if i and i.is_Integer:
return self.rewrite(cos)._eval_nseries(x, n=n, logx=logx)
return Function._eval_nseries(self, x, n=n, logx=logx)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if isinstance(arg, log):
I = S.ImaginaryUnit
x = arg.args[0]
return I*(x**-I - x**I)/(x**-I + x**I)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
re, im = self._as_real_imag(deep=deep, **hints)
if im:
denom = cos(2*re) + cosh(2*im)
return (sin(2*re)/denom, sinh(2*im)/denom)
else:
return (self.func(re), S.Zero)
def _eval_expand_trig(self, **hints):
from sympy import im, re
arg = self.args[0]
x = None
if arg.is_Add:
from sympy import symmetric_poly
n = len(arg.args)
TX = []
for x in arg.args:
tx = tan(x, evaluate=False)._eval_expand_trig()
TX.append(tx)
Yg = numbered_symbols('Y')
Y = [ next(Yg) for i in range(n) ]
p = [0, 0]
for i in range(n + 1):
p[1 - i % 2] += symmetric_poly(i, Y)*(-1)**((i % 4)//2)
return (p[0]/p[1]).subs(list(zip(Y, TX)))
else:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff.is_Integer and coeff > 1:
I = S.ImaginaryUnit
z = Symbol('dummy', real=True)
P = ((1 + I*z)**coeff).expand()
return (im(P)/re(P)).subs([(z, tan(terms))])
return tan(arg)
def _eval_rewrite_as_exp(self, arg, **kwargs):
I = S.ImaginaryUnit
if isinstance(arg, TrigonometricFunction) or isinstance(arg, HyperbolicFunction):
arg = arg.func(arg.args[0]).rewrite(exp)
neg_exp, pos_exp = exp(-arg*I), exp(arg*I)
return I*(neg_exp - pos_exp)/(neg_exp + pos_exp)
def _eval_rewrite_as_sin(self, x, **kwargs):
return 2*sin(x)**2/sin(2*x)
def _eval_rewrite_as_cos(self, x, **kwargs):
return cos(x - S.Pi / 2, evaluate=False) / cos(x)
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return sin(arg)/cos(arg)
def _eval_rewrite_as_cot(self, arg, **kwargs):
return 1/cot(arg)
def _eval_rewrite_as_sec(self, arg, **kwargs):
sin_in_sec_form = sin(arg)._eval_rewrite_as_sec(arg)
cos_in_sec_form = cos(arg)._eval_rewrite_as_sec(arg)
return sin_in_sec_form / cos_in_sec_form
def _eval_rewrite_as_csc(self, arg, **kwargs):
sin_in_csc_form = sin(arg)._eval_rewrite_as_csc(arg)
cos_in_csc_form = cos(arg)._eval_rewrite_as_csc(arg)
return sin_in_csc_form / cos_in_csc_form
def _eval_rewrite_as_pow(self, arg, **kwargs):
y = self.rewrite(cos).rewrite(pow)
if y.has(cos):
return None
return y
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
y = self.rewrite(cos).rewrite(sqrt)
if y.has(cos):
return None
return y
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
def _eval_is_real(self):
return self.args[0].is_real
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_imaginary:
return True
class cot(TrigonometricFunction):
"""
The cotangent function.
Returns the cotangent of x (measured in radians).
Notes
=====
See :func:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import cot, pi
>>> from sympy.abc import x
>>> cot(x**2).diff(x)
2*x*(-cot(x**2)**2 - 1)
>>> cot(1).diff(x)
0
>>> cot(pi/12)
sqrt(3) + 2
See Also
========
sin, csc, cos, sec, tan
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Cot
"""
def period(self, symbol=None):
return self._period(pi, symbol)
def fdiff(self, argindex=1):
if argindex == 1:
return S.NegativeOne - self**2
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return acot
@classmethod
def eval(cls, arg):
from sympy.calculus.util import AccumBounds
if arg.is_Number:
if arg is S.NaN:
return S.NaN
if arg is S.Zero:
return S.ComplexInfinity
if arg is S.ComplexInfinity:
return S.NaN
if isinstance(arg, AccumBounds):
return -tan(arg + S.Pi/2)
if arg.could_extract_minus_sign():
return -cls(-arg)
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return -S.ImaginaryUnit * coth(i_coeff)
pi_coeff = _pi_coeff(arg, 2)
if pi_coeff is not None:
if pi_coeff.is_integer:
return S.ComplexInfinity
if not pi_coeff.is_Rational:
narg = pi_coeff*S.Pi
if narg != arg:
return cls(narg)
return None
if pi_coeff.is_Rational:
if pi_coeff.q > 2 and not pi_coeff.q % 2:
narg = pi_coeff*S.Pi*2
cresult, sresult = cos(narg), cos(narg - S.Pi/2)
if not isinstance(cresult, cos) \
and not isinstance(sresult, cos):
return (1 + cresult)/sresult
table2 = {
12: (3, 4),
20: (4, 5),
30: (5, 6),
15: (6, 10),
24: (6, 8),
40: (8, 10),
60: (20, 30),
120: (40, 60)
}
q = pi_coeff.q
p = pi_coeff.p % q
if q in table2:
nvala, nvalb = cls(p*S.Pi/table2[q][0]), cls(p*S.Pi/table2[q][1])
if None == nvala or None == nvalb:
return None
return (1 + nvala*nvalb)/(nvalb - nvala)
narg = (((pi_coeff + S.Half) % 1) - S.Half)*S.Pi
# see cos() to specify which expressions should be
# expanded automatically in terms of radicals
cresult, sresult = cos(narg), cos(narg - S.Pi/2)
if not isinstance(cresult, cos) \
and not isinstance(sresult, cos):
if sresult == 0:
return S.ComplexInfinity
return cresult / sresult
if narg != arg:
return cls(narg)
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
cotm = cot(m)
if cotm is S.ComplexInfinity:
return cot(x)
else: # cotm == 0
return -tan(x)
if isinstance(arg, acot):
return arg.args[0]
if isinstance(arg, atan):
x = arg.args[0]
return 1 / x
if isinstance(arg, atan2):
y, x = arg.args
return x/y
if isinstance(arg, asin):
x = arg.args[0]
return sqrt(1 - x**2) / x
if isinstance(arg, acos):
x = arg.args[0]
return x / sqrt(1 - x**2)
if isinstance(arg, acsc):
x = arg.args[0]
return sqrt(1 - 1 / x**2) * x
if isinstance(arg, asec):
x = arg.args[0]
return 1 / (sqrt(1 - 1 / x**2) * x)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
from sympy import bernoulli
if n == 0:
return 1 / sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
B = bernoulli(n + 1)
F = factorial(n + 1)
return (-1)**((n + 1)//2) * 2**(n + 1) * B/F * x**n
def _eval_nseries(self, x, n, logx):
i = self.args[0].limit(x, 0)/S.Pi
if i and i.is_Integer:
return self.rewrite(cos)._eval_nseries(x, n=n, logx=logx)
return self.rewrite(tan)._eval_nseries(x, n=n, logx=logx)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
re, im = self._as_real_imag(deep=deep, **hints)
if im:
denom = cos(2*re) - cosh(2*im)
return (-sin(2*re)/denom, -sinh(2*im)/denom)
else:
return (self.func(re), S.Zero)
def _eval_rewrite_as_exp(self, arg, **kwargs):
I = S.ImaginaryUnit
if isinstance(arg, TrigonometricFunction) or isinstance(arg, HyperbolicFunction):
arg = arg.func(arg.args[0]).rewrite(exp)
neg_exp, pos_exp = exp(-arg*I), exp(arg*I)
return I*(pos_exp + neg_exp)/(pos_exp - neg_exp)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if isinstance(arg, log):
I = S.ImaginaryUnit
x = arg.args[0]
return -I*(x**-I + x**I)/(x**-I - x**I)
def _eval_rewrite_as_sin(self, x, **kwargs):
return sin(2*x)/(2*(sin(x)**2))
def _eval_rewrite_as_cos(self, x, **kwargs):
return cos(x) / cos(x - S.Pi / 2, evaluate=False)
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return cos(arg)/sin(arg)
def _eval_rewrite_as_tan(self, arg, **kwargs):
return 1/tan(arg)
def _eval_rewrite_as_sec(self, arg, **kwargs):
cos_in_sec_form = cos(arg)._eval_rewrite_as_sec(arg)
sin_in_sec_form = sin(arg)._eval_rewrite_as_sec(arg)
return cos_in_sec_form / sin_in_sec_form
def _eval_rewrite_as_csc(self, arg, **kwargs):
cos_in_csc_form = cos(arg)._eval_rewrite_as_csc(arg)
sin_in_csc_form = sin(arg)._eval_rewrite_as_csc(arg)
return cos_in_csc_form / sin_in_csc_form
def _eval_rewrite_as_pow(self, arg, **kwargs):
y = self.rewrite(cos).rewrite(pow)
if y.has(cos):
return None
return y
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
y = self.rewrite(cos).rewrite(sqrt)
if y.has(cos):
return None
return y
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return 1/arg
else:
return self.func(arg)
def _eval_is_real(self):
return self.args[0].is_real
def _eval_expand_trig(self, **hints):
from sympy import im, re
arg = self.args[0]
x = None
if arg.is_Add:
from sympy import symmetric_poly
n = len(arg.args)
CX = []
for x in arg.args:
cx = cot(x, evaluate=False)._eval_expand_trig()
CX.append(cx)
Yg = numbered_symbols('Y')
Y = [ next(Yg) for i in range(n) ]
p = [0, 0]
for i in range(n, -1, -1):
p[(n - i) % 2] += symmetric_poly(i, Y)*(-1)**(((n - i) % 4)//2)
return (p[0]/p[1]).subs(list(zip(Y, CX)))
else:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff.is_Integer and coeff > 1:
I = S.ImaginaryUnit
z = Symbol('dummy', real=True)
P = ((z + I)**coeff).expand()
return (re(P)/im(P)).subs([(z, cot(terms))])
return cot(arg)
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_imaginary:
return True
def _eval_subs(self, old, new):
if self == old:
return new
arg = self.args[0]
argnew = arg.subs(old, new)
if arg != argnew and (argnew/S.Pi).is_integer:
return S.ComplexInfinity
return cot(argnew)
class ReciprocalTrigonometricFunction(TrigonometricFunction):
"""Base class for reciprocal functions of trigonometric functions. """
_reciprocal_of = None # mandatory, to be defined in subclass
# _is_even and _is_odd are used for correct evaluation of csc(-x), sec(-x)
# TODO refactor into TrigonometricFunction common parts of
# trigonometric functions eval() like even/odd, func(x+2*k*pi), etc.
_is_even = None # optional, to be defined in subclass
_is_odd = None # optional, to be defined in subclass
@classmethod
def eval(cls, arg):
if arg.could_extract_minus_sign():
if cls._is_even:
return cls(-arg)
if cls._is_odd:
return -cls(-arg)
pi_coeff = _pi_coeff(arg)
if (pi_coeff is not None
and not (2*pi_coeff).is_integer
and pi_coeff.is_Rational):
q = pi_coeff.q
p = pi_coeff.p % (2*q)
if p > q:
narg = (pi_coeff - 1)*S.Pi
return -cls(narg)
if 2*p > q:
narg = (1 - pi_coeff)*S.Pi
if cls._is_odd:
return cls(narg)
elif cls._is_even:
return -cls(narg)
if hasattr(arg, 'inverse') and arg.inverse() == cls:
return arg.args[0]
t = cls._reciprocal_of.eval(arg)
if t is None:
return t
elif any(isinstance(i, cos) for i in (t, -t)):
return (1/t).rewrite(sec)
elif any(isinstance(i, sin) for i in (t, -t)):
return (1/t).rewrite(csc)
else:
return 1/t
def _call_reciprocal(self, method_name, *args, **kwargs):
# Calls method_name on _reciprocal_of
o = self._reciprocal_of(self.args[0])
return getattr(o, method_name)(*args, **kwargs)
def _calculate_reciprocal(self, method_name, *args, **kwargs):
# If calling method_name on _reciprocal_of returns a value != None
# then return the reciprocal of that value
t = self._call_reciprocal(method_name, *args, **kwargs)
return 1/t if t is not None else t
def _rewrite_reciprocal(self, method_name, arg):
# Special handling for rewrite functions. If reciprocal rewrite returns
# unmodified expression, then return None
t = self._call_reciprocal(method_name, arg)
if t is not None and t != self._reciprocal_of(arg):
return 1/t
def _period(self, symbol):
f = self.args[0]
return self._reciprocal_of(f).period(symbol)
def fdiff(self, argindex=1):
return -self._calculate_reciprocal("fdiff", argindex)/self**2
def _eval_rewrite_as_exp(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_exp", arg)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_Pow", arg)
def _eval_rewrite_as_sin(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_sin", arg)
def _eval_rewrite_as_cos(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_cos", arg)
def _eval_rewrite_as_tan(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_tan", arg)
def _eval_rewrite_as_pow(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_pow", arg)
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_sqrt", arg)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
return (1/self._reciprocal_of(self.args[0])).as_real_imag(deep,
**hints)
def _eval_expand_trig(self, **hints):
return self._calculate_reciprocal("_eval_expand_trig", **hints)
def _eval_is_real(self):
return self._reciprocal_of(self.args[0])._eval_is_real()
def _eval_as_leading_term(self, x):
return (1/self._reciprocal_of(self.args[0]))._eval_as_leading_term(x)
def _eval_is_finite(self):
return (1/self._reciprocal_of(self.args[0])).is_finite
def _eval_nseries(self, x, n, logx):
return (1/self._reciprocal_of(self.args[0]))._eval_nseries(x, n, logx)
class sec(ReciprocalTrigonometricFunction):
"""
The secant function.
Returns the secant of x (measured in radians).
Notes
=====
See :func:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import sec
>>> from sympy.abc import x
>>> sec(x**2).diff(x)
2*x*tan(x**2)*sec(x**2)
>>> sec(1).diff(x)
0
See Also
========
sin, csc, cos, tan, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Sec
"""
_reciprocal_of = cos
_is_even = True
def period(self, symbol=None):
return self._period(symbol)
def _eval_rewrite_as_cot(self, arg, **kwargs):
cot_half_sq = cot(arg/2)**2
return (cot_half_sq + 1)/(cot_half_sq - 1)
def _eval_rewrite_as_cos(self, arg, **kwargs):
return (1/cos(arg))
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return sin(arg)/(cos(arg)*sin(arg))
def _eval_rewrite_as_sin(self, arg, **kwargs):
return (1 / cos(arg)._eval_rewrite_as_sin(arg))
def _eval_rewrite_as_tan(self, arg, **kwargs):
return (1 / cos(arg)._eval_rewrite_as_tan(arg))
def _eval_rewrite_as_csc(self, arg, **kwargs):
return csc(pi / 2 - arg, evaluate=False)
def fdiff(self, argindex=1):
if argindex == 1:
return tan(self.args[0])*sec(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
# Reference Formula:
# http://functions.wolfram.com/ElementaryFunctions/Sec/06/01/02/01/
from sympy.functions.combinatorial.numbers import euler
if n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
k = n//2
return (-1)**k*euler(2*k)/factorial(2*k)*x**(2*k)
class csc(ReciprocalTrigonometricFunction):
"""
The cosecant function.
Returns the cosecant of x (measured in radians).
Notes
=====
See :func:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import csc
>>> from sympy.abc import x
>>> csc(x**2).diff(x)
-2*x*cot(x**2)*csc(x**2)
>>> csc(1).diff(x)
0
See Also
========
sin, cos, sec, tan, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Csc
"""
_reciprocal_of = sin
_is_odd = True
def period(self, symbol=None):
return self._period(symbol)
def _eval_rewrite_as_sin(self, arg, **kwargs):
return (1/sin(arg))
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return cos(arg)/(sin(arg)*cos(arg))
def _eval_rewrite_as_cot(self, arg, **kwargs):
cot_half = cot(arg/2)
return (1 + cot_half**2)/(2*cot_half)
def _eval_rewrite_as_cos(self, arg, **kwargs):
return (1 / sin(arg)._eval_rewrite_as_cos(arg))
def _eval_rewrite_as_sec(self, arg, **kwargs):
return sec(pi / 2 - arg, evaluate=False)
def _eval_rewrite_as_tan(self, arg, **kwargs):
return (1 / sin(arg)._eval_rewrite_as_tan(arg))
def fdiff(self, argindex=1):
if argindex == 1:
return -cot(self.args[0])*csc(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
from sympy import bernoulli
if n == 0:
return 1/sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
k = n//2 + 1
return ((-1)**(k - 1)*2*(2**(2*k - 1) - 1)*
bernoulli(2*k)*x**(2*k - 1)/factorial(2*k))
class sinc(Function):
r"""Represents unnormalized sinc function
Examples
========
>>> from sympy import sinc, oo, jn, Product, Symbol
>>> from sympy.abc import x
>>> sinc(x)
sinc(x)
* Automated Evaluation
>>> sinc(0)
1
>>> sinc(oo)
0
* Differentiation
>>> sinc(x).diff()
(x*cos(x) - sin(x))/x**2
* Series Expansion
>>> sinc(x).series()
1 - x**2/6 + x**4/120 + O(x**6)
* As zero'th order spherical Bessel Function
>>> sinc(x).rewrite(jn)
jn(0, x)
References
==========
.. [1] https://en.wikipedia.org/wiki/Sinc_function
"""
def fdiff(self, argindex=1):
x = self.args[0]
if argindex == 1:
return (x*cos(x) - sin(x)) / x**2
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
if arg.is_zero:
return S.One
if arg.is_Number:
if arg in [S.Infinity, -S.Infinity]:
return S.Zero
elif arg is S.NaN:
return S.NaN
if arg is S.ComplexInfinity:
return S.NaN
if arg.could_extract_minus_sign():
return cls(-arg)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_integer:
if fuzzy_not(arg.is_zero):
return S.Zero
elif (2*pi_coeff).is_integer:
return S.NegativeOne**(pi_coeff - S.Half) / arg
def _eval_nseries(self, x, n, logx):
x = self.args[0]
return (sin(x)/x)._eval_nseries(x, n, logx)
def _eval_rewrite_as_jn(self, arg, **kwargs):
from sympy.functions.special.bessel import jn
return jn(0, arg)
def _eval_rewrite_as_sin(self, arg, **kwargs):
return Piecewise((sin(arg)/arg, Ne(arg, 0)), (1, True))
###############################################################################
########################### TRIGONOMETRIC INVERSES ############################
###############################################################################
class InverseTrigonometricFunction(Function):
"""Base class for inverse trigonometric functions."""
pass
class asin(InverseTrigonometricFunction):
"""
The inverse sine function.
Returns the arcsine of x in radians.
Notes
=====
asin(x) will evaluate automatically in the cases oo, -oo, 0, 1,
-1 and for some instances when the result is a rational multiple
of pi (see the eval class method).
Examples
========
>>> from sympy import asin, oo, pi
>>> asin(1)
pi/2
>>> asin(-1)
-pi/2
See Also
========
sin, csc, cos, sec, tan, cot
acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] http://dlmf.nist.gov/4.23
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSin
"""
def fdiff(self, argindex=1):
if argindex == 1:
return 1/sqrt(1 - self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
def _eval_is_positive(self):
return self._eval_is_real() and self.args[0].is_positive
def _eval_is_negative(self):
return self._eval_is_real() and self.args[0].is_negative
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.NegativeInfinity * S.ImaginaryUnit
elif arg is S.NegativeInfinity:
return S.Infinity * S.ImaginaryUnit
elif arg is S.Zero:
return S.Zero
elif arg is S.One:
return S.Pi / 2
elif arg is S.NegativeOne:
return -S.Pi / 2
if arg is S.ComplexInfinity:
return S.ComplexInfinity
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_number:
cst_table = {
sqrt(3)/2: 3,
-sqrt(3)/2: -3,
sqrt(2)/2: 4,
-sqrt(2)/2: -4,
1/sqrt(2): 4,
-1/sqrt(2): -4,
sqrt((5 - sqrt(5))/8): 5,
-sqrt((5 - sqrt(5))/8): -5,
S.Half: 6,
-S.Half: -6,
sqrt(2 - sqrt(2))/2: 8,
-sqrt(2 - sqrt(2))/2: -8,
(sqrt(5) - 1)/4: 10,
(1 - sqrt(5))/4: -10,
(sqrt(3) - 1)/sqrt(2**3): 12,
(1 - sqrt(3))/sqrt(2**3): -12,
(sqrt(5) + 1)/4: S(10)/3,
-(sqrt(5) + 1)/4: -S(10)/3
}
if arg in cst_table:
return S.Pi / cst_table[arg]
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * asinh(i_coeff)
if isinstance(arg, sin):
ang = arg.args[0]
if ang.is_comparable:
ang %= 2*pi # restrict to [0,2*pi)
if ang > pi: # restrict to (-pi,pi]
ang = pi - ang
# restrict to [-pi/2,pi/2]
if ang > pi/2:
ang = pi - ang
if ang < -pi/2:
ang = -pi - ang
return ang
if isinstance(arg, cos): # acos(x) + asin(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
return pi/2 - acos(arg)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) >= 2 and n > 2:
p = previous_terms[-2]
return p * (n - 2)**2/(n*(n - 1)) * x**2
else:
k = (n - 1) // 2
R = RisingFactorial(S.Half, k)
F = factorial(k)
return R / F * x**n / n
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
def _eval_rewrite_as_acos(self, x, **kwargs):
return S.Pi/2 - acos(x)
def _eval_rewrite_as_atan(self, x, **kwargs):
return 2*atan(x/(1 + sqrt(1 - x**2)))
def _eval_rewrite_as_log(self, x, **kwargs):
return -S.ImaginaryUnit*log(S.ImaginaryUnit*x + sqrt(1 - x**2))
def _eval_rewrite_as_acot(self, arg, **kwargs):
return 2*acot((1 + sqrt(1 - arg**2))/arg)
def _eval_rewrite_as_asec(self, arg, **kwargs):
return S.Pi/2 - asec(1/arg)
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return acsc(1/arg)
def _eval_is_real(self):
x = self.args[0]
return x.is_real and (1 - abs(x)).is_nonnegative
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return sin
class acos(InverseTrigonometricFunction):
"""
The inverse cosine function.
Returns the arc cosine of x (measured in radians).
Notes
=====
``acos(x)`` will evaluate automatically in the cases
``oo``, ``-oo``, ``0``, ``1``, ``-1``.
``acos(zoo)`` evaluates to ``zoo``
(see note in :py:class`sympy.functions.elementary.trigonometric.asec`)
Examples
========
>>> from sympy import acos, oo, pi
>>> acos(1)
0
>>> acos(0)
pi/2
>>> acos(oo)
oo*I
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] http://dlmf.nist.gov/4.23
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCos
"""
def fdiff(self, argindex=1):
if argindex == 1:
return -1/sqrt(1 - self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity * S.ImaginaryUnit
elif arg is S.NegativeInfinity:
return S.NegativeInfinity * S.ImaginaryUnit
elif arg is S.Zero:
return S.Pi / 2
elif arg is S.One:
return S.Zero
elif arg is S.NegativeOne:
return S.Pi
if arg is S.ComplexInfinity:
return S.ComplexInfinity
if arg.is_number:
cst_table = {
S.Half: S.Pi/3,
-S.Half: 2*S.Pi/3,
sqrt(2)/2: S.Pi/4,
-sqrt(2)/2: 3*S.Pi/4,
1/sqrt(2): S.Pi/4,
-1/sqrt(2): 3*S.Pi/4,
sqrt(3)/2: S.Pi/6,
-sqrt(3)/2: 5*S.Pi/6,
}
if arg in cst_table:
return cst_table[arg]
if isinstance(arg, cos):
ang = arg.args[0]
if ang.is_comparable:
ang %= 2*pi # restrict to [0,2*pi)
if ang > pi: # restrict to [0,pi]
ang = 2*pi - ang
return ang
if isinstance(arg, sin): # acos(x) + asin(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
return pi/2 - asin(arg)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return S.Pi / 2
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) >= 2 and n > 2:
p = previous_terms[-2]
return p * (n - 2)**2/(n*(n - 1)) * x**2
else:
k = (n - 1) // 2
R = RisingFactorial(S.Half, k)
F = factorial(k)
return -R / F * x**n / n
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
def _eval_is_real(self):
x = self.args[0]
return x.is_real and (1 - abs(x)).is_nonnegative
def _eval_is_nonnegative(self):
return self._eval_is_real()
def _eval_nseries(self, x, n, logx):
return self._eval_rewrite_as_log(self.args[0])._eval_nseries(x, n, logx)
def _eval_rewrite_as_log(self, x, **kwargs):
return S.Pi/2 + S.ImaginaryUnit * \
log(S.ImaginaryUnit * x + sqrt(1 - x**2))
def _eval_rewrite_as_asin(self, x, **kwargs):
return S.Pi/2 - asin(x)
def _eval_rewrite_as_atan(self, x, **kwargs):
return atan(sqrt(1 - x**2)/x) + (S.Pi/2)*(1 - x*sqrt(1/x**2))
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return cos
def _eval_rewrite_as_acot(self, arg, **kwargs):
return S.Pi/2 - 2*acot((1 + sqrt(1 - arg**2))/arg)
def _eval_rewrite_as_asec(self, arg, **kwargs):
return asec(1/arg)
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return S.Pi/2 - acsc(1/arg)
def _eval_conjugate(self):
z = self.args[0]
r = self.func(self.args[0].conjugate())
if z.is_real is False:
return r
elif z.is_real and (z + 1).is_nonnegative and (z - 1).is_nonpositive:
return r
class atan(InverseTrigonometricFunction):
"""
The inverse tangent function.
Returns the arc tangent of x (measured in radians).
Notes
=====
atan(x) will evaluate automatically in the cases
oo, -oo, 0, 1, -1.
Examples
========
>>> from sympy import atan, oo, pi
>>> atan(0)
0
>>> atan(1)
pi/4
>>> atan(oo)
pi/2
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, acos, asec, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] http://dlmf.nist.gov/4.23
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcTan
"""
def fdiff(self, argindex=1):
if argindex == 1:
return 1/(1 + self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
def _eval_is_positive(self):
return self.args[0].is_positive
def _eval_is_nonnegative(self):
return self.args[0].is_nonnegative
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Pi / 2
elif arg is S.NegativeInfinity:
return -S.Pi / 2
elif arg is S.Zero:
return S.Zero
elif arg is S.One:
return S.Pi / 4
elif arg is S.NegativeOne:
return -S.Pi / 4
if arg is S.ComplexInfinity:
from sympy.calculus.util import AccumBounds
return AccumBounds(-S.Pi/2, S.Pi/2)
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_number:
cst_table = {
sqrt(3)/3: 6,
-sqrt(3)/3: -6,
1/sqrt(3): 6,
-1/sqrt(3): -6,
sqrt(3): 3,
-sqrt(3): -3,
(1 + sqrt(2)): S(8)/3,
-(1 + sqrt(2)): S(8)/3,
(sqrt(2) - 1): 8,
(1 - sqrt(2)): -8,
sqrt((5 + 2*sqrt(5))): S(5)/2,
-sqrt((5 + 2*sqrt(5))): -S(5)/2,
(2 - sqrt(3)): 12,
-(2 - sqrt(3)): -12
}
if arg in cst_table:
return S.Pi / cst_table[arg]
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * atanh(i_coeff)
if isinstance(arg, tan):
ang = arg.args[0]
if ang.is_comparable:
ang %= pi # restrict to [0,pi)
if ang > pi/2: # restrict to [-pi/2,pi/2]
ang -= pi
return ang
if isinstance(arg, cot): # atan(x) + acot(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
ang = pi/2 - acot(arg)
if ang > pi/2: # restrict to [-pi/2,pi/2]
ang -= pi
return ang
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
return (-1)**((n - 1)//2) * x**n / n
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
def _eval_is_real(self):
return self.args[0].is_real
def _eval_rewrite_as_log(self, x, **kwargs):
return S.ImaginaryUnit/2 * (log(S(1) - S.ImaginaryUnit * x)
- log(S(1) + S.ImaginaryUnit * x))
def _eval_aseries(self, n, args0, x, logx):
if args0[0] == S.Infinity:
return (S.Pi/2 - atan(1/self.args[0]))._eval_nseries(x, n, logx)
elif args0[0] == S.NegativeInfinity:
return (-S.Pi/2 - atan(1/self.args[0]))._eval_nseries(x, n, logx)
else:
return super(atan, self)._eval_aseries(n, args0, x, logx)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return tan
def _eval_rewrite_as_asin(self, arg, **kwargs):
return sqrt(arg**2)/arg*(S.Pi/2 - asin(1/sqrt(1 + arg**2)))
def _eval_rewrite_as_acos(self, arg, **kwargs):
return sqrt(arg**2)/arg*acos(1/sqrt(1 + arg**2))
def _eval_rewrite_as_acot(self, arg, **kwargs):
return acot(1/arg)
def _eval_rewrite_as_asec(self, arg, **kwargs):
return sqrt(arg**2)/arg*asec(sqrt(1 + arg**2))
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return sqrt(arg**2)/arg*(S.Pi/2 - acsc(sqrt(1 + arg**2)))
class acot(InverseTrigonometricFunction):
"""
The inverse cotangent function.
Returns the arc cotangent of x (measured in radians).
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, acos, asec, atan, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] http://dlmf.nist.gov/4.23
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCot
"""
def fdiff(self, argindex=1):
if argindex == 1:
return -1 / (1 + self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
def _eval_is_positive(self):
return self.args[0].is_nonnegative
def _eval_is_negative(self):
return self.args[0].is_negative
def _eval_is_real(self):
return self.args[0].is_real
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Zero
elif arg is S.NegativeInfinity:
return S.Zero
elif arg is S.Zero:
return S.Pi/ 2
elif arg is S.One:
return S.Pi / 4
elif arg is S.NegativeOne:
return -S.Pi / 4
if arg is S.ComplexInfinity:
return S.Zero
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_number:
cst_table = {
sqrt(3)/3: 3,
-sqrt(3)/3: -3,
1/sqrt(3): 3,
-1/sqrt(3): -3,
sqrt(3): 6,
-sqrt(3): -6,
(1 + sqrt(2)): 8,
-(1 + sqrt(2)): -8,
(1 - sqrt(2)): -S(8)/3,
(sqrt(2) - 1): S(8)/3,
sqrt(5 + 2*sqrt(5)): 10,
-sqrt(5 + 2*sqrt(5)): -10,
(2 + sqrt(3)): 12,
-(2 + sqrt(3)): -12,
(2 - sqrt(3)): S(12)/5,
-(2 - sqrt(3)): -S(12)/5,
}
if arg in cst_table:
return S.Pi / cst_table[arg]
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return -S.ImaginaryUnit * acoth(i_coeff)
if isinstance(arg, cot):
ang = arg.args[0]
if ang.is_comparable:
ang %= pi # restrict to [0,pi)
if ang > pi/2: # restrict to (-pi/2,pi/2]
ang -= pi;
return ang
if isinstance(arg, tan): # atan(x) + acot(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
ang = pi/2 - atan(arg)
if ang > pi/2: # restrict to (-pi/2,pi/2]
ang -= pi
return ang
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return S.Pi / 2 # FIX THIS
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
return (-1)**((n + 1)//2) * x**n / n
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
def _eval_aseries(self, n, args0, x, logx):
if args0[0] == S.Infinity:
return (S.Pi/2 - acot(1/self.args[0]))._eval_nseries(x, n, logx)
elif args0[0] == S.NegativeInfinity:
return (3*S.Pi/2 - acot(1/self.args[0]))._eval_nseries(x, n, logx)
else:
return super(atan, self)._eval_aseries(n, args0, x, logx)
def _eval_rewrite_as_log(self, x, **kwargs):
return S.ImaginaryUnit/2 * (log(1 - S.ImaginaryUnit/x)
- log(1 + S.ImaginaryUnit/x))
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return cot
def _eval_rewrite_as_asin(self, arg, **kwargs):
return (arg*sqrt(1/arg**2)*
(S.Pi/2 - asin(sqrt(-arg**2)/sqrt(-arg**2 - 1))))
def _eval_rewrite_as_acos(self, arg, **kwargs):
return arg*sqrt(1/arg**2)*acos(sqrt(-arg**2)/sqrt(-arg**2 - 1))
def _eval_rewrite_as_atan(self, arg, **kwargs):
return atan(1/arg)
def _eval_rewrite_as_asec(self, arg, **kwargs):
return arg*sqrt(1/arg**2)*asec(sqrt((1 + arg**2)/arg**2))
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return arg*sqrt(1/arg**2)*(S.Pi/2 - acsc(sqrt((1 + arg**2)/arg**2)))
class asec(InverseTrigonometricFunction):
r"""
The inverse secant function.
Returns the arc secant of x (measured in radians).
Notes
=====
``asec(x)`` will evaluate automatically in the cases
``oo``, ``-oo``, ``0``, ``1``, ``-1``.
``asec(x)`` has branch cut in the interval [-1, 1]. For complex arguments,
it can be defined [4]_ as
.. math::
sec^{-1}(z) = -i*(log(\sqrt{1 - z^2} + 1) / z)
At ``x = 0``, for positive branch cut, the limit evaluates to ``zoo``. For
negative branch cut, the limit
.. math::
\lim_{z \to 0}-i*(log(-\sqrt{1 - z^2} + 1) / z)
simplifies to :math:`-i*log(z/2 + O(z^3))` which ultimately evaluates to
``zoo``.
As ``asex(x)`` = ``asec(1/x)``, a similar argument can be given for
``acos(x)``.
Examples
========
>>> from sympy import asec, oo, pi
>>> asec(1)
0
>>> asec(-1)
pi
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, acos, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] http://dlmf.nist.gov/4.23
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSec
.. [4] http://reference.wolfram.com/language/ref/ArcSec.html
"""
@classmethod
def eval(cls, arg):
if arg.is_zero:
return S.ComplexInfinity
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.One:
return S.Zero
elif arg is S.NegativeOne:
return S.Pi
if arg in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
return S.Pi/2
if isinstance(arg, sec):
ang = arg.args[0]
if ang.is_comparable:
ang %= 2*pi # restrict to [0,2*pi)
if ang > pi: # restrict to [0,pi]
ang = 2*pi - ang
return ang
if isinstance(arg, csc): # asec(x) + acsc(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
return pi/2 - acsc(arg)
def fdiff(self, argindex=1):
if argindex == 1:
return 1/(self.args[0]**2*sqrt(1 - 1/self.args[0]**2))
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return sec
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if Order(1,x).contains(arg):
return log(arg)
else:
return self.func(arg)
def _eval_is_real(self):
x = self.args[0]
if x.is_real is False:
return False
return fuzzy_or(((x - 1).is_nonnegative, (-x - 1).is_nonnegative))
def _eval_rewrite_as_log(self, arg, **kwargs):
return S.Pi/2 + S.ImaginaryUnit*log(S.ImaginaryUnit/arg + sqrt(1 - 1/arg**2))
def _eval_rewrite_as_asin(self, arg, **kwargs):
return S.Pi/2 - asin(1/arg)
def _eval_rewrite_as_acos(self, arg, **kwargs):
return acos(1/arg)
def _eval_rewrite_as_atan(self, arg, **kwargs):
return sqrt(arg**2)/arg*(-S.Pi/2 + 2*atan(arg + sqrt(arg**2 - 1)))
def _eval_rewrite_as_acot(self, arg, **kwargs):
return sqrt(arg**2)/arg*(-S.Pi/2 + 2*acot(arg - sqrt(arg**2 - 1)))
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return S.Pi/2 - acsc(arg)
class acsc(InverseTrigonometricFunction):
"""
The inverse cosecant function.
Returns the arc cosecant of x (measured in radians).
Notes
=====
acsc(x) will evaluate automatically in the cases
oo, -oo, 0, 1, -1.
Examples
========
>>> from sympy import acsc, oo, pi
>>> acsc(1)
pi/2
>>> acsc(-1)
-pi/2
See Also
========
sin, csc, cos, sec, tan, cot
asin, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] http://dlmf.nist.gov/4.23
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCsc
"""
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.One:
return S.Pi/2
elif arg is S.NegativeOne:
return -S.Pi/2
if arg in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
return S.Zero
if isinstance(arg, csc):
ang = arg.args[0]
if ang.is_comparable:
ang %= 2*pi # restrict to [0,2*pi)
if ang > pi: # restrict to (-pi,pi]
ang = pi - ang
# restrict to [-pi/2,pi/2]
if ang > pi/2:
ang = pi - ang
if ang < -pi/2:
ang = -pi - ang
return ang
if isinstance(arg, sec): # asec(x) + acsc(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
return pi/2 - asec(arg)
def fdiff(self, argindex=1):
if argindex == 1:
return -1/(self.args[0]**2*sqrt(1 - 1/self.args[0]**2))
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return csc
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if Order(1,x).contains(arg):
return log(arg)
else:
return self.func(arg)
def _eval_rewrite_as_log(self, arg, **kwargs):
return -S.ImaginaryUnit*log(S.ImaginaryUnit/arg + sqrt(1 - 1/arg**2))
def _eval_rewrite_as_asin(self, arg, **kwargs):
return asin(1/arg)
def _eval_rewrite_as_acos(self, arg, **kwargs):
return S.Pi/2 - acos(1/arg)
def _eval_rewrite_as_atan(self, arg, **kwargs):
return sqrt(arg**2)/arg*(S.Pi/2 - atan(sqrt(arg**2 - 1)))
def _eval_rewrite_as_acot(self, arg, **kwargs):
return sqrt(arg**2)/arg*(S.Pi/2 - acot(1/sqrt(arg**2 - 1)))
def _eval_rewrite_as_asec(self, arg, **kwargs):
return S.Pi/2 - asec(arg)
class atan2(InverseTrigonometricFunction):
r"""
The function ``atan2(y, x)`` computes `\operatorname{atan}(y/x)` taking
two arguments `y` and `x`. Signs of both `y` and `x` are considered to
determine the appropriate quadrant of `\operatorname{atan}(y/x)`.
The range is `(-\pi, \pi]`. The complete definition reads as follows:
.. math::
\operatorname{atan2}(y, x) =
\begin{cases}
\arctan\left(\frac y x\right) & \qquad x > 0 \\
\arctan\left(\frac y x\right) + \pi& \qquad y \ge 0 , x < 0 \\
\arctan\left(\frac y x\right) - \pi& \qquad y < 0 , x < 0 \\
+\frac{\pi}{2} & \qquad y > 0 , x = 0 \\
-\frac{\pi}{2} & \qquad y < 0 , x = 0 \\
\text{undefined} & \qquad y = 0, x = 0
\end{cases}
Attention: Note the role reversal of both arguments. The `y`-coordinate
is the first argument and the `x`-coordinate the second.
Examples
========
Going counter-clock wise around the origin we find the
following angles:
>>> from sympy import atan2
>>> atan2(0, 1)
0
>>> atan2(1, 1)
pi/4
>>> atan2(1, 0)
pi/2
>>> atan2(1, -1)
3*pi/4
>>> atan2(0, -1)
pi
>>> atan2(-1, -1)
-3*pi/4
>>> atan2(-1, 0)
-pi/2
>>> atan2(-1, 1)
-pi/4
which are all correct. Compare this to the results of the ordinary
`\operatorname{atan}` function for the point `(x, y) = (-1, 1)`
>>> from sympy import atan, S
>>> atan(S(1) / -1)
-pi/4
>>> atan2(1, -1)
3*pi/4
where only the `\operatorname{atan2}` function reurns what we expect.
We can differentiate the function with respect to both arguments:
>>> from sympy import diff
>>> from sympy.abc import x, y
>>> diff(atan2(y, x), x)
-y/(x**2 + y**2)
>>> diff(atan2(y, x), y)
x/(x**2 + y**2)
We can express the `\operatorname{atan2}` function in terms of
complex logarithms:
>>> from sympy import log
>>> atan2(y, x).rewrite(log)
-I*log((x + I*y)/sqrt(x**2 + y**2))
and in terms of `\operatorname(atan)`:
>>> from sympy import atan
>>> atan2(y, x).rewrite(atan)
2*atan(y/(x + sqrt(x**2 + y**2)))
but note that this form is undefined on the negative real axis.
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, acos, asec, atan, acot
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] https://en.wikipedia.org/wiki/Atan2
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcTan2
"""
@classmethod
def eval(cls, y, x):
from sympy import Heaviside, im, re
if x is S.NegativeInfinity:
if y.is_zero:
# Special case y = 0 because we define Heaviside(0) = 1/2
return S.Pi
return 2*S.Pi*(Heaviside(re(y))) - S.Pi
elif x is S.Infinity:
return S.Zero
elif x.is_imaginary and y.is_imaginary and x.is_number and y.is_number:
x = im(x)
y = im(y)
if x.is_real and y.is_real:
if x.is_positive:
return atan(y / x)
elif x.is_negative:
if y.is_negative:
return atan(y / x) - S.Pi
elif y.is_nonnegative:
return atan(y / x) + S.Pi
elif x.is_zero:
if y.is_positive:
return S.Pi/2
elif y.is_negative:
return -S.Pi/2
elif y.is_zero:
return S.NaN
if y.is_zero and x.is_real and fuzzy_not(x.is_zero):
return S.Pi * (S.One - Heaviside(x))
if x.is_number and y.is_number:
return -S.ImaginaryUnit*log(
(x + S.ImaginaryUnit*y)/sqrt(x**2 + y**2))
def _eval_rewrite_as_log(self, y, x, **kwargs):
return -S.ImaginaryUnit*log((x + S.ImaginaryUnit*y) / sqrt(x**2 + y**2))
def _eval_rewrite_as_atan(self, y, x, **kwargs):
return 2*atan(y / (sqrt(x**2 + y**2) + x))
def _eval_rewrite_as_arg(self, y, x, **kwargs):
from sympy import arg
if x.is_real and y.is_real:
return arg(x + y*S.ImaginaryUnit)
I = S.ImaginaryUnit
n = x + I*y
d = x**2 + y**2
return arg(n/sqrt(d)) - I*log(abs(n)/sqrt(abs(d)))
def _eval_is_real(self):
return self.args[0].is_real and self.args[1].is_real
def _eval_conjugate(self):
return self.func(self.args[0].conjugate(), self.args[1].conjugate())
def fdiff(self, argindex):
y, x = self.args
if argindex == 1:
# Diff wrt y
return x/(x**2 + y**2)
elif argindex == 2:
# Diff wrt x
return -y/(x**2 + y**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_evalf(self, prec):
y, x = self.args
if x.is_real and y.is_real:
super(atan2, self)._eval_evalf(prec)
|
955620fdafbef7175b8698b37e7faf1d8026618da13154c6f93ce7f9bd45fe04
|
from __future__ import print_function, division
from sympy.core import Function, S, sympify
from sympy.core.add import Add
from sympy.core.containers import Tuple
from sympy.core.operations import LatticeOp, ShortCircuit
from sympy.core.function import (Application, Lambda,
ArgumentIndexError)
from sympy.core.evaluate import global_evaluate
from sympy.core.expr import Expr
from sympy.core.mod import Mod
from sympy.core.mul import Mul
from sympy.core.numbers import Rational
from sympy.core.power import Pow
from sympy.core.relational import Eq, Relational
from sympy.core.singleton import Singleton
from sympy.core.symbol import Dummy
from sympy.core.rules import Transform
from sympy.core.compatibility import with_metaclass, range
from sympy.core.logic import fuzzy_and, fuzzy_or, _torf
from sympy.logic.boolalg import And, Or
def _minmax_as_Piecewise(op, *args):
# helper for Min/Max rewrite as Piecewise
from sympy.functions.elementary.piecewise import Piecewise
ec = []
for i, a in enumerate(args):
c = []
for j in range(i + 1, len(args)):
c.append(Relational(a, args[j], op))
ec.append((a, And(*c)))
return Piecewise(*ec)
class IdentityFunction(with_metaclass(Singleton, Lambda)):
"""
The identity function
Examples
========
>>> from sympy import Id, Symbol
>>> x = Symbol('x')
>>> Id(x)
x
"""
def __new__(cls):
from sympy.sets.sets import FiniteSet
x = Dummy('x')
#construct "by hand" to avoid infinite loop
obj = Expr.__new__(cls, Tuple(x), x)
obj.nargs = FiniteSet(1)
return obj
Id = S.IdentityFunction
###############################################################################
############################# ROOT and SQUARE ROOT FUNCTION ###################
###############################################################################
def sqrt(arg, evaluate=None):
"""The square root function
sqrt(x) -> Returns the principal square root of x.
The parameter evaluate determines if the expression should be evaluated.
If None, its value is taken from global_evaluate
Examples
========
>>> from sympy import sqrt, Symbol
>>> x = Symbol('x')
>>> sqrt(x)
sqrt(x)
>>> sqrt(x)**2
x
Note that sqrt(x**2) does not simplify to x.
>>> sqrt(x**2)
sqrt(x**2)
This is because the two are not equal to each other in general.
For example, consider x == -1:
>>> from sympy import Eq
>>> Eq(sqrt(x**2), x).subs(x, -1)
False
This is because sqrt computes the principal square root, so the square may
put the argument in a different branch. This identity does hold if x is
positive:
>>> y = Symbol('y', positive=True)
>>> sqrt(y**2)
y
You can force this simplification by using the powdenest() function with
the force option set to True:
>>> from sympy import powdenest
>>> sqrt(x**2)
sqrt(x**2)
>>> powdenest(sqrt(x**2), force=True)
x
To get both branches of the square root you can use the rootof function:
>>> from sympy import rootof
>>> [rootof(x**2-3,i) for i in (0,1)]
[-sqrt(3), sqrt(3)]
See Also
========
sympy.polys.rootoftools.rootof, root, real_root
References
==========
.. [1] https://en.wikipedia.org/wiki/Square_root
.. [2] https://en.wikipedia.org/wiki/Principal_value
"""
# arg = sympify(arg) is handled by Pow
return Pow(arg, S.Half, evaluate=evaluate)
def cbrt(arg, evaluate=None):
"""This function computes the principal cube root of `arg`, so
it's just a shortcut for `arg**Rational(1, 3)`.
The parameter evaluate determines if the expression should be evaluated.
If None, its value is taken from global_evaluate.
Examples
========
>>> from sympy import cbrt, Symbol
>>> x = Symbol('x')
>>> cbrt(x)
x**(1/3)
>>> cbrt(x)**3
x
Note that cbrt(x**3) does not simplify to x.
>>> cbrt(x**3)
(x**3)**(1/3)
This is because the two are not equal to each other in general.
For example, consider `x == -1`:
>>> from sympy import Eq
>>> Eq(cbrt(x**3), x).subs(x, -1)
False
This is because cbrt computes the principal cube root, this
identity does hold if `x` is positive:
>>> y = Symbol('y', positive=True)
>>> cbrt(y**3)
y
See Also
========
sympy.polys.rootoftools.rootof, root, real_root
References
==========
* https://en.wikipedia.org/wiki/Cube_root
* https://en.wikipedia.org/wiki/Principal_value
"""
return Pow(arg, Rational(1, 3), evaluate=evaluate)
def root(arg, n, k=0, evaluate=None):
"""root(x, n, k) -> Returns the k-th n-th root of x, defaulting to the
principal root (k=0).
The parameter evaluate determines if the expression should be evaluated.
If None, its value is taken from global_evaluate.
Examples
========
>>> from sympy import root, Rational
>>> from sympy.abc import x, n
>>> root(x, 2)
sqrt(x)
>>> root(x, 3)
x**(1/3)
>>> root(x, n)
x**(1/n)
>>> root(x, -Rational(2, 3))
x**(-3/2)
To get the k-th n-th root, specify k:
>>> root(-2, 3, 2)
-(-1)**(2/3)*2**(1/3)
To get all n n-th roots you can use the rootof function.
The following examples show the roots of unity for n
equal 2, 3 and 4:
>>> from sympy import rootof, I
>>> [rootof(x**2 - 1, i) for i in range(2)]
[-1, 1]
>>> [rootof(x**3 - 1,i) for i in range(3)]
[1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2]
>>> [rootof(x**4 - 1,i) for i in range(4)]
[-1, 1, -I, I]
SymPy, like other symbolic algebra systems, returns the
complex root of negative numbers. This is the principal
root and differs from the text-book result that one might
be expecting. For example, the cube root of -8 does not
come back as -2:
>>> root(-8, 3)
2*(-1)**(1/3)
The real_root function can be used to either make the principal
result real (or simply to return the real root directly):
>>> from sympy import real_root
>>> real_root(_)
-2
>>> real_root(-32, 5)
-2
Alternatively, the n//2-th n-th root of a negative number can be
computed with root:
>>> root(-32, 5, 5//2)
-2
See Also
========
sympy.polys.rootoftools.rootof
sympy.core.power.integer_nthroot
sqrt, real_root
References
==========
* https://en.wikipedia.org/wiki/Square_root
* https://en.wikipedia.org/wiki/Real_root
* https://en.wikipedia.org/wiki/Root_of_unity
* https://en.wikipedia.org/wiki/Principal_value
* http://mathworld.wolfram.com/CubeRoot.html
"""
n = sympify(n)
if k:
return Mul(Pow(arg, S.One/n, evaluate=evaluate), S.NegativeOne**(2*k/n), evaluate=evaluate)
return Pow(arg, 1/n, evaluate=evaluate)
def real_root(arg, n=None, evaluate=None):
"""Return the real nth-root of arg if possible. If n is omitted then
all instances of (-n)**(1/odd) will be changed to -n**(1/odd); this
will only create a real root of a principal root -- the presence of
other factors may cause the result to not be real.
The parameter evaluate determines if the expression should be evaluated.
If None, its value is taken from global_evaluate.
Examples
========
>>> from sympy import root, real_root, Rational
>>> from sympy.abc import x, n
>>> real_root(-8, 3)
-2
>>> root(-8, 3)
2*(-1)**(1/3)
>>> real_root(_)
-2
If one creates a non-principal root and applies real_root, the
result will not be real (so use with caution):
>>> root(-8, 3, 2)
-2*(-1)**(2/3)
>>> real_root(_)
-2*(-1)**(2/3)
See Also
========
sympy.polys.rootoftools.rootof
sympy.core.power.integer_nthroot
root, sqrt
"""
from sympy.functions.elementary.complexes import Abs, im, sign
from sympy.functions.elementary.piecewise import Piecewise
if n is not None:
return Piecewise(
(root(arg, n, evaluate=evaluate), Or(Eq(n, S.One), Eq(n, S.NegativeOne))),
(Mul(sign(arg), root(Abs(arg), n, evaluate=evaluate), evaluate=evaluate),
And(Eq(im(arg), S.Zero), Eq(Mod(n, 2), S.One))),
(root(arg, n, evaluate=evaluate), True))
rv = sympify(arg)
n1pow = Transform(lambda x: -(-x.base)**x.exp,
lambda x:
x.is_Pow and
x.base.is_negative and
x.exp.is_Rational and
x.exp.p == 1 and x.exp.q % 2)
return rv.xreplace(n1pow)
###############################################################################
############################# MINIMUM and MAXIMUM #############################
###############################################################################
class MinMaxBase(Expr, LatticeOp):
def __new__(cls, *args, **assumptions):
evaluate = assumptions.pop('evaluate', True)
args = (sympify(arg) for arg in args)
# first standard filter, for cls.zero and cls.identity
# also reshape Max(a, Max(b, c)) to Max(a, b, c)
if evaluate:
try:
args = frozenset(cls._new_args_filter(args))
except ShortCircuit:
return cls.zero
else:
args = frozenset(args)
if evaluate:
# remove redundant args that are easily identified
args = cls._collapse_arguments(args, **assumptions)
# find local zeros
args = cls._find_localzeros(args, **assumptions)
if not args:
return cls.identity
if len(args) == 1:
return list(args).pop()
# base creation
_args = frozenset(args)
obj = Expr.__new__(cls, _args, **assumptions)
obj._argset = _args
return obj
@classmethod
def _collapse_arguments(cls, args, **assumptions):
"""Remove redundant args.
Examples
========
>>> from sympy import Min, Max
>>> from sympy.abc import a, b, c, d, e
Any arg in parent that appears in any
parent-like function in any of the flat args
of parent can be removed from that sub-arg:
>>> Min(a, Max(b, Min(a, c, d)))
Min(a, Max(b, Min(c, d)))
If the arg of parent appears in an opposite-than parent
function in any of the flat args of parent that function
can be replaced with the arg:
>>> Min(a, Max(b, Min(c, d, Max(a, e))))
Min(a, Max(b, Min(a, c, d)))
"""
from sympy.utilities.iterables import ordered
from sympy.simplify.simplify import walk
if not args:
return args
args = list(ordered(args))
if cls == Min:
other = Max
else:
other = Min
# find global comparable max of Max and min of Min if a new
# value is being introduced in these args at position 0 of
# the ordered args
if args[0].is_number:
sifted = mins, maxs = [], []
for i in args:
for v in walk(i, Min, Max):
if v.args[0].is_comparable:
sifted[isinstance(v, Max)].append(v)
small = Min.identity
for i in mins:
v = i.args[0]
if v.is_number and (v < small) == True:
small = v
big = Max.identity
for i in maxs:
v = i.args[0]
if v.is_number and (v > big) == True:
big = v
# at the point when this function is called from __new__,
# there may be more than one numeric arg present since
# local zeros have not been handled yet, so look through
# more than the first arg
if cls == Min:
for i in range(len(args)):
if not args[i].is_number:
break
if (args[i] < small) == True:
small = args[i]
elif cls == Max:
for i in range(len(args)):
if not args[i].is_number:
break
if (args[i] > big) == True:
big = args[i]
T = None
if cls == Min:
if small != Min.identity:
other = Max
T = small
elif big != Max.identity:
other = Min
T = big
if T is not None:
# remove numerical redundancy
for i in range(len(args)):
a = args[i]
if isinstance(a, other):
a0 = a.args[0]
if ((a0 > T) if other == Max else (a0 < T)) == True:
args[i] = cls.identity
# remove redundant symbolic args
def do(ai, a):
if not isinstance(ai, (Min, Max)):
return ai
cond = a in ai.args
if not cond:
return ai.func(*[do(i, a) for i in ai.args],
evaluate=False)
if isinstance(ai, cls):
return ai.func(*[do(i, a) for i in ai.args if i != a],
evaluate=False)
return a
for i, a in enumerate(args):
args[i + 1:] = [do(ai, a) for ai in args[i + 1:]]
# factor out common elements as for
# Min(Max(x, y), Max(x, z)) -> Max(x, Min(y, z))
# and vice versa when swapping Min/Max -- do this only for the
# easy case where all functions contain something in common;
# trying to find some optimal subset of args to modify takes
# too long
if len(args) > 1:
common = None
remove = []
sets = []
for i in range(len(args)):
a = args[i]
if not isinstance(a, other):
continue
s = set(a.args)
common = s if common is None else (common & s)
if not common:
break
sets.append(s)
remove.append(i)
if common:
sets = filter(None, [s - common for s in sets])
sets = [other(*s, evaluate=False) for s in sets]
for i in reversed(remove):
args.pop(i)
oargs = [cls(*sets)] if sets else []
oargs.extend(common)
args.append(other(*oargs, evaluate=False))
return args
@classmethod
def _new_args_filter(cls, arg_sequence):
"""
Generator filtering args.
first standard filter, for cls.zero and cls.identity.
Also reshape Max(a, Max(b, c)) to Max(a, b, c),
and check arguments for comparability
"""
for arg in arg_sequence:
# pre-filter, checking comparability of arguments
if not isinstance(arg, Expr) or arg.is_real is False or (
arg.is_number and
not arg.is_comparable):
raise ValueError("The argument '%s' is not comparable." % arg)
if arg == cls.zero:
raise ShortCircuit(arg)
elif arg == cls.identity:
continue
elif arg.func == cls:
for x in arg.args:
yield x
else:
yield arg
@classmethod
def _find_localzeros(cls, values, **options):
"""
Sequentially allocate values to localzeros.
When a value is identified as being more extreme than another member it
replaces that member; if this is never true, then the value is simply
appended to the localzeros.
"""
localzeros = set()
for v in values:
is_newzero = True
localzeros_ = list(localzeros)
for z in localzeros_:
if id(v) == id(z):
is_newzero = False
else:
con = cls._is_connected(v, z)
if con:
is_newzero = False
if con is True or con == cls:
localzeros.remove(z)
localzeros.update([v])
if is_newzero:
localzeros.update([v])
return localzeros
@classmethod
def _is_connected(cls, x, y):
"""
Check if x and y are connected somehow.
"""
from sympy.core.exprtools import factor_terms
def hit(v, t, f):
if not v.is_Relational:
return t if v else f
for i in range(2):
if x == y:
return True
r = hit(x >= y, Max, Min)
if r is not None:
return r
r = hit(y <= x, Max, Min)
if r is not None:
return r
r = hit(x <= y, Min, Max)
if r is not None:
return r
r = hit(y >= x, Min, Max)
if r is not None:
return r
# simplification can be expensive, so be conservative
# in what is attempted
x = factor_terms(x - y)
y = S.Zero
return False
def _eval_derivative(self, s):
# f(x).diff(s) -> x.diff(s) * f.fdiff(1)(s)
i = 0
l = []
for a in self.args:
i += 1
da = a.diff(s)
if da is S.Zero:
continue
try:
df = self.fdiff(i)
except ArgumentIndexError:
df = Function.fdiff(self, i)
l.append(df * da)
return Add(*l)
def _eval_rewrite_as_Abs(self, *args, **kwargs):
from sympy.functions.elementary.complexes import Abs
s = (args[0] + self.func(*args[1:]))/2
d = abs(args[0] - self.func(*args[1:]))/2
return (s + d if isinstance(self, Max) else s - d).rewrite(Abs)
def evalf(self, prec=None, **options):
return self.func(*[a.evalf(prec, **options) for a in self.args])
n = evalf
_eval_is_algebraic = lambda s: _torf(i.is_algebraic for i in s.args)
_eval_is_antihermitian = lambda s: _torf(i.is_antihermitian for i in s.args)
_eval_is_commutative = lambda s: _torf(i.is_commutative for i in s.args)
_eval_is_complex = lambda s: _torf(i.is_complex for i in s.args)
_eval_is_composite = lambda s: _torf(i.is_composite for i in s.args)
_eval_is_even = lambda s: _torf(i.is_even for i in s.args)
_eval_is_finite = lambda s: _torf(i.is_finite for i in s.args)
_eval_is_hermitian = lambda s: _torf(i.is_hermitian for i in s.args)
_eval_is_imaginary = lambda s: _torf(i.is_imaginary for i in s.args)
_eval_is_infinite = lambda s: _torf(i.is_infinite for i in s.args)
_eval_is_integer = lambda s: _torf(i.is_integer for i in s.args)
_eval_is_irrational = lambda s: _torf(i.is_irrational for i in s.args)
_eval_is_negative = lambda s: _torf(i.is_negative for i in s.args)
_eval_is_noninteger = lambda s: _torf(i.is_noninteger for i in s.args)
_eval_is_nonnegative = lambda s: _torf(i.is_nonnegative for i in s.args)
_eval_is_nonpositive = lambda s: _torf(i.is_nonpositive for i in s.args)
_eval_is_nonzero = lambda s: _torf(i.is_nonzero for i in s.args)
_eval_is_odd = lambda s: _torf(i.is_odd for i in s.args)
_eval_is_polar = lambda s: _torf(i.is_polar for i in s.args)
_eval_is_positive = lambda s: _torf(i.is_positive for i in s.args)
_eval_is_prime = lambda s: _torf(i.is_prime for i in s.args)
_eval_is_rational = lambda s: _torf(i.is_rational for i in s.args)
_eval_is_real = lambda s: _torf(i.is_real for i in s.args)
_eval_is_transcendental = lambda s: _torf(i.is_transcendental for i in s.args)
_eval_is_zero = lambda s: _torf(i.is_zero for i in s.args)
class Max(MinMaxBase, Application):
"""
Return, if possible, the maximum value of the list.
When number of arguments is equal one, then
return this argument.
When number of arguments is equal two, then
return, if possible, the value from (a, b) that is >= the other.
In common case, when the length of list greater than 2, the task
is more complicated. Return only the arguments, which are greater
than others, if it is possible to determine directional relation.
If is not possible to determine such a relation, return a partially
evaluated result.
Assumptions are used to make the decision too.
Also, only comparable arguments are permitted.
It is named ``Max`` and not ``max`` to avoid conflicts
with the built-in function ``max``.
Examples
========
>>> from sympy import Max, Symbol, oo
>>> from sympy.abc import x, y
>>> p = Symbol('p', positive=True)
>>> n = Symbol('n', negative=True)
>>> Max(x, -2) #doctest: +SKIP
Max(x, -2)
>>> Max(x, -2).subs(x, 3)
3
>>> Max(p, -2)
p
>>> Max(x, y)
Max(x, y)
>>> Max(x, y) == Max(y, x)
True
>>> Max(x, Max(y, z)) #doctest: +SKIP
Max(x, y, z)
>>> Max(n, 8, p, 7, -oo) #doctest: +SKIP
Max(8, p)
>>> Max (1, x, oo)
oo
* Algorithm
The task can be considered as searching of supremums in the
directed complete partial orders [1]_.
The source values are sequentially allocated by the isolated subsets
in which supremums are searched and result as Max arguments.
If the resulted supremum is single, then it is returned.
The isolated subsets are the sets of values which are only the comparable
with each other in the current set. E.g. natural numbers are comparable with
each other, but not comparable with the `x` symbol. Another example: the
symbol `x` with negative assumption is comparable with a natural number.
Also there are "least" elements, which are comparable with all others,
and have a zero property (maximum or minimum for all elements). E.g. `oo`.
In case of it the allocation operation is terminated and only this value is
returned.
Assumption:
- if A > B > C then A > C
- if A == B then B can be removed
References
==========
.. [1] https://en.wikipedia.org/wiki/Directed_complete_partial_order
.. [2] https://en.wikipedia.org/wiki/Lattice_%28order%29
See Also
========
Min : find minimum values
"""
zero = S.Infinity
identity = S.NegativeInfinity
def fdiff( self, argindex ):
from sympy import Heaviside
n = len(self.args)
if 0 < argindex and argindex <= n:
argindex -= 1
if n == 2:
return Heaviside(self.args[argindex] - self.args[1 - argindex])
newargs = tuple([self.args[i] for i in range(n) if i != argindex])
return Heaviside(self.args[argindex] - Max(*newargs))
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_Heaviside(self, *args, **kwargs):
from sympy import Heaviside
return Add(*[j*Mul(*[Heaviside(j - i) for i in args if i!=j]) \
for j in args])
def _eval_rewrite_as_Piecewise(self, *args, **kwargs):
return _minmax_as_Piecewise('>=', *args)
def _eval_is_positive(self):
return fuzzy_or(a.is_positive for a in self.args)
def _eval_is_nonnegative(self):
return fuzzy_or(a.is_nonnegative for a in self.args)
def _eval_is_negative(self):
return fuzzy_and(a.is_negative for a in self.args)
class Min(MinMaxBase, Application):
"""
Return, if possible, the minimum value of the list.
It is named ``Min`` and not ``min`` to avoid conflicts
with the built-in function ``min``.
Examples
========
>>> from sympy import Min, Symbol, oo
>>> from sympy.abc import x, y
>>> p = Symbol('p', positive=True)
>>> n = Symbol('n', negative=True)
>>> Min(x, -2) #doctest: +SKIP
Min(x, -2)
>>> Min(x, -2).subs(x, 3)
-2
>>> Min(p, -3)
-3
>>> Min(x, y) #doctest: +SKIP
Min(x, y)
>>> Min(n, 8, p, -7, p, oo) #doctest: +SKIP
Min(n, -7)
See Also
========
Max : find maximum values
"""
zero = S.NegativeInfinity
identity = S.Infinity
def fdiff( self, argindex ):
from sympy import Heaviside
n = len(self.args)
if 0 < argindex and argindex <= n:
argindex -= 1
if n == 2:
return Heaviside( self.args[1-argindex] - self.args[argindex] )
newargs = tuple([ self.args[i] for i in range(n) if i != argindex])
return Heaviside( Min(*newargs) - self.args[argindex] )
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_Heaviside(self, *args, **kwargs):
from sympy import Heaviside
return Add(*[j*Mul(*[Heaviside(i-j) for i in args if i!=j]) \
for j in args])
def _eval_rewrite_as_Piecewise(self, *args, **kwargs):
return _minmax_as_Piecewise('<=', *args)
def _eval_is_positive(self):
return fuzzy_and(a.is_positive for a in self.args)
def _eval_is_nonnegative(self):
return fuzzy_and(a.is_nonnegative for a in self.args)
def _eval_is_negative(self):
return fuzzy_or(a.is_negative for a in self.args)
|
533bbb2b271e74c66e7d38370eb890f5e253ea99098d923966ce2e88b00444bf
|
from __future__ import print_function, division
from sympy.core import S, sympify, cacheit
from sympy.core.add import Add
from sympy.core.function import Function, ArgumentIndexError, _coeff_isneg
from sympy.functions.combinatorial.factorials import factorial, RisingFactorial
from sympy.functions.elementary.exponential import exp, log
from sympy.functions.elementary.miscellaneous import sqrt
def _rewrite_hyperbolics_as_exp(expr):
expr = sympify(expr)
return expr.xreplace({h: h.rewrite(exp)
for h in expr.atoms(HyperbolicFunction)})
###############################################################################
########################### HYPERBOLIC FUNCTIONS ##############################
###############################################################################
class HyperbolicFunction(Function):
"""
Base class for hyperbolic functions.
See Also
========
sinh, cosh, tanh, coth
"""
unbranched = True
def _peeloff_ipi(arg):
"""
Split ARG into two parts, a "rest" and a multiple of I*pi/2.
This assumes ARG to be an Add.
The multiple of I*pi returned in the second position is always a Rational.
Examples
========
>>> from sympy.functions.elementary.hyperbolic import _peeloff_ipi as peel
>>> from sympy import pi, I
>>> from sympy.abc import x, y
>>> peel(x + I*pi/2)
(x, I*pi/2)
>>> peel(x + I*2*pi/3 + I*pi*y)
(x + I*pi*y + I*pi/6, I*pi/2)
"""
for a in Add.make_args(arg):
if a == S.Pi*S.ImaginaryUnit:
K = S.One
break
elif a.is_Mul:
K, p = a.as_two_terms()
if p == S.Pi*S.ImaginaryUnit and K.is_Rational:
break
else:
return arg, S.Zero
m1 = (K % S.Half)*S.Pi*S.ImaginaryUnit
m2 = K*S.Pi*S.ImaginaryUnit - m1
return arg - m2, m2
class sinh(HyperbolicFunction):
r"""
The hyperbolic sine function, `\frac{e^x - e^{-x}}{2}`.
* sinh(x) -> Returns the hyperbolic sine of x
See Also
========
cosh, tanh, asinh
"""
def fdiff(self, argindex=1):
"""
Returns the first derivative of this function.
"""
if argindex == 1:
return cosh(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return asinh
@classmethod
def eval(cls, arg):
from sympy import sin
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.NegativeInfinity
elif arg is S.Zero:
return S.Zero
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * sin(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
if arg.is_Add:
x, m = _peeloff_ipi(arg)
if m:
return sinh(m)*cosh(x) + cosh(m)*sinh(x)
if arg.func == asinh:
return arg.args[0]
if arg.func == acosh:
x = arg.args[0]
return sqrt(x - 1) * sqrt(x + 1)
if arg.func == atanh:
x = arg.args[0]
return x/sqrt(1 - x**2)
if arg.func == acoth:
x = arg.args[0]
return 1/(sqrt(x - 1) * sqrt(x + 1))
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
"""
Returns the next term in the Taylor series expansion.
"""
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2:
p = previous_terms[-2]
return p * x**2 / (n*(n - 1))
else:
return x**(n) / factorial(n)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
"""
Returns this function as a complex coordinate.
"""
from sympy import cos, sin
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
return (sinh(re)*cos(im), cosh(re)*sin(im))
def _eval_expand_complex(self, deep=True, **hints):
re_part, im_part = self.as_real_imag(deep=deep, **hints)
return re_part + im_part*S.ImaginaryUnit
def _eval_expand_trig(self, deep=True, **hints):
if deep:
arg = self.args[0].expand(deep, **hints)
else:
arg = self.args[0]
x = None
if arg.is_Add: # TODO, implement more if deep stuff here
x, y = arg.as_two_terms()
else:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff is not S.One and coeff.is_Integer and terms is not S.One:
x = terms
y = (coeff - 1)*x
if x is not None:
return (sinh(x)*cosh(y) + sinh(y)*cosh(x)).expand(trig=True)
return sinh(arg)
def _eval_rewrite_as_tractable(self, arg, **kwargs):
return (exp(arg) - exp(-arg)) / 2
def _eval_rewrite_as_exp(self, arg, **kwargs):
return (exp(arg) - exp(-arg)) / 2
def _eval_rewrite_as_cosh(self, arg, **kwargs):
return -S.ImaginaryUnit*cosh(arg + S.Pi*S.ImaginaryUnit/2)
def _eval_rewrite_as_tanh(self, arg, **kwargs):
tanh_half = tanh(S.Half*arg)
return 2*tanh_half/(1 - tanh_half**2)
def _eval_rewrite_as_coth(self, arg, **kwargs):
coth_half = coth(S.Half*arg)
return 2*coth_half/(coth_half**2 - 1)
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
def _eval_is_real(self):
if self.args[0].is_real:
return True
def _eval_is_positive(self):
if self.args[0].is_real:
return self.args[0].is_positive
def _eval_is_negative(self):
if self.args[0].is_real:
return self.args[0].is_negative
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_imaginary:
return True
class cosh(HyperbolicFunction):
r"""
The hyperbolic cosine function, `\frac{e^x + e^{-x}}{2}`.
* cosh(x) -> Returns the hyperbolic cosine of x
See Also
========
sinh, tanh, acosh
"""
def fdiff(self, argindex=1):
if argindex == 1:
return sinh(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
from sympy import cos
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Infinity
elif arg is S.Zero:
return S.One
elif arg.is_negative:
return cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return cos(i_coeff)
else:
if _coeff_isneg(arg):
return cls(-arg)
if arg.is_Add:
x, m = _peeloff_ipi(arg)
if m:
return cosh(m)*cosh(x) + sinh(m)*sinh(x)
if arg.func == asinh:
return sqrt(1 + arg.args[0]**2)
if arg.func == acosh:
return arg.args[0]
if arg.func == atanh:
return 1/sqrt(1 - arg.args[0]**2)
if arg.func == acoth:
x = arg.args[0]
return x/(sqrt(x - 1) * sqrt(x + 1))
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2:
p = previous_terms[-2]
return p * x**2 / (n*(n - 1))
else:
return x**(n)/factorial(n)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
from sympy import cos, sin
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
return (cosh(re)*cos(im), sinh(re)*sin(im))
def _eval_expand_complex(self, deep=True, **hints):
re_part, im_part = self.as_real_imag(deep=deep, **hints)
return re_part + im_part*S.ImaginaryUnit
def _eval_expand_trig(self, deep=True, **hints):
if deep:
arg = self.args[0].expand(deep, **hints)
else:
arg = self.args[0]
x = None
if arg.is_Add: # TODO, implement more if deep stuff here
x, y = arg.as_two_terms()
else:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff is not S.One and coeff.is_Integer and terms is not S.One:
x = terms
y = (coeff - 1)*x
if x is not None:
return (cosh(x)*cosh(y) + sinh(x)*sinh(y)).expand(trig=True)
return cosh(arg)
def _eval_rewrite_as_tractable(self, arg, **kwargs):
return (exp(arg) + exp(-arg)) / 2
def _eval_rewrite_as_exp(self, arg, **kwargs):
return (exp(arg) + exp(-arg)) / 2
def _eval_rewrite_as_sinh(self, arg, **kwargs):
return -S.ImaginaryUnit*sinh(arg + S.Pi*S.ImaginaryUnit/2)
def _eval_rewrite_as_tanh(self, arg, **kwargs):
tanh_half = tanh(S.Half*arg)**2
return (1 + tanh_half)/(1 - tanh_half)
def _eval_rewrite_as_coth(self, arg, **kwargs):
coth_half = coth(S.Half*arg)**2
return (coth_half + 1)/(coth_half - 1)
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return S.One
else:
return self.func(arg)
def _eval_is_positive(self):
if self.args[0].is_real:
return True
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_imaginary:
return True
class tanh(HyperbolicFunction):
r"""
The hyperbolic tangent function, `\frac{\sinh(x)}{\cosh(x)}`.
* tanh(x) -> Returns the hyperbolic tangent of x
See Also
========
sinh, cosh, atanh
"""
def fdiff(self, argindex=1):
if argindex == 1:
return S.One - tanh(self.args[0])**2
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return atanh
@classmethod
def eval(cls, arg):
from sympy import tan
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.One
elif arg is S.NegativeInfinity:
return S.NegativeOne
elif arg is S.Zero:
return S.Zero
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
if _coeff_isneg(i_coeff):
return -S.ImaginaryUnit * tan(-i_coeff)
return S.ImaginaryUnit * tan(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
if arg.is_Add:
x, m = _peeloff_ipi(arg)
if m:
tanhm = tanh(m)
if tanhm is S.ComplexInfinity:
return coth(x)
else: # tanhm == 0
return tanh(x)
if arg.func == asinh:
x = arg.args[0]
return x/sqrt(1 + x**2)
if arg.func == acosh:
x = arg.args[0]
return sqrt(x - 1) * sqrt(x + 1) / x
if arg.func == atanh:
return arg.args[0]
if arg.func == acoth:
return 1/arg.args[0]
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
from sympy import bernoulli
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
a = 2**(n + 1)
B = bernoulli(n + 1)
F = factorial(n + 1)
return a*(a - 1) * B/F * x**n
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
from sympy import cos, sin
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
denom = sinh(re)**2 + cos(im)**2
return (sinh(re)*cosh(re)/denom, sin(im)*cos(im)/denom)
def _eval_rewrite_as_tractable(self, arg, **kwargs):
neg_exp, pos_exp = exp(-arg), exp(arg)
return (pos_exp - neg_exp)/(pos_exp + neg_exp)
def _eval_rewrite_as_exp(self, arg, **kwargs):
neg_exp, pos_exp = exp(-arg), exp(arg)
return (pos_exp - neg_exp)/(pos_exp + neg_exp)
def _eval_rewrite_as_sinh(self, arg, **kwargs):
return S.ImaginaryUnit*sinh(arg)/sinh(S.Pi*S.ImaginaryUnit/2 - arg)
def _eval_rewrite_as_cosh(self, arg, **kwargs):
return S.ImaginaryUnit*cosh(S.Pi*S.ImaginaryUnit/2 - arg)/cosh(arg)
def _eval_rewrite_as_coth(self, arg, **kwargs):
return 1/coth(arg)
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
def _eval_is_real(self):
if self.args[0].is_real:
return True
def _eval_is_positive(self):
if self.args[0].is_real:
return self.args[0].is_positive
def _eval_is_negative(self):
if self.args[0].is_real:
return self.args[0].is_negative
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_real:
return True
class coth(HyperbolicFunction):
r"""
The hyperbolic cotangent function, `\frac{\cosh(x)}{\sinh(x)}`.
* coth(x) -> Returns the hyperbolic cotangent of x
"""
def fdiff(self, argindex=1):
if argindex == 1:
return -1/sinh(self.args[0])**2
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return acoth
@classmethod
def eval(cls, arg):
from sympy import cot
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.One
elif arg is S.NegativeInfinity:
return S.NegativeOne
elif arg is S.Zero:
return S.ComplexInfinity
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
if _coeff_isneg(i_coeff):
return S.ImaginaryUnit * cot(-i_coeff)
return -S.ImaginaryUnit * cot(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
if arg.is_Add:
x, m = _peeloff_ipi(arg)
if m:
cothm = coth(m)
if cothm is S.ComplexInfinity:
return coth(x)
else: # cothm == 0
return tanh(x)
if arg.func == asinh:
x = arg.args[0]
return sqrt(1 + x**2)/x
if arg.func == acosh:
x = arg.args[0]
return x/(sqrt(x - 1) * sqrt(x + 1))
if arg.func == atanh:
return 1/arg.args[0]
if arg.func == acoth:
return arg.args[0]
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
from sympy import bernoulli
if n == 0:
return 1 / sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
B = bernoulli(n + 1)
F = factorial(n + 1)
return 2**(n + 1) * B/F * x**n
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
from sympy import cos, sin
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
denom = sinh(re)**2 + sin(im)**2
return (sinh(re)*cosh(re)/denom, -sin(im)*cos(im)/denom)
def _eval_rewrite_as_tractable(self, arg, **kwargs):
neg_exp, pos_exp = exp(-arg), exp(arg)
return (pos_exp + neg_exp)/(pos_exp - neg_exp)
def _eval_rewrite_as_exp(self, arg, **kwargs):
neg_exp, pos_exp = exp(-arg), exp(arg)
return (pos_exp + neg_exp)/(pos_exp - neg_exp)
def _eval_rewrite_as_sinh(self, arg, **kwargs):
return -S.ImaginaryUnit*sinh(S.Pi*S.ImaginaryUnit/2 - arg)/sinh(arg)
def _eval_rewrite_as_cosh(self, arg, **kwargs):
return -S.ImaginaryUnit*cosh(arg)/cosh(S.Pi*S.ImaginaryUnit/2 - arg)
def _eval_rewrite_as_tanh(self, arg, **kwargs):
return 1/tanh(arg)
def _eval_is_positive(self):
if self.args[0].is_real:
return self.args[0].is_positive
def _eval_is_negative(self):
if self.args[0].is_real:
return self.args[0].is_negative
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return 1/arg
else:
return self.func(arg)
class ReciprocalHyperbolicFunction(HyperbolicFunction):
"""Base class for reciprocal functions of hyperbolic functions. """
#To be defined in class
_reciprocal_of = None
_is_even = None
_is_odd = None
@classmethod
def eval(cls, arg):
if arg.could_extract_minus_sign():
if cls._is_even:
return cls(-arg)
if cls._is_odd:
return -cls(-arg)
t = cls._reciprocal_of.eval(arg)
if hasattr(arg, 'inverse') and arg.inverse() == cls:
return arg.args[0]
return 1/t if t is not None else t
def _call_reciprocal(self, method_name, *args, **kwargs):
# Calls method_name on _reciprocal_of
o = self._reciprocal_of(self.args[0])
return getattr(o, method_name)(*args, **kwargs)
def _calculate_reciprocal(self, method_name, *args, **kwargs):
# If calling method_name on _reciprocal_of returns a value != None
# then return the reciprocal of that value
t = self._call_reciprocal(method_name, *args, **kwargs)
return 1/t if t is not None else t
def _rewrite_reciprocal(self, method_name, arg):
# Special handling for rewrite functions. If reciprocal rewrite returns
# unmodified expression, then return None
t = self._call_reciprocal(method_name, arg)
if t is not None and t != self._reciprocal_of(arg):
return 1/t
def _eval_rewrite_as_exp(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_exp", arg)
def _eval_rewrite_as_tractable(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_tractable", arg)
def _eval_rewrite_as_tanh(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_tanh", arg)
def _eval_rewrite_as_coth(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_coth", arg)
def as_real_imag(self, deep = True, **hints):
return (1 / self._reciprocal_of(self.args[0])).as_real_imag(deep, **hints)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def _eval_expand_complex(self, deep=True, **hints):
re_part, im_part = self.as_real_imag(deep=True, **hints)
return re_part + S.ImaginaryUnit*im_part
def _eval_as_leading_term(self, x):
return (1/self._reciprocal_of(self.args[0]))._eval_as_leading_term(x)
def _eval_is_real(self):
return self._reciprocal_of(self.args[0]).is_real
def _eval_is_finite(self):
return (1/self._reciprocal_of(self.args[0])).is_finite
class csch(ReciprocalHyperbolicFunction):
r"""
The hyperbolic cosecant function, `\frac{2}{e^x - e^{-x}}`
* csch(x) -> Returns the hyperbolic cosecant of x
See Also
========
sinh, cosh, tanh, sech, asinh, acosh
"""
_reciprocal_of = sinh
_is_odd = True
def fdiff(self, argindex=1):
"""
Returns the first derivative of this function
"""
if argindex == 1:
return -coth(self.args[0]) * csch(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
"""
Returns the next term in the Taylor series expansion
"""
from sympy import bernoulli
if n == 0:
return 1/sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
B = bernoulli(n + 1)
F = factorial(n + 1)
return 2 * (1 - 2**n) * B/F * x**n
def _eval_rewrite_as_cosh(self, arg, **kwargs):
return S.ImaginaryUnit / cosh(arg + S.ImaginaryUnit * S.Pi / 2)
def _eval_is_positive(self):
if self.args[0].is_real:
return self.args[0].is_positive
def _eval_is_negative(self):
if self.args[0].is_real:
return self.args[0].is_negative
def _sage_(self):
import sage.all as sage
return sage.csch(self.args[0]._sage_())
class sech(ReciprocalHyperbolicFunction):
r"""
The hyperbolic secant function, `\frac{2}{e^x + e^{-x}}`
* sech(x) -> Returns the hyperbolic secant of x
See Also
========
sinh, cosh, tanh, coth, csch, asinh, acosh
"""
_reciprocal_of = cosh
_is_even = True
def fdiff(self, argindex=1):
if argindex == 1:
return - tanh(self.args[0])*sech(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
from sympy.functions.combinatorial.numbers import euler
if n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
return euler(n) / factorial(n) * x**(n)
def _eval_rewrite_as_sinh(self, arg, **kwargs):
return S.ImaginaryUnit / sinh(arg + S.ImaginaryUnit * S.Pi /2)
def _eval_is_positive(self):
if self.args[0].is_real:
return True
def _sage_(self):
import sage.all as sage
return sage.sech(self.args[0]._sage_())
###############################################################################
############################# HYPERBOLIC INVERSES #############################
###############################################################################
class InverseHyperbolicFunction(Function):
"""Base class for inverse hyperbolic functions."""
pass
class asinh(InverseHyperbolicFunction):
"""
The inverse hyperbolic sine function.
* asinh(x) -> Returns the inverse hyperbolic sine of x
See Also
========
acosh, atanh, sinh
"""
def fdiff(self, argindex=1):
if argindex == 1:
return 1/sqrt(self.args[0]**2 + 1)
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
from sympy import asin
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.NegativeInfinity
elif arg is S.Zero:
return S.Zero
elif arg is S.One:
return log(sqrt(2) + 1)
elif arg is S.NegativeOne:
return log(sqrt(2) - 1)
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.ComplexInfinity
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * asin(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) >= 2 and n > 2:
p = previous_terms[-2]
return -p * (n - 2)**2/(n*(n - 1)) * x**2
else:
k = (n - 1) // 2
R = RisingFactorial(S.Half, k)
F = factorial(k)
return (-1)**k * R / F * x**n / n
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
def _eval_rewrite_as_log(self, x, **kwargs):
return log(x + sqrt(x**2 + 1))
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return sinh
class acosh(InverseHyperbolicFunction):
"""
The inverse hyperbolic cosine function.
* acosh(x) -> Returns the inverse hyperbolic cosine of x
See Also
========
asinh, atanh, cosh
"""
def fdiff(self, argindex=1):
if argindex == 1:
return 1/sqrt(self.args[0]**2 - 1)
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Infinity
elif arg is S.Zero:
return S.Pi*S.ImaginaryUnit / 2
elif arg is S.One:
return S.Zero
elif arg is S.NegativeOne:
return S.Pi*S.ImaginaryUnit
if arg.is_number:
cst_table = {
S.ImaginaryUnit: log(S.ImaginaryUnit*(1 + sqrt(2))),
-S.ImaginaryUnit: log(-S.ImaginaryUnit*(1 + sqrt(2))),
S.Half: S.Pi/3,
-S.Half: 2*S.Pi/3,
sqrt(2)/2: S.Pi/4,
-sqrt(2)/2: 3*S.Pi/4,
1/sqrt(2): S.Pi/4,
-1/sqrt(2): 3*S.Pi/4,
sqrt(3)/2: S.Pi/6,
-sqrt(3)/2: 5*S.Pi/6,
(sqrt(3) - 1)/sqrt(2**3): 5*S.Pi/12,
-(sqrt(3) - 1)/sqrt(2**3): 7*S.Pi/12,
sqrt(2 + sqrt(2))/2: S.Pi/8,
-sqrt(2 + sqrt(2))/2: 7*S.Pi/8,
sqrt(2 - sqrt(2))/2: 3*S.Pi/8,
-sqrt(2 - sqrt(2))/2: 5*S.Pi/8,
(1 + sqrt(3))/(2*sqrt(2)): S.Pi/12,
-(1 + sqrt(3))/(2*sqrt(2)): 11*S.Pi/12,
(sqrt(5) + 1)/4: S.Pi/5,
-(sqrt(5) + 1)/4: 4*S.Pi/5
}
if arg in cst_table:
if arg.is_real:
return cst_table[arg]*S.ImaginaryUnit
return cst_table[arg]
if arg is S.ComplexInfinity:
return S.ComplexInfinity
if arg == S.ImaginaryUnit*S.Infinity:
return S.Infinity + S.ImaginaryUnit*S.Pi/2
if arg == -S.ImaginaryUnit*S.Infinity:
return S.Infinity - S.ImaginaryUnit*S.Pi/2
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return S.Pi*S.ImaginaryUnit / 2
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) >= 2 and n > 2:
p = previous_terms[-2]
return p * (n - 2)**2/(n*(n - 1)) * x**2
else:
k = (n - 1) // 2
R = RisingFactorial(S.Half, k)
F = factorial(k)
return -R / F * S.ImaginaryUnit * x**n / n
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return S.ImaginaryUnit*S.Pi/2
else:
return self.func(arg)
def _eval_rewrite_as_log(self, x, **kwargs):
return log(x + sqrt(x + 1) * sqrt(x - 1))
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return cosh
class atanh(InverseHyperbolicFunction):
"""
The inverse hyperbolic tangent function.
* atanh(x) -> Returns the inverse hyperbolic tangent of x
See Also
========
asinh, acosh, tanh
"""
def fdiff(self, argindex=1):
if argindex == 1:
return 1/(1 - self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
from sympy import atan
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Zero:
return S.Zero
elif arg is S.One:
return S.Infinity
elif arg is S.NegativeOne:
return S.NegativeInfinity
elif arg is S.Infinity:
return -S.ImaginaryUnit * atan(arg)
elif arg is S.NegativeInfinity:
return S.ImaginaryUnit * atan(-arg)
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
from sympy.calculus.util import AccumBounds
return S.ImaginaryUnit*AccumBounds(-S.Pi/2, S.Pi/2)
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * atan(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
return x**n / n
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
def _eval_rewrite_as_log(self, x, **kwargs):
return (log(1 + x) - log(1 - x)) / 2
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return tanh
class acoth(InverseHyperbolicFunction):
"""
The inverse hyperbolic cotangent function.
* acoth(x) -> Returns the inverse hyperbolic cotangent of x
"""
def fdiff(self, argindex=1):
if argindex == 1:
return 1/(1 - self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
from sympy import acot
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Zero
elif arg is S.NegativeInfinity:
return S.Zero
elif arg is S.Zero:
return S.Pi*S.ImaginaryUnit / 2
elif arg is S.One:
return S.Infinity
elif arg is S.NegativeOne:
return S.NegativeInfinity
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.Zero
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return -S.ImaginaryUnit * acot(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return S.Pi*S.ImaginaryUnit / 2
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
return x**n / n
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return S.ImaginaryUnit*S.Pi/2
else:
return self.func(arg)
def _eval_rewrite_as_log(self, x, **kwargs):
return (log(1 + 1/x) - log(1 - 1/x)) / 2
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return coth
class asech(InverseHyperbolicFunction):
"""
The inverse hyperbolic secant function.
* asech(x) -> Returns the inverse hyperbolic secant of x
Examples
========
>>> from sympy import asech, sqrt, S
>>> from sympy.abc import x
>>> asech(x).diff(x)
-1/(x*sqrt(1 - x**2))
>>> asech(1).diff(x)
0
>>> asech(1)
0
>>> asech(S(2))
I*pi/3
>>> asech(-sqrt(2))
3*I*pi/4
>>> asech((sqrt(6) - sqrt(2)))
I*pi/12
See Also
========
asinh, atanh, cosh, acoth
References
==========
.. [1] https://en.wikipedia.org/wiki/Hyperbolic_function
.. [2] http://dlmf.nist.gov/4.37
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSech/
"""
def fdiff(self, argindex=1):
if argindex == 1:
z = self.args[0]
return -1/(z*sqrt(1 - z**2))
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Pi*S.ImaginaryUnit / 2
elif arg is S.NegativeInfinity:
return S.Pi*S.ImaginaryUnit / 2
elif arg is S.Zero:
return S.Infinity
elif arg is S.One:
return S.Zero
elif arg is S.NegativeOne:
return S.Pi*S.ImaginaryUnit
if arg.is_number:
cst_table = {
S.ImaginaryUnit: - (S.Pi*S.ImaginaryUnit / 2) + log(1 + sqrt(2)),
-S.ImaginaryUnit: (S.Pi*S.ImaginaryUnit / 2) + log(1 + sqrt(2)),
(sqrt(6) - sqrt(2)): S.Pi / 12,
(sqrt(2) - sqrt(6)): 11*S.Pi / 12,
sqrt(2 - 2/sqrt(5)): S.Pi / 10,
-sqrt(2 - 2/sqrt(5)): 9*S.Pi / 10,
2 / sqrt(2 + sqrt(2)): S.Pi / 8,
-2 / sqrt(2 + sqrt(2)): 7*S.Pi / 8,
2 / sqrt(3): S.Pi / 6,
-2 / sqrt(3): 5*S.Pi / 6,
(sqrt(5) - 1): S.Pi / 5,
(1 - sqrt(5)): 4*S.Pi / 5,
sqrt(2): S.Pi / 4,
-sqrt(2): 3*S.Pi / 4,
sqrt(2 + 2/sqrt(5)): 3*S.Pi / 10,
-sqrt(2 + 2/sqrt(5)): 7*S.Pi / 10,
S(2): S.Pi / 3,
-S(2): 2*S.Pi / 3,
sqrt(2*(2 + sqrt(2))): 3*S.Pi / 8,
-sqrt(2*(2 + sqrt(2))): 5*S.Pi / 8,
(1 + sqrt(5)): 2*S.Pi / 5,
(-1 - sqrt(5)): 3*S.Pi / 5,
(sqrt(6) + sqrt(2)): 5*S.Pi / 12,
(-sqrt(6) - sqrt(2)): 7*S.Pi / 12,
}
if arg in cst_table:
if arg.is_real:
return cst_table[arg]*S.ImaginaryUnit
return cst_table[arg]
if arg is S.ComplexInfinity:
from sympy.calculus.util import AccumBounds
return S.ImaginaryUnit*AccumBounds(-S.Pi/2, S.Pi/2)
@staticmethod
@cacheit
def expansion_term(n, x, *previous_terms):
if n == 0:
return log(2 / x)
elif n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2 and n > 2:
p = previous_terms[-2]
return p * (n - 1)**2 // (n // 2)**2 * x**2 / 4
else:
k = n // 2
R = RisingFactorial(S.Half , k) * n
F = factorial(k) * n // 2 * n // 2
return -1 * R / F * x**n / 4
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return sech
def _eval_rewrite_as_log(self, arg, **kwargs):
return log(1/arg + sqrt(1/arg - 1) * sqrt(1/arg + 1))
class acsch(InverseHyperbolicFunction):
"""
The inverse hyperbolic cosecant function.
* acsch(x) -> Returns the inverse hyperbolic cosecant of x
Examples
========
>>> from sympy import acsch, sqrt, S
>>> from sympy.abc import x
>>> acsch(x).diff(x)
-1/(x**2*sqrt(1 + x**(-2)))
>>> acsch(1).diff(x)
0
>>> acsch(1)
log(1 + sqrt(2))
>>> acsch(S.ImaginaryUnit)
-I*pi/2
>>> acsch(-2*S.ImaginaryUnit)
I*pi/6
>>> acsch(S.ImaginaryUnit*(sqrt(6) - sqrt(2)))
-5*I*pi/12
References
==========
.. [1] https://en.wikipedia.org/wiki/Hyperbolic_function
.. [2] http://dlmf.nist.gov/4.37
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCsch/
"""
def fdiff(self, argindex=1):
if argindex == 1:
z = self.args[0]
return -1/(z**2*sqrt(1 + 1/z**2))
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Zero
elif arg is S.NegativeInfinity:
return S.Zero
elif arg is S.Zero:
return S.ComplexInfinity
elif arg is S.One:
return log(1 + sqrt(2))
elif arg is S.NegativeOne:
return - log(1 + sqrt(2))
if arg.is_number:
cst_table = {
S.ImaginaryUnit: -S.Pi / 2,
S.ImaginaryUnit*(sqrt(2) + sqrt(6)): -S.Pi / 12,
S.ImaginaryUnit*(1 + sqrt(5)): -S.Pi / 10,
S.ImaginaryUnit*2 / sqrt(2 - sqrt(2)): -S.Pi / 8,
S.ImaginaryUnit*2: -S.Pi / 6,
S.ImaginaryUnit*sqrt(2 + 2/sqrt(5)): -S.Pi / 5,
S.ImaginaryUnit*sqrt(2): -S.Pi / 4,
S.ImaginaryUnit*(sqrt(5)-1): -3*S.Pi / 10,
S.ImaginaryUnit*2 / sqrt(3): -S.Pi / 3,
S.ImaginaryUnit*2 / sqrt(2 + sqrt(2)): -3*S.Pi / 8,
S.ImaginaryUnit*sqrt(2 - 2/sqrt(5)): -2*S.Pi / 5,
S.ImaginaryUnit*(sqrt(6) - sqrt(2)): -5*S.Pi / 12,
S(2): -S.ImaginaryUnit*log((1+sqrt(5))/2),
}
if arg in cst_table:
return cst_table[arg]*S.ImaginaryUnit
if arg is S.ComplexInfinity:
return S.Zero
if _coeff_isneg(arg):
return -cls(-arg)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return csch
def _eval_rewrite_as_log(self, arg, **kwargs):
return log(1/arg + sqrt(1/arg**2 + 1))
|
f98ae3d0f9dab85e63ecda7031978683a0cc0806f38266e01d401b21eafcaafc
|
from __future__ import print_function, division
from sympy.core import S, Add, Mul, sympify, Symbol, Dummy, Basic
from sympy.core.expr import Expr
from sympy.core.exprtools import factor_terms
from sympy.core.function import (Function, Derivative, ArgumentIndexError,
AppliedUndef)
from sympy.core.logic import fuzzy_not, fuzzy_or
from sympy.core.numbers import pi, I, oo
from sympy.core.relational import Eq
from sympy.functions.elementary.exponential import exp, exp_polar, log
from sympy.functions.elementary.integers import ceiling
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import atan, atan2
###############################################################################
######################### REAL and IMAGINARY PARTS ############################
###############################################################################
class re(Function):
"""
Returns real part of expression. This function performs only
elementary analysis and so it will fail to decompose properly
more complicated expressions. If completely simplified result
is needed then use Basic.as_real_imag() or perform complex
expansion on instance of this function.
Examples
========
>>> from sympy import re, im, I, E
>>> from sympy.abc import x, y
>>> re(2*E)
2*E
>>> re(2*I + 17)
17
>>> re(2*I)
0
>>> re(im(x) + x*I + 2)
2
See Also
========
im
"""
is_real = True
unbranched = True # implicitly works on the projection to C
@classmethod
def eval(cls, arg):
if arg is S.NaN:
return S.NaN
elif arg is S.ComplexInfinity:
return S.NaN
elif arg.is_real:
return arg
elif arg.is_imaginary or (S.ImaginaryUnit*arg).is_real:
return S.Zero
elif arg.is_Matrix:
return arg.as_real_imag()[0]
elif arg.is_Function and isinstance(arg, conjugate):
return re(arg.args[0])
else:
included, reverted, excluded = [], [], []
args = Add.make_args(arg)
for term in args:
coeff = term.as_coefficient(S.ImaginaryUnit)
if coeff is not None:
if not coeff.is_real:
reverted.append(coeff)
elif not term.has(S.ImaginaryUnit) and term.is_real:
excluded.append(term)
else:
# Try to do some advanced expansion. If
# impossible, don't try to do re(arg) again
# (because this is what we are trying to do now).
real_imag = term.as_real_imag(ignore=arg)
if real_imag:
excluded.append(real_imag[0])
else:
included.append(term)
if len(args) != len(included):
a, b, c = (Add(*xs) for xs in [included, reverted, excluded])
return cls(a) - im(b) + c
def as_real_imag(self, deep=True, **hints):
"""
Returns the real number with a zero imaginary part.
"""
return (self, S.Zero)
def _eval_derivative(self, x):
if x.is_real or self.args[0].is_real:
return re(Derivative(self.args[0], x, evaluate=True))
if x.is_imaginary or self.args[0].is_imaginary:
return -S.ImaginaryUnit \
* im(Derivative(self.args[0], x, evaluate=True))
def _eval_rewrite_as_im(self, arg, **kwargs):
return self.args[0] - S.ImaginaryUnit*im(self.args[0])
def _eval_is_algebraic(self):
return self.args[0].is_algebraic
def _eval_is_zero(self):
# is_imaginary implies nonzero
return fuzzy_or([self.args[0].is_imaginary, self.args[0].is_zero])
def _sage_(self):
import sage.all as sage
return sage.real_part(self.args[0]._sage_())
class im(Function):
"""
Returns imaginary part of expression. This function performs only
elementary analysis and so it will fail to decompose properly more
complicated expressions. If completely simplified result is needed then
use Basic.as_real_imag() or perform complex expansion on instance of
this function.
Examples
========
>>> from sympy import re, im, E, I
>>> from sympy.abc import x, y
>>> im(2*E)
0
>>> re(2*I + 17)
17
>>> im(x*I)
re(x)
>>> im(re(x) + y)
im(y)
See Also
========
re
"""
is_real = True
unbranched = True # implicitly works on the projection to C
@classmethod
def eval(cls, arg):
if arg is S.NaN:
return S.NaN
elif arg is S.ComplexInfinity:
return S.NaN
elif arg.is_real:
return S.Zero
elif arg.is_imaginary or (S.ImaginaryUnit*arg).is_real:
return -S.ImaginaryUnit * arg
elif arg.is_Matrix:
return arg.as_real_imag()[1]
elif arg.is_Function and isinstance(arg, conjugate):
return -im(arg.args[0])
else:
included, reverted, excluded = [], [], []
args = Add.make_args(arg)
for term in args:
coeff = term.as_coefficient(S.ImaginaryUnit)
if coeff is not None:
if not coeff.is_real:
reverted.append(coeff)
else:
excluded.append(coeff)
elif term.has(S.ImaginaryUnit) or not term.is_real:
# Try to do some advanced expansion. If
# impossible, don't try to do im(arg) again
# (because this is what we are trying to do now).
real_imag = term.as_real_imag(ignore=arg)
if real_imag:
excluded.append(real_imag[1])
else:
included.append(term)
if len(args) != len(included):
a, b, c = (Add(*xs) for xs in [included, reverted, excluded])
return cls(a) + re(b) + c
def as_real_imag(self, deep=True, **hints):
"""
Return the imaginary part with a zero real part.
Examples
========
>>> from sympy.functions import im
>>> from sympy import I
>>> im(2 + 3*I).as_real_imag()
(3, 0)
"""
return (self, S.Zero)
def _eval_derivative(self, x):
if x.is_real or self.args[0].is_real:
return im(Derivative(self.args[0], x, evaluate=True))
if x.is_imaginary or self.args[0].is_imaginary:
return -S.ImaginaryUnit \
* re(Derivative(self.args[0], x, evaluate=True))
def _sage_(self):
import sage.all as sage
return sage.imag_part(self.args[0]._sage_())
def _eval_rewrite_as_re(self, arg, **kwargs):
return -S.ImaginaryUnit*(self.args[0] - re(self.args[0]))
def _eval_is_algebraic(self):
return self.args[0].is_algebraic
def _eval_is_zero(self):
return self.args[0].is_real
###############################################################################
############### SIGN, ABSOLUTE VALUE, ARGUMENT and CONJUGATION ################
###############################################################################
class sign(Function):
"""
Returns the complex sign of an expression:
If the expression is real the sign will be:
* 1 if expression is positive
* 0 if expression is equal to zero
* -1 if expression is negative
If the expression is imaginary the sign will be:
* I if im(expression) is positive
* -I if im(expression) is negative
Otherwise an unevaluated expression will be returned. When evaluated, the
result (in general) will be ``cos(arg(expr)) + I*sin(arg(expr))``.
Examples
========
>>> from sympy.functions import sign
>>> from sympy.core.numbers import I
>>> sign(-1)
-1
>>> sign(0)
0
>>> sign(-3*I)
-I
>>> sign(1 + I)
sign(1 + I)
>>> _.evalf()
0.707106781186548 + 0.707106781186548*I
See Also
========
Abs, conjugate
"""
is_finite = True
is_complex = True
def doit(self, **hints):
if self.args[0].is_zero is False:
return self.args[0] / Abs(self.args[0])
return self
@classmethod
def eval(cls, arg):
# handle what we can
if arg.is_Mul:
c, args = arg.as_coeff_mul()
unk = []
s = sign(c)
for a in args:
if a.is_negative:
s = -s
elif a.is_positive:
pass
else:
ai = im(a)
if a.is_imaginary and ai.is_comparable: # i.e. a = I*real
s *= S.ImaginaryUnit
if ai.is_negative:
# can't use sign(ai) here since ai might not be
# a Number
s = -s
else:
unk.append(a)
if c is S.One and len(unk) == len(args):
return None
return s * cls(arg._new_rawargs(*unk))
if arg is S.NaN:
return S.NaN
if arg.is_zero: # it may be an Expr that is zero
return S.Zero
if arg.is_positive:
return S.One
if arg.is_negative:
return S.NegativeOne
if arg.is_Function:
if isinstance(arg, sign):
return arg
if arg.is_imaginary:
if arg.is_Pow and arg.exp is S.Half:
# we catch this because non-trivial sqrt args are not expanded
# e.g. sqrt(1-sqrt(2)) --x--> to I*sqrt(sqrt(2) - 1)
return S.ImaginaryUnit
arg2 = -S.ImaginaryUnit * arg
if arg2.is_positive:
return S.ImaginaryUnit
if arg2.is_negative:
return -S.ImaginaryUnit
def _eval_Abs(self):
if fuzzy_not(self.args[0].is_zero):
return S.One
def _eval_conjugate(self):
return sign(conjugate(self.args[0]))
def _eval_derivative(self, x):
if self.args[0].is_real:
from sympy.functions.special.delta_functions import DiracDelta
return 2 * Derivative(self.args[0], x, evaluate=True) \
* DiracDelta(self.args[0])
elif self.args[0].is_imaginary:
from sympy.functions.special.delta_functions import DiracDelta
return 2 * Derivative(self.args[0], x, evaluate=True) \
* DiracDelta(-S.ImaginaryUnit * self.args[0])
def _eval_is_nonnegative(self):
if self.args[0].is_nonnegative:
return True
def _eval_is_nonpositive(self):
if self.args[0].is_nonpositive:
return True
def _eval_is_imaginary(self):
return self.args[0].is_imaginary
def _eval_is_integer(self):
return self.args[0].is_real
def _eval_is_zero(self):
return self.args[0].is_zero
def _eval_power(self, other):
if (
fuzzy_not(self.args[0].is_zero) and
other.is_integer and
other.is_even
):
return S.One
def _sage_(self):
import sage.all as sage
return sage.sgn(self.args[0]._sage_())
def _eval_rewrite_as_Piecewise(self, arg, **kwargs):
if arg.is_real:
return Piecewise((1, arg > 0), (-1, arg < 0), (0, True))
def _eval_rewrite_as_Heaviside(self, arg, **kwargs):
from sympy.functions.special.delta_functions import Heaviside
if arg.is_real:
return Heaviside(arg)*2-1
def _eval_simplify(self, ratio, measure, rational, inverse):
return self.func(self.args[0].factor())
class Abs(Function):
"""
Return the absolute value of the argument.
This is an extension of the built-in function abs() to accept symbolic
values. If you pass a SymPy expression to the built-in abs(), it will
pass it automatically to Abs().
Examples
========
>>> from sympy import Abs, Symbol, S
>>> Abs(-1)
1
>>> x = Symbol('x', real=True)
>>> Abs(-x)
Abs(x)
>>> Abs(x**2)
x**2
>>> abs(-x) # The Python built-in
Abs(x)
Note that the Python built-in will return either an Expr or int depending on
the argument::
>>> type(abs(-1))
<... 'int'>
>>> type(abs(S.NegativeOne))
<class 'sympy.core.numbers.One'>
Abs will always return a sympy object.
See Also
========
sign, conjugate
"""
is_real = True
is_negative = False
unbranched = True
def fdiff(self, argindex=1):
"""
Get the first derivative of the argument to Abs().
Examples
========
>>> from sympy.abc import x
>>> from sympy.functions import Abs
>>> Abs(-x).fdiff()
sign(x)
"""
if argindex == 1:
return sign(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
from sympy.simplify.simplify import signsimp
from sympy.core.function import expand_mul
if hasattr(arg, '_eval_Abs'):
obj = arg._eval_Abs()
if obj is not None:
return obj
if not isinstance(arg, Expr):
raise TypeError("Bad argument type for Abs(): %s" % type(arg))
# handle what we can
arg = signsimp(arg, evaluate=False)
if arg.is_Mul:
known = []
unk = []
for t in arg.args:
tnew = cls(t)
if isinstance(tnew, cls):
unk.append(tnew.args[0])
else:
known.append(tnew)
known = Mul(*known)
unk = cls(Mul(*unk), evaluate=False) if unk else S.One
return known*unk
if arg is S.NaN:
return S.NaN
if arg is S.ComplexInfinity:
return S.Infinity
if arg.is_Pow:
base, exponent = arg.as_base_exp()
if base.is_real:
if exponent.is_integer:
if exponent.is_even:
return arg
if base is S.NegativeOne:
return S.One
if isinstance(base, cls) and exponent is S.NegativeOne:
return arg
return Abs(base)**exponent
if base.is_nonnegative:
return base**re(exponent)
if base.is_negative:
return (-base)**re(exponent)*exp(-S.Pi*im(exponent))
return
elif not base.has(Symbol): # complex base
# express base**exponent as exp(exponent*log(base))
a, b = log(base).as_real_imag()
z = a + I*b
return exp(re(exponent*z))
if isinstance(arg, exp):
return exp(re(arg.args[0]))
if isinstance(arg, AppliedUndef):
return
if arg.is_Add and arg.has(S.Infinity, S.NegativeInfinity):
if any(a.is_infinite for a in arg.as_real_imag()):
return S.Infinity
if arg.is_zero:
return S.Zero
if arg.is_nonnegative:
return arg
if arg.is_nonpositive:
return -arg
if arg.is_imaginary:
arg2 = -S.ImaginaryUnit * arg
if arg2.is_nonnegative:
return arg2
# reject result if all new conjugates are just wrappers around
# an expression that was already in the arg
conj = signsimp(arg.conjugate(), evaluate=False)
new_conj = conj.atoms(conjugate) - arg.atoms(conjugate)
if new_conj and all(arg.has(i.args[0]) for i in new_conj):
return
if arg != conj and arg != -conj:
ignore = arg.atoms(Abs)
abs_free_arg = arg.xreplace({i: Dummy(real=True) for i in ignore})
unk = [a for a in abs_free_arg.free_symbols if a.is_real is None]
if not unk or not all(conj.has(conjugate(u)) for u in unk):
return sqrt(expand_mul(arg*conj))
def _eval_is_integer(self):
if self.args[0].is_real:
return self.args[0].is_integer
def _eval_is_nonzero(self):
return fuzzy_not(self._args[0].is_zero)
def _eval_is_zero(self):
return self._args[0].is_zero
def _eval_is_positive(self):
is_z = self.is_zero
if is_z is not None:
return not is_z
def _eval_is_rational(self):
if self.args[0].is_real:
return self.args[0].is_rational
def _eval_is_even(self):
if self.args[0].is_real:
return self.args[0].is_even
def _eval_is_odd(self):
if self.args[0].is_real:
return self.args[0].is_odd
def _eval_is_algebraic(self):
return self.args[0].is_algebraic
def _eval_power(self, exponent):
if self.args[0].is_real and exponent.is_integer:
if exponent.is_even:
return self.args[0]**exponent
elif exponent is not S.NegativeOne and exponent.is_Integer:
return self.args[0]**(exponent - 1)*self
return
def _eval_nseries(self, x, n, logx):
direction = self.args[0].leadterm(x)[0]
s = self.args[0]._eval_nseries(x, n=n, logx=logx)
when = Eq(direction, 0)
return Piecewise(
((s.subs(direction, 0)), when),
(sign(direction)*s, True),
)
def _sage_(self):
import sage.all as sage
return sage.abs_symbolic(self.args[0]._sage_())
def _eval_derivative(self, x):
if self.args[0].is_real or self.args[0].is_imaginary:
return Derivative(self.args[0], x, evaluate=True) \
* sign(conjugate(self.args[0]))
rv = (re(self.args[0]) * Derivative(re(self.args[0]), x,
evaluate=True) + im(self.args[0]) * Derivative(im(self.args[0]),
x, evaluate=True)) / Abs(self.args[0])
return rv.rewrite(sign)
def _eval_rewrite_as_Heaviside(self, arg, **kwargs):
# Note this only holds for real arg (since Heaviside is not defined
# for complex arguments).
from sympy.functions.special.delta_functions import Heaviside
if arg.is_real:
return arg*(Heaviside(arg) - Heaviside(-arg))
def _eval_rewrite_as_Piecewise(self, arg, **kwargs):
if arg.is_real:
return Piecewise((arg, arg >= 0), (-arg, True))
def _eval_rewrite_as_sign(self, arg, **kwargs):
return arg/sign(arg)
class arg(Function):
"""
Returns the argument (in radians) of a complex number. For a positive
number, the argument is always 0.
Examples
========
>>> from sympy.functions import arg
>>> from sympy import I, sqrt
>>> arg(2.0)
0
>>> arg(I)
pi/2
>>> arg(sqrt(2) + I*sqrt(2))
pi/4
"""
is_real = True
is_finite = True
@classmethod
def eval(cls, arg):
if isinstance(arg, exp_polar):
return periodic_argument(arg, oo)
if not arg.is_Atom:
c, arg_ = factor_terms(arg).as_coeff_Mul()
if arg_.is_Mul:
arg_ = Mul(*[a if (sign(a) not in (-1, 1)) else
sign(a) for a in arg_.args])
arg_ = sign(c)*arg_
else:
arg_ = arg
if arg_.atoms(AppliedUndef):
return
x, y = arg_.as_real_imag()
rv = atan2(y, x)
if rv.is_number:
return rv
if arg_ != arg:
return cls(arg_, evaluate=False)
def _eval_derivative(self, t):
x, y = self.args[0].as_real_imag()
return (x * Derivative(y, t, evaluate=True) - y *
Derivative(x, t, evaluate=True)) / (x**2 + y**2)
def _eval_rewrite_as_atan2(self, arg, **kwargs):
x, y = self.args[0].as_real_imag()
return atan2(y, x)
class conjugate(Function):
"""
Returns the `complex conjugate` Ref[1] of an argument.
In mathematics, the complex conjugate of a complex number
is given by changing the sign of the imaginary part.
Thus, the conjugate of the complex number
:math:`a + ib` (where a and b are real numbers) is :math:`a - ib`
Examples
========
>>> from sympy import conjugate, I
>>> conjugate(2)
2
>>> conjugate(I)
-I
See Also
========
sign, Abs
References
==========
.. [1] https://en.wikipedia.org/wiki/Complex_conjugation
"""
@classmethod
def eval(cls, arg):
obj = arg._eval_conjugate()
if obj is not None:
return obj
def _eval_Abs(self):
return Abs(self.args[0], evaluate=True)
def _eval_adjoint(self):
return transpose(self.args[0])
def _eval_conjugate(self):
return self.args[0]
def _eval_derivative(self, x):
if x.is_real:
return conjugate(Derivative(self.args[0], x, evaluate=True))
elif x.is_imaginary:
return -conjugate(Derivative(self.args[0], x, evaluate=True))
def _eval_transpose(self):
return adjoint(self.args[0])
def _eval_is_algebraic(self):
return self.args[0].is_algebraic
class transpose(Function):
"""
Linear map transposition.
"""
@classmethod
def eval(cls, arg):
obj = arg._eval_transpose()
if obj is not None:
return obj
def _eval_adjoint(self):
return conjugate(self.args[0])
def _eval_conjugate(self):
return adjoint(self.args[0])
def _eval_transpose(self):
return self.args[0]
class adjoint(Function):
"""
Conjugate transpose or Hermite conjugation.
"""
@classmethod
def eval(cls, arg):
obj = arg._eval_adjoint()
if obj is not None:
return obj
obj = arg._eval_transpose()
if obj is not None:
return conjugate(obj)
def _eval_adjoint(self):
return self.args[0]
def _eval_conjugate(self):
return transpose(self.args[0])
def _eval_transpose(self):
return conjugate(self.args[0])
def _latex(self, printer, exp=None, *args):
arg = printer._print(self.args[0])
tex = r'%s^{\dagger}' % arg
if exp:
tex = r'\left(%s\right)^{%s}' % (tex, printer._print(exp))
return tex
def _pretty(self, printer, *args):
from sympy.printing.pretty.stringpict import prettyForm
pform = printer._print(self.args[0], *args)
if printer._use_unicode:
pform = pform**prettyForm(u'\N{DAGGER}')
else:
pform = pform**prettyForm('+')
return pform
###############################################################################
############### HANDLING OF POLAR NUMBERS #####################################
###############################################################################
class polar_lift(Function):
"""
Lift argument to the Riemann surface of the logarithm, using the
standard branch.
>>> from sympy import Symbol, polar_lift, I
>>> p = Symbol('p', polar=True)
>>> x = Symbol('x')
>>> polar_lift(4)
4*exp_polar(0)
>>> polar_lift(-4)
4*exp_polar(I*pi)
>>> polar_lift(-I)
exp_polar(-I*pi/2)
>>> polar_lift(I + 2)
polar_lift(2 + I)
>>> polar_lift(4*x)
4*polar_lift(x)
>>> polar_lift(4*p)
4*p
See Also
========
sympy.functions.elementary.exponential.exp_polar
periodic_argument
"""
is_polar = True
is_comparable = False # Cannot be evalf'd.
@classmethod
def eval(cls, arg):
from sympy.functions.elementary.complexes import arg as argument
if arg.is_number:
ar = argument(arg)
# In general we want to affirm that something is known,
# e.g. `not ar.has(argument) and not ar.has(atan)`
# but for now we will just be more restrictive and
# see that it has evaluated to one of the known values.
if ar in (0, pi/2, -pi/2, pi):
return exp_polar(I*ar)*abs(arg)
if arg.is_Mul:
args = arg.args
else:
args = [arg]
included = []
excluded = []
positive = []
for arg in args:
if arg.is_polar:
included += [arg]
elif arg.is_positive:
positive += [arg]
else:
excluded += [arg]
if len(excluded) < len(args):
if excluded:
return Mul(*(included + positive))*polar_lift(Mul(*excluded))
elif included:
return Mul(*(included + positive))
else:
return Mul(*positive)*exp_polar(0)
def _eval_evalf(self, prec):
""" Careful! any evalf of polar numbers is flaky """
return self.args[0]._eval_evalf(prec)
def _eval_Abs(self):
return Abs(self.args[0], evaluate=True)
class periodic_argument(Function):
"""
Represent the argument on a quotient of the Riemann surface of the
logarithm. That is, given a period P, always return a value in
(-P/2, P/2], by using exp(P*I) == 1.
>>> from sympy import exp, exp_polar, periodic_argument, unbranched_argument
>>> from sympy import I, pi
>>> unbranched_argument(exp(5*I*pi))
pi
>>> unbranched_argument(exp_polar(5*I*pi))
5*pi
>>> periodic_argument(exp_polar(5*I*pi), 2*pi)
pi
>>> periodic_argument(exp_polar(5*I*pi), 3*pi)
-pi
>>> periodic_argument(exp_polar(5*I*pi), pi)
0
See Also
========
sympy.functions.elementary.exponential.exp_polar
polar_lift : Lift argument to the Riemann surface of the logarithm
principal_branch
"""
@classmethod
def _getunbranched(cls, ar):
if ar.is_Mul:
args = ar.args
else:
args = [ar]
unbranched = 0
for a in args:
if not a.is_polar:
unbranched += arg(a)
elif isinstance(a, exp_polar):
unbranched += a.exp.as_real_imag()[1]
elif a.is_Pow:
re, im = a.exp.as_real_imag()
unbranched += re*unbranched_argument(
a.base) + im*log(abs(a.base))
elif isinstance(a, polar_lift):
unbranched += arg(a.args[0])
else:
return None
return unbranched
@classmethod
def eval(cls, ar, period):
# Our strategy is to evaluate the argument on the Riemann surface of the
# logarithm, and then reduce.
# NOTE evidently this means it is a rather bad idea to use this with
# period != 2*pi and non-polar numbers.
if not period.is_positive:
return None
if period == oo and isinstance(ar, principal_branch):
return periodic_argument(*ar.args)
if isinstance(ar, polar_lift) and period >= 2*pi:
return periodic_argument(ar.args[0], period)
if ar.is_Mul:
newargs = [x for x in ar.args if not x.is_positive]
if len(newargs) != len(ar.args):
return periodic_argument(Mul(*newargs), period)
unbranched = cls._getunbranched(ar)
if unbranched is None:
return None
if unbranched.has(periodic_argument, atan2, atan):
return None
if period == oo:
return unbranched
if period != oo:
n = ceiling(unbranched/period - S(1)/2)*period
if not n.has(ceiling):
return unbranched - n
def _eval_evalf(self, prec):
z, period = self.args
if period == oo:
unbranched = periodic_argument._getunbranched(z)
if unbranched is None:
return self
return unbranched._eval_evalf(prec)
ub = periodic_argument(z, oo)._eval_evalf(prec)
return (ub - ceiling(ub/period - S(1)/2)*period)._eval_evalf(prec)
def unbranched_argument(arg):
return periodic_argument(arg, oo)
class principal_branch(Function):
"""
Represent a polar number reduced to its principal branch on a quotient
of the Riemann surface of the logarithm.
This is a function of two arguments. The first argument is a polar
number `z`, and the second one a positive real number of infinity, `p`.
The result is "z mod exp_polar(I*p)".
>>> from sympy import exp_polar, principal_branch, oo, I, pi
>>> from sympy.abc import z
>>> principal_branch(z, oo)
z
>>> principal_branch(exp_polar(2*pi*I)*3, 2*pi)
3*exp_polar(0)
>>> principal_branch(exp_polar(2*pi*I)*3*z, 2*pi)
3*principal_branch(z, 2*pi)
See Also
========
sympy.functions.elementary.exponential.exp_polar
polar_lift : Lift argument to the Riemann surface of the logarithm
periodic_argument
"""
is_polar = True
is_comparable = False # cannot always be evalf'd
@classmethod
def eval(self, x, period):
from sympy import oo, exp_polar, I, Mul, polar_lift, Symbol
if isinstance(x, polar_lift):
return principal_branch(x.args[0], period)
if period == oo:
return x
ub = periodic_argument(x, oo)
barg = periodic_argument(x, period)
if ub != barg and not ub.has(periodic_argument) \
and not barg.has(periodic_argument):
pl = polar_lift(x)
def mr(expr):
if not isinstance(expr, Symbol):
return polar_lift(expr)
return expr
pl = pl.replace(polar_lift, mr)
# Recompute unbranched argument
ub = periodic_argument(pl, oo)
if not pl.has(polar_lift):
if ub != barg:
res = exp_polar(I*(barg - ub))*pl
else:
res = pl
if not res.is_polar and not res.has(exp_polar):
res *= exp_polar(0)
return res
if not x.free_symbols:
c, m = x, ()
else:
c, m = x.as_coeff_mul(*x.free_symbols)
others = []
for y in m:
if y.is_positive:
c *= y
else:
others += [y]
m = tuple(others)
arg = periodic_argument(c, period)
if arg.has(periodic_argument):
return None
if arg.is_number and (unbranched_argument(c) != arg or
(arg == 0 and m != () and c != 1)):
if arg == 0:
return abs(c)*principal_branch(Mul(*m), period)
return principal_branch(exp_polar(I*arg)*Mul(*m), period)*abs(c)
if arg.is_number and ((abs(arg) < period/2) == True or arg == period/2) \
and m == ():
return exp_polar(arg*I)*abs(c)
def _eval_evalf(self, prec):
from sympy import exp, pi, I
z, period = self.args
p = periodic_argument(z, period)._eval_evalf(prec)
if abs(p) > pi or p == -pi:
return self # Cannot evalf for this argument.
return (abs(z)*exp(I*p))._eval_evalf(prec)
def _polarify(eq, lift, pause=False):
from sympy import Integral
if eq.is_polar:
return eq
if eq.is_number and not pause:
return polar_lift(eq)
if isinstance(eq, Symbol) and not pause and lift:
return polar_lift(eq)
elif eq.is_Atom:
return eq
elif eq.is_Add:
r = eq.func(*[_polarify(arg, lift, pause=True) for arg in eq.args])
if lift:
return polar_lift(r)
return r
elif eq.is_Function:
return eq.func(*[_polarify(arg, lift, pause=False) for arg in eq.args])
elif isinstance(eq, Integral):
# Don't lift the integration variable
func = _polarify(eq.function, lift, pause=pause)
limits = []
for limit in eq.args[1:]:
var = _polarify(limit[0], lift=False, pause=pause)
rest = _polarify(limit[1:], lift=lift, pause=pause)
limits.append((var,) + rest)
return Integral(*((func,) + tuple(limits)))
else:
return eq.func(*[_polarify(arg, lift, pause=pause)
if isinstance(arg, Expr) else arg for arg in eq.args])
def polarify(eq, subs=True, lift=False):
"""
Turn all numbers in eq into their polar equivalents (under the standard
choice of argument).
Note that no attempt is made to guess a formal convention of adding
polar numbers, expressions like 1 + x will generally not be altered.
Note also that this function does not promote exp(x) to exp_polar(x).
If ``subs`` is True, all symbols which are not already polar will be
substituted for polar dummies; in this case the function behaves much
like posify.
If ``lift`` is True, both addition statements and non-polar symbols are
changed to their polar_lift()ed versions.
Note that lift=True implies subs=False.
>>> from sympy import polarify, sin, I
>>> from sympy.abc import x, y
>>> expr = (-x)**y
>>> expr.expand()
(-x)**y
>>> polarify(expr)
((_x*exp_polar(I*pi))**_y, {_x: x, _y: y})
>>> polarify(expr)[0].expand()
_x**_y*exp_polar(_y*I*pi)
>>> polarify(x, lift=True)
polar_lift(x)
>>> polarify(x*(1+y), lift=True)
polar_lift(x)*polar_lift(y + 1)
Adds are treated carefully:
>>> polarify(1 + sin((1 + I)*x))
(sin(_x*polar_lift(1 + I)) + 1, {_x: x})
"""
if lift:
subs = False
eq = _polarify(sympify(eq), lift)
if not subs:
return eq
reps = {s: Dummy(s.name, polar=True) for s in eq.free_symbols}
eq = eq.subs(reps)
return eq, {r: s for s, r in reps.items()}
def _unpolarify(eq, exponents_only, pause=False):
if not isinstance(eq, Basic) or eq.is_Atom:
return eq
if not pause:
if isinstance(eq, exp_polar):
return exp(_unpolarify(eq.exp, exponents_only))
if isinstance(eq, principal_branch) and eq.args[1] == 2*pi:
return _unpolarify(eq.args[0], exponents_only)
if (
eq.is_Add or eq.is_Mul or eq.is_Boolean or
eq.is_Relational and (
eq.rel_op in ('==', '!=') and 0 in eq.args or
eq.rel_op not in ('==', '!='))
):
return eq.func(*[_unpolarify(x, exponents_only) for x in eq.args])
if isinstance(eq, polar_lift):
return _unpolarify(eq.args[0], exponents_only)
if eq.is_Pow:
expo = _unpolarify(eq.exp, exponents_only)
base = _unpolarify(eq.base, exponents_only,
not (expo.is_integer and not pause))
return base**expo
if eq.is_Function and getattr(eq.func, 'unbranched', False):
return eq.func(*[_unpolarify(x, exponents_only, exponents_only)
for x in eq.args])
return eq.func(*[_unpolarify(x, exponents_only, True) for x in eq.args])
def unpolarify(eq, subs={}, exponents_only=False):
"""
If p denotes the projection from the Riemann surface of the logarithm to
the complex line, return a simplified version eq' of `eq` such that
p(eq') == p(eq).
Also apply the substitution subs in the end. (This is a convenience, since
``unpolarify``, in a certain sense, undoes polarify.)
>>> from sympy import unpolarify, polar_lift, sin, I
>>> unpolarify(polar_lift(I + 2))
2 + I
>>> unpolarify(sin(polar_lift(I + 7)))
sin(7 + I)
"""
if isinstance(eq, bool):
return eq
eq = sympify(eq)
if subs != {}:
return unpolarify(eq.subs(subs))
changed = True
pause = False
if exponents_only:
pause = True
while changed:
changed = False
res = _unpolarify(eq, exponents_only, pause)
if res != eq:
changed = True
eq = res
if isinstance(res, bool):
return res
# Finally, replacing Exp(0) by 1 is always correct.
# So is polar_lift(0) -> 0.
return res.subs({exp_polar(0): 1, polar_lift(0): 0})
# /cyclic/
from sympy.core import basic as _
_.abs_ = Abs
del _
|
e3cd421afd85fea80933d3253a16b3bfbbb1d6105410454583c90d29ba96c052
|
"""Hypergeometric and Meijer G-functions"""
from __future__ import print_function, division
from sympy.core import S, I, pi, oo, zoo, ilcm, Mod
from sympy.core.function import Function, Derivative, ArgumentIndexError
from sympy.core.compatibility import reduce, range
from sympy.core.containers import Tuple
from sympy.core.mul import Mul
from sympy.core.symbol import Dummy
from sympy.functions import (sqrt, exp, log, sin, cos, asin, atan,
sinh, cosh, asinh, acosh, atanh, acoth, Abs)
from sympy.utilities.iterables import default_sort_key
class TupleArg(Tuple):
def limit(self, x, xlim, dir='+'):
""" Compute limit x->xlim.
"""
from sympy.series.limits import limit
return TupleArg(*[limit(f, x, xlim, dir) for f in self.args])
# TODO should __new__ accept **options?
# TODO should constructors should check if parameters are sensible?
def _prep_tuple(v):
"""
Turn an iterable argument V into a Tuple and unpolarify, since both
hypergeometric and meijer g-functions are unbranched in their parameters.
Examples
========
>>> from sympy.functions.special.hyper import _prep_tuple
>>> _prep_tuple([1, 2, 3])
(1, 2, 3)
>>> _prep_tuple((4, 5))
(4, 5)
>>> _prep_tuple((7, 8, 9))
(7, 8, 9)
"""
from sympy import unpolarify
return TupleArg(*[unpolarify(x) for x in v])
class TupleParametersBase(Function):
""" Base class that takes care of differentiation, when some of
the arguments are actually tuples. """
# This is not deduced automatically since there are Tuples as arguments.
is_commutative = True
def _eval_derivative(self, s):
try:
res = 0
if self.args[0].has(s) or self.args[1].has(s):
for i, p in enumerate(self._diffargs):
m = self._diffargs[i].diff(s)
if m != 0:
res += self.fdiff((1, i))*m
return res + self.fdiff(3)*self.args[2].diff(s)
except (ArgumentIndexError, NotImplementedError):
return Derivative(self, s)
class hyper(TupleParametersBase):
r"""
The (generalized) hypergeometric function is defined by a series where
the ratios of successive terms are a rational function of the summation
index. When convergent, it is continued analytically to the largest
possible domain.
The hypergeometric function depends on two vectors of parameters, called
the numerator parameters :math:`a_p`, and the denominator parameters
:math:`b_q`. It also has an argument :math:`z`. The series definition is
.. math ::
{}_pF_q\left(\begin{matrix} a_1, \cdots, a_p \\ b_1, \cdots, b_q \end{matrix}
\middle| z \right)
= \sum_{n=0}^\infty \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n}
\frac{z^n}{n!},
where :math:`(a)_n = (a)(a+1)\cdots(a+n-1)` denotes the rising factorial.
If one of the :math:`b_q` is a non-positive integer then the series is
undefined unless one of the `a_p` is a larger (i.e. smaller in
magnitude) non-positive integer. If none of the :math:`b_q` is a
non-positive integer and one of the :math:`a_p` is a non-positive
integer, then the series reduces to a polynomial. To simplify the
following discussion, we assume that none of the :math:`a_p` or
:math:`b_q` is a non-positive integer. For more details, see the
references.
The series converges for all :math:`z` if :math:`p \le q`, and thus
defines an entire single-valued function in this case. If :math:`p =
q+1` the series converges for :math:`|z| < 1`, and can be continued
analytically into a half-plane. If :math:`p > q+1` the series is
divergent for all :math:`z`.
Note: The hypergeometric function constructor currently does *not* check
if the parameters actually yield a well-defined function.
Examples
========
The parameters :math:`a_p` and :math:`b_q` can be passed as arbitrary
iterables, for example:
>>> from sympy.functions import hyper
>>> from sympy.abc import x, n, a
>>> hyper((1, 2, 3), [3, 4], x)
hyper((1, 2, 3), (3, 4), x)
There is also pretty printing (it looks better using unicode):
>>> from sympy import pprint
>>> pprint(hyper((1, 2, 3), [3, 4], x), use_unicode=False)
_
|_ /1, 2, 3 | \
| | | x|
3 2 \ 3, 4 | /
The parameters must always be iterables, even if they are vectors of
length one or zero:
>>> hyper((1, ), [], x)
hyper((1,), (), x)
But of course they may be variables (but if they depend on x then you
should not expect much implemented functionality):
>>> hyper((n, a), (n**2,), x)
hyper((n, a), (n**2,), x)
The hypergeometric function generalizes many named special functions.
The function hyperexpand() tries to express a hypergeometric function
using named special functions.
For example:
>>> from sympy import hyperexpand
>>> hyperexpand(hyper([], [], x))
exp(x)
You can also use expand_func:
>>> from sympy import expand_func
>>> expand_func(x*hyper([1, 1], [2], -x))
log(x + 1)
More examples:
>>> from sympy import S
>>> hyperexpand(hyper([], [S(1)/2], -x**2/4))
cos(x)
>>> hyperexpand(x*hyper([S(1)/2, S(1)/2], [S(3)/2], x**2))
asin(x)
We can also sometimes hyperexpand parametric functions:
>>> from sympy.abc import a
>>> hyperexpand(hyper([-a], [], x))
(1 - x)**a
See Also
========
sympy.simplify.hyperexpand
sympy.functions.special.gamma_functions.gamma
meijerg
References
==========
.. [1] Luke, Y. L. (1969), The Special Functions and Their Approximations,
Volume 1
.. [2] https://en.wikipedia.org/wiki/Generalized_hypergeometric_function
"""
def __new__(cls, ap, bq, z):
# TODO should we check convergence conditions?
return Function.__new__(cls, _prep_tuple(ap), _prep_tuple(bq), z)
@classmethod
def eval(cls, ap, bq, z):
from sympy import unpolarify
if len(ap) <= len(bq) or (len(ap) == len(bq) + 1 and (Abs(z) <= 1) == True):
nz = unpolarify(z)
if z != nz:
return hyper(ap, bq, nz)
def fdiff(self, argindex=3):
if argindex != 3:
raise ArgumentIndexError(self, argindex)
nap = Tuple(*[a + 1 for a in self.ap])
nbq = Tuple(*[b + 1 for b in self.bq])
fac = Mul(*self.ap)/Mul(*self.bq)
return fac*hyper(nap, nbq, self.argument)
def _eval_expand_func(self, **hints):
from sympy import gamma, hyperexpand
if len(self.ap) == 2 and len(self.bq) == 1 and self.argument == 1:
a, b = self.ap
c = self.bq[0]
return gamma(c)*gamma(c - a - b)/gamma(c - a)/gamma(c - b)
return hyperexpand(self)
def _eval_rewrite_as_Sum(self, ap, bq, z, **kwargs):
from sympy.functions import factorial, RisingFactorial, Piecewise
from sympy import Sum
n = Dummy("n", integer=True)
rfap = Tuple(*[RisingFactorial(a, n) for a in ap])
rfbq = Tuple(*[RisingFactorial(b, n) for b in bq])
coeff = Mul(*rfap) / Mul(*rfbq)
return Piecewise((Sum(coeff * z**n / factorial(n), (n, 0, oo)),
self.convergence_statement), (self, True))
@property
def argument(self):
""" Argument of the hypergeometric function. """
return self.args[2]
@property
def ap(self):
""" Numerator parameters of the hypergeometric function. """
return Tuple(*self.args[0])
@property
def bq(self):
""" Denominator parameters of the hypergeometric function. """
return Tuple(*self.args[1])
@property
def _diffargs(self):
return self.ap + self.bq
@property
def eta(self):
""" A quantity related to the convergence of the series. """
return sum(self.ap) - sum(self.bq)
@property
def radius_of_convergence(self):
"""
Compute the radius of convergence of the defining series.
Note that even if this is not oo, the function may still be evaluated
outside of the radius of convergence by analytic continuation. But if
this is zero, then the function is not actually defined anywhere else.
>>> from sympy.functions import hyper
>>> from sympy.abc import z
>>> hyper((1, 2), [3], z).radius_of_convergence
1
>>> hyper((1, 2, 3), [4], z).radius_of_convergence
0
>>> hyper((1, 2), (3, 4), z).radius_of_convergence
oo
"""
if any(a.is_integer and (a <= 0) == True for a in self.ap + self.bq):
aints = [a for a in self.ap if a.is_Integer and (a <= 0) == True]
bints = [a for a in self.bq if a.is_Integer and (a <= 0) == True]
if len(aints) < len(bints):
return S(0)
popped = False
for b in bints:
cancelled = False
while aints:
a = aints.pop()
if a >= b:
cancelled = True
break
popped = True
if not cancelled:
return S(0)
if aints or popped:
# There are still non-positive numerator parameters.
# This is a polynomial.
return oo
if len(self.ap) == len(self.bq) + 1:
return S(1)
elif len(self.ap) <= len(self.bq):
return oo
else:
return S(0)
@property
def convergence_statement(self):
""" Return a condition on z under which the series converges. """
from sympy import And, Or, re, Ne, oo
R = self.radius_of_convergence
if R == 0:
return False
if R == oo:
return True
# The special functions and their approximations, page 44
e = self.eta
z = self.argument
c1 = And(re(e) < 0, abs(z) <= 1)
c2 = And(0 <= re(e), re(e) < 1, abs(z) <= 1, Ne(z, 1))
c3 = And(re(e) >= 1, abs(z) < 1)
return Or(c1, c2, c3)
def _eval_simplify(self, ratio, measure, rational, inverse):
from sympy.simplify.hyperexpand import hyperexpand
return hyperexpand(self)
def _sage_(self):
import sage.all as sage
ap = [arg._sage_() for arg in self.args[0]]
bq = [arg._sage_() for arg in self.args[1]]
return sage.hypergeometric(ap, bq, self.argument._sage_())
class meijerg(TupleParametersBase):
r"""
The Meijer G-function is defined by a Mellin-Barnes type integral that
resembles an inverse Mellin transform. It generalizes the hypergeometric
functions.
The Meijer G-function depends on four sets of parameters. There are
"*numerator parameters*"
:math:`a_1, \ldots, a_n` and :math:`a_{n+1}, \ldots, a_p`, and there are
"*denominator parameters*"
:math:`b_1, \ldots, b_m` and :math:`b_{m+1}, \ldots, b_q`.
Confusingly, it is traditionally denoted as follows (note the position
of `m`, `n`, `p`, `q`, and how they relate to the lengths of the four
parameter vectors):
.. math ::
G_{p,q}^{m,n} \left(\begin{matrix}a_1, \cdots, a_n & a_{n+1}, \cdots, a_p \\
b_1, \cdots, b_m & b_{m+1}, \cdots, b_q
\end{matrix} \middle| z \right).
However, in sympy the four parameter vectors are always available
separately (see examples), so that there is no need to keep track of the
decorating sub- and super-scripts on the G symbol.
The G function is defined as the following integral:
.. math ::
\frac{1}{2 \pi i} \int_L \frac{\prod_{j=1}^m \Gamma(b_j - s)
\prod_{j=1}^n \Gamma(1 - a_j + s)}{\prod_{j=m+1}^q \Gamma(1- b_j +s)
\prod_{j=n+1}^p \Gamma(a_j - s)} z^s \mathrm{d}s,
where :math:`\Gamma(z)` is the gamma function. There are three possible
contours which we will not describe in detail here (see the references).
If the integral converges along more than one of them the definitions
agree. The contours all separate the poles of :math:`\Gamma(1-a_j+s)`
from the poles of :math:`\Gamma(b_k-s)`, so in particular the G function
is undefined if :math:`a_j - b_k \in \mathbb{Z}_{>0}` for some
:math:`j \le n` and :math:`k \le m`.
The conditions under which one of the contours yields a convergent integral
are complicated and we do not state them here, see the references.
Note: Currently the Meijer G-function constructor does *not* check any
convergence conditions.
Examples
========
You can pass the parameters either as four separate vectors:
>>> from sympy.functions import meijerg
>>> from sympy.abc import x, a
>>> from sympy.core.containers import Tuple
>>> from sympy import pprint
>>> pprint(meijerg((1, 2), (a, 4), (5,), [], x), use_unicode=False)
__1, 2 /1, 2 a, 4 | \
/__ | | x|
\_|4, 1 \ 5 | /
or as two nested vectors:
>>> pprint(meijerg([(1, 2), (3, 4)], ([5], Tuple()), x), use_unicode=False)
__1, 2 /1, 2 3, 4 | \
/__ | | x|
\_|4, 1 \ 5 | /
As with the hypergeometric function, the parameters may be passed as
arbitrary iterables. Vectors of length zero and one also have to be
passed as iterables. The parameters need not be constants, but if they
depend on the argument then not much implemented functionality should be
expected.
All the subvectors of parameters are available:
>>> from sympy import pprint
>>> g = meijerg([1], [2], [3], [4], x)
>>> pprint(g, use_unicode=False)
__1, 1 /1 2 | \
/__ | | x|
\_|2, 2 \3 4 | /
>>> g.an
(1,)
>>> g.ap
(1, 2)
>>> g.aother
(2,)
>>> g.bm
(3,)
>>> g.bq
(3, 4)
>>> g.bother
(4,)
The Meijer G-function generalizes the hypergeometric functions.
In some cases it can be expressed in terms of hypergeometric functions,
using Slater's theorem. For example:
>>> from sympy import hyperexpand
>>> from sympy.abc import a, b, c
>>> hyperexpand(meijerg([a], [], [c], [b], x), allow_hyper=True)
x**c*gamma(-a + c + 1)*hyper((-a + c + 1,),
(-b + c + 1,), -x)/gamma(-b + c + 1)
Thus the Meijer G-function also subsumes many named functions as special
cases. You can use expand_func or hyperexpand to (try to) rewrite a
Meijer G-function in terms of named special functions. For example:
>>> from sympy import expand_func, S
>>> expand_func(meijerg([[],[]], [[0],[]], -x))
exp(x)
>>> hyperexpand(meijerg([[],[]], [[S(1)/2],[0]], (x/2)**2))
sin(x)/sqrt(pi)
See Also
========
hyper
sympy.simplify.hyperexpand
References
==========
.. [1] Luke, Y. L. (1969), The Special Functions and Their Approximations,
Volume 1
.. [2] https://en.wikipedia.org/wiki/Meijer_G-function
"""
def __new__(cls, *args):
if len(args) == 5:
args = [(args[0], args[1]), (args[2], args[3]), args[4]]
if len(args) != 3:
raise TypeError("args must be either as, as', bs, bs', z or "
"as, bs, z")
def tr(p):
if len(p) != 2:
raise TypeError("wrong argument")
return TupleArg(_prep_tuple(p[0]), _prep_tuple(p[1]))
arg0, arg1 = tr(args[0]), tr(args[1])
if Tuple(arg0, arg1).has(oo, zoo, -oo):
raise ValueError("G-function parameters must be finite")
if any((a - b).is_Integer and a - b > 0
for a in arg0[0] for b in arg1[0]):
raise ValueError("no parameter a1, ..., an may differ from "
"any b1, ..., bm by a positive integer")
# TODO should we check convergence conditions?
return Function.__new__(cls, arg0, arg1, args[2])
def fdiff(self, argindex=3):
if argindex != 3:
return self._diff_wrt_parameter(argindex[1])
if len(self.an) >= 1:
a = list(self.an)
a[0] -= 1
G = meijerg(a, self.aother, self.bm, self.bother, self.argument)
return 1/self.argument * ((self.an[0] - 1)*self + G)
elif len(self.bm) >= 1:
b = list(self.bm)
b[0] += 1
G = meijerg(self.an, self.aother, b, self.bother, self.argument)
return 1/self.argument * (self.bm[0]*self - G)
else:
return S.Zero
def _diff_wrt_parameter(self, idx):
# Differentiation wrt a parameter can only be done in very special
# cases. In particular, if we want to differentiate with respect to
# `a`, all other gamma factors have to reduce to rational functions.
#
# Let MT denote mellin transform. Suppose T(-s) is the gamma factor
# appearing in the definition of G. Then
#
# MT(log(z)G(z)) = d/ds T(s) = d/da T(s) + ...
#
# Thus d/da G(z) = log(z)G(z) - ...
# The ... can be evaluated as a G function under the above conditions,
# the formula being most easily derived by using
#
# d Gamma(s + n) Gamma(s + n) / 1 1 1 \
# -- ------------ = ------------ | - + ---- + ... + --------- |
# ds Gamma(s) Gamma(s) \ s s + 1 s + n - 1 /
#
# which follows from the difference equation of the digamma function.
# (There is a similar equation for -n instead of +n).
# We first figure out how to pair the parameters.
an = list(self.an)
ap = list(self.aother)
bm = list(self.bm)
bq = list(self.bother)
if idx < len(an):
an.pop(idx)
else:
idx -= len(an)
if idx < len(ap):
ap.pop(idx)
else:
idx -= len(ap)
if idx < len(bm):
bm.pop(idx)
else:
bq.pop(idx - len(bm))
pairs1 = []
pairs2 = []
for l1, l2, pairs in [(an, bq, pairs1), (ap, bm, pairs2)]:
while l1:
x = l1.pop()
found = None
for i, y in enumerate(l2):
if not Mod((x - y).simplify(), 1):
found = i
break
if found is None:
raise NotImplementedError('Derivative not expressible '
'as G-function?')
y = l2[i]
l2.pop(i)
pairs.append((x, y))
# Now build the result.
res = log(self.argument)*self
for a, b in pairs1:
sign = 1
n = a - b
base = b
if n < 0:
sign = -1
n = b - a
base = a
for k in range(n):
res -= sign*meijerg(self.an + (base + k + 1,), self.aother,
self.bm, self.bother + (base + k + 0,),
self.argument)
for a, b in pairs2:
sign = 1
n = b - a
base = a
if n < 0:
sign = -1
n = a - b
base = b
for k in range(n):
res -= sign*meijerg(self.an, self.aother + (base + k + 1,),
self.bm + (base + k + 0,), self.bother,
self.argument)
return res
def get_period(self):
"""
Return a number P such that G(x*exp(I*P)) == G(x).
>>> from sympy.functions.special.hyper import meijerg
>>> from sympy.abc import z
>>> from sympy import pi, S
>>> meijerg([1], [], [], [], z).get_period()
2*pi
>>> meijerg([pi], [], [], [], z).get_period()
oo
>>> meijerg([1, 2], [], [], [], z).get_period()
oo
>>> meijerg([1,1], [2], [1, S(1)/2, S(1)/3], [1], z).get_period()
12*pi
"""
# This follows from slater's theorem.
def compute(l):
# first check that no two differ by an integer
for i, b in enumerate(l):
if not b.is_Rational:
return oo
for j in range(i + 1, len(l)):
if not Mod((b - l[j]).simplify(), 1):
return oo
return reduce(ilcm, (x.q for x in l), 1)
beta = compute(self.bm)
alpha = compute(self.an)
p, q = len(self.ap), len(self.bq)
if p == q:
if beta == oo or alpha == oo:
return oo
return 2*pi*ilcm(alpha, beta)
elif p < q:
return 2*pi*beta
else:
return 2*pi*alpha
def _eval_expand_func(self, **hints):
from sympy import hyperexpand
return hyperexpand(self)
def _eval_evalf(self, prec):
# The default code is insufficient for polar arguments.
# mpmath provides an optional argument "r", which evaluates
# G(z**(1/r)). I am not sure what its intended use is, but we hijack it
# here in the following way: to evaluate at a number z of |argument|
# less than (say) n*pi, we put r=1/n, compute z' = root(z, n)
# (carefully so as not to loose the branch information), and evaluate
# G(z'**(1/r)) = G(z'**n) = G(z).
from sympy.functions import exp_polar, ceiling
from sympy import Expr
import mpmath
znum = self.argument._eval_evalf(prec)
if znum.has(exp_polar):
znum, branch = znum.as_coeff_mul(exp_polar)
if len(branch) != 1:
return
branch = branch[0].args[0]/I
else:
branch = S(0)
n = ceiling(abs(branch/S.Pi)) + 1
znum = znum**(S(1)/n)*exp(I*branch / n)
# Convert all args to mpf or mpc
try:
[z, r, ap, bq] = [arg._to_mpmath(prec)
for arg in [znum, 1/n, self.args[0], self.args[1]]]
except ValueError:
return
with mpmath.workprec(prec):
v = mpmath.meijerg(ap, bq, z, r)
return Expr._from_mpmath(v, prec)
def integrand(self, s):
""" Get the defining integrand D(s). """
from sympy import gamma
return self.argument**s \
* Mul(*(gamma(b - s) for b in self.bm)) \
* Mul(*(gamma(1 - a + s) for a in self.an)) \
/ Mul(*(gamma(1 - b + s) for b in self.bother)) \
/ Mul(*(gamma(a - s) for a in self.aother))
@property
def argument(self):
""" Argument of the Meijer G-function. """
return self.args[2]
@property
def an(self):
""" First set of numerator parameters. """
return Tuple(*self.args[0][0])
@property
def ap(self):
""" Combined numerator parameters. """
return Tuple(*(self.args[0][0] + self.args[0][1]))
@property
def aother(self):
""" Second set of numerator parameters. """
return Tuple(*self.args[0][1])
@property
def bm(self):
""" First set of denominator parameters. """
return Tuple(*self.args[1][0])
@property
def bq(self):
""" Combined denominator parameters. """
return Tuple(*(self.args[1][0] + self.args[1][1]))
@property
def bother(self):
""" Second set of denominator parameters. """
return Tuple(*self.args[1][1])
@property
def _diffargs(self):
return self.ap + self.bq
@property
def nu(self):
""" A quantity related to the convergence region of the integral,
c.f. references. """
return sum(self.bq) - sum(self.ap)
@property
def delta(self):
""" A quantity related to the convergence region of the integral,
c.f. references. """
return len(self.bm) + len(self.an) - S(len(self.ap) + len(self.bq))/2
@property
def is_number(self):
""" Returns true if expression has numeric data only. """
return not self.free_symbols
class HyperRep(Function):
"""
A base class for "hyper representation functions".
This is used exclusively in hyperexpand(), but fits more logically here.
pFq is branched at 1 if p == q+1. For use with slater-expansion, we want
define an "analytic continuation" to all polar numbers, which is
continuous on circles and on the ray t*exp_polar(I*pi). Moreover, we want
a "nice" expression for the various cases.
This base class contains the core logic, concrete derived classes only
supply the actual functions.
"""
@classmethod
def eval(cls, *args):
from sympy import unpolarify
newargs = tuple(map(unpolarify, args[:-1])) + args[-1:]
if args != newargs:
return cls(*newargs)
@classmethod
def _expr_small(cls, x):
""" An expression for F(x) which holds for |x| < 1. """
raise NotImplementedError
@classmethod
def _expr_small_minus(cls, x):
""" An expression for F(-x) which holds for |x| < 1. """
raise NotImplementedError
@classmethod
def _expr_big(cls, x, n):
""" An expression for F(exp_polar(2*I*pi*n)*x), |x| > 1. """
raise NotImplementedError
@classmethod
def _expr_big_minus(cls, x, n):
""" An expression for F(exp_polar(2*I*pi*n + pi*I)*x), |x| > 1. """
raise NotImplementedError
def _eval_rewrite_as_nonrep(self, *args, **kwargs):
from sympy import Piecewise
x, n = self.args[-1].extract_branch_factor(allow_half=True)
minus = False
newargs = self.args[:-1] + (x,)
if not n.is_Integer:
minus = True
n -= S(1)/2
newerargs = newargs + (n,)
if minus:
small = self._expr_small_minus(*newargs)
big = self._expr_big_minus(*newerargs)
else:
small = self._expr_small(*newargs)
big = self._expr_big(*newerargs)
if big == small:
return small
return Piecewise((big, abs(x) > 1), (small, True))
def _eval_rewrite_as_nonrepsmall(self, *args, **kwargs):
x, n = self.args[-1].extract_branch_factor(allow_half=True)
args = self.args[:-1] + (x,)
if not n.is_Integer:
return self._expr_small_minus(*args)
return self._expr_small(*args)
class HyperRep_power1(HyperRep):
""" Return a representative for hyper([-a], [], z) == (1 - z)**a. """
@classmethod
def _expr_small(cls, a, x):
return (1 - x)**a
@classmethod
def _expr_small_minus(cls, a, x):
return (1 + x)**a
@classmethod
def _expr_big(cls, a, x, n):
if a.is_integer:
return cls._expr_small(a, x)
return (x - 1)**a*exp((2*n - 1)*pi*I*a)
@classmethod
def _expr_big_minus(cls, a, x, n):
if a.is_integer:
return cls._expr_small_minus(a, x)
return (1 + x)**a*exp(2*n*pi*I*a)
class HyperRep_power2(HyperRep):
""" Return a representative for hyper([a, a - 1/2], [2*a], z). """
@classmethod
def _expr_small(cls, a, x):
return 2**(2*a - 1)*(1 + sqrt(1 - x))**(1 - 2*a)
@classmethod
def _expr_small_minus(cls, a, x):
return 2**(2*a - 1)*(1 + sqrt(1 + x))**(1 - 2*a)
@classmethod
def _expr_big(cls, a, x, n):
sgn = -1
if n.is_odd:
sgn = 1
n -= 1
return 2**(2*a - 1)*(1 + sgn*I*sqrt(x - 1))**(1 - 2*a) \
*exp(-2*n*pi*I*a)
@classmethod
def _expr_big_minus(cls, a, x, n):
sgn = 1
if n.is_odd:
sgn = -1
return sgn*2**(2*a - 1)*(sqrt(1 + x) + sgn)**(1 - 2*a)*exp(-2*pi*I*a*n)
class HyperRep_log1(HyperRep):
""" Represent -z*hyper([1, 1], [2], z) == log(1 - z). """
@classmethod
def _expr_small(cls, x):
return log(1 - x)
@classmethod
def _expr_small_minus(cls, x):
return log(1 + x)
@classmethod
def _expr_big(cls, x, n):
return log(x - 1) + (2*n - 1)*pi*I
@classmethod
def _expr_big_minus(cls, x, n):
return log(1 + x) + 2*n*pi*I
class HyperRep_atanh(HyperRep):
""" Represent hyper([1/2, 1], [3/2], z) == atanh(sqrt(z))/sqrt(z). """
@classmethod
def _expr_small(cls, x):
return atanh(sqrt(x))/sqrt(x)
def _expr_small_minus(cls, x):
return atan(sqrt(x))/sqrt(x)
def _expr_big(cls, x, n):
if n.is_even:
return (acoth(sqrt(x)) + I*pi/2)/sqrt(x)
else:
return (acoth(sqrt(x)) - I*pi/2)/sqrt(x)
def _expr_big_minus(cls, x, n):
if n.is_even:
return atan(sqrt(x))/sqrt(x)
else:
return (atan(sqrt(x)) - pi)/sqrt(x)
class HyperRep_asin1(HyperRep):
""" Represent hyper([1/2, 1/2], [3/2], z) == asin(sqrt(z))/sqrt(z). """
@classmethod
def _expr_small(cls, z):
return asin(sqrt(z))/sqrt(z)
@classmethod
def _expr_small_minus(cls, z):
return asinh(sqrt(z))/sqrt(z)
@classmethod
def _expr_big(cls, z, n):
return S(-1)**n*((S(1)/2 - n)*pi/sqrt(z) + I*acosh(sqrt(z))/sqrt(z))
@classmethod
def _expr_big_minus(cls, z, n):
return S(-1)**n*(asinh(sqrt(z))/sqrt(z) + n*pi*I/sqrt(z))
class HyperRep_asin2(HyperRep):
""" Represent hyper([1, 1], [3/2], z) == asin(sqrt(z))/sqrt(z)/sqrt(1-z). """
# TODO this can be nicer
@classmethod
def _expr_small(cls, z):
return HyperRep_asin1._expr_small(z) \
/HyperRep_power1._expr_small(S(1)/2, z)
@classmethod
def _expr_small_minus(cls, z):
return HyperRep_asin1._expr_small_minus(z) \
/HyperRep_power1._expr_small_minus(S(1)/2, z)
@classmethod
def _expr_big(cls, z, n):
return HyperRep_asin1._expr_big(z, n) \
/HyperRep_power1._expr_big(S(1)/2, z, n)
@classmethod
def _expr_big_minus(cls, z, n):
return HyperRep_asin1._expr_big_minus(z, n) \
/HyperRep_power1._expr_big_minus(S(1)/2, z, n)
class HyperRep_sqrts1(HyperRep):
""" Return a representative for hyper([-a, 1/2 - a], [1/2], z). """
@classmethod
def _expr_small(cls, a, z):
return ((1 - sqrt(z))**(2*a) + (1 + sqrt(z))**(2*a))/2
@classmethod
def _expr_small_minus(cls, a, z):
return (1 + z)**a*cos(2*a*atan(sqrt(z)))
@classmethod
def _expr_big(cls, a, z, n):
if n.is_even:
return ((sqrt(z) + 1)**(2*a)*exp(2*pi*I*n*a) +
(sqrt(z) - 1)**(2*a)*exp(2*pi*I*(n - 1)*a))/2
else:
n -= 1
return ((sqrt(z) - 1)**(2*a)*exp(2*pi*I*a*(n + 1)) +
(sqrt(z) + 1)**(2*a)*exp(2*pi*I*a*n))/2
@classmethod
def _expr_big_minus(cls, a, z, n):
if n.is_even:
return (1 + z)**a*exp(2*pi*I*n*a)*cos(2*a*atan(sqrt(z)))
else:
return (1 + z)**a*exp(2*pi*I*n*a)*cos(2*a*atan(sqrt(z)) - 2*pi*a)
class HyperRep_sqrts2(HyperRep):
""" Return a representative for
sqrt(z)/2*[(1-sqrt(z))**2a - (1 + sqrt(z))**2a]
== -2*z/(2*a+1) d/dz hyper([-a - 1/2, -a], [1/2], z)"""
@classmethod
def _expr_small(cls, a, z):
return sqrt(z)*((1 - sqrt(z))**(2*a) - (1 + sqrt(z))**(2*a))/2
@classmethod
def _expr_small_minus(cls, a, z):
return sqrt(z)*(1 + z)**a*sin(2*a*atan(sqrt(z)))
@classmethod
def _expr_big(cls, a, z, n):
if n.is_even:
return sqrt(z)/2*((sqrt(z) - 1)**(2*a)*exp(2*pi*I*a*(n - 1)) -
(sqrt(z) + 1)**(2*a)*exp(2*pi*I*a*n))
else:
n -= 1
return sqrt(z)/2*((sqrt(z) - 1)**(2*a)*exp(2*pi*I*a*(n + 1)) -
(sqrt(z) + 1)**(2*a)*exp(2*pi*I*a*n))
def _expr_big_minus(cls, a, z, n):
if n.is_even:
return (1 + z)**a*exp(2*pi*I*n*a)*sqrt(z)*sin(2*a*atan(sqrt(z)))
else:
return (1 + z)**a*exp(2*pi*I*n*a)*sqrt(z) \
*sin(2*a*atan(sqrt(z)) - 2*pi*a)
class HyperRep_log2(HyperRep):
""" Represent log(1/2 + sqrt(1 - z)/2) == -z/4*hyper([3/2, 1, 1], [2, 2], z) """
@classmethod
def _expr_small(cls, z):
return log(S(1)/2 + sqrt(1 - z)/2)
@classmethod
def _expr_small_minus(cls, z):
return log(S(1)/2 + sqrt(1 + z)/2)
@classmethod
def _expr_big(cls, z, n):
if n.is_even:
return (n - S(1)/2)*pi*I + log(sqrt(z)/2) + I*asin(1/sqrt(z))
else:
return (n - S(1)/2)*pi*I + log(sqrt(z)/2) - I*asin(1/sqrt(z))
def _expr_big_minus(cls, z, n):
if n.is_even:
return pi*I*n + log(S(1)/2 + sqrt(1 + z)/2)
else:
return pi*I*n + log(sqrt(1 + z)/2 - S(1)/2)
class HyperRep_cosasin(HyperRep):
""" Represent hyper([a, -a], [1/2], z) == cos(2*a*asin(sqrt(z))). """
# Note there are many alternative expressions, e.g. as powers of a sum of
# square roots.
@classmethod
def _expr_small(cls, a, z):
return cos(2*a*asin(sqrt(z)))
@classmethod
def _expr_small_minus(cls, a, z):
return cosh(2*a*asinh(sqrt(z)))
@classmethod
def _expr_big(cls, a, z, n):
return cosh(2*a*acosh(sqrt(z)) + a*pi*I*(2*n - 1))
@classmethod
def _expr_big_minus(cls, a, z, n):
return cosh(2*a*asinh(sqrt(z)) + 2*a*pi*I*n)
class HyperRep_sinasin(HyperRep):
""" Represent 2*a*z*hyper([1 - a, 1 + a], [3/2], z)
== sqrt(z)/sqrt(1-z)*sin(2*a*asin(sqrt(z))) """
@classmethod
def _expr_small(cls, a, z):
return sqrt(z)/sqrt(1 - z)*sin(2*a*asin(sqrt(z)))
@classmethod
def _expr_small_minus(cls, a, z):
return -sqrt(z)/sqrt(1 + z)*sinh(2*a*asinh(sqrt(z)))
@classmethod
def _expr_big(cls, a, z, n):
return -1/sqrt(1 - 1/z)*sinh(2*a*acosh(sqrt(z)) + a*pi*I*(2*n - 1))
@classmethod
def _expr_big_minus(cls, a, z, n):
return -1/sqrt(1 + 1/z)*sinh(2*a*asinh(sqrt(z)) + 2*a*pi*I*n)
class appellf1(Function):
r"""
This is the Appell hypergeometric function of two variables as:
.. math ::
F_1(a,b_1,b_2,c,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
\frac{(a)_{m+n} (b_1)_m (b_2)_n}{(c)_{m+n}}
\frac{x^m y^n}{m! n!}.
References
==========
.. [1] https://en.wikipedia.org/wiki/Appell_series
.. [2] http://functions.wolfram.com/HypergeometricFunctions/AppellF1/
"""
@classmethod
def eval(cls, a, b1, b2, c, x, y):
if default_sort_key(b1) > default_sort_key(b2):
b1, b2 = b2, b1
x, y = y, x
return cls(a, b1, b2, c, x, y)
elif b1 == b2 and default_sort_key(x) > default_sort_key(y):
x, y = y, x
return cls(a, b1, b2, c, x, y)
if x == 0 and y == 0:
return S.One
def fdiff(self, argindex=5):
a, b1, b2, c, x, y = self.args
if argindex == 5:
return (a*b1/c)*appellf1(a + 1, b1 + 1, b2, c + 1, x, y)
elif argindex == 6:
return (a*b2/c)*appellf1(a + 1, b1, b2 + 1, c + 1, x, y)
elif argindex in (1, 2, 3, 4):
return Derivative(self, self.args[argindex-1])
else:
raise ArgumentIndexError(self, argindex)
|
46aa0b8a1733b5e32d4252e94938474bb3593e6a39ccc90bfa3f5656fe42679c
|
from __future__ import print_function, division
from sympy.core import Add, S, sympify, oo, pi, Dummy, expand_func
from sympy.core.compatibility import range
from sympy.core.function import Function, ArgumentIndexError
from sympy.core.numbers import Rational
from sympy.core.power import Pow
from .zeta_functions import zeta
from .error_functions import erf, erfc
from sympy.functions.elementary.exponential import exp, log
from sympy.functions.elementary.integers import ceiling, floor
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import sin, cos, cot
from sympy.functions.combinatorial.numbers import bernoulli, harmonic
from sympy.functions.combinatorial.factorials import factorial, rf, RisingFactorial
###############################################################################
############################ COMPLETE GAMMA FUNCTION ##########################
###############################################################################
class gamma(Function):
r"""
The gamma function
.. math::
\Gamma(x) := \int^{\infty}_{0} t^{x-1} e^{-t} \mathrm{d}t.
The ``gamma`` function implements the function which passes through the
values of the factorial function, i.e. `\Gamma(n) = (n - 1)!` when n is
an integer. More general, `\Gamma(z)` is defined in the whole complex
plane except at the negative integers where there are simple poles.
Examples
========
>>> from sympy import S, I, pi, oo, gamma
>>> from sympy.abc import x
Several special values are known:
>>> gamma(1)
1
>>> gamma(4)
6
>>> gamma(S(3)/2)
sqrt(pi)/2
The Gamma function obeys the mirror symmetry:
>>> from sympy import conjugate
>>> conjugate(gamma(x))
gamma(conjugate(x))
Differentiation with respect to x is supported:
>>> from sympy import diff
>>> diff(gamma(x), x)
gamma(x)*polygamma(0, x)
Series expansion is also supported:
>>> from sympy import series
>>> series(gamma(x), x, 0, 3)
1/x - EulerGamma + x*(EulerGamma**2/2 + pi**2/12) + x**2*(-EulerGamma*pi**2/12 + polygamma(2, 1)/6 - EulerGamma**3/6) + O(x**3)
We can numerically evaluate the gamma function to arbitrary precision
on the whole complex plane:
>>> gamma(pi).evalf(40)
2.288037795340032417959588909060233922890
>>> gamma(1+I).evalf(20)
0.49801566811835604271 - 0.15494982830181068512*I
See Also
========
lowergamma: Lower incomplete gamma function.
uppergamma: Upper incomplete gamma function.
polygamma: Polygamma function.
loggamma: Log Gamma function.
digamma: Digamma function.
trigamma: Trigamma function.
sympy.functions.special.beta_functions.beta: Euler Beta function.
References
==========
.. [1] https://en.wikipedia.org/wiki/Gamma_function
.. [2] http://dlmf.nist.gov/5
.. [3] http://mathworld.wolfram.com/GammaFunction.html
.. [4] http://functions.wolfram.com/GammaBetaErf/Gamma/
"""
unbranched = True
def fdiff(self, argindex=1):
if argindex == 1:
return self.func(self.args[0])*polygamma(0, self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg.is_Integer:
if arg.is_positive:
return factorial(arg - 1)
else:
return S.ComplexInfinity
elif arg.is_Rational:
if arg.q == 2:
n = abs(arg.p) // arg.q
if arg.is_positive:
k, coeff = n, S.One
else:
n = k = n + 1
if n & 1 == 0:
coeff = S.One
else:
coeff = S.NegativeOne
for i in range(3, 2*k, 2):
coeff *= i
if arg.is_positive:
return coeff*sqrt(S.Pi) / 2**n
else:
return 2**n*sqrt(S.Pi) / coeff
def _eval_expand_func(self, **hints):
arg = self.args[0]
if arg.is_Rational:
if abs(arg.p) > arg.q:
x = Dummy('x')
n = arg.p // arg.q
p = arg.p - n*arg.q
return self.func(x + n)._eval_expand_func().subs(x, Rational(p, arg.q))
if arg.is_Add:
coeff, tail = arg.as_coeff_add()
if coeff and coeff.q != 1:
intpart = floor(coeff)
tail = (coeff - intpart,) + tail
coeff = intpart
tail = arg._new_rawargs(*tail, reeval=False)
return self.func(tail)*RisingFactorial(tail, coeff)
return self.func(*self.args)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def _eval_is_real(self):
x = self.args[0]
if x.is_positive or x.is_noninteger:
return True
def _eval_is_positive(self):
x = self.args[0]
if x.is_positive:
return True
elif x.is_noninteger:
return floor(x).is_even
def _eval_rewrite_as_tractable(self, z, **kwargs):
return exp(loggamma(z))
def _eval_rewrite_as_factorial(self, z, **kwargs):
return factorial(z - 1)
def _eval_nseries(self, x, n, logx):
x0 = self.args[0].limit(x, 0)
if not (x0.is_Integer and x0 <= 0):
return super(gamma, self)._eval_nseries(x, n, logx)
t = self.args[0] - x0
return (self.func(t + 1)/rf(self.args[0], -x0 + 1))._eval_nseries(x, n, logx)
def _sage_(self):
import sage.all as sage
return sage.gamma(self.args[0]._sage_())
###############################################################################
################## LOWER and UPPER INCOMPLETE GAMMA FUNCTIONS #################
###############################################################################
class lowergamma(Function):
r"""
The lower incomplete gamma function.
It can be defined as the meromorphic continuation of
.. math::
\gamma(s, x) := \int_0^x t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \Gamma(s, x).
This can be shown to be the same as
.. math::
\gamma(s, x) = \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),
where :math:`{}_1F_1` is the (confluent) hypergeometric function.
Examples
========
>>> from sympy import lowergamma, S
>>> from sympy.abc import s, x
>>> lowergamma(s, x)
lowergamma(s, x)
>>> lowergamma(3, x)
-2*(x**2/2 + x + 1)*exp(-x) + 2
>>> lowergamma(-S(1)/2, x)
-2*sqrt(pi)*erf(sqrt(x)) - 2*exp(-x)/sqrt(x)
See Also
========
gamma: Gamma function.
uppergamma: Upper incomplete gamma function.
polygamma: Polygamma function.
loggamma: Log Gamma function.
digamma: Digamma function.
trigamma: Trigamma function.
sympy.functions.special.beta_functions.beta: Euler Beta function.
References
==========
.. [1] https://en.wikipedia.org/wiki/Incomplete_gamma_function#Lower_incomplete_Gamma_function
.. [2] Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5,
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
.. [3] http://dlmf.nist.gov/8
.. [4] http://functions.wolfram.com/GammaBetaErf/Gamma2/
.. [5] http://functions.wolfram.com/GammaBetaErf/Gamma3/
"""
def fdiff(self, argindex=2):
from sympy import meijerg, unpolarify
if argindex == 2:
a, z = self.args
return exp(-unpolarify(z))*z**(a - 1)
elif argindex == 1:
a, z = self.args
return gamma(a)*digamma(a) - log(z)*uppergamma(a, z) \
- meijerg([], [1, 1], [0, 0, a], [], z)
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, a, x):
# For lack of a better place, we use this one to extract branching
# information. The following can be
# found in the literature (c/f references given above), albeit scattered:
# 1) For fixed x != 0, lowergamma(s, x) is an entire function of s
# 2) For fixed positive integers s, lowergamma(s, x) is an entire
# function of x.
# 3) For fixed non-positive integers s,
# lowergamma(s, exp(I*2*pi*n)*x) =
# 2*pi*I*n*(-1)**(-s)/factorial(-s) + lowergamma(s, x)
# (this follows from lowergamma(s, x).diff(x) = x**(s-1)*exp(-x)).
# 4) For fixed non-integral s,
# lowergamma(s, x) = x**s*gamma(s)*lowergamma_unbranched(s, x),
# where lowergamma_unbranched(s, x) is an entire function (in fact
# of both s and x), i.e.
# lowergamma(s, exp(2*I*pi*n)*x) = exp(2*pi*I*n*a)*lowergamma(a, x)
from sympy import unpolarify, I
if x == 0:
return S.Zero
nx, n = x.extract_branch_factor()
if a.is_integer and a.is_positive:
nx = unpolarify(x)
if nx != x:
return lowergamma(a, nx)
elif a.is_integer and a.is_nonpositive:
if n != 0:
return 2*pi*I*n*(-1)**(-a)/factorial(-a) + lowergamma(a, nx)
elif n != 0:
return exp(2*pi*I*n*a)*lowergamma(a, nx)
# Special values.
if a.is_Number:
if a is S.One:
return S.One - exp(-x)
elif a is S.Half:
return sqrt(pi)*erf(sqrt(x))
elif a.is_Integer or (2*a).is_Integer:
b = a - 1
if b.is_positive:
if a.is_integer:
return factorial(b) - exp(-x) * factorial(b) * Add(*[x ** k / factorial(k) for k in range(a)])
else:
return gamma(a) * (lowergamma(S.Half, x)/sqrt(pi) - exp(-x) * Add(*[x**(k-S.Half) / gamma(S.Half+k) for k in range(1, a+S.Half)]))
if not a.is_Integer:
return (-1)**(S.Half - a) * pi*erf(sqrt(x)) / gamma(1-a) + exp(-x) * Add(*[x**(k+a-1)*gamma(a) / gamma(a+k) for k in range(1, S(3)/2-a)])
def _eval_evalf(self, prec):
from mpmath import mp, workprec
from sympy import Expr
if all(x.is_number for x in self.args):
a = self.args[0]._to_mpmath(prec)
z = self.args[1]._to_mpmath(prec)
with workprec(prec):
res = mp.gammainc(a, 0, z)
return Expr._from_mpmath(res, prec)
else:
return self
def _eval_conjugate(self):
z = self.args[1]
if not z in (S.Zero, S.NegativeInfinity):
return self.func(self.args[0].conjugate(), z.conjugate())
def _eval_rewrite_as_uppergamma(self, s, x, **kwargs):
return gamma(s) - uppergamma(s, x)
def _eval_rewrite_as_expint(self, s, x, **kwargs):
from sympy import expint
if s.is_integer and s.is_nonpositive:
return self
return self.rewrite(uppergamma).rewrite(expint)
class uppergamma(Function):
r"""
The upper incomplete gamma function.
It can be defined as the meromorphic continuation of
.. math::
\Gamma(s, x) := \int_x^\infty t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \gamma(s, x).
where `\gamma(s, x)` is the lower incomplete gamma function,
:class:`lowergamma`. This can be shown to be the same as
.. math::
\Gamma(s, x) = \Gamma(s) - \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),
where :math:`{}_1F_1` is the (confluent) hypergeometric function.
The upper incomplete gamma function is also essentially equivalent to the
generalized exponential integral:
.. math::
\operatorname{E}_{n}(x) = \int_{1}^{\infty}{\frac{e^{-xt}}{t^n} \, dt} = x^{n-1}\Gamma(1-n,x).
Examples
========
>>> from sympy import uppergamma, S
>>> from sympy.abc import s, x
>>> uppergamma(s, x)
uppergamma(s, x)
>>> uppergamma(3, x)
2*(x**2/2 + x + 1)*exp(-x)
>>> uppergamma(-S(1)/2, x)
-2*sqrt(pi)*erfc(sqrt(x)) + 2*exp(-x)/sqrt(x)
>>> uppergamma(-2, x)
expint(3, x)/x**2
See Also
========
gamma: Gamma function.
lowergamma: Lower incomplete gamma function.
polygamma: Polygamma function.
loggamma: Log Gamma function.
digamma: Digamma function.
trigamma: Trigamma function.
sympy.functions.special.beta_functions.beta: Euler Beta function.
References
==========
.. [1] https://en.wikipedia.org/wiki/Incomplete_gamma_function#Upper_incomplete_Gamma_function
.. [2] Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5,
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
.. [3] http://dlmf.nist.gov/8
.. [4] http://functions.wolfram.com/GammaBetaErf/Gamma2/
.. [5] http://functions.wolfram.com/GammaBetaErf/Gamma3/
.. [6] https://en.wikipedia.org/wiki/Exponential_integral#Relation_with_other_functions
"""
def fdiff(self, argindex=2):
from sympy import meijerg, unpolarify
if argindex == 2:
a, z = self.args
return -exp(-unpolarify(z))*z**(a - 1)
elif argindex == 1:
a, z = self.args
return uppergamma(a, z)*log(z) + meijerg([], [1, 1], [0, 0, a], [], z)
else:
raise ArgumentIndexError(self, argindex)
def _eval_evalf(self, prec):
from mpmath import mp, workprec
from sympy import Expr
if all(x.is_number for x in self.args):
a = self.args[0]._to_mpmath(prec)
z = self.args[1]._to_mpmath(prec)
with workprec(prec):
res = mp.gammainc(a, z, mp.inf)
return Expr._from_mpmath(res, prec)
return self
@classmethod
def eval(cls, a, z):
from sympy import unpolarify, I, expint
if z.is_Number:
if z is S.NaN:
return S.NaN
elif z is S.Infinity:
return S.Zero
elif z is S.Zero:
# TODO: Holds only for Re(a) > 0:
return gamma(a)
# We extract branching information here. C/f lowergamma.
nx, n = z.extract_branch_factor()
if a.is_integer and (a > 0) == True:
nx = unpolarify(z)
if z != nx:
return uppergamma(a, nx)
elif a.is_integer and (a <= 0) == True:
if n != 0:
return -2*pi*I*n*(-1)**(-a)/factorial(-a) + uppergamma(a, nx)
elif n != 0:
return gamma(a)*(1 - exp(2*pi*I*n*a)) + exp(2*pi*I*n*a)*uppergamma(a, nx)
# Special values.
if a.is_Number:
if a is S.One:
return exp(-z)
elif a is S.Half:
return sqrt(pi)*erfc(sqrt(z))
elif a.is_Integer or (2*a).is_Integer:
b = a - 1
if b.is_positive:
if a.is_integer:
return exp(-z) * factorial(b) * Add(*[z**k / factorial(k) for k in range(a)])
else:
return gamma(a) * erfc(sqrt(z)) + (-1)**(a - S(3)/2) * exp(-z) * sqrt(z) * Add(*[gamma(-S.Half - k) * (-z)**k / gamma(1-a) for k in range(a - S.Half)])
elif b.is_Integer:
return expint(-b, z)*unpolarify(z)**(b + 1)
if not a.is_Integer:
return (-1)**(S.Half - a) * pi*erfc(sqrt(z))/gamma(1-a) - z**a * exp(-z) * Add(*[z**k * gamma(a) / gamma(a+k+1) for k in range(S.Half - a)])
def _eval_conjugate(self):
z = self.args[1]
if not z in (S.Zero, S.NegativeInfinity):
return self.func(self.args[0].conjugate(), z.conjugate())
def _eval_rewrite_as_lowergamma(self, s, x, **kwargs):
return gamma(s) - lowergamma(s, x)
def _eval_rewrite_as_expint(self, s, x, **kwargs):
from sympy import expint
return expint(1 - s, x)*x**s
def _sage_(self):
import sage.all as sage
return sage.gamma(self.args[0]._sage_(), self.args[1]._sage_())
###############################################################################
###################### POLYGAMMA and LOGGAMMA FUNCTIONS #######################
###############################################################################
class polygamma(Function):
r"""
The function ``polygamma(n, z)`` returns ``log(gamma(z)).diff(n + 1)``.
It is a meromorphic function on `\mathbb{C}` and defined as the (n+1)-th
derivative of the logarithm of the gamma function:
.. math::
\psi^{(n)} (z) := \frac{\mathrm{d}^{n+1}}{\mathrm{d} z^{n+1}} \log\Gamma(z).
Examples
========
Several special values are known:
>>> from sympy import S, polygamma
>>> polygamma(0, 1)
-EulerGamma
>>> polygamma(0, 1/S(2))
-2*log(2) - EulerGamma
>>> polygamma(0, 1/S(3))
-log(3) - sqrt(3)*pi/6 - EulerGamma - log(sqrt(3))
>>> polygamma(0, 1/S(4))
-pi/2 - log(4) - log(2) - EulerGamma
>>> polygamma(0, 2)
1 - EulerGamma
>>> polygamma(0, 23)
19093197/5173168 - EulerGamma
>>> from sympy import oo, I
>>> polygamma(0, oo)
oo
>>> polygamma(0, -oo)
oo
>>> polygamma(0, I*oo)
oo
>>> polygamma(0, -I*oo)
oo
Differentiation with respect to x is supported:
>>> from sympy import Symbol, diff
>>> x = Symbol("x")
>>> diff(polygamma(0, x), x)
polygamma(1, x)
>>> diff(polygamma(0, x), x, 2)
polygamma(2, x)
>>> diff(polygamma(0, x), x, 3)
polygamma(3, x)
>>> diff(polygamma(1, x), x)
polygamma(2, x)
>>> diff(polygamma(1, x), x, 2)
polygamma(3, x)
>>> diff(polygamma(2, x), x)
polygamma(3, x)
>>> diff(polygamma(2, x), x, 2)
polygamma(4, x)
>>> n = Symbol("n")
>>> diff(polygamma(n, x), x)
polygamma(n + 1, x)
>>> diff(polygamma(n, x), x, 2)
polygamma(n + 2, x)
We can rewrite polygamma functions in terms of harmonic numbers:
>>> from sympy import harmonic
>>> polygamma(0, x).rewrite(harmonic)
harmonic(x - 1) - EulerGamma
>>> polygamma(2, x).rewrite(harmonic)
2*harmonic(x - 1, 3) - 2*zeta(3)
>>> ni = Symbol("n", integer=True)
>>> polygamma(ni, x).rewrite(harmonic)
(-1)**(n + 1)*(-harmonic(x - 1, n + 1) + zeta(n + 1))*factorial(n)
See Also
========
gamma: Gamma function.
lowergamma: Lower incomplete gamma function.
uppergamma: Upper incomplete gamma function.
loggamma: Log Gamma function.
digamma: Digamma function.
trigamma: Trigamma function.
sympy.functions.special.beta_functions.beta: Euler Beta function.
References
==========
.. [1] https://en.wikipedia.org/wiki/Polygamma_function
.. [2] http://mathworld.wolfram.com/PolygammaFunction.html
.. [3] http://functions.wolfram.com/GammaBetaErf/PolyGamma/
.. [4] http://functions.wolfram.com/GammaBetaErf/PolyGamma2/
"""
def fdiff(self, argindex=2):
if argindex == 2:
n, z = self.args[:2]
return polygamma(n + 1, z)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_positive(self):
if self.args[1].is_positive and (self.args[0] > 0) == True:
return self.args[0].is_odd
def _eval_is_negative(self):
if self.args[1].is_positive and (self.args[0] > 0) == True:
return self.args[0].is_even
def _eval_is_real(self):
return self.args[0].is_real
def _eval_aseries(self, n, args0, x, logx):
from sympy import Order
if args0[1] != oo or not \
(self.args[0].is_Integer and self.args[0].is_nonnegative):
return super(polygamma, self)._eval_aseries(n, args0, x, logx)
z = self.args[1]
N = self.args[0]
if N == 0:
# digamma function series
# Abramowitz & Stegun, p. 259, 6.3.18
r = log(z) - 1/(2*z)
o = None
if n < 2:
o = Order(1/z, x)
else:
m = ceiling((n + 1)//2)
l = [bernoulli(2*k) / (2*k*z**(2*k)) for k in range(1, m)]
r -= Add(*l)
o = Order(1/z**(2*m), x)
return r._eval_nseries(x, n, logx) + o
else:
# proper polygamma function
# Abramowitz & Stegun, p. 260, 6.4.10
# We return terms to order higher than O(x**n) on purpose
# -- otherwise we would not be able to return any terms for
# quite a long time!
fac = gamma(N)
e0 = fac + N*fac/(2*z)
m = ceiling((n + 1)//2)
for k in range(1, m):
fac = fac*(2*k + N - 1)*(2*k + N - 2) / ((2*k)*(2*k - 1))
e0 += bernoulli(2*k)*fac/z**(2*k)
o = Order(1/z**(2*m), x)
if n == 0:
o = Order(1/z, x)
elif n == 1:
o = Order(1/z**2, x)
r = e0._eval_nseries(z, n, logx) + o
return (-1 * (-1/z)**N * r)._eval_nseries(x, n, logx)
@classmethod
def eval(cls, n, z):
n, z = list(map(sympify, (n, z)))
from sympy import unpolarify
if n.is_integer:
if n.is_nonnegative:
nz = unpolarify(z)
if z != nz:
return polygamma(n, nz)
if n == -1:
return loggamma(z)
else:
if z.is_Number:
if z is S.NaN:
return S.NaN
elif z is S.Infinity:
if n.is_Number:
if n is S.Zero:
return S.Infinity
else:
return S.Zero
elif z.is_Integer:
if z.is_nonpositive:
return S.ComplexInfinity
else:
if n is S.Zero:
return -S.EulerGamma + harmonic(z - 1, 1)
elif n.is_odd:
return (-1)**(n + 1)*factorial(n)*zeta(n + 1, z)
if n == 0:
if z is S.NaN:
return S.NaN
elif z.is_Rational:
p, q = z.as_numer_denom()
# only expand for small denominators to avoid creating long expressions
if q <= 5:
return expand_func(polygamma(n, z, evaluate=False))
elif z in (S.Infinity, S.NegativeInfinity):
return S.Infinity
else:
t = z.extract_multiplicatively(S.ImaginaryUnit)
if t in (S.Infinity, S.NegativeInfinity):
return S.Infinity
# TODO n == 1 also can do some rational z
def _eval_expand_func(self, **hints):
n, z = self.args
if n.is_Integer and n.is_nonnegative:
if z.is_Add:
coeff = z.args[0]
if coeff.is_Integer:
e = -(n + 1)
if coeff > 0:
tail = Add(*[Pow(
z - i, e) for i in range(1, int(coeff) + 1)])
else:
tail = -Add(*[Pow(
z + i, e) for i in range(0, int(-coeff))])
return polygamma(n, z - coeff) + (-1)**n*factorial(n)*tail
elif z.is_Mul:
coeff, z = z.as_two_terms()
if coeff.is_Integer and coeff.is_positive:
tail = [ polygamma(n, z + Rational(
i, coeff)) for i in range(0, int(coeff)) ]
if n == 0:
return Add(*tail)/coeff + log(coeff)
else:
return Add(*tail)/coeff**(n + 1)
z *= coeff
if n == 0 and z.is_Rational:
p, q = z.as_numer_denom()
# Reference:
# Values of the polygamma functions at rational arguments, J. Choi, 2007
part_1 = -S.EulerGamma - pi * cot(p * pi / q) / 2 - log(q) + Add(
*[cos(2 * k * pi * p / q) * log(2 * sin(k * pi / q)) for k in range(1, q)])
if z > 0:
n = floor(z)
z0 = z - n
return part_1 + Add(*[1 / (z0 + k) for k in range(n)])
elif z < 0:
n = floor(1 - z)
z0 = z + n
return part_1 - Add(*[1 / (z0 - 1 - k) for k in range(n)])
return polygamma(n, z)
def _eval_rewrite_as_zeta(self, n, z, **kwargs):
if n >= S.One:
return (-1)**(n + 1)*factorial(n)*zeta(n + 1, z)
else:
return self
def _eval_rewrite_as_harmonic(self, n, z, **kwargs):
if n.is_integer:
if n == S.Zero:
return harmonic(z - 1) - S.EulerGamma
else:
return S.NegativeOne**(n+1) * factorial(n) * (zeta(n+1) - harmonic(z-1, n+1))
def _eval_as_leading_term(self, x):
from sympy import Order
n, z = [a.as_leading_term(x) for a in self.args]
o = Order(z, x)
if n == 0 and o.contains(1/x):
return o.getn() * log(x)
else:
return self.func(n, z)
class loggamma(Function):
r"""
The ``loggamma`` function implements the logarithm of the
gamma function i.e, `\log\Gamma(x)`.
Examples
========
Several special values are known. For numerical integral
arguments we have:
>>> from sympy import loggamma
>>> loggamma(-2)
oo
>>> loggamma(0)
oo
>>> loggamma(1)
0
>>> loggamma(2)
0
>>> loggamma(3)
log(2)
and for symbolic values:
>>> from sympy import Symbol
>>> n = Symbol("n", integer=True, positive=True)
>>> loggamma(n)
log(gamma(n))
>>> loggamma(-n)
oo
for half-integral values:
>>> from sympy import S, pi
>>> loggamma(S(5)/2)
log(3*sqrt(pi)/4)
>>> loggamma(n/2)
log(2**(1 - n)*sqrt(pi)*gamma(n)/gamma(n/2 + 1/2))
and general rational arguments:
>>> from sympy import expand_func
>>> L = loggamma(S(16)/3)
>>> expand_func(L).doit()
-5*log(3) + loggamma(1/3) + log(4) + log(7) + log(10) + log(13)
>>> L = loggamma(S(19)/4)
>>> expand_func(L).doit()
-4*log(4) + loggamma(3/4) + log(3) + log(7) + log(11) + log(15)
>>> L = loggamma(S(23)/7)
>>> expand_func(L).doit()
-3*log(7) + log(2) + loggamma(2/7) + log(9) + log(16)
The loggamma function has the following limits towards infinity:
>>> from sympy import oo
>>> loggamma(oo)
oo
>>> loggamma(-oo)
zoo
The loggamma function obeys the mirror symmetry
if `x \in \mathbb{C} \setminus \{-\infty, 0\}`:
>>> from sympy.abc import x
>>> from sympy import conjugate
>>> conjugate(loggamma(x))
loggamma(conjugate(x))
Differentiation with respect to x is supported:
>>> from sympy import diff
>>> diff(loggamma(x), x)
polygamma(0, x)
Series expansion is also supported:
>>> from sympy import series
>>> series(loggamma(x), x, 0, 4)
-log(x) - EulerGamma*x + pi**2*x**2/12 + x**3*polygamma(2, 1)/6 + O(x**4)
We can numerically evaluate the gamma function to arbitrary precision
on the whole complex plane:
>>> from sympy import I
>>> loggamma(5).evalf(30)
3.17805383034794561964694160130
>>> loggamma(I).evalf(20)
-0.65092319930185633889 - 1.8724366472624298171*I
See Also
========
gamma: Gamma function.
lowergamma: Lower incomplete gamma function.
uppergamma: Upper incomplete gamma function.
polygamma: Polygamma function.
digamma: Digamma function.
trigamma: Trigamma function.
sympy.functions.special.beta_functions.beta: Euler Beta function.
References
==========
.. [1] https://en.wikipedia.org/wiki/Gamma_function
.. [2] http://dlmf.nist.gov/5
.. [3] http://mathworld.wolfram.com/LogGammaFunction.html
.. [4] http://functions.wolfram.com/GammaBetaErf/LogGamma/
"""
@classmethod
def eval(cls, z):
z = sympify(z)
if z.is_integer:
if z.is_nonpositive:
return S.Infinity
elif z.is_positive:
return log(gamma(z))
elif z.is_rational:
p, q = z.as_numer_denom()
# Half-integral values:
if p.is_positive and q == 2:
return log(sqrt(S.Pi) * 2**(1 - p) * gamma(p) / gamma((p + 1)*S.Half))
if z is S.Infinity:
return S.Infinity
elif abs(z) is S.Infinity:
return S.ComplexInfinity
if z is S.NaN:
return S.NaN
def _eval_expand_func(self, **hints):
from sympy import Sum
z = self.args[0]
if z.is_Rational:
p, q = z.as_numer_denom()
# General rational arguments (u + p/q)
# Split z as n + p/q with p < q
n = p // q
p = p - n*q
if p.is_positive and q.is_positive and p < q:
k = Dummy("k")
if n.is_positive:
return loggamma(p / q) - n*log(q) + Sum(log((k - 1)*q + p), (k, 1, n))
elif n.is_negative:
return loggamma(p / q) - n*log(q) + S.Pi*S.ImaginaryUnit*n - Sum(log(k*q - p), (k, 1, -n))
elif n.is_zero:
return loggamma(p / q)
return self
def _eval_nseries(self, x, n, logx=None):
x0 = self.args[0].limit(x, 0)
if x0 is S.Zero:
f = self._eval_rewrite_as_intractable(*self.args)
return f._eval_nseries(x, n, logx)
return super(loggamma, self)._eval_nseries(x, n, logx)
def _eval_aseries(self, n, args0, x, logx):
from sympy import Order
if args0[0] != oo:
return super(loggamma, self)._eval_aseries(n, args0, x, logx)
z = self.args[0]
m = min(n, ceiling((n + S(1))/2))
r = log(z)*(z - S(1)/2) - z + log(2*pi)/2
l = [bernoulli(2*k) / (2*k*(2*k - 1)*z**(2*k - 1)) for k in range(1, m)]
o = None
if m == 0:
o = Order(1, x)
else:
o = Order(1/z**(2*m - 1), x)
# It is very inefficient to first add the order and then do the nseries
return (r + Add(*l))._eval_nseries(x, n, logx) + o
def _eval_rewrite_as_intractable(self, z, **kwargs):
return log(gamma(z))
def _eval_is_real(self):
return self.args[0].is_real
def _eval_conjugate(self):
z = self.args[0]
if not z in (S.Zero, S.NegativeInfinity):
return self.func(z.conjugate())
def fdiff(self, argindex=1):
if argindex == 1:
return polygamma(0, self.args[0])
else:
raise ArgumentIndexError(self, argindex)
def _sage_(self):
import sage.all as sage
return sage.log_gamma(self.args[0]._sage_())
def digamma(x):
r"""
The digamma function is the first derivative of the loggamma function i.e,
.. math::
\psi(x) := \frac{\mathrm{d}}{\mathrm{d} z} \log\Gamma(z)
= \frac{\Gamma'(z)}{\Gamma(z) }
In this case, ``digamma(z) = polygamma(0, z)``.
See Also
========
gamma: Gamma function.
lowergamma: Lower incomplete gamma function.
uppergamma: Upper incomplete gamma function.
polygamma: Polygamma function.
loggamma: Log Gamma function.
trigamma: Trigamma function.
sympy.functions.special.beta_functions.beta: Euler Beta function.
References
==========
.. [1] https://en.wikipedia.org/wiki/Digamma_function
.. [2] http://mathworld.wolfram.com/DigammaFunction.html
.. [3] http://functions.wolfram.com/GammaBetaErf/PolyGamma2/
"""
return polygamma(0, x)
def trigamma(x):
r"""
The trigamma function is the second derivative of the loggamma function i.e,
.. math::
\psi^{(1)}(z) := \frac{\mathrm{d}^{2}}{\mathrm{d} z^{2}} \log\Gamma(z).
In this case, ``trigamma(z) = polygamma(1, z)``.
See Also
========
gamma: Gamma function.
lowergamma: Lower incomplete gamma function.
uppergamma: Upper incomplete gamma function.
polygamma: Polygamma function.
loggamma: Log Gamma function.
digamma: Digamma function.
sympy.functions.special.beta_functions.beta: Euler Beta function.
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigamma_function
.. [2] http://mathworld.wolfram.com/TrigammaFunction.html
.. [3] http://functions.wolfram.com/GammaBetaErf/PolyGamma2/
"""
return polygamma(1, x)
|
0270258d69e5b935182a9889754cd4dcf915cf5330398e597b15c3a812e6a674
|
from __future__ import print_function, division
from sympy.core import S, sympify
from sympy.core.compatibility import range
from sympy.functions import Piecewise, piecewise_fold
from sympy.sets.sets import Interval
from sympy.core.cache import lru_cache
def _add_splines(c, b1, d, b2):
"""Construct c*b1 + d*b2."""
if b1 == S.Zero or c == S.Zero:
rv = piecewise_fold(d * b2)
elif b2 == S.Zero or d == S.Zero:
rv = piecewise_fold(c * b1)
else:
new_args = []
# Just combining the Piecewise without any fancy optimization
p1 = piecewise_fold(c * b1)
p2 = piecewise_fold(d * b2)
# Search all Piecewise arguments except (0, True)
p2args = list(p2.args[:-1])
# This merging algorithm assumes the conditions in
# p1 and p2 are sorted
for arg in p1.args[:-1]:
# Conditional of Piecewise are And objects
# the args of the And object is a tuple of two
# Relational objects the numerical value is in the .rhs
# of the Relational object
expr = arg.expr
cond = arg.cond
lower = cond.args[0].rhs
# Check p2 for matching conditions that can be merged
for i, arg2 in enumerate(p2args):
expr2 = arg2.expr
cond2 = arg2.cond
lower_2 = cond2.args[0].rhs
upper_2 = cond2.args[1].rhs
if cond2 == cond:
# Conditions match, join expressions
expr += expr2
# Remove matching element
del p2args[i]
# No need to check the rest
break
elif lower_2 < lower and upper_2 <= lower:
# Check if arg2 condition smaller than arg1,
# add to new_args by itself (no match expected
# in p1)
new_args.append(arg2)
del p2args[i]
break
# Checked all, add expr and cond
new_args.append((expr, cond))
# Add remaining items from p2args
new_args.extend(p2args)
# Add final (0, True)
new_args.append((0, True))
rv = Piecewise(*new_args)
return rv.expand()
@lru_cache(maxsize=128)
def bspline_basis(d, knots, n, x):
"""The `n`-th B-spline at `x` of degree `d` with knots.
B-Splines are piecewise polynomials of degree `d` [1]_. They are
defined on a set of knots, which is a sequence of integers or
floats.
The 0th degree splines have a value of one on a single interval:
>>> from sympy import bspline_basis
>>> from sympy.abc import x
>>> d = 0
>>> knots = tuple(range(5))
>>> bspline_basis(d, knots, 0, x)
Piecewise((1, (x >= 0) & (x <= 1)), (0, True))
For a given ``(d, knots)`` there are ``len(knots)-d-1`` B-splines
defined, that are indexed by ``n`` (starting at 0).
Here is an example of a cubic B-spline:
>>> bspline_basis(3, tuple(range(5)), 0, x)
Piecewise((x**3/6, (x >= 0) & (x <= 1)),
(-x**3/2 + 2*x**2 - 2*x + 2/3,
(x >= 1) & (x <= 2)),
(x**3/2 - 4*x**2 + 10*x - 22/3,
(x >= 2) & (x <= 3)),
(-x**3/6 + 2*x**2 - 8*x + 32/3,
(x >= 3) & (x <= 4)),
(0, True))
By repeating knot points, you can introduce discontinuities in the
B-splines and their derivatives:
>>> d = 1
>>> knots = (0, 0, 2, 3, 4)
>>> bspline_basis(d, knots, 0, x)
Piecewise((1 - x/2, (x >= 0) & (x <= 2)), (0, True))
It is quite time consuming to construct and evaluate B-splines. If
you need to evaluate a B-splines many times, it is best to
lambdify them first:
>>> from sympy import lambdify
>>> d = 3
>>> knots = tuple(range(10))
>>> b0 = bspline_basis(d, knots, 0, x)
>>> f = lambdify(x, b0)
>>> y = f(0.5)
See Also
========
bsplines_basis_set
References
==========
.. [1] https://en.wikipedia.org/wiki/B-spline
"""
knots = tuple(sympify(k) for k in knots)
d = int(d)
n = int(n)
n_knots = len(knots)
n_intervals = n_knots - 1
if n + d + 1 > n_intervals:
raise ValueError("n + d + 1 must not exceed len(knots) - 1")
if d == 0:
result = Piecewise(
(S.One, Interval(knots[n], knots[n + 1]).contains(x)), (0, True)
)
elif d > 0:
denom = knots[n + d + 1] - knots[n + 1]
if denom != S.Zero:
B = (knots[n + d + 1] - x) / denom
b2 = bspline_basis(d - 1, knots, n + 1, x)
else:
b2 = B = S.Zero
denom = knots[n + d] - knots[n]
if denom != S.Zero:
A = (x - knots[n]) / denom
b1 = bspline_basis(d - 1, knots, n, x)
else:
b1 = A = S.Zero
result = _add_splines(A, b1, B, b2)
else:
raise ValueError("degree must be non-negative: %r" % n)
return result
def bspline_basis_set(d, knots, x):
"""Return the ``len(knots)-d-1`` B-splines at ``x`` of degree ``d``
with ``knots``.
This function returns a list of Piecewise polynomials that are the
``len(knots)-d-1`` B-splines of degree ``d`` for the given knots.
This function calls ``bspline_basis(d, knots, n, x)`` for different
values of ``n``.
Examples
========
>>> from sympy import bspline_basis_set
>>> from sympy.abc import x
>>> d = 2
>>> knots = range(5)
>>> splines = bspline_basis_set(d, knots, x)
>>> splines
[Piecewise((x**2/2, (x >= 0) & (x <= 1)),
(-x**2 + 3*x - 3/2, (x >= 1) & (x <= 2)),
(x**2/2 - 3*x + 9/2, (x >= 2) & (x <= 3)),
(0, True)),
Piecewise((x**2/2 - x + 1/2, (x >= 1) & (x <= 2)),
(-x**2 + 5*x - 11/2, (x >= 2) & (x <= 3)),
(x**2/2 - 4*x + 8, (x >= 3) & (x <= 4)),
(0, True))]
See Also
========
bsplines_basis
"""
n_splines = len(knots) - d - 1
return [bspline_basis(d, tuple(knots), i, x) for i in range(n_splines)]
def interpolating_spline(d, x, X, Y):
"""Return spline of degree ``d``, passing through the given ``X``
and ``Y`` values.
This function returns a piecewise function such that each part is
a polynomial of degree not greater than ``d``. The value of ``d``
must be 1 or greater and the values of ``X`` must be strictly
increasing.
Examples
========
>>> from sympy import interpolating_spline
>>> from sympy.abc import x
>>> interpolating_spline(1, x, [1, 2, 4, 7], [3, 6, 5, 7])
Piecewise((3*x, (x >= 1) & (x <= 2)),
(7 - x/2, (x >= 2) & (x <= 4)),
(2*x/3 + 7/3, (x >= 4) & (x <= 7)))
>>> interpolating_spline(3, x, [-2, 0, 1, 3, 4], [4, 2, 1, 1, 3])
Piecewise((7*x**3/117 + 7*x**2/117 - 131*x/117 + 2, (x >= -2) & (x <= 1)),
(10*x**3/117 - 2*x**2/117 - 122*x/117 + 77/39, (x >= 1) & (x <= 4)))
See Also
========
bsplines_basis_set, sympy.polys.specialpolys.interpolating_poly
"""
from sympy import symbols, Number, Dummy, Rational
from sympy.solvers.solveset import linsolve
from sympy.matrices.dense import Matrix
# Input sanitization
d = sympify(d)
if not (d.is_Integer and d.is_positive):
raise ValueError("Spline degree must be a positive integer, not %s." % d)
if len(X) != len(Y):
raise ValueError("Number of X and Y coordinates must be the same.")
if len(X) < d + 1:
raise ValueError("Degree must be less than the number of control points.")
if not all(a < b for a, b in zip(X, X[1:])):
raise ValueError("The x-coordinates must be strictly increasing.")
# Evaluating knots value
if d.is_odd:
j = (d + 1) // 2
interior_knots = X[j:-j]
else:
j = d // 2
interior_knots = [
Rational(a + b, 2) for a, b in zip(X[j : -j - 1], X[j + 1 : -j])
]
knots = [X[0]] * (d + 1) + list(interior_knots) + [X[-1]] * (d + 1)
basis = bspline_basis_set(d, knots, x)
A = [[b.subs(x, v) for b in basis] for v in X]
coeff = linsolve((Matrix(A), Matrix(Y)), symbols("c0:{}".format(len(X)), cls=Dummy))
coeff = list(coeff)[0]
intervals = set([c for b in basis for (e, c) in b.args if c != True])
# Sorting the intervals
# ival contains the end-points of each interval
ival = [e.atoms(Number) for e in intervals]
ival = [list(sorted(e))[0] for e in ival]
com = zip(ival, intervals)
com = sorted(com, key=lambda x: x[0])
intervals = [y for x, y in com]
basis_dicts = [dict((c, e) for (e, c) in b.args) for b in basis]
spline = []
for i in intervals:
piece = sum(
[c * d.get(i, S.Zero) for (c, d) in zip(coeff, basis_dicts)], S.Zero
)
spline.append((piece, i))
return Piecewise(*spline)
|
6b61864451605e2be24e4756223a4b3967c2c65fb4a39d92499e7141b95f3bfb
|
""" Riemann zeta and related function. """
from __future__ import print_function, division
from sympy.core import Function, S, sympify, pi, I
from sympy.core.compatibility import range
from sympy.core.function import ArgumentIndexError
from sympy.functions.combinatorial.numbers import bernoulli, factorial, harmonic
from sympy.functions.elementary.exponential import log, exp_polar
from sympy.functions.elementary.miscellaneous import sqrt
###############################################################################
###################### LERCH TRANSCENDENT #####################################
###############################################################################
class lerchphi(Function):
r"""
Lerch transcendent (Lerch phi function).
For :math:`\operatorname{Re}(a) > 0`, `|z| < 1` and `s \in \mathbb{C}`, the
Lerch transcendent is defined as
.. math :: \Phi(z, s, a) = \sum_{n=0}^\infty \frac{z^n}{(n + a)^s},
where the standard branch of the argument is used for :math:`n + a`,
and by analytic continuation for other values of the parameters.
A commonly used related function is the Lerch zeta function, defined by
.. math:: L(q, s, a) = \Phi(e^{2\pi i q}, s, a).
**Analytic Continuation and Branching Behavior**
It can be shown that
.. math:: \Phi(z, s, a) = z\Phi(z, s, a+1) + a^{-s}.
This provides the analytic continuation to `\operatorname{Re}(a) \le 0`.
Assume now `\operatorname{Re}(a) > 0`. The integral representation
.. math:: \Phi_0(z, s, a) = \int_0^\infty \frac{t^{s-1} e^{-at}}{1 - ze^{-t}}
\frac{\mathrm{d}t}{\Gamma(s)}
provides an analytic continuation to :math:`\mathbb{C} - [1, \infty)`.
Finally, for :math:`x \in (1, \infty)` we find
.. math:: \lim_{\epsilon \to 0^+} \Phi_0(x + i\epsilon, s, a)
-\lim_{\epsilon \to 0^+} \Phi_0(x - i\epsilon, s, a)
= \frac{2\pi i \log^{s-1}{x}}{x^a \Gamma(s)},
using the standard branch for both :math:`\log{x}` and
:math:`\log{\log{x}}` (a branch of :math:`\log{\log{x}}` is needed to
evaluate :math:`\log{x}^{s-1}`).
This concludes the analytic continuation. The Lerch transcendent is thus
branched at :math:`z \in \{0, 1, \infty\}` and
:math:`a \in \mathbb{Z}_{\le 0}`. For fixed :math:`z, a` outside these
branch points, it is an entire function of :math:`s`.
See Also
========
polylog, zeta
References
==========
.. [1] Bateman, H.; Erdelyi, A. (1953), Higher Transcendental Functions,
Vol. I, New York: McGraw-Hill. Section 1.11.
.. [2] http://dlmf.nist.gov/25.14
.. [3] https://en.wikipedia.org/wiki/Lerch_transcendent
Examples
========
The Lerch transcendent is a fairly general function, for this reason it does
not automatically evaluate to simpler functions. Use expand_func() to
achieve this.
If :math:`z=1`, the Lerch transcendent reduces to the Hurwitz zeta function:
>>> from sympy import lerchphi, expand_func
>>> from sympy.abc import z, s, a
>>> expand_func(lerchphi(1, s, a))
zeta(s, a)
More generally, if :math:`z` is a root of unity, the Lerch transcendent
reduces to a sum of Hurwitz zeta functions:
>>> expand_func(lerchphi(-1, s, a))
2**(-s)*zeta(s, a/2) - 2**(-s)*zeta(s, a/2 + 1/2)
If :math:`a=1`, the Lerch transcendent reduces to the polylogarithm:
>>> expand_func(lerchphi(z, s, 1))
polylog(s, z)/z
More generally, if :math:`a` is rational, the Lerch transcendent reduces
to a sum of polylogarithms:
>>> from sympy import S
>>> expand_func(lerchphi(z, s, S(1)/2))
2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -
polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))
>>> expand_func(lerchphi(z, s, S(3)/2))
-2**s/z + 2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -
polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))/z
The derivatives with respect to :math:`z` and :math:`a` can be computed in
closed form:
>>> lerchphi(z, s, a).diff(z)
(-a*lerchphi(z, s, a) + lerchphi(z, s - 1, a))/z
>>> lerchphi(z, s, a).diff(a)
-s*lerchphi(z, s + 1, a)
"""
def _eval_expand_func(self, **hints):
from sympy import exp, I, floor, Add, Poly, Dummy, exp_polar, unpolarify
z, s, a = self.args
if z == 1:
return zeta(s, a)
if s.is_Integer and s <= 0:
t = Dummy('t')
p = Poly((t + a)**(-s), t)
start = 1/(1 - t)
res = S(0)
for c in reversed(p.all_coeffs()):
res += c*start
start = t*start.diff(t)
return res.subs(t, z)
if a.is_Rational:
# See section 18 of
# Kelly B. Roach. Hypergeometric Function Representations.
# In: Proceedings of the 1997 International Symposium on Symbolic and
# Algebraic Computation, pages 205-211, New York, 1997. ACM.
# TODO should something be polarified here?
add = S(0)
mul = S(1)
# First reduce a to the interaval (0, 1]
if a > 1:
n = floor(a)
if n == a:
n -= 1
a -= n
mul = z**(-n)
add = Add(*[-z**(k - n)/(a + k)**s for k in range(n)])
elif a <= 0:
n = floor(-a) + 1
a += n
mul = z**n
add = Add(*[z**(n - 1 - k)/(a - k - 1)**s for k in range(n)])
m, n = S([a.p, a.q])
zet = exp_polar(2*pi*I/n)
root = z**(1/n)
return add + mul*n**(s - 1)*Add(
*[polylog(s, zet**k*root)._eval_expand_func(**hints)
/ (unpolarify(zet)**k*root)**m for k in range(n)])
# TODO use minpoly instead of ad-hoc methods when issue 5888 is fixed
if isinstance(z, exp) and (z.args[0]/(pi*I)).is_Rational or z in [-1, I, -I]:
# TODO reference?
if z == -1:
p, q = S([1, 2])
elif z == I:
p, q = S([1, 4])
elif z == -I:
p, q = S([-1, 4])
else:
arg = z.args[0]/(2*pi*I)
p, q = S([arg.p, arg.q])
return Add(*[exp(2*pi*I*k*p/q)/q**s*zeta(s, (k + a)/q)
for k in range(q)])
return lerchphi(z, s, a)
def fdiff(self, argindex=1):
z, s, a = self.args
if argindex == 3:
return -s*lerchphi(z, s + 1, a)
elif argindex == 1:
return (lerchphi(z, s - 1, a) - a*lerchphi(z, s, a))/z
else:
raise ArgumentIndexError
def _eval_rewrite_helper(self, z, s, a, target):
res = self._eval_expand_func()
if res.has(target):
return res
else:
return self
def _eval_rewrite_as_zeta(self, z, s, a, **kwargs):
return self._eval_rewrite_helper(z, s, a, zeta)
def _eval_rewrite_as_polylog(self, z, s, a, **kwargs):
return self._eval_rewrite_helper(z, s, a, polylog)
###############################################################################
###################### POLYLOGARITHM ##########################################
###############################################################################
class polylog(Function):
r"""
Polylogarithm function.
For :math:`|z| < 1` and :math:`s \in \mathbb{C}`, the polylogarithm is
defined by
.. math:: \operatorname{Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s},
where the standard branch of the argument is used for :math:`n`. It admits
an analytic continuation which is branched at :math:`z=1` (notably not on the
sheet of initial definition), :math:`z=0` and :math:`z=\infty`.
The name polylogarithm comes from the fact that for :math:`s=1`, the
polylogarithm is related to the ordinary logarithm (see examples), and that
.. math:: \operatorname{Li}_{s+1}(z) =
\int_0^z \frac{\operatorname{Li}_s(t)}{t} \mathrm{d}t.
The polylogarithm is a special case of the Lerch transcendent:
.. math:: \operatorname{Li}_{s}(z) = z \Phi(z, s, 1)
See Also
========
zeta, lerchphi
Examples
========
For :math:`z \in \{0, 1, -1\}`, the polylogarithm is automatically expressed
using other functions:
>>> from sympy import polylog
>>> from sympy.abc import s
>>> polylog(s, 0)
0
>>> polylog(s, 1)
zeta(s)
>>> polylog(s, -1)
-dirichlet_eta(s)
If :math:`s` is a negative integer, :math:`0` or :math:`1`, the
polylogarithm can be expressed using elementary functions. This can be
done using expand_func():
>>> from sympy import expand_func
>>> from sympy.abc import z
>>> expand_func(polylog(1, z))
-log(1 - z)
>>> expand_func(polylog(0, z))
z/(1 - z)
The derivative with respect to :math:`z` can be computed in closed form:
>>> polylog(s, z).diff(z)
polylog(s - 1, z)/z
The polylogarithm can be expressed in terms of the lerch transcendent:
>>> from sympy import lerchphi
>>> polylog(s, z).rewrite(lerchphi)
z*lerchphi(z, s, 1)
"""
@classmethod
def eval(cls, s, z):
s, z = sympify((s, z))
if z == 1:
return zeta(s)
elif z == -1:
return -dirichlet_eta(s)
elif z == 0:
return S.Zero
elif s == 2:
if z == S.Half:
return pi**2/12 - log(2)**2/2
elif z == 2:
return pi**2/4 - I*pi*log(2)
elif z == -(sqrt(5) - 1)/2:
return -pi**2/15 + log((sqrt(5)-1)/2)**2/2
elif z == -(sqrt(5) + 1)/2:
return -pi**2/10 - log((sqrt(5)+1)/2)**2
elif z == (3 - sqrt(5))/2:
return pi**2/15 - log((sqrt(5)-1)/2)**2
elif z == (sqrt(5) - 1)/2:
return pi**2/10 - log((sqrt(5)-1)/2)**2
# For s = 0 or -1 use explicit formulas to evaluate, but
# automatically expanding polylog(1, z) to -log(1-z) seems undesirable
# for summation methods based on hypergeometric functions
elif s == 0:
return z/(1 - z)
elif s == -1:
return z/(1 - z)**2
# polylog is branched, but not over the unit disk
from sympy.functions.elementary.complexes import (Abs, unpolarify,
polar_lift)
if z.has(exp_polar, polar_lift) and (Abs(z) <= S.One) == True:
return cls(s, unpolarify(z))
def fdiff(self, argindex=1):
s, z = self.args
if argindex == 2:
return polylog(s - 1, z)/z
raise ArgumentIndexError
def _eval_rewrite_as_lerchphi(self, s, z, **kwargs):
return z*lerchphi(z, s, 1)
def _eval_expand_func(self, **hints):
from sympy import log, expand_mul, Dummy
s, z = self.args
if s == 1:
return -log(1 - z)
if s.is_Integer and s <= 0:
u = Dummy('u')
start = u/(1 - u)
for _ in range(-s):
start = u*start.diff(u)
return expand_mul(start).subs(u, z)
return polylog(s, z)
###############################################################################
###################### HURWITZ GENERALIZED ZETA FUNCTION ######################
###############################################################################
class zeta(Function):
r"""
Hurwitz zeta function (or Riemann zeta function).
For `\operatorname{Re}(a) > 0` and `\operatorname{Re}(s) > 1`, this function is defined as
.. math:: \zeta(s, a) = \sum_{n=0}^\infty \frac{1}{(n + a)^s},
where the standard choice of argument for :math:`n + a` is used. For fixed
:math:`a` with `\operatorname{Re}(a) > 0` the Hurwitz zeta function admits a
meromorphic continuation to all of :math:`\mathbb{C}`, it is an unbranched
function with a simple pole at :math:`s = 1`.
Analytic continuation to other :math:`a` is possible under some circumstances,
but this is not typically done.
The Hurwitz zeta function is a special case of the Lerch transcendent:
.. math:: \zeta(s, a) = \Phi(1, s, a).
This formula defines an analytic continuation for all possible values of
:math:`s` and :math:`a` (also `\operatorname{Re}(a) < 0`), see the documentation of
:class:`lerchphi` for a description of the branching behavior.
If no value is passed for :math:`a`, by this function assumes a default value
of :math:`a = 1`, yielding the Riemann zeta function.
See Also
========
dirichlet_eta, lerchphi, polylog
References
==========
.. [1] http://dlmf.nist.gov/25.11
.. [2] https://en.wikipedia.org/wiki/Hurwitz_zeta_function
Examples
========
For :math:`a = 1` the Hurwitz zeta function reduces to the famous Riemann
zeta function:
.. math:: \zeta(s, 1) = \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.
>>> from sympy import zeta
>>> from sympy.abc import s
>>> zeta(s, 1)
zeta(s)
>>> zeta(s)
zeta(s)
The Riemann zeta function can also be expressed using the Dirichlet eta
function:
>>> from sympy import dirichlet_eta
>>> zeta(s).rewrite(dirichlet_eta)
dirichlet_eta(s)/(1 - 2**(1 - s))
The Riemann zeta function at positive even integer and negative odd integer
values is related to the Bernoulli numbers:
>>> zeta(2)
pi**2/6
>>> zeta(4)
pi**4/90
>>> zeta(-1)
-1/12
The specific formulae are:
.. math:: \zeta(2n) = (-1)^{n+1} \frac{B_{2n} (2\pi)^{2n}}{2(2n)!}
.. math:: \zeta(-n) = -\frac{B_{n+1}}{n+1}
At negative even integers the Riemann zeta function is zero:
>>> zeta(-4)
0
No closed-form expressions are known at positive odd integers, but
numerical evaluation is possible:
>>> zeta(3).n()
1.20205690315959
The derivative of :math:`\zeta(s, a)` with respect to :math:`a` is easily
computed:
>>> from sympy.abc import a
>>> zeta(s, a).diff(a)
-s*zeta(s + 1, a)
However the derivative with respect to :math:`s` has no useful closed form
expression:
>>> zeta(s, a).diff(s)
Derivative(zeta(s, a), s)
The Hurwitz zeta function can be expressed in terms of the Lerch transcendent,
:class:`sympy.functions.special.lerchphi`:
>>> from sympy import lerchphi
>>> zeta(s, a).rewrite(lerchphi)
lerchphi(1, s, a)
"""
@classmethod
def eval(cls, z, a_=None):
if a_ is None:
z, a = list(map(sympify, (z, 1)))
else:
z, a = list(map(sympify, (z, a_)))
if a.is_Number:
if a is S.NaN:
return S.NaN
elif a is S.One and a_ is not None:
return cls(z)
# TODO Should a == 0 return S.NaN as well?
if z.is_Number:
if z is S.NaN:
return S.NaN
elif z is S.Infinity:
return S.One
elif z is S.Zero:
return S.Half - a
elif z is S.One:
return S.ComplexInfinity
if z.is_integer:
if a.is_Integer:
if z.is_negative:
zeta = (-1)**z * bernoulli(-z + 1)/(-z + 1)
elif z.is_even and z.is_positive:
B, F = bernoulli(z), factorial(z)
zeta = ((-1)**(z/2+1) * 2**(z - 1) * B * pi**z) / F
else:
return
if a.is_negative:
return zeta + harmonic(abs(a), z)
else:
return zeta - harmonic(a - 1, z)
def _eval_rewrite_as_dirichlet_eta(self, s, a=1, **kwargs):
if a != 1:
return self
s = self.args[0]
return dirichlet_eta(s)/(1 - 2**(1 - s))
def _eval_rewrite_as_lerchphi(self, s, a=1, **kwargs):
return lerchphi(1, s, a)
def _eval_is_finite(self):
arg_is_one = (self.args[0] - 1).is_zero
if arg_is_one is not None:
return not arg_is_one
def fdiff(self, argindex=1):
if len(self.args) == 2:
s, a = self.args
else:
s, a = self.args + (1,)
if argindex == 2:
return -s*zeta(s + 1, a)
else:
raise ArgumentIndexError
class dirichlet_eta(Function):
r"""
Dirichlet eta function.
For `\operatorname{Re}(s) > 0`, this function is defined as
.. math:: \eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}.
It admits a unique analytic continuation to all of :math:`\mathbb{C}`.
It is an entire, unbranched function.
See Also
========
zeta
References
==========
.. [1] https://en.wikipedia.org/wiki/Dirichlet_eta_function
Examples
========
The Dirichlet eta function is closely related to the Riemann zeta function:
>>> from sympy import dirichlet_eta, zeta
>>> from sympy.abc import s
>>> dirichlet_eta(s).rewrite(zeta)
(1 - 2**(1 - s))*zeta(s)
"""
@classmethod
def eval(cls, s):
if s == 1:
return log(2)
z = zeta(s)
if not z.has(zeta):
return (1 - 2**(1 - s))*z
def _eval_rewrite_as_zeta(self, s, **kwargs):
return (1 - 2**(1 - s)) * zeta(s)
class stieltjes(Function):
r"""Represents Stieltjes constants, :math:`\gamma_{k}` that occur in
Laurent Series expansion of the Riemann zeta function.
Examples
========
>>> from sympy import stieltjes
>>> from sympy.abc import n, m
>>> stieltjes(n)
stieltjes(n)
zero'th stieltjes constant
>>> stieltjes(0)
EulerGamma
>>> stieltjes(0, 1)
EulerGamma
For generalized stieltjes constants
>>> stieltjes(n, m)
stieltjes(n, m)
Constants are only defined for integers >= 0
>>> stieltjes(-1)
zoo
References
==========
.. [1] https://en.wikipedia.org/wiki/Stieltjes_constants
"""
@classmethod
def eval(cls, n, a=None):
n = sympify(n)
if a is not None:
a = sympify(a)
if a is S.NaN:
return S.NaN
if a.is_Integer and a.is_nonpositive:
return S.ComplexInfinity
if n.is_Number:
if n is S.NaN:
return S.NaN
elif n < 0:
return S.ComplexInfinity
elif not n.is_Integer:
return S.ComplexInfinity
elif n == 0 and a in [None, 1]:
return S.EulerGamma
|
e24bb1fda8aaa21996b28490e1123d4fa197899d7fb4d4b4e12fd97e81b6c995
|
""" This module contains various functions that are special cases
of incomplete gamma functions. It should probably be renamed. """
from __future__ import print_function, division
from sympy.core import Add, S, sympify, cacheit, pi, I
from sympy.core.compatibility import range
from sympy.core.function import Function, ArgumentIndexError
from sympy.core.symbol import Symbol
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt, root
from sympy.functions.elementary.exponential import exp, log
from sympy.functions.elementary.complexes import polar_lift
from sympy.functions.elementary.hyperbolic import cosh, sinh
from sympy.functions.elementary.trigonometric import cos, sin, sinc
from sympy.functions.special.hyper import hyper, meijerg
# TODO series expansions
# TODO see the "Note:" in Ei
###############################################################################
################################ ERROR FUNCTION ###############################
###############################################################################
class erf(Function):
r"""
The Gauss error function. This function is defined as:
.. math ::
\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \mathrm{d}t.
Examples
========
>>> from sympy import I, oo, erf
>>> from sympy.abc import z
Several special values are known:
>>> erf(0)
0
>>> erf(oo)
1
>>> erf(-oo)
-1
>>> erf(I*oo)
oo*I
>>> erf(-I*oo)
-oo*I
In general one can pull out factors of -1 and I from the argument:
>>> erf(-z)
-erf(z)
The error function obeys the mirror symmetry:
>>> from sympy import conjugate
>>> conjugate(erf(z))
erf(conjugate(z))
Differentiation with respect to z is supported:
>>> from sympy import diff
>>> diff(erf(z), z)
2*exp(-z**2)/sqrt(pi)
We can numerically evaluate the error function to arbitrary precision
on the whole complex plane:
>>> erf(4).evalf(30)
0.999999984582742099719981147840
>>> erf(-4*I).evalf(30)
-1296959.73071763923152794095062*I
See Also
========
erfc: Complementary error function.
erfi: Imaginary error function.
erf2: Two-argument error function.
erfinv: Inverse error function.
erfcinv: Inverse Complementary error function.
erf2inv: Inverse two-argument error function.
References
==========
.. [1] https://en.wikipedia.org/wiki/Error_function
.. [2] http://dlmf.nist.gov/7
.. [3] http://mathworld.wolfram.com/Erf.html
.. [4] http://functions.wolfram.com/GammaBetaErf/Erf
"""
unbranched = True
def fdiff(self, argindex=1):
if argindex == 1:
return 2*exp(-self.args[0]**2)/sqrt(S.Pi)
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return erfinv
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.One
elif arg is S.NegativeInfinity:
return S.NegativeOne
elif arg is S.Zero:
return S.Zero
if isinstance(arg, erfinv):
return arg.args[0]
if isinstance(arg, erfcinv):
return S.One - arg.args[0]
if isinstance(arg, erf2inv) and arg.args[0] is S.Zero:
return arg.args[1]
# Try to pull out factors of I
t = arg.extract_multiplicatively(S.ImaginaryUnit)
if t is S.Infinity or t is S.NegativeInfinity:
return arg
# Try to pull out factors of -1
if arg.could_extract_minus_sign():
return -cls(-arg)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
k = floor((n - 1)/S(2))
if len(previous_terms) > 2:
return -previous_terms[-2] * x**2 * (n - 2)/(n*k)
else:
return 2*(-1)**k * x**n/(n*factorial(k)*sqrt(S.Pi))
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def _eval_is_real(self):
return self.args[0].is_real
def _eval_rewrite_as_uppergamma(self, z, **kwargs):
from sympy import uppergamma
return sqrt(z**2)/z*(S.One - uppergamma(S.Half, z**2)/sqrt(S.Pi))
def _eval_rewrite_as_fresnels(self, z, **kwargs):
arg = (S.One - S.ImaginaryUnit)*z/sqrt(pi)
return (S.One + S.ImaginaryUnit)*(fresnelc(arg) - I*fresnels(arg))
def _eval_rewrite_as_fresnelc(self, z, **kwargs):
arg = (S.One - S.ImaginaryUnit)*z/sqrt(pi)
return (S.One + S.ImaginaryUnit)*(fresnelc(arg) - I*fresnels(arg))
def _eval_rewrite_as_meijerg(self, z, **kwargs):
return z/sqrt(pi)*meijerg([S.Half], [], [0], [-S.Half], z**2)
def _eval_rewrite_as_hyper(self, z, **kwargs):
return 2*z/sqrt(pi)*hyper([S.Half], [3*S.Half], -z**2)
def _eval_rewrite_as_expint(self, z, **kwargs):
return sqrt(z**2)/z - z*expint(S.Half, z**2)/sqrt(S.Pi)
def _eval_rewrite_as_tractable(self, z, **kwargs):
return S.One - _erfs(z)*exp(-z**2)
def _eval_rewrite_as_erfc(self, z, **kwargs):
return S.One - erfc(z)
def _eval_rewrite_as_erfi(self, z, **kwargs):
return -I*erfi(I*z)
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return 2*x/sqrt(pi)
else:
return self.func(arg)
def as_real_imag(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
x, y = self.args[0].expand(deep, **hints).as_real_imag()
else:
x, y = self.args[0].as_real_imag()
sq = -y**2/x**2
re = S.Half*(self.func(x + x*sqrt(sq)) + self.func(x - x*sqrt(sq)))
im = x/(2*y) * sqrt(sq) * (self.func(x - x*sqrt(sq)) -
self.func(x + x*sqrt(sq)))
return (re, im)
class erfc(Function):
r"""
Complementary Error Function. The function is defined as:
.. math ::
\mathrm{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} \mathrm{d}t
Examples
========
>>> from sympy import I, oo, erfc
>>> from sympy.abc import z
Several special values are known:
>>> erfc(0)
1
>>> erfc(oo)
0
>>> erfc(-oo)
2
>>> erfc(I*oo)
-oo*I
>>> erfc(-I*oo)
oo*I
The error function obeys the mirror symmetry:
>>> from sympy import conjugate
>>> conjugate(erfc(z))
erfc(conjugate(z))
Differentiation with respect to z is supported:
>>> from sympy import diff
>>> diff(erfc(z), z)
-2*exp(-z**2)/sqrt(pi)
It also follows
>>> erfc(-z)
2 - erfc(z)
We can numerically evaluate the complementary error function to arbitrary precision
on the whole complex plane:
>>> erfc(4).evalf(30)
0.0000000154172579002800188521596734869
>>> erfc(4*I).evalf(30)
1.0 - 1296959.73071763923152794095062*I
See Also
========
erf: Gaussian error function.
erfi: Imaginary error function.
erf2: Two-argument error function.
erfinv: Inverse error function.
erfcinv: Inverse Complementary error function.
erf2inv: Inverse two-argument error function.
References
==========
.. [1] https://en.wikipedia.org/wiki/Error_function
.. [2] http://dlmf.nist.gov/7
.. [3] http://mathworld.wolfram.com/Erfc.html
.. [4] http://functions.wolfram.com/GammaBetaErf/Erfc
"""
unbranched = True
def fdiff(self, argindex=1):
if argindex == 1:
return -2*exp(-self.args[0]**2)/sqrt(S.Pi)
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return erfcinv
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Zero
elif arg is S.Zero:
return S.One
if isinstance(arg, erfinv):
return S.One - arg.args[0]
if isinstance(arg, erfcinv):
return arg.args[0]
# Try to pull out factors of I
t = arg.extract_multiplicatively(S.ImaginaryUnit)
if t is S.Infinity or t is S.NegativeInfinity:
return -arg
# Try to pull out factors of -1
if arg.could_extract_minus_sign():
return S(2) - cls(-arg)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return S.One
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
k = floor((n - 1)/S(2))
if len(previous_terms) > 2:
return -previous_terms[-2] * x**2 * (n - 2)/(n*k)
else:
return -2*(-1)**k * x**n/(n*factorial(k)*sqrt(S.Pi))
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def _eval_is_real(self):
return self.args[0].is_real
def _eval_rewrite_as_tractable(self, z, **kwargs):
return self.rewrite(erf).rewrite("tractable", deep=True)
def _eval_rewrite_as_erf(self, z, **kwargs):
return S.One - erf(z)
def _eval_rewrite_as_erfi(self, z, **kwargs):
return S.One + I*erfi(I*z)
def _eval_rewrite_as_fresnels(self, z, **kwargs):
arg = (S.One - S.ImaginaryUnit)*z/sqrt(pi)
return S.One - (S.One + S.ImaginaryUnit)*(fresnelc(arg) - I*fresnels(arg))
def _eval_rewrite_as_fresnelc(self, z, **kwargs):
arg = (S.One-S.ImaginaryUnit)*z/sqrt(pi)
return S.One - (S.One + S.ImaginaryUnit)*(fresnelc(arg) - I*fresnels(arg))
def _eval_rewrite_as_meijerg(self, z, **kwargs):
return S.One - z/sqrt(pi)*meijerg([S.Half], [], [0], [-S.Half], z**2)
def _eval_rewrite_as_hyper(self, z, **kwargs):
return S.One - 2*z/sqrt(pi)*hyper([S.Half], [3*S.Half], -z**2)
def _eval_rewrite_as_uppergamma(self, z, **kwargs):
from sympy import uppergamma
return S.One - sqrt(z**2)/z*(S.One - uppergamma(S.Half, z**2)/sqrt(S.Pi))
def _eval_rewrite_as_expint(self, z, **kwargs):
return S.One - sqrt(z**2)/z + z*expint(S.Half, z**2)/sqrt(S.Pi)
def _eval_expand_func(self, **hints):
return self.rewrite(erf)
def _eval_as_leading_term(self, x):
from sympy import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return S.One
else:
return self.func(arg)
def as_real_imag(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
x, y = self.args[0].expand(deep, **hints).as_real_imag()
else:
x, y = self.args[0].as_real_imag()
sq = -y**2/x**2
re = S.Half*(self.func(x + x*sqrt(sq)) + self.func(x - x*sqrt(sq)))
im = x/(2*y) * sqrt(sq) * (self.func(x - x*sqrt(sq)) -
self.func(x + x*sqrt(sq)))
return (re, im)
class erfi(Function):
r"""
Imaginary error function. The function erfi is defined as:
.. math ::
\mathrm{erfi}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{t^2} \mathrm{d}t
Examples
========
>>> from sympy import I, oo, erfi
>>> from sympy.abc import z
Several special values are known:
>>> erfi(0)
0
>>> erfi(oo)
oo
>>> erfi(-oo)
-oo
>>> erfi(I*oo)
I
>>> erfi(-I*oo)
-I
In general one can pull out factors of -1 and I from the argument:
>>> erfi(-z)
-erfi(z)
>>> from sympy import conjugate
>>> conjugate(erfi(z))
erfi(conjugate(z))
Differentiation with respect to z is supported:
>>> from sympy import diff
>>> diff(erfi(z), z)
2*exp(z**2)/sqrt(pi)
We can numerically evaluate the imaginary error function to arbitrary precision
on the whole complex plane:
>>> erfi(2).evalf(30)
18.5648024145755525987042919132
>>> erfi(-2*I).evalf(30)
-0.995322265018952734162069256367*I
See Also
========
erf: Gaussian error function.
erfc: Complementary error function.
erf2: Two-argument error function.
erfinv: Inverse error function.
erfcinv: Inverse Complementary error function.
erf2inv: Inverse two-argument error function.
References
==========
.. [1] https://en.wikipedia.org/wiki/Error_function
.. [2] http://mathworld.wolfram.com/Erfi.html
.. [3] http://functions.wolfram.com/GammaBetaErf/Erfi
"""
unbranched = True
def fdiff(self, argindex=1):
if argindex == 1:
return 2*exp(self.args[0]**2)/sqrt(S.Pi)
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, z):
if z.is_Number:
if z is S.NaN:
return S.NaN
elif z is S.Zero:
return S.Zero
elif z is S.Infinity:
return S.Infinity
# Try to pull out factors of -1
if z.could_extract_minus_sign():
return -cls(-z)
# Try to pull out factors of I
nz = z.extract_multiplicatively(I)
if nz is not None:
if nz is S.Infinity:
return I
if isinstance(nz, erfinv):
return I*nz.args[0]
if isinstance(nz, erfcinv):
return I*(S.One - nz.args[0])
if isinstance(nz, erf2inv) and nz.args[0] is S.Zero:
return I*nz.args[1]
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
k = floor((n - 1)/S(2))
if len(previous_terms) > 2:
return previous_terms[-2] * x**2 * (n - 2)/(n*k)
else:
return 2 * x**n/(n*factorial(k)*sqrt(S.Pi))
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def _eval_is_real(self):
return self.args[0].is_real
def _eval_rewrite_as_tractable(self, z, **kwargs):
return self.rewrite(erf).rewrite("tractable", deep=True)
def _eval_rewrite_as_erf(self, z, **kwargs):
return -I*erf(I*z)
def _eval_rewrite_as_erfc(self, z, **kwargs):
return I*erfc(I*z) - I
def _eval_rewrite_as_fresnels(self, z, **kwargs):
arg = (S.One + S.ImaginaryUnit)*z/sqrt(pi)
return (S.One - S.ImaginaryUnit)*(fresnelc(arg) - I*fresnels(arg))
def _eval_rewrite_as_fresnelc(self, z, **kwargs):
arg = (S.One + S.ImaginaryUnit)*z/sqrt(pi)
return (S.One - S.ImaginaryUnit)*(fresnelc(arg) - I*fresnels(arg))
def _eval_rewrite_as_meijerg(self, z, **kwargs):
return z/sqrt(pi)*meijerg([S.Half], [], [0], [-S.Half], -z**2)
def _eval_rewrite_as_hyper(self, z, **kwargs):
return 2*z/sqrt(pi)*hyper([S.Half], [3*S.Half], z**2)
def _eval_rewrite_as_uppergamma(self, z, **kwargs):
from sympy import uppergamma
return sqrt(-z**2)/z*(uppergamma(S.Half, -z**2)/sqrt(S.Pi) - S.One)
def _eval_rewrite_as_expint(self, z, **kwargs):
return sqrt(-z**2)/z - z*expint(S.Half, -z**2)/sqrt(S.Pi)
def _eval_expand_func(self, **hints):
return self.rewrite(erf)
def as_real_imag(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
x, y = self.args[0].expand(deep, **hints).as_real_imag()
else:
x, y = self.args[0].as_real_imag()
sq = -y**2/x**2
re = S.Half*(self.func(x + x*sqrt(sq)) + self.func(x - x*sqrt(sq)))
im = x/(2*y) * sqrt(sq) * (self.func(x - x*sqrt(sq)) -
self.func(x + x*sqrt(sq)))
return (re, im)
class erf2(Function):
r"""
Two-argument error function. This function is defined as:
.. math ::
\mathrm{erf2}(x, y) = \frac{2}{\sqrt{\pi}} \int_x^y e^{-t^2} \mathrm{d}t
Examples
========
>>> from sympy import I, oo, erf2
>>> from sympy.abc import x, y
Several special values are known:
>>> erf2(0, 0)
0
>>> erf2(x, x)
0
>>> erf2(x, oo)
1 - erf(x)
>>> erf2(x, -oo)
-erf(x) - 1
>>> erf2(oo, y)
erf(y) - 1
>>> erf2(-oo, y)
erf(y) + 1
In general one can pull out factors of -1:
>>> erf2(-x, -y)
-erf2(x, y)
The error function obeys the mirror symmetry:
>>> from sympy import conjugate
>>> conjugate(erf2(x, y))
erf2(conjugate(x), conjugate(y))
Differentiation with respect to x, y is supported:
>>> from sympy import diff
>>> diff(erf2(x, y), x)
-2*exp(-x**2)/sqrt(pi)
>>> diff(erf2(x, y), y)
2*exp(-y**2)/sqrt(pi)
See Also
========
erf: Gaussian error function.
erfc: Complementary error function.
erfi: Imaginary error function.
erfinv: Inverse error function.
erfcinv: Inverse Complementary error function.
erf2inv: Inverse two-argument error function.
References
==========
.. [1] http://functions.wolfram.com/GammaBetaErf/Erf2/
"""
def fdiff(self, argindex):
x, y = self.args
if argindex == 1:
return -2*exp(-x**2)/sqrt(S.Pi)
elif argindex == 2:
return 2*exp(-y**2)/sqrt(S.Pi)
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, x, y):
I = S.Infinity
N = S.NegativeInfinity
O = S.Zero
if x is S.NaN or y is S.NaN:
return S.NaN
elif x == y:
return S.Zero
elif (x is I or x is N or x is O) or (y is I or y is N or y is O):
return erf(y) - erf(x)
if isinstance(y, erf2inv) and y.args[0] == x:
return y.args[1]
#Try to pull out -1 factor
sign_x = x.could_extract_minus_sign()
sign_y = y.could_extract_minus_sign()
if (sign_x and sign_y):
return -cls(-x, -y)
elif (sign_x or sign_y):
return erf(y)-erf(x)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate(), self.args[1].conjugate())
def _eval_is_real(self):
return self.args[0].is_real and self.args[1].is_real
def _eval_rewrite_as_erf(self, x, y, **kwargs):
return erf(y) - erf(x)
def _eval_rewrite_as_erfc(self, x, y, **kwargs):
return erfc(x) - erfc(y)
def _eval_rewrite_as_erfi(self, x, y, **kwargs):
return I*(erfi(I*x)-erfi(I*y))
def _eval_rewrite_as_fresnels(self, x, y, **kwargs):
return erf(y).rewrite(fresnels) - erf(x).rewrite(fresnels)
def _eval_rewrite_as_fresnelc(self, x, y, **kwargs):
return erf(y).rewrite(fresnelc) - erf(x).rewrite(fresnelc)
def _eval_rewrite_as_meijerg(self, x, y, **kwargs):
return erf(y).rewrite(meijerg) - erf(x).rewrite(meijerg)
def _eval_rewrite_as_hyper(self, x, y, **kwargs):
return erf(y).rewrite(hyper) - erf(x).rewrite(hyper)
def _eval_rewrite_as_uppergamma(self, x, y, **kwargs):
from sympy import uppergamma
return (sqrt(y**2)/y*(S.One - uppergamma(S.Half, y**2)/sqrt(S.Pi)) -
sqrt(x**2)/x*(S.One - uppergamma(S.Half, x**2)/sqrt(S.Pi)))
def _eval_rewrite_as_expint(self, x, y, **kwargs):
return erf(y).rewrite(expint) - erf(x).rewrite(expint)
def _eval_expand_func(self, **hints):
return self.rewrite(erf)
class erfinv(Function):
r"""
Inverse Error Function. The erfinv function is defined as:
.. math ::
\mathrm{erf}(x) = y \quad \Rightarrow \quad \mathrm{erfinv}(y) = x
Examples
========
>>> from sympy import I, oo, erfinv
>>> from sympy.abc import x
Several special values are known:
>>> erfinv(0)
0
>>> erfinv(1)
oo
Differentiation with respect to x is supported:
>>> from sympy import diff
>>> diff(erfinv(x), x)
sqrt(pi)*exp(erfinv(x)**2)/2
We can numerically evaluate the inverse error function to arbitrary precision
on [-1, 1]:
>>> erfinv(0.2).evalf(30)
0.179143454621291692285822705344
See Also
========
erf: Gaussian error function.
erfc: Complementary error function.
erfi: Imaginary error function.
erf2: Two-argument error function.
erfcinv: Inverse Complementary error function.
erf2inv: Inverse two-argument error function.
References
==========
.. [1] https://en.wikipedia.org/wiki/Error_function#Inverse_functions
.. [2] http://functions.wolfram.com/GammaBetaErf/InverseErf/
"""
def fdiff(self, argindex =1):
if argindex == 1:
return sqrt(S.Pi)*exp(self.func(self.args[0])**2)*S.Half
else :
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return erf
@classmethod
def eval(cls, z):
if z is S.NaN:
return S.NaN
elif z is S.NegativeOne:
return S.NegativeInfinity
elif z is S.Zero:
return S.Zero
elif z is S.One:
return S.Infinity
if isinstance(z, erf) and z.args[0].is_real:
return z.args[0]
# Try to pull out factors of -1
nz = z.extract_multiplicatively(-1)
if nz is not None and (isinstance(nz, erf) and (nz.args[0]).is_real):
return -nz.args[0]
def _eval_rewrite_as_erfcinv(self, z, **kwargs):
return erfcinv(1-z)
class erfcinv (Function):
r"""
Inverse Complementary Error Function. The erfcinv function is defined as:
.. math ::
\mathrm{erfc}(x) = y \quad \Rightarrow \quad \mathrm{erfcinv}(y) = x
Examples
========
>>> from sympy import I, oo, erfcinv
>>> from sympy.abc import x
Several special values are known:
>>> erfcinv(1)
0
>>> erfcinv(0)
oo
Differentiation with respect to x is supported:
>>> from sympy import diff
>>> diff(erfcinv(x), x)
-sqrt(pi)*exp(erfcinv(x)**2)/2
See Also
========
erf: Gaussian error function.
erfc: Complementary error function.
erfi: Imaginary error function.
erf2: Two-argument error function.
erfinv: Inverse error function.
erf2inv: Inverse two-argument error function.
References
==========
.. [1] https://en.wikipedia.org/wiki/Error_function#Inverse_functions
.. [2] http://functions.wolfram.com/GammaBetaErf/InverseErfc/
"""
def fdiff(self, argindex =1):
if argindex == 1:
return -sqrt(S.Pi)*exp(self.func(self.args[0])**2)*S.Half
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return erfc
@classmethod
def eval(cls, z):
if z is S.NaN:
return S.NaN
elif z is S.Zero:
return S.Infinity
elif z is S.One:
return S.Zero
elif z == 2:
return S.NegativeInfinity
def _eval_rewrite_as_erfinv(self, z, **kwargs):
return erfinv(1-z)
class erf2inv(Function):
r"""
Two-argument Inverse error function. The erf2inv function is defined as:
.. math ::
\mathrm{erf2}(x, w) = y \quad \Rightarrow \quad \mathrm{erf2inv}(x, y) = w
Examples
========
>>> from sympy import I, oo, erf2inv, erfinv, erfcinv
>>> from sympy.abc import x, y
Several special values are known:
>>> erf2inv(0, 0)
0
>>> erf2inv(1, 0)
1
>>> erf2inv(0, 1)
oo
>>> erf2inv(0, y)
erfinv(y)
>>> erf2inv(oo, y)
erfcinv(-y)
Differentiation with respect to x and y is supported:
>>> from sympy import diff
>>> diff(erf2inv(x, y), x)
exp(-x**2 + erf2inv(x, y)**2)
>>> diff(erf2inv(x, y), y)
sqrt(pi)*exp(erf2inv(x, y)**2)/2
See Also
========
erf: Gaussian error function.
erfc: Complementary error function.
erfi: Imaginary error function.
erf2: Two-argument error function.
erfinv: Inverse error function.
erfcinv: Inverse complementary error function.
References
==========
.. [1] http://functions.wolfram.com/GammaBetaErf/InverseErf2/
"""
def fdiff(self, argindex):
x, y = self.args
if argindex == 1:
return exp(self.func(x,y)**2-x**2)
elif argindex == 2:
return sqrt(S.Pi)*S.Half*exp(self.func(x,y)**2)
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, x, y):
if x is S.NaN or y is S.NaN:
return S.NaN
elif x is S.Zero and y is S.Zero:
return S.Zero
elif x is S.Zero and y is S.One:
return S.Infinity
elif x is S.One and y is S.Zero:
return S.One
elif x is S.Zero:
return erfinv(y)
elif x is S.Infinity:
return erfcinv(-y)
elif y is S.Zero:
return x
elif y is S.Infinity:
return erfinv(x)
###############################################################################
#################### EXPONENTIAL INTEGRALS ####################################
###############################################################################
class Ei(Function):
r"""
The classical exponential integral.
For use in SymPy, this function is defined as
.. math:: \operatorname{Ei}(x) = \sum_{n=1}^\infty \frac{x^n}{n\, n!}
+ \log(x) + \gamma,
where `\gamma` is the Euler-Mascheroni constant.
If `x` is a polar number, this defines an analytic function on the
Riemann surface of the logarithm. Otherwise this defines an analytic
function in the cut plane `\mathbb{C} \setminus (-\infty, 0]`.
**Background**
The name *exponential integral* comes from the following statement:
.. math:: \operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t
If the integral is interpreted as a Cauchy principal value, this statement
holds for `x > 0` and `\operatorname{Ei}(x)` as defined above.
Examples
========
>>> from sympy import Ei, polar_lift, exp_polar, I, pi
>>> from sympy.abc import x
>>> Ei(-1)
Ei(-1)
This yields a real value:
>>> Ei(-1).n(chop=True)
-0.219383934395520
On the other hand the analytic continuation is not real:
>>> Ei(polar_lift(-1)).n(chop=True)
-0.21938393439552 + 3.14159265358979*I
The exponential integral has a logarithmic branch point at the origin:
>>> Ei(x*exp_polar(2*I*pi))
Ei(x) + 2*I*pi
Differentiation is supported:
>>> Ei(x).diff(x)
exp(x)/x
The exponential integral is related to many other special functions.
For example:
>>> from sympy import uppergamma, expint, Shi
>>> Ei(x).rewrite(expint)
-expint(1, x*exp_polar(I*pi)) - I*pi
>>> Ei(x).rewrite(Shi)
Chi(x) + Shi(x)
See Also
========
expint: Generalised exponential integral.
E1: Special case of the generalised exponential integral.
li: Logarithmic integral.
Li: Offset logarithmic integral.
Si: Sine integral.
Ci: Cosine integral.
Shi: Hyperbolic sine integral.
Chi: Hyperbolic cosine integral.
sympy.functions.special.gamma_functions.uppergamma: Upper incomplete gamma function.
References
==========
.. [1] http://dlmf.nist.gov/6.6
.. [2] https://en.wikipedia.org/wiki/Exponential_integral
.. [3] Abramowitz & Stegun, section 5: http://people.math.sfu.ca/~cbm/aands/page_228.htm
"""
@classmethod
def eval(cls, z):
if z is S.Zero:
return S.NegativeInfinity
elif z is S.Infinity:
return S.Infinity
elif z is S.NegativeInfinity:
return S.Zero
nz, n = z.extract_branch_factor()
if n:
return Ei(nz) + 2*I*pi*n
def fdiff(self, argindex=1):
from sympy import unpolarify
arg = unpolarify(self.args[0])
if argindex == 1:
return exp(arg)/arg
else:
raise ArgumentIndexError(self, argindex)
def _eval_evalf(self, prec):
if (self.args[0]/polar_lift(-1)).is_positive:
return Function._eval_evalf(self, prec) + (I*pi)._eval_evalf(prec)
return Function._eval_evalf(self, prec)
def _eval_rewrite_as_uppergamma(self, z, **kwargs):
from sympy import uppergamma
# XXX this does not currently work usefully because uppergamma
# immediately turns into expint
return -uppergamma(0, polar_lift(-1)*z) - I*pi
def _eval_rewrite_as_expint(self, z, **kwargs):
return -expint(1, polar_lift(-1)*z) - I*pi
def _eval_rewrite_as_li(self, z, **kwargs):
if isinstance(z, log):
return li(z.args[0])
# TODO:
# Actually it only holds that:
# Ei(z) = li(exp(z))
# for -pi < imag(z) <= pi
return li(exp(z))
def _eval_rewrite_as_Si(self, z, **kwargs):
return Shi(z) + Chi(z)
_eval_rewrite_as_Ci = _eval_rewrite_as_Si
_eval_rewrite_as_Chi = _eval_rewrite_as_Si
_eval_rewrite_as_Shi = _eval_rewrite_as_Si
def _eval_rewrite_as_tractable(self, z, **kwargs):
return exp(z) * _eis(z)
def _eval_nseries(self, x, n, logx):
x0 = self.args[0].limit(x, 0)
if x0 is S.Zero:
f = self._eval_rewrite_as_Si(*self.args)
return f._eval_nseries(x, n, logx)
return super(Ei, self)._eval_nseries(x, n, logx)
class expint(Function):
r"""
Generalized exponential integral.
This function is defined as
.. math:: \operatorname{E}_\nu(z) = z^{\nu - 1} \Gamma(1 - \nu, z),
where `\Gamma(1 - \nu, z)` is the upper incomplete gamma function
(``uppergamma``).
Hence for :math:`z` with positive real part we have
.. math:: \operatorname{E}_\nu(z)
= \int_1^\infty \frac{e^{-zt}}{z^\nu} \mathrm{d}t,
which explains the name.
The representation as an incomplete gamma function provides an analytic
continuation for :math:`\operatorname{E}_\nu(z)`. If :math:`\nu` is a
non-positive integer the exponential integral is thus an unbranched
function of :math:`z`, otherwise there is a branch point at the origin.
Refer to the incomplete gamma function documentation for details of the
branching behavior.
Examples
========
>>> from sympy import expint, S
>>> from sympy.abc import nu, z
Differentiation is supported. Differentiation with respect to z explains
further the name: for integral orders, the exponential integral is an
iterated integral of the exponential function.
>>> expint(nu, z).diff(z)
-expint(nu - 1, z)
Differentiation with respect to nu has no classical expression:
>>> expint(nu, z).diff(nu)
-z**(nu - 1)*meijerg(((), (1, 1)), ((0, 0, 1 - nu), ()), z)
At non-postive integer orders, the exponential integral reduces to the
exponential function:
>>> expint(0, z)
exp(-z)/z
>>> expint(-1, z)
exp(-z)/z + exp(-z)/z**2
At half-integers it reduces to error functions:
>>> expint(S(1)/2, z)
sqrt(pi)*erfc(sqrt(z))/sqrt(z)
At positive integer orders it can be rewritten in terms of exponentials
and expint(1, z). Use expand_func() to do this:
>>> from sympy import expand_func
>>> expand_func(expint(5, z))
z**4*expint(1, z)/24 + (-z**3 + z**2 - 2*z + 6)*exp(-z)/24
The generalised exponential integral is essentially equivalent to the
incomplete gamma function:
>>> from sympy import uppergamma
>>> expint(nu, z).rewrite(uppergamma)
z**(nu - 1)*uppergamma(1 - nu, z)
As such it is branched at the origin:
>>> from sympy import exp_polar, pi, I
>>> expint(4, z*exp_polar(2*pi*I))
I*pi*z**3/3 + expint(4, z)
>>> expint(nu, z*exp_polar(2*pi*I))
z**(nu - 1)*(exp(2*I*pi*nu) - 1)*gamma(1 - nu) + expint(nu, z)
See Also
========
Ei: Another related function called exponential integral.
E1: The classical case, returns expint(1, z).
li: Logarithmic integral.
Li: Offset logarithmic integral.
Si: Sine integral.
Ci: Cosine integral.
Shi: Hyperbolic sine integral.
Chi: Hyperbolic cosine integral.
sympy.functions.special.gamma_functions.uppergamma
References
==========
.. [1] http://dlmf.nist.gov/8.19
.. [2] http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/
.. [3] https://en.wikipedia.org/wiki/Exponential_integral
"""
@classmethod
def eval(cls, nu, z):
from sympy import (unpolarify, expand_mul, uppergamma, exp, gamma,
factorial)
nu2 = unpolarify(nu)
if nu != nu2:
return expint(nu2, z)
if nu.is_Integer and nu <= 0 or (not nu.is_Integer and (2*nu).is_Integer):
return unpolarify(expand_mul(z**(nu - 1)*uppergamma(1 - nu, z)))
# Extract branching information. This can be deduced from what is
# explained in lowergamma.eval().
z, n = z.extract_branch_factor()
if n == 0:
return
if nu.is_integer:
if not nu > 0:
return
return expint(nu, z) \
- 2*pi*I*n*(-1)**(nu - 1)/factorial(nu - 1)*unpolarify(z)**(nu - 1)
else:
return (exp(2*I*pi*nu*n) - 1)*z**(nu - 1)*gamma(1 - nu) + expint(nu, z)
def fdiff(self, argindex):
from sympy import meijerg
nu, z = self.args
if argindex == 1:
return -z**(nu - 1)*meijerg([], [1, 1], [0, 0, 1 - nu], [], z)
elif argindex == 2:
return -expint(nu - 1, z)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_uppergamma(self, nu, z, **kwargs):
from sympy import uppergamma
return z**(nu - 1)*uppergamma(1 - nu, z)
def _eval_rewrite_as_Ei(self, nu, z, **kwargs):
from sympy import exp_polar, unpolarify, exp, factorial
if nu == 1:
return -Ei(z*exp_polar(-I*pi)) - I*pi
elif nu.is_Integer and nu > 1:
# DLMF, 8.19.7
x = -unpolarify(z)
return x**(nu - 1)/factorial(nu - 1)*E1(z).rewrite(Ei) + \
exp(x)/factorial(nu - 1) * \
Add(*[factorial(nu - k - 2)*x**k for k in range(nu - 1)])
else:
return self
def _eval_expand_func(self, **hints):
return self.rewrite(Ei).rewrite(expint, **hints)
def _eval_rewrite_as_Si(self, nu, z, **kwargs):
if nu != 1:
return self
return Shi(z) - Chi(z)
_eval_rewrite_as_Ci = _eval_rewrite_as_Si
_eval_rewrite_as_Chi = _eval_rewrite_as_Si
_eval_rewrite_as_Shi = _eval_rewrite_as_Si
def _eval_nseries(self, x, n, logx):
if not self.args[0].has(x):
nu = self.args[0]
if nu == 1:
f = self._eval_rewrite_as_Si(*self.args)
return f._eval_nseries(x, n, logx)
elif nu.is_Integer and nu > 1:
f = self._eval_rewrite_as_Ei(*self.args)
return f._eval_nseries(x, n, logx)
return super(expint, self)._eval_nseries(x, n, logx)
def _sage_(self):
import sage.all as sage
return sage.exp_integral_e(self.args[0]._sage_(), self.args[1]._sage_())
def E1(z):
"""
Classical case of the generalized exponential integral.
This is equivalent to ``expint(1, z)``.
See Also
========
Ei: Exponential integral.
expint: Generalised exponential integral.
li: Logarithmic integral.
Li: Offset logarithmic integral.
Si: Sine integral.
Ci: Cosine integral.
Shi: Hyperbolic sine integral.
Chi: Hyperbolic cosine integral.
"""
return expint(1, z)
class li(Function):
r"""
The classical logarithmic integral.
For the use in SymPy, this function is defined as
.. math:: \operatorname{li}(x) = \int_0^x \frac{1}{\log(t)} \mathrm{d}t \,.
Examples
========
>>> from sympy import I, oo, li
>>> from sympy.abc import z
Several special values are known:
>>> li(0)
0
>>> li(1)
-oo
>>> li(oo)
oo
Differentiation with respect to z is supported:
>>> from sympy import diff
>>> diff(li(z), z)
1/log(z)
Defining the `li` function via an integral:
The logarithmic integral can also be defined in terms of Ei:
>>> from sympy import Ei
>>> li(z).rewrite(Ei)
Ei(log(z))
>>> diff(li(z).rewrite(Ei), z)
1/log(z)
We can numerically evaluate the logarithmic integral to arbitrary precision
on the whole complex plane (except the singular points):
>>> li(2).evalf(30)
1.04516378011749278484458888919
>>> li(2*I).evalf(30)
1.0652795784357498247001125598 + 3.08346052231061726610939702133*I
We can even compute Soldner's constant by the help of mpmath:
>>> from mpmath import findroot
>>> findroot(li, 2)
1.45136923488338
Further transformations include rewriting `li` in terms of
the trigonometric integrals `Si`, `Ci`, `Shi` and `Chi`:
>>> from sympy import Si, Ci, Shi, Chi
>>> li(z).rewrite(Si)
-log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z))
>>> li(z).rewrite(Ci)
-log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z))
>>> li(z).rewrite(Shi)
-log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))
>>> li(z).rewrite(Chi)
-log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))
See Also
========
Li: Offset logarithmic integral.
Ei: Exponential integral.
expint: Generalised exponential integral.
E1: Special case of the generalised exponential integral.
Si: Sine integral.
Ci: Cosine integral.
Shi: Hyperbolic sine integral.
Chi: Hyperbolic cosine integral.
References
==========
.. [1] https://en.wikipedia.org/wiki/Logarithmic_integral
.. [2] http://mathworld.wolfram.com/LogarithmicIntegral.html
.. [3] http://dlmf.nist.gov/6
.. [4] http://mathworld.wolfram.com/SoldnersConstant.html
"""
@classmethod
def eval(cls, z):
if z is S.Zero:
return S.Zero
elif z is S.One:
return S.NegativeInfinity
elif z is S.Infinity:
return S.Infinity
def fdiff(self, argindex=1):
arg = self.args[0]
if argindex == 1:
return S.One / log(arg)
else:
raise ArgumentIndexError(self, argindex)
def _eval_conjugate(self):
z = self.args[0]
# Exclude values on the branch cut (-oo, 0)
if not (z.is_real and z.is_negative):
return self.func(z.conjugate())
def _eval_rewrite_as_Li(self, z, **kwargs):
return Li(z) + li(2)
def _eval_rewrite_as_Ei(self, z, **kwargs):
return Ei(log(z))
def _eval_rewrite_as_uppergamma(self, z, **kwargs):
from sympy import uppergamma
return (-uppergamma(0, -log(z)) +
S.Half*(log(log(z)) - log(S.One/log(z))) - log(-log(z)))
def _eval_rewrite_as_Si(self, z, **kwargs):
return (Ci(I*log(z)) - I*Si(I*log(z)) -
S.Half*(log(S.One/log(z)) - log(log(z))) - log(I*log(z)))
_eval_rewrite_as_Ci = _eval_rewrite_as_Si
def _eval_rewrite_as_Shi(self, z, **kwargs):
return (Chi(log(z)) - Shi(log(z)) - S.Half*(log(S.One/log(z)) - log(log(z))))
_eval_rewrite_as_Chi = _eval_rewrite_as_Shi
def _eval_rewrite_as_hyper(self, z, **kwargs):
return (log(z)*hyper((1, 1), (2, 2), log(z)) +
S.Half*(log(log(z)) - log(S.One/log(z))) + S.EulerGamma)
def _eval_rewrite_as_meijerg(self, z, **kwargs):
return (-log(-log(z)) - S.Half*(log(S.One/log(z)) - log(log(z)))
- meijerg(((), (1,)), ((0, 0), ()), -log(z)))
def _eval_rewrite_as_tractable(self, z, **kwargs):
return z * _eis(log(z))
class Li(Function):
r"""
The offset logarithmic integral.
For the use in SymPy, this function is defined as
.. math:: \operatorname{Li}(x) = \operatorname{li}(x) - \operatorname{li}(2)
Examples
========
>>> from sympy import I, oo, Li
>>> from sympy.abc import z
The following special value is known:
>>> Li(2)
0
Differentiation with respect to z is supported:
>>> from sympy import diff
>>> diff(Li(z), z)
1/log(z)
The shifted logarithmic integral can be written in terms of `li(z)`:
>>> from sympy import li
>>> Li(z).rewrite(li)
li(z) - li(2)
We can numerically evaluate the logarithmic integral to arbitrary precision
on the whole complex plane (except the singular points):
>>> Li(2).evalf(30)
0
>>> Li(4).evalf(30)
1.92242131492155809316615998938
See Also
========
li: Logarithmic integral.
Ei: Exponential integral.
expint: Generalised exponential integral.
E1: Special case of the generalised exponential integral.
Si: Sine integral.
Ci: Cosine integral.
Shi: Hyperbolic sine integral.
Chi: Hyperbolic cosine integral.
References
==========
.. [1] https://en.wikipedia.org/wiki/Logarithmic_integral
.. [2] http://mathworld.wolfram.com/LogarithmicIntegral.html
.. [3] http://dlmf.nist.gov/6
"""
@classmethod
def eval(cls, z):
if z is S.Infinity:
return S.Infinity
elif z is 2*S.One:
return S.Zero
def fdiff(self, argindex=1):
arg = self.args[0]
if argindex == 1:
return S.One / log(arg)
else:
raise ArgumentIndexError(self, argindex)
def _eval_evalf(self, prec):
return self.rewrite(li).evalf(prec)
def _eval_rewrite_as_li(self, z, **kwargs):
return li(z) - li(2)
def _eval_rewrite_as_tractable(self, z, **kwargs):
return self.rewrite(li).rewrite("tractable", deep=True)
###############################################################################
#################### TRIGONOMETRIC INTEGRALS ##################################
###############################################################################
class TrigonometricIntegral(Function):
""" Base class for trigonometric integrals. """
@classmethod
def eval(cls, z):
if z == 0:
return cls._atzero
elif z is S.Infinity:
return cls._atinf()
elif z is S.NegativeInfinity:
return cls._atneginf()
nz = z.extract_multiplicatively(polar_lift(I))
if nz is None and cls._trigfunc(0) == 0:
nz = z.extract_multiplicatively(I)
if nz is not None:
return cls._Ifactor(nz, 1)
nz = z.extract_multiplicatively(polar_lift(-I))
if nz is not None:
return cls._Ifactor(nz, -1)
nz = z.extract_multiplicatively(polar_lift(-1))
if nz is None and cls._trigfunc(0) == 0:
nz = z.extract_multiplicatively(-1)
if nz is not None:
return cls._minusfactor(nz)
nz, n = z.extract_branch_factor()
if n == 0 and nz == z:
return
return 2*pi*I*n*cls._trigfunc(0) + cls(nz)
def fdiff(self, argindex=1):
from sympy import unpolarify
arg = unpolarify(self.args[0])
if argindex == 1:
return self._trigfunc(arg)/arg
def _eval_rewrite_as_Ei(self, z, **kwargs):
return self._eval_rewrite_as_expint(z).rewrite(Ei)
def _eval_rewrite_as_uppergamma(self, z, **kwargs):
from sympy import uppergamma
return self._eval_rewrite_as_expint(z).rewrite(uppergamma)
def _eval_nseries(self, x, n, logx):
# NOTE this is fairly inefficient
from sympy import log, EulerGamma, Pow
n += 1
if self.args[0].subs(x, 0) != 0:
return super(TrigonometricIntegral, self)._eval_nseries(x, n, logx)
baseseries = self._trigfunc(x)._eval_nseries(x, n, logx)
if self._trigfunc(0) != 0:
baseseries -= 1
baseseries = baseseries.replace(Pow, lambda t, n: t**n/n, simultaneous=False)
if self._trigfunc(0) != 0:
baseseries += EulerGamma + log(x)
return baseseries.subs(x, self.args[0])._eval_nseries(x, n, logx)
class Si(TrigonometricIntegral):
r"""
Sine integral.
This function is defined by
.. math:: \operatorname{Si}(z) = \int_0^z \frac{\sin{t}}{t} \mathrm{d}t.
It is an entire function.
Examples
========
>>> from sympy import Si
>>> from sympy.abc import z
The sine integral is an antiderivative of sin(z)/z:
>>> Si(z).diff(z)
sin(z)/z
It is unbranched:
>>> from sympy import exp_polar, I, pi
>>> Si(z*exp_polar(2*I*pi))
Si(z)
Sine integral behaves much like ordinary sine under multiplication by ``I``:
>>> Si(I*z)
I*Shi(z)
>>> Si(-z)
-Si(z)
It can also be expressed in terms of exponential integrals, but beware
that the latter is branched:
>>> from sympy import expint
>>> Si(z).rewrite(expint)
-I*(-expint(1, z*exp_polar(-I*pi/2))/2 +
expint(1, z*exp_polar(I*pi/2))/2) + pi/2
It can be rewritten in the form of sinc function (By definition)
>>> from sympy import sinc
>>> Si(z).rewrite(sinc)
Integral(sinc(t), (t, 0, z))
See Also
========
Ci: Cosine integral.
Shi: Hyperbolic sine integral.
Chi: Hyperbolic cosine integral.
Ei: Exponential integral.
expint: Generalised exponential integral.
sinc: unnormalized sinc function
E1: Special case of the generalised exponential integral.
li: Logarithmic integral.
Li: Offset logarithmic integral.
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_integral
"""
_trigfunc = sin
_atzero = S(0)
@classmethod
def _atinf(cls):
return pi*S.Half
@classmethod
def _atneginf(cls):
return -pi*S.Half
@classmethod
def _minusfactor(cls, z):
return -Si(z)
@classmethod
def _Ifactor(cls, z, sign):
return I*Shi(z)*sign
def _eval_rewrite_as_expint(self, z, **kwargs):
# XXX should we polarify z?
return pi/2 + (E1(polar_lift(I)*z) - E1(polar_lift(-I)*z))/2/I
def _eval_rewrite_as_sinc(self, z, **kwargs):
from sympy import Integral
t = Symbol('t', Dummy=True)
return Integral(sinc(t), (t, 0, z))
def _sage_(self):
import sage.all as sage
return sage.sin_integral(self.args[0]._sage_())
class Ci(TrigonometricIntegral):
r"""
Cosine integral.
This function is defined for positive `x` by
.. math:: \operatorname{Ci}(x) = \gamma + \log{x}
+ \int_0^x \frac{\cos{t} - 1}{t} \mathrm{d}t
= -\int_x^\infty \frac{\cos{t}}{t} \mathrm{d}t,
where `\gamma` is the Euler-Mascheroni constant.
We have
.. math:: \operatorname{Ci}(z) =
-\frac{\operatorname{E}_1\left(e^{i\pi/2} z\right)
+ \operatorname{E}_1\left(e^{-i \pi/2} z\right)}{2}
which holds for all polar `z` and thus provides an analytic
continuation to the Riemann surface of the logarithm.
The formula also holds as stated
for `z \in \mathbb{C}` with `\Re(z) > 0`.
By lifting to the principal branch we obtain an analytic function on the
cut complex plane.
Examples
========
>>> from sympy import Ci
>>> from sympy.abc import z
The cosine integral is a primitive of `\cos(z)/z`:
>>> Ci(z).diff(z)
cos(z)/z
It has a logarithmic branch point at the origin:
>>> from sympy import exp_polar, I, pi
>>> Ci(z*exp_polar(2*I*pi))
Ci(z) + 2*I*pi
The cosine integral behaves somewhat like ordinary `\cos` under multiplication by `i`:
>>> from sympy import polar_lift
>>> Ci(polar_lift(I)*z)
Chi(z) + I*pi/2
>>> Ci(polar_lift(-1)*z)
Ci(z) + I*pi
It can also be expressed in terms of exponential integrals:
>>> from sympy import expint
>>> Ci(z).rewrite(expint)
-expint(1, z*exp_polar(-I*pi/2))/2 - expint(1, z*exp_polar(I*pi/2))/2
See Also
========
Si: Sine integral.
Shi: Hyperbolic sine integral.
Chi: Hyperbolic cosine integral.
Ei: Exponential integral.
expint: Generalised exponential integral.
E1: Special case of the generalised exponential integral.
li: Logarithmic integral.
Li: Offset logarithmic integral.
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_integral
"""
_trigfunc = cos
_atzero = S.ComplexInfinity
@classmethod
def _atinf(cls):
return S.Zero
@classmethod
def _atneginf(cls):
return I*pi
@classmethod
def _minusfactor(cls, z):
return Ci(z) + I*pi
@classmethod
def _Ifactor(cls, z, sign):
return Chi(z) + I*pi/2*sign
def _eval_rewrite_as_expint(self, z, **kwargs):
return -(E1(polar_lift(I)*z) + E1(polar_lift(-I)*z))/2
def _sage_(self):
import sage.all as sage
return sage.cos_integral(self.args[0]._sage_())
class Shi(TrigonometricIntegral):
r"""
Sinh integral.
This function is defined by
.. math:: \operatorname{Shi}(z) = \int_0^z \frac{\sinh{t}}{t} \mathrm{d}t.
It is an entire function.
Examples
========
>>> from sympy import Shi
>>> from sympy.abc import z
The Sinh integral is a primitive of `\sinh(z)/z`:
>>> Shi(z).diff(z)
sinh(z)/z
It is unbranched:
>>> from sympy import exp_polar, I, pi
>>> Shi(z*exp_polar(2*I*pi))
Shi(z)
The `\sinh` integral behaves much like ordinary `\sinh` under multiplication by `i`:
>>> Shi(I*z)
I*Si(z)
>>> Shi(-z)
-Shi(z)
It can also be expressed in terms of exponential integrals, but beware
that the latter is branched:
>>> from sympy import expint
>>> Shi(z).rewrite(expint)
expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2
See Also
========
Si: Sine integral.
Ci: Cosine integral.
Chi: Hyperbolic cosine integral.
Ei: Exponential integral.
expint: Generalised exponential integral.
E1: Special case of the generalised exponential integral.
li: Logarithmic integral.
Li: Offset logarithmic integral.
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_integral
"""
_trigfunc = sinh
_atzero = S(0)
@classmethod
def _atinf(cls):
return S.Infinity
@classmethod
def _atneginf(cls):
return S.NegativeInfinity
@classmethod
def _minusfactor(cls, z):
return -Shi(z)
@classmethod
def _Ifactor(cls, z, sign):
return I*Si(z)*sign
def _eval_rewrite_as_expint(self, z, **kwargs):
from sympy import exp_polar
# XXX should we polarify z?
return (E1(z) - E1(exp_polar(I*pi)*z))/2 - I*pi/2
def _sage_(self):
import sage.all as sage
return sage.sinh_integral(self.args[0]._sage_())
class Chi(TrigonometricIntegral):
r"""
Cosh integral.
This function is defined for positive :math:`x` by
.. math:: \operatorname{Chi}(x) = \gamma + \log{x}
+ \int_0^x \frac{\cosh{t} - 1}{t} \mathrm{d}t,
where :math:`\gamma` is the Euler-Mascheroni constant.
We have
.. math:: \operatorname{Chi}(z) = \operatorname{Ci}\left(e^{i \pi/2}z\right)
- i\frac{\pi}{2},
which holds for all polar :math:`z` and thus provides an analytic
continuation to the Riemann surface of the logarithm.
By lifting to the principal branch we obtain an analytic function on the
cut complex plane.
Examples
========
>>> from sympy import Chi
>>> from sympy.abc import z
The `\cosh` integral is a primitive of `\cosh(z)/z`:
>>> Chi(z).diff(z)
cosh(z)/z
It has a logarithmic branch point at the origin:
>>> from sympy import exp_polar, I, pi
>>> Chi(z*exp_polar(2*I*pi))
Chi(z) + 2*I*pi
The `\cosh` integral behaves somewhat like ordinary `\cosh` under multiplication by `i`:
>>> from sympy import polar_lift
>>> Chi(polar_lift(I)*z)
Ci(z) + I*pi/2
>>> Chi(polar_lift(-1)*z)
Chi(z) + I*pi
It can also be expressed in terms of exponential integrals:
>>> from sympy import expint
>>> Chi(z).rewrite(expint)
-expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2
See Also
========
Si: Sine integral.
Ci: Cosine integral.
Shi: Hyperbolic sine integral.
Ei: Exponential integral.
expint: Generalised exponential integral.
E1: Special case of the generalised exponential integral.
li: Logarithmic integral.
Li: Offset logarithmic integral.
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_integral
"""
_trigfunc = cosh
_atzero = S.ComplexInfinity
@classmethod
def _atinf(cls):
return S.Infinity
@classmethod
def _atneginf(cls):
return S.Infinity
@classmethod
def _minusfactor(cls, z):
return Chi(z) + I*pi
@classmethod
def _Ifactor(cls, z, sign):
return Ci(z) + I*pi/2*sign
def _eval_rewrite_as_expint(self, z, **kwargs):
from sympy import exp_polar
return -I*pi/2 - (E1(z) + E1(exp_polar(I*pi)*z))/2
def _sage_(self):
import sage.all as sage
return sage.cosh_integral(self.args[0]._sage_())
###############################################################################
#################### FRESNEL INTEGRALS ########################################
###############################################################################
class FresnelIntegral(Function):
""" Base class for the Fresnel integrals."""
unbranched = True
@classmethod
def eval(cls, z):
# Value at zero
if z is S.Zero:
return S(0)
# Try to pull out factors of -1 and I
prefact = S.One
newarg = z
changed = False
nz = newarg.extract_multiplicatively(-1)
if nz is not None:
prefact = -prefact
newarg = nz
changed = True
nz = newarg.extract_multiplicatively(I)
if nz is not None:
prefact = cls._sign*I*prefact
newarg = nz
changed = True
if changed:
return prefact*cls(newarg)
# Values at positive infinities signs
# if any were extracted automatically
if z is S.Infinity:
return S.Half
elif z is I*S.Infinity:
return cls._sign*I*S.Half
def fdiff(self, argindex=1):
if argindex == 1:
return self._trigfunc(S.Half*pi*self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_real(self):
return self.args[0].is_real
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def _as_real_imag(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
return (re, im)
def as_real_imag(self, deep=True, **hints):
# Fresnel S
# http://functions.wolfram.com/06.32.19.0003.01
# http://functions.wolfram.com/06.32.19.0006.01
# Fresnel C
# http://functions.wolfram.com/06.33.19.0003.01
# http://functions.wolfram.com/06.33.19.0006.01
x, y = self._as_real_imag(deep=deep, **hints)
sq = -y**2/x**2
re = S.Half*(self.func(x + x*sqrt(sq)) + self.func(x - x*sqrt(sq)))
im = x/(2*y) * sqrt(sq) * (self.func(x - x*sqrt(sq)) -
self.func(x + x*sqrt(sq)))
return (re, im)
class fresnels(FresnelIntegral):
r"""
Fresnel integral S.
This function is defined by
.. math:: \operatorname{S}(z) = \int_0^z \sin{\frac{\pi}{2} t^2} \mathrm{d}t.
It is an entire function.
Examples
========
>>> from sympy import I, oo, fresnels
>>> from sympy.abc import z
Several special values are known:
>>> fresnels(0)
0
>>> fresnels(oo)
1/2
>>> fresnels(-oo)
-1/2
>>> fresnels(I*oo)
-I/2
>>> fresnels(-I*oo)
I/2
In general one can pull out factors of -1 and `i` from the argument:
>>> fresnels(-z)
-fresnels(z)
>>> fresnels(I*z)
-I*fresnels(z)
The Fresnel S integral obeys the mirror symmetry
`\overline{S(z)} = S(\bar{z})`:
>>> from sympy import conjugate
>>> conjugate(fresnels(z))
fresnels(conjugate(z))
Differentiation with respect to `z` is supported:
>>> from sympy import diff
>>> diff(fresnels(z), z)
sin(pi*z**2/2)
Defining the Fresnel functions via an integral
>>> from sympy import integrate, pi, sin, gamma, expand_func
>>> integrate(sin(pi*z**2/2), z)
3*fresnels(z)*gamma(3/4)/(4*gamma(7/4))
>>> expand_func(integrate(sin(pi*z**2/2), z))
fresnels(z)
We can numerically evaluate the Fresnel integral to arbitrary precision
on the whole complex plane:
>>> fresnels(2).evalf(30)
0.343415678363698242195300815958
>>> fresnels(-2*I).evalf(30)
0.343415678363698242195300815958*I
See Also
========
fresnelc: Fresnel cosine integral.
References
==========
.. [1] https://en.wikipedia.org/wiki/Fresnel_integral
.. [2] http://dlmf.nist.gov/7
.. [3] http://mathworld.wolfram.com/FresnelIntegrals.html
.. [4] http://functions.wolfram.com/GammaBetaErf/FresnelS
.. [5] The converging factors for the fresnel integrals
by John W. Wrench Jr. and Vicki Alley
"""
_trigfunc = sin
_sign = -S.One
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 1:
p = previous_terms[-1]
return (-pi**2*x**4*(4*n - 1)/(8*n*(2*n + 1)*(4*n + 3))) * p
else:
return x**3 * (-x**4)**n * (S(2)**(-2*n - 1)*pi**(2*n + 1)) / ((4*n + 3)*factorial(2*n + 1))
def _eval_rewrite_as_erf(self, z, **kwargs):
return (S.One + I)/4 * (erf((S.One + I)/2*sqrt(pi)*z) - I*erf((S.One - I)/2*sqrt(pi)*z))
def _eval_rewrite_as_hyper(self, z, **kwargs):
return pi*z**3/6 * hyper([S(3)/4], [S(3)/2, S(7)/4], -pi**2*z**4/16)
def _eval_rewrite_as_meijerg(self, z, **kwargs):
return (pi*z**(S(9)/4) / (sqrt(2)*(z**2)**(S(3)/4)*(-z)**(S(3)/4))
* meijerg([], [1], [S(3)/4], [S(1)/4, 0], -pi**2*z**4/16))
def _eval_aseries(self, n, args0, x, logx):
from sympy import Order
point = args0[0]
# Expansion at oo and -oo
if point in [S.Infinity, -S.Infinity]:
z = self.args[0]
# expansion of S(x) = S1(x*sqrt(pi/2)), see reference[5] page 1-8
# as only real infinities are dealt with, sin and cos are O(1)
p = [(-1)**k * factorial(4*k + 1) /
(2**(2*k + 2) * z**(4*k + 3) * 2**(2*k)*factorial(2*k))
for k in range(0, n) if 4*k + 3 < n]
q = [1/(2*z)] + [(-1)**k * factorial(4*k - 1) /
(2**(2*k + 1) * z**(4*k + 1) * 2**(2*k - 1)*factorial(2*k - 1))
for k in range(1, n) if 4*k + 1 < n]
p = [-sqrt(2/pi)*t for t in p]
q = [-sqrt(2/pi)*t for t in q]
s = 1 if point is S.Infinity else -1
# The expansion at oo is 1/2 + some odd powers of z
# To get the expansion at -oo, replace z by -z and flip the sign
# The result -1/2 + the same odd powers of z as before.
return s*S.Half + (sin(z**2)*Add(*p) + cos(z**2)*Add(*q)
).subs(x, sqrt(2/pi)*x) + Order(1/z**n, x)
# All other points are not handled
return super(fresnels, self)._eval_aseries(n, args0, x, logx)
class fresnelc(FresnelIntegral):
r"""
Fresnel integral C.
This function is defined by
.. math:: \operatorname{C}(z) = \int_0^z \cos{\frac{\pi}{2} t^2} \mathrm{d}t.
It is an entire function.
Examples
========
>>> from sympy import I, oo, fresnelc
>>> from sympy.abc import z
Several special values are known:
>>> fresnelc(0)
0
>>> fresnelc(oo)
1/2
>>> fresnelc(-oo)
-1/2
>>> fresnelc(I*oo)
I/2
>>> fresnelc(-I*oo)
-I/2
In general one can pull out factors of -1 and `i` from the argument:
>>> fresnelc(-z)
-fresnelc(z)
>>> fresnelc(I*z)
I*fresnelc(z)
The Fresnel C integral obeys the mirror symmetry
`\overline{C(z)} = C(\bar{z})`:
>>> from sympy import conjugate
>>> conjugate(fresnelc(z))
fresnelc(conjugate(z))
Differentiation with respect to `z` is supported:
>>> from sympy import diff
>>> diff(fresnelc(z), z)
cos(pi*z**2/2)
Defining the Fresnel functions via an integral
>>> from sympy import integrate, pi, cos, gamma, expand_func
>>> integrate(cos(pi*z**2/2), z)
fresnelc(z)*gamma(1/4)/(4*gamma(5/4))
>>> expand_func(integrate(cos(pi*z**2/2), z))
fresnelc(z)
We can numerically evaluate the Fresnel integral to arbitrary precision
on the whole complex plane:
>>> fresnelc(2).evalf(30)
0.488253406075340754500223503357
>>> fresnelc(-2*I).evalf(30)
-0.488253406075340754500223503357*I
See Also
========
fresnels: Fresnel sine integral.
References
==========
.. [1] https://en.wikipedia.org/wiki/Fresnel_integral
.. [2] http://dlmf.nist.gov/7
.. [3] http://mathworld.wolfram.com/FresnelIntegrals.html
.. [4] http://functions.wolfram.com/GammaBetaErf/FresnelC
.. [5] The converging factors for the fresnel integrals
by John W. Wrench Jr. and Vicki Alley
"""
_trigfunc = cos
_sign = S.One
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 1:
p = previous_terms[-1]
return (-pi**2*x**4*(4*n - 3)/(8*n*(2*n - 1)*(4*n + 1))) * p
else:
return x * (-x**4)**n * (S(2)**(-2*n)*pi**(2*n)) / ((4*n + 1)*factorial(2*n))
def _eval_rewrite_as_erf(self, z, **kwargs):
return (S.One - I)/4 * (erf((S.One + I)/2*sqrt(pi)*z) + I*erf((S.One - I)/2*sqrt(pi)*z))
def _eval_rewrite_as_hyper(self, z, **kwargs):
return z * hyper([S.One/4], [S.One/2, S(5)/4], -pi**2*z**4/16)
def _eval_rewrite_as_meijerg(self, z, **kwargs):
return (pi*z**(S(3)/4) / (sqrt(2)*root(z**2, 4)*root(-z, 4))
* meijerg([], [1], [S(1)/4], [S(3)/4, 0], -pi**2*z**4/16))
def _eval_aseries(self, n, args0, x, logx):
from sympy import Order
point = args0[0]
# Expansion at oo
if point in [S.Infinity, -S.Infinity]:
z = self.args[0]
# expansion of C(x) = C1(x*sqrt(pi/2)), see reference[5] page 1-8
# as only real infinities are dealt with, sin and cos are O(1)
p = [(-1)**k * factorial(4*k + 1) /
(2**(2*k + 2) * z**(4*k + 3) * 2**(2*k)*factorial(2*k))
for k in range(0, n) if 4*k + 3 < n]
q = [1/(2*z)] + [(-1)**k * factorial(4*k - 1) /
(2**(2*k + 1) * z**(4*k + 1) * 2**(2*k - 1)*factorial(2*k - 1))
for k in range(1, n) if 4*k + 1 < n]
p = [-sqrt(2/pi)*t for t in p]
q = [ sqrt(2/pi)*t for t in q]
s = 1 if point is S.Infinity else -1
# The expansion at oo is 1/2 + some odd powers of z
# To get the expansion at -oo, replace z by -z and flip the sign
# The result -1/2 + the same odd powers of z as before.
return s*S.Half + (cos(z**2)*Add(*p) + sin(z**2)*Add(*q)
).subs(x, sqrt(2/pi)*x) + Order(1/z**n, x)
# All other points are not handled
return super(fresnelc, self)._eval_aseries(n, args0, x, logx)
###############################################################################
#################### HELPER FUNCTIONS #########################################
###############################################################################
class _erfs(Function):
"""
Helper function to make the `\\mathrm{erf}(z)` function
tractable for the Gruntz algorithm.
"""
def _eval_aseries(self, n, args0, x, logx):
from sympy import Order
point = args0[0]
# Expansion at oo
if point is S.Infinity:
z = self.args[0]
l = [ 1/sqrt(S.Pi) * factorial(2*k)*(-S(
4))**(-k)/factorial(k) * (1/z)**(2*k + 1) for k in range(0, n) ]
o = Order(1/z**(2*n + 1), x)
# It is very inefficient to first add the order and then do the nseries
return (Add(*l))._eval_nseries(x, n, logx) + o
# Expansion at I*oo
t = point.extract_multiplicatively(S.ImaginaryUnit)
if t is S.Infinity:
z = self.args[0]
# TODO: is the series really correct?
l = [ 1/sqrt(S.Pi) * factorial(2*k)*(-S(
4))**(-k)/factorial(k) * (1/z)**(2*k + 1) for k in range(0, n) ]
o = Order(1/z**(2*n + 1), x)
# It is very inefficient to first add the order and then do the nseries
return (Add(*l))._eval_nseries(x, n, logx) + o
# All other points are not handled
return super(_erfs, self)._eval_aseries(n, args0, x, logx)
def fdiff(self, argindex=1):
if argindex == 1:
z = self.args[0]
return -2/sqrt(S.Pi) + 2*z*_erfs(z)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_intractable(self, z, **kwargs):
return (S.One - erf(z))*exp(z**2)
class _eis(Function):
"""
Helper function to make the `\\mathrm{Ei}(z)` and `\\mathrm{li}(z)` functions
tractable for the Gruntz algorithm.
"""
def _eval_aseries(self, n, args0, x, logx):
from sympy import Order
if args0[0] != S.Infinity:
return super(_erfs, self)._eval_aseries(n, args0, x, logx)
z = self.args[0]
l = [ factorial(k) * (1/z)**(k + 1) for k in range(0, n) ]
o = Order(1/z**(n + 1), x)
# It is very inefficient to first add the order and then do the nseries
return (Add(*l))._eval_nseries(x, n, logx) + o
def fdiff(self, argindex=1):
if argindex == 1:
z = self.args[0]
return S.One / z - _eis(z)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_intractable(self, z, **kwargs):
return exp(-z)*Ei(z)
def _eval_nseries(self, x, n, logx):
x0 = self.args[0].limit(x, 0)
if x0 is S.Zero:
f = self._eval_rewrite_as_intractable(*self.args)
return f._eval_nseries(x, n, logx)
return super(_eis, self)._eval_nseries(x, n, logx)
|
09f74c1c4f777fa8f267ce25c3500b85b69757238c634b3161ca98a442c56ef1
|
"""
This module mainly implements special orthogonal polynomials.
See also functions.combinatorial.numbers which contains some
combinatorial polynomials.
"""
from __future__ import print_function, division
from sympy.core import Rational
from sympy.core.function import Function, ArgumentIndexError
from sympy.core.singleton import S
from sympy.core.symbol import Dummy
from sympy.functions.combinatorial.factorials import binomial, factorial, RisingFactorial
from sympy.functions.elementary.complexes import re
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import cos
from sympy.functions.special.gamma_functions import gamma
from sympy.functions.special.hyper import hyper
from sympy.polys.orthopolys import (
jacobi_poly,
gegenbauer_poly,
chebyshevt_poly,
chebyshevu_poly,
laguerre_poly,
hermite_poly,
legendre_poly
)
_x = Dummy('x')
class OrthogonalPolynomial(Function):
"""Base class for orthogonal polynomials.
"""
@classmethod
def _eval_at_order(cls, n, x):
if n.is_integer and n >= 0:
return cls._ortho_poly(int(n), _x).subs(_x, x)
def _eval_conjugate(self):
return self.func(self.args[0], self.args[1].conjugate())
#----------------------------------------------------------------------------
# Jacobi polynomials
#
class jacobi(OrthogonalPolynomial):
r"""
Jacobi polynomial :math:`P_n^{\left(\alpha, \beta\right)}(x)`
jacobi(n, alpha, beta, x) gives the nth Jacobi polynomial
in x, :math:`P_n^{\left(\alpha, \beta\right)}(x)`.
The Jacobi polynomials are orthogonal on :math:`[-1, 1]` with respect
to the weight :math:`\left(1-x\right)^\alpha \left(1+x\right)^\beta`.
Examples
========
>>> from sympy import jacobi, S, conjugate, diff
>>> from sympy.abc import n,a,b,x
>>> jacobi(0, a, b, x)
1
>>> jacobi(1, a, b, x)
a/2 - b/2 + x*(a/2 + b/2 + 1)
>>> jacobi(2, a, b, x) # doctest:+SKIP
(a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 + x**2*(a**2/8 + a*b/4 + 7*a/8 +
b**2/8 + 7*b/8 + 3/2) + x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - 1/2)
>>> jacobi(n, a, b, x)
jacobi(n, a, b, x)
>>> jacobi(n, a, a, x)
RisingFactorial(a + 1, n)*gegenbauer(n,
a + 1/2, x)/RisingFactorial(2*a + 1, n)
>>> jacobi(n, 0, 0, x)
legendre(n, x)
>>> jacobi(n, S(1)/2, S(1)/2, x)
RisingFactorial(3/2, n)*chebyshevu(n, x)/factorial(n + 1)
>>> jacobi(n, -S(1)/2, -S(1)/2, x)
RisingFactorial(1/2, n)*chebyshevt(n, x)/factorial(n)
>>> jacobi(n, a, b, -x)
(-1)**n*jacobi(n, b, a, x)
>>> jacobi(n, a, b, 0)
2**(-n)*gamma(a + n + 1)*hyper((-b - n, -n), (a + 1,), -1)/(factorial(n)*gamma(a + 1))
>>> jacobi(n, a, b, 1)
RisingFactorial(a + 1, n)/factorial(n)
>>> conjugate(jacobi(n, a, b, x))
jacobi(n, conjugate(a), conjugate(b), conjugate(x))
>>> diff(jacobi(n,a,b,x), x)
(a/2 + b/2 + n/2 + 1/2)*jacobi(n - 1, a + 1, b + 1, x)
See Also
========
gegenbauer,
chebyshevt_root, chebyshevu, chebyshevu_root,
legendre, assoc_legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly,
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials
.. [2] http://mathworld.wolfram.com/JacobiPolynomial.html
.. [3] http://functions.wolfram.com/Polynomials/JacobiP/
"""
@classmethod
def eval(cls, n, a, b, x):
# Simplify to other polynomials
# P^{a, a}_n(x)
if a == b:
if a == -S.Half:
return RisingFactorial(S.Half, n) / factorial(n) * chebyshevt(n, x)
elif a == S.Zero:
return legendre(n, x)
elif a == S.Half:
return RisingFactorial(3*S.Half, n) / factorial(n + 1) * chebyshevu(n, x)
else:
return RisingFactorial(a + 1, n) / RisingFactorial(2*a + 1, n) * gegenbauer(n, a + S.Half, x)
elif b == -a:
# P^{a, -a}_n(x)
return gamma(n + a + 1) / gamma(n + 1) * (1 + x)**(a/2) / (1 - x)**(a/2) * assoc_legendre(n, -a, x)
elif a == -b:
# P^{-b, b}_n(x)
return gamma(n - b + 1) / gamma(n + 1) * (1 - x)**(b/2) / (1 + x)**(b/2) * assoc_legendre(n, b, x)
if not n.is_Number:
# Symbolic result P^{a,b}_n(x)
# P^{a,b}_n(-x) ---> (-1)**n * P^{b,a}_n(-x)
if x.could_extract_minus_sign():
return S.NegativeOne**n * jacobi(n, b, a, -x)
# We can evaluate for some special values of x
if x == S.Zero:
return (2**(-n) * gamma(a + n + 1) / (gamma(a + 1) * factorial(n)) *
hyper([-b - n, -n], [a + 1], -1))
if x == S.One:
return RisingFactorial(a + 1, n) / factorial(n)
elif x == S.Infinity:
if n.is_positive:
# Make sure a+b+2*n \notin Z
if (a + b + 2*n).is_integer:
raise ValueError("Error. a + b + 2*n should not be an integer.")
return RisingFactorial(a + b + n + 1, n) * S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
return jacobi_poly(n, a, b, x)
def fdiff(self, argindex=4):
from sympy import Sum
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt a
n, a, b, x = self.args
k = Dummy("k")
f1 = 1 / (a + b + n + k + 1)
f2 = ((a + b + 2*k + 1) * RisingFactorial(b + k + 1, n - k) /
((n - k) * RisingFactorial(a + b + k + 1, n - k)))
return Sum(f1 * (jacobi(n, a, b, x) + f2*jacobi(k, a, b, x)), (k, 0, n - 1))
elif argindex == 3:
# Diff wrt b
n, a, b, x = self.args
k = Dummy("k")
f1 = 1 / (a + b + n + k + 1)
f2 = (-1)**(n - k) * ((a + b + 2*k + 1) * RisingFactorial(a + k + 1, n - k) /
((n - k) * RisingFactorial(a + b + k + 1, n - k)))
return Sum(f1 * (jacobi(n, a, b, x) + f2*jacobi(k, a, b, x)), (k, 0, n - 1))
elif argindex == 4:
# Diff wrt x
n, a, b, x = self.args
return S.Half * (a + b + n + 1) * jacobi(n - 1, a + 1, b + 1, x)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, a, b, x, **kwargs):
from sympy import Sum
# Make sure n \in N
if n.is_negative or n.is_integer is False:
raise ValueError("Error: n should be a non-negative integer.")
k = Dummy("k")
kern = (RisingFactorial(-n, k) * RisingFactorial(a + b + n + 1, k) * RisingFactorial(a + k + 1, n - k) /
factorial(k) * ((1 - x)/2)**k)
return 1 / factorial(n) * Sum(kern, (k, 0, n))
def _eval_conjugate(self):
n, a, b, x = self.args
return self.func(n, a.conjugate(), b.conjugate(), x.conjugate())
def jacobi_normalized(n, a, b, x):
r"""
Jacobi polynomial :math:`P_n^{\left(\alpha, \beta\right)}(x)`
jacobi_normalized(n, alpha, beta, x) gives the nth Jacobi polynomial
in x, :math:`P_n^{\left(\alpha, \beta\right)}(x)`.
The Jacobi polynomials are orthogonal on :math:`[-1, 1]` with respect
to the weight :math:`\left(1-x\right)^\alpha \left(1+x\right)^\beta`.
This functions returns the polynomials normilzed:
.. math::
\int_{-1}^{1}
P_m^{\left(\alpha, \beta\right)}(x)
P_n^{\left(\alpha, \beta\right)}(x)
(1-x)^{\alpha} (1+x)^{\beta} \mathrm{d}x
= \delta_{m,n}
Examples
========
>>> from sympy import jacobi_normalized
>>> from sympy.abc import n,a,b,x
>>> jacobi_normalized(n, a, b, x)
jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)/((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1)))
See Also
========
gegenbauer,
chebyshevt_root, chebyshevu, chebyshevu_root,
legendre, assoc_legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly,
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials
.. [2] http://mathworld.wolfram.com/JacobiPolynomial.html
.. [3] http://functions.wolfram.com/Polynomials/JacobiP/
"""
nfactor = (S(2)**(a + b + 1) * (gamma(n + a + 1) * gamma(n + b + 1))
/ (2*n + a + b + 1) / (factorial(n) * gamma(n + a + b + 1)))
return jacobi(n, a, b, x) / sqrt(nfactor)
#----------------------------------------------------------------------------
# Gegenbauer polynomials
#
class gegenbauer(OrthogonalPolynomial):
r"""
Gegenbauer polynomial :math:`C_n^{\left(\alpha\right)}(x)`
gegenbauer(n, alpha, x) gives the nth Gegenbauer polynomial
in x, :math:`C_n^{\left(\alpha\right)}(x)`.
The Gegenbauer polynomials are orthogonal on :math:`[-1, 1]` with
respect to the weight :math:`\left(1-x^2\right)^{\alpha-\frac{1}{2}}`.
Examples
========
>>> from sympy import gegenbauer, conjugate, diff
>>> from sympy.abc import n,a,x
>>> gegenbauer(0, a, x)
1
>>> gegenbauer(1, a, x)
2*a*x
>>> gegenbauer(2, a, x)
-a + x**2*(2*a**2 + 2*a)
>>> gegenbauer(3, a, x)
x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)
>>> gegenbauer(n, a, x)
gegenbauer(n, a, x)
>>> gegenbauer(n, a, -x)
(-1)**n*gegenbauer(n, a, x)
>>> gegenbauer(n, a, 0)
2**n*sqrt(pi)*gamma(a + n/2)/(gamma(a)*gamma(1/2 - n/2)*gamma(n + 1))
>>> gegenbauer(n, a, 1)
gamma(2*a + n)/(gamma(2*a)*gamma(n + 1))
>>> conjugate(gegenbauer(n, a, x))
gegenbauer(n, conjugate(a), conjugate(x))
>>> diff(gegenbauer(n, a, x), x)
2*a*gegenbauer(n - 1, a + 1, x)
See Also
========
jacobi,
chebyshevt_root, chebyshevu, chebyshevu_root,
legendre, assoc_legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Gegenbauer_polynomials
.. [2] http://mathworld.wolfram.com/GegenbauerPolynomial.html
.. [3] http://functions.wolfram.com/Polynomials/GegenbauerC3/
"""
@classmethod
def eval(cls, n, a, x):
# For negative n the polynomials vanish
# See http://functions.wolfram.com/Polynomials/GegenbauerC3/03/01/03/0012/
if n.is_negative:
return S.Zero
# Some special values for fixed a
if a == S.Half:
return legendre(n, x)
elif a == S.One:
return chebyshevu(n, x)
elif a == S.NegativeOne:
return S.Zero
if not n.is_Number:
# Handle this before the general sign extraction rule
if x == S.NegativeOne:
if (re(a) > S.Half) == True:
return S.ComplexInfinity
else:
# No sec function available yet
#return (cos(S.Pi*(a+n)) * sec(S.Pi*a) * gamma(2*a+n) /
# (gamma(2*a) * gamma(n+1)))
return None
# Symbolic result C^a_n(x)
# C^a_n(-x) ---> (-1)**n * C^a_n(x)
if x.could_extract_minus_sign():
return S.NegativeOne**n * gegenbauer(n, a, -x)
# We can evaluate for some special values of x
if x == S.Zero:
return (2**n * sqrt(S.Pi) * gamma(a + S.Half*n) /
(gamma((1 - n)/2) * gamma(n + 1) * gamma(a)) )
if x == S.One:
return gamma(2*a + n) / (gamma(2*a) * gamma(n + 1))
elif x == S.Infinity:
if n.is_positive:
return RisingFactorial(a, n) * S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
return gegenbauer_poly(n, a, x)
def fdiff(self, argindex=3):
from sympy import Sum
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt a
n, a, x = self.args
k = Dummy("k")
factor1 = 2 * (1 + (-1)**(n - k)) * (k + a) / ((k +
n + 2*a) * (n - k))
factor2 = 2*(k + 1) / ((k + 2*a) * (2*k + 2*a + 1)) + \
2 / (k + n + 2*a)
kern = factor1*gegenbauer(k, a, x) + factor2*gegenbauer(n, a, x)
return Sum(kern, (k, 0, n - 1))
elif argindex == 3:
# Diff wrt x
n, a, x = self.args
return 2*a*gegenbauer(n - 1, a + 1, x)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, a, x, **kwargs):
from sympy import Sum
k = Dummy("k")
kern = ((-1)**k * RisingFactorial(a, n - k) * (2*x)**(n - 2*k) /
(factorial(k) * factorial(n - 2*k)))
return Sum(kern, (k, 0, floor(n/2)))
def _eval_conjugate(self):
n, a, x = self.args
return self.func(n, a.conjugate(), x.conjugate())
#----------------------------------------------------------------------------
# Chebyshev polynomials of first and second kind
#
class chebyshevt(OrthogonalPolynomial):
r"""
Chebyshev polynomial of the first kind, :math:`T_n(x)`
chebyshevt(n, x) gives the nth Chebyshev polynomial (of the first
kind) in x, :math:`T_n(x)`.
The Chebyshev polynomials of the first kind are orthogonal on
:math:`[-1, 1]` with respect to the weight :math:`\frac{1}{\sqrt{1-x^2}}`.
Examples
========
>>> from sympy import chebyshevt, chebyshevu, diff
>>> from sympy.abc import n,x
>>> chebyshevt(0, x)
1
>>> chebyshevt(1, x)
x
>>> chebyshevt(2, x)
2*x**2 - 1
>>> chebyshevt(n, x)
chebyshevt(n, x)
>>> chebyshevt(n, -x)
(-1)**n*chebyshevt(n, x)
>>> chebyshevt(-n, x)
chebyshevt(n, x)
>>> chebyshevt(n, 0)
cos(pi*n/2)
>>> chebyshevt(n, -1)
(-1)**n
>>> diff(chebyshevt(n, x), x)
n*chebyshevu(n - 1, x)
See Also
========
jacobi, gegenbauer,
chebyshevt_root, chebyshevu, chebyshevu_root,
legendre, assoc_legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial
.. [2] http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
.. [3] http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
.. [4] http://functions.wolfram.com/Polynomials/ChebyshevT/
.. [5] http://functions.wolfram.com/Polynomials/ChebyshevU/
"""
_ortho_poly = staticmethod(chebyshevt_poly)
@classmethod
def eval(cls, n, x):
if not n.is_Number:
# Symbolic result T_n(x)
# T_n(-x) ---> (-1)**n * T_n(x)
if x.could_extract_minus_sign():
return S.NegativeOne**n * chebyshevt(n, -x)
# T_{-n}(x) ---> T_n(x)
if n.could_extract_minus_sign():
return chebyshevt(-n, x)
# We can evaluate for some special values of x
if x == S.Zero:
return cos(S.Half * S.Pi * n)
if x == S.One:
return S.One
elif x == S.Infinity:
return S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
if n.is_negative:
# T_{-n}(x) == T_n(x)
return cls._eval_at_order(-n, x)
else:
return cls._eval_at_order(n, x)
def fdiff(self, argindex=2):
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt x
n, x = self.args
return n * chebyshevu(n - 1, x)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
from sympy import Sum
k = Dummy("k")
kern = binomial(n, 2*k) * (x**2 - 1)**k * x**(n - 2*k)
return Sum(kern, (k, 0, floor(n/2)))
class chebyshevu(OrthogonalPolynomial):
r"""
Chebyshev polynomial of the second kind, :math:`U_n(x)`
chebyshevu(n, x) gives the nth Chebyshev polynomial of the second
kind in x, :math:`U_n(x)`.
The Chebyshev polynomials of the second kind are orthogonal on
:math:`[-1, 1]` with respect to the weight :math:`\sqrt{1-x^2}`.
Examples
========
>>> from sympy import chebyshevt, chebyshevu, diff
>>> from sympy.abc import n,x
>>> chebyshevu(0, x)
1
>>> chebyshevu(1, x)
2*x
>>> chebyshevu(2, x)
4*x**2 - 1
>>> chebyshevu(n, x)
chebyshevu(n, x)
>>> chebyshevu(n, -x)
(-1)**n*chebyshevu(n, x)
>>> chebyshevu(-n, x)
-chebyshevu(n - 2, x)
>>> chebyshevu(n, 0)
cos(pi*n/2)
>>> chebyshevu(n, 1)
n + 1
>>> diff(chebyshevu(n, x), x)
(-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1)
See Also
========
jacobi, gegenbauer,
chebyshevt, chebyshevt_root, chebyshevu_root,
legendre, assoc_legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial
.. [2] http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
.. [3] http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
.. [4] http://functions.wolfram.com/Polynomials/ChebyshevT/
.. [5] http://functions.wolfram.com/Polynomials/ChebyshevU/
"""
_ortho_poly = staticmethod(chebyshevu_poly)
@classmethod
def eval(cls, n, x):
if not n.is_Number:
# Symbolic result U_n(x)
# U_n(-x) ---> (-1)**n * U_n(x)
if x.could_extract_minus_sign():
return S.NegativeOne**n * chebyshevu(n, -x)
# U_{-n}(x) ---> -U_{n-2}(x)
if n.could_extract_minus_sign():
if n == S.NegativeOne:
return S.Zero
else:
return -chebyshevu(-n - 2, x)
# We can evaluate for some special values of x
if x == S.Zero:
return cos(S.Half * S.Pi * n)
if x == S.One:
return S.One + n
elif x == S.Infinity:
return S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
if n.is_negative:
# U_{-n}(x) ---> -U_{n-2}(x)
if n == S.NegativeOne:
return S.Zero
else:
return -cls._eval_at_order(-n - 2, x)
else:
return cls._eval_at_order(n, x)
def fdiff(self, argindex=2):
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt x
n, x = self.args
return ((n + 1) * chebyshevt(n + 1, x) - x * chebyshevu(n, x)) / (x**2 - 1)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
from sympy import Sum
k = Dummy("k")
kern = S.NegativeOne**k * factorial(
n - k) * (2*x)**(n - 2*k) / (factorial(k) * factorial(n - 2*k))
return Sum(kern, (k, 0, floor(n/2)))
class chebyshevt_root(Function):
r"""
chebyshev_root(n, k) returns the kth root (indexed from zero) of
the nth Chebyshev polynomial of the first kind; that is, if
0 <= k < n, chebyshevt(n, chebyshevt_root(n, k)) == 0.
Examples
========
>>> from sympy import chebyshevt, chebyshevt_root
>>> chebyshevt_root(3, 2)
-sqrt(3)/2
>>> chebyshevt(3, chebyshevt_root(3, 2))
0
See Also
========
jacobi, gegenbauer,
chebyshevt, chebyshevu, chebyshevu_root,
legendre, assoc_legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
"""
@classmethod
def eval(cls, n, k):
if not ((0 <= k) and (k < n)):
raise ValueError("must have 0 <= k < n, "
"got k = %s and n = %s" % (k, n))
return cos(S.Pi*(2*k + 1)/(2*n))
class chebyshevu_root(Function):
r"""
chebyshevu_root(n, k) returns the kth root (indexed from zero) of the
nth Chebyshev polynomial of the second kind; that is, if 0 <= k < n,
chebyshevu(n, chebyshevu_root(n, k)) == 0.
Examples
========
>>> from sympy import chebyshevu, chebyshevu_root
>>> chebyshevu_root(3, 2)
-sqrt(2)/2
>>> chebyshevu(3, chebyshevu_root(3, 2))
0
See Also
========
chebyshevt, chebyshevt_root, chebyshevu,
legendre, assoc_legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
"""
@classmethod
def eval(cls, n, k):
if not ((0 <= k) and (k < n)):
raise ValueError("must have 0 <= k < n, "
"got k = %s and n = %s" % (k, n))
return cos(S.Pi*(k + 1)/(n + 1))
#----------------------------------------------------------------------------
# Legendre polynomials and Associated Legendre polynomials
#
class legendre(OrthogonalPolynomial):
r"""
legendre(n, x) gives the nth Legendre polynomial of x, :math:`P_n(x)`
The Legendre polynomials are orthogonal on [-1, 1] with respect to
the constant weight 1. They satisfy :math:`P_n(1) = 1` for all n; further,
:math:`P_n` is odd for odd n and even for even n.
Examples
========
>>> from sympy import legendre, diff
>>> from sympy.abc import x, n
>>> legendre(0, x)
1
>>> legendre(1, x)
x
>>> legendre(2, x)
3*x**2/2 - 1/2
>>> legendre(n, x)
legendre(n, x)
>>> diff(legendre(n,x), x)
n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1)
See Also
========
jacobi, gegenbauer,
chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
assoc_legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Legendre_polynomial
.. [2] http://mathworld.wolfram.com/LegendrePolynomial.html
.. [3] http://functions.wolfram.com/Polynomials/LegendreP/
.. [4] http://functions.wolfram.com/Polynomials/LegendreP2/
"""
_ortho_poly = staticmethod(legendre_poly)
@classmethod
def eval(cls, n, x):
if not n.is_Number:
# Symbolic result L_n(x)
# L_n(-x) ---> (-1)**n * L_n(x)
if x.could_extract_minus_sign():
return S.NegativeOne**n * legendre(n, -x)
# L_{-n}(x) ---> L_{n-1}(x)
if n.could_extract_minus_sign():
return legendre(-n - S.One, x)
# We can evaluate for some special values of x
if x == S.Zero:
return sqrt(S.Pi)/(gamma(S.Half - n/2)*gamma(S.One + n/2))
elif x == S.One:
return S.One
elif x == S.Infinity:
return S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial;
# L_{-n}(x) ---> L_{n-1}(x)
if n.is_negative:
n = -n - S.One
return cls._eval_at_order(n, x)
def fdiff(self, argindex=2):
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt x
# Find better formula, this is unsuitable for x = 1
n, x = self.args
return n/(x**2 - 1)*(x*legendre(n, x) - legendre(n - 1, x))
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
from sympy import Sum
k = Dummy("k")
kern = (-1)**k*binomial(n, k)**2*((1 + x)/2)**(n - k)*((1 - x)/2)**k
return Sum(kern, (k, 0, n))
class assoc_legendre(Function):
r"""
assoc_legendre(n,m, x) gives :math:`P_n^m(x)`, where n and m are
the degree and order or an expression which is related to the nth
order Legendre polynomial, :math:`P_n(x)` in the following manner:
.. math::
P_n^m(x) = (-1)^m (1 - x^2)^{\frac{m}{2}}
\frac{\mathrm{d}^m P_n(x)}{\mathrm{d} x^m}
Associated Legendre polynomial are orthogonal on [-1, 1] with:
- weight = 1 for the same m, and different n.
- weight = 1/(1-x**2) for the same n, and different m.
Examples
========
>>> from sympy import assoc_legendre
>>> from sympy.abc import x, m, n
>>> assoc_legendre(0,0, x)
1
>>> assoc_legendre(1,0, x)
x
>>> assoc_legendre(1,1, x)
-sqrt(1 - x**2)
>>> assoc_legendre(n,m,x)
assoc_legendre(n, m, x)
See Also
========
jacobi, gegenbauer,
chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Associated_Legendre_polynomials
.. [2] http://mathworld.wolfram.com/LegendrePolynomial.html
.. [3] http://functions.wolfram.com/Polynomials/LegendreP/
.. [4] http://functions.wolfram.com/Polynomials/LegendreP2/
"""
@classmethod
def _eval_at_order(cls, n, m):
P = legendre_poly(n, _x, polys=True).diff((_x, m))
return (-1)**m * (1 - _x**2)**Rational(m, 2) * P.as_expr()
@classmethod
def eval(cls, n, m, x):
if m.could_extract_minus_sign():
# P^{-m}_n ---> F * P^m_n
return S.NegativeOne**(-m) * (factorial(m + n)/factorial(n - m)) * assoc_legendre(n, -m, x)
if m == 0:
# P^0_n ---> L_n
return legendre(n, x)
if x == 0:
return 2**m*sqrt(S.Pi) / (gamma((1 - m - n)/2)*gamma(1 - (m - n)/2))
if n.is_Number and m.is_Number and n.is_integer and m.is_integer:
if n.is_negative:
raise ValueError("%s : 1st index must be nonnegative integer (got %r)" % (cls, n))
if abs(m) > n:
raise ValueError("%s : abs('2nd index') must be <= '1st index' (got %r, %r)" % (cls, n, m))
return cls._eval_at_order(int(n), abs(int(m))).subs(_x, x)
def fdiff(self, argindex=3):
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt m
raise ArgumentIndexError(self, argindex)
elif argindex == 3:
# Diff wrt x
# Find better formula, this is unsuitable for x = 1
n, m, x = self.args
return 1/(x**2 - 1)*(x*n*assoc_legendre(n, m, x) - (m + n)*assoc_legendre(n - 1, m, x))
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, m, x, **kwargs):
from sympy import Sum
k = Dummy("k")
kern = factorial(2*n - 2*k)/(2**n*factorial(n - k)*factorial(
k)*factorial(n - 2*k - m))*(-1)**k*x**(n - m - 2*k)
return (1 - x**2)**(m/2) * Sum(kern, (k, 0, floor((n - m)*S.Half)))
def _eval_conjugate(self):
n, m, x = self.args
return self.func(n, m.conjugate(), x.conjugate())
#----------------------------------------------------------------------------
# Hermite polynomials
#
class hermite(OrthogonalPolynomial):
r"""
hermite(n, x) gives the nth Hermite polynomial in x, :math:`H_n(x)`
The Hermite polynomials are orthogonal on :math:`(-\infty, \infty)`
with respect to the weight :math:`\exp\left(-x^2\right)`.
Examples
========
>>> from sympy import hermite, diff
>>> from sympy.abc import x, n
>>> hermite(0, x)
1
>>> hermite(1, x)
2*x
>>> hermite(2, x)
4*x**2 - 2
>>> hermite(n, x)
hermite(n, x)
>>> diff(hermite(n,x), x)
2*n*hermite(n - 1, x)
>>> hermite(n, -x)
(-1)**n*hermite(n, x)
See Also
========
jacobi, gegenbauer,
chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
legendre, assoc_legendre,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Hermite_polynomial
.. [2] http://mathworld.wolfram.com/HermitePolynomial.html
.. [3] http://functions.wolfram.com/Polynomials/HermiteH/
"""
_ortho_poly = staticmethod(hermite_poly)
@classmethod
def eval(cls, n, x):
if not n.is_Number:
# Symbolic result H_n(x)
# H_n(-x) ---> (-1)**n * H_n(x)
if x.could_extract_minus_sign():
return S.NegativeOne**n * hermite(n, -x)
# We can evaluate for some special values of x
if x == S.Zero:
return 2**n * sqrt(S.Pi) / gamma((S.One - n)/2)
elif x == S.Infinity:
return S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
if n.is_negative:
raise ValueError(
"The index n must be nonnegative integer (got %r)" % n)
else:
return cls._eval_at_order(n, x)
def fdiff(self, argindex=2):
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt x
n, x = self.args
return 2*n*hermite(n - 1, x)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
from sympy import Sum
k = Dummy("k")
kern = (-1)**k / (factorial(k)*factorial(n - 2*k)) * (2*x)**(n - 2*k)
return factorial(n)*Sum(kern, (k, 0, floor(n/2)))
#----------------------------------------------------------------------------
# Laguerre polynomials
#
class laguerre(OrthogonalPolynomial):
r"""
Returns the nth Laguerre polynomial in x, :math:`L_n(x)`.
Parameters
==========
n : int
Degree of Laguerre polynomial. Must be ``n >= 0``.
Examples
========
>>> from sympy import laguerre, diff
>>> from sympy.abc import x, n
>>> laguerre(0, x)
1
>>> laguerre(1, x)
1 - x
>>> laguerre(2, x)
x**2/2 - 2*x + 1
>>> laguerre(3, x)
-x**3/6 + 3*x**2/2 - 3*x + 1
>>> laguerre(n, x)
laguerre(n, x)
>>> diff(laguerre(n, x), x)
-assoc_laguerre(n - 1, 1, x)
See Also
========
jacobi, gegenbauer,
chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
legendre, assoc_legendre,
hermite,
assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial
.. [2] http://mathworld.wolfram.com/LaguerrePolynomial.html
.. [3] http://functions.wolfram.com/Polynomials/LaguerreL/
.. [4] http://functions.wolfram.com/Polynomials/LaguerreL3/
"""
_ortho_poly = staticmethod(laguerre_poly)
@classmethod
def eval(cls, n, x):
if not n.is_Number:
# Symbolic result L_n(x)
# L_{n}(-x) ---> exp(-x) * L_{-n-1}(x)
# L_{-n}(x) ---> exp(x) * L_{n-1}(-x)
if n.could_extract_minus_sign():
return exp(x) * laguerre(n - 1, -x)
# We can evaluate for some special values of x
if x == S.Zero:
return S.One
elif x == S.NegativeInfinity:
return S.Infinity
elif x == S.Infinity:
return S.NegativeOne**n * S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
if n.is_negative:
raise ValueError(
"The index n must be nonnegative integer (got %r)" % n)
else:
return cls._eval_at_order(n, x)
def fdiff(self, argindex=2):
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt x
n, x = self.args
return -assoc_laguerre(n - 1, 1, x)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
from sympy import Sum
# Make sure n \in N_0
if n.is_negative or n.is_integer is False:
raise ValueError("Error: n should be a non-negative integer.")
k = Dummy("k")
kern = RisingFactorial(-n, k) / factorial(k)**2 * x**k
return Sum(kern, (k, 0, n))
class assoc_laguerre(OrthogonalPolynomial):
r"""
Returns the nth generalized Laguerre polynomial in x, :math:`L_n(x)`.
Parameters
==========
n : int
Degree of Laguerre polynomial. Must be ``n >= 0``.
alpha : Expr
Arbitrary expression. For ``alpha=0`` regular Laguerre
polynomials will be generated.
Examples
========
>>> from sympy import laguerre, assoc_laguerre, diff
>>> from sympy.abc import x, n, a
>>> assoc_laguerre(0, a, x)
1
>>> assoc_laguerre(1, a, x)
a - x + 1
>>> assoc_laguerre(2, a, x)
a**2/2 + 3*a/2 + x**2/2 + x*(-a - 2) + 1
>>> assoc_laguerre(3, a, x)
a**3/6 + a**2 + 11*a/6 - x**3/6 + x**2*(a/2 + 3/2) +
x*(-a**2/2 - 5*a/2 - 3) + 1
>>> assoc_laguerre(n, a, 0)
binomial(a + n, a)
>>> assoc_laguerre(n, a, x)
assoc_laguerre(n, a, x)
>>> assoc_laguerre(n, 0, x)
laguerre(n, x)
>>> diff(assoc_laguerre(n, a, x), x)
-assoc_laguerre(n - 1, a + 1, x)
>>> diff(assoc_laguerre(n, a, x), a)
Sum(assoc_laguerre(_k, a, x)/(-a + n), (_k, 0, n - 1))
See Also
========
jacobi, gegenbauer,
chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
legendre, assoc_legendre,
hermite,
laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial#Assoc_laguerre_polynomials
.. [2] http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html
.. [3] http://functions.wolfram.com/Polynomials/LaguerreL/
.. [4] http://functions.wolfram.com/Polynomials/LaguerreL3/
"""
@classmethod
def eval(cls, n, alpha, x):
# L_{n}^{0}(x) ---> L_{n}(x)
if alpha == S.Zero:
return laguerre(n, x)
if not n.is_Number:
# We can evaluate for some special values of x
if x == S.Zero:
return binomial(n + alpha, alpha)
elif x == S.Infinity and n > S.Zero:
return S.NegativeOne**n * S.Infinity
elif x == S.NegativeInfinity and n > S.Zero:
return S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
if n.is_negative:
raise ValueError(
"The index n must be nonnegative integer (got %r)" % n)
else:
return laguerre_poly(n, x, alpha)
def fdiff(self, argindex=3):
from sympy import Sum
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt alpha
n, alpha, x = self.args
k = Dummy("k")
return Sum(assoc_laguerre(k, alpha, x) / (n - alpha), (k, 0, n - 1))
elif argindex == 3:
# Diff wrt x
n, alpha, x = self.args
return -assoc_laguerre(n - 1, alpha + 1, x)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
from sympy import Sum
# Make sure n \in N_0
if n.is_negative or n.is_integer is False:
raise ValueError("Error: n should be a non-negative integer.")
k = Dummy("k")
kern = RisingFactorial(
-n, k) / (gamma(k + alpha + 1) * factorial(k)) * x**k
return gamma(n + alpha + 1) / factorial(n) * Sum(kern, (k, 0, n))
def _eval_conjugate(self):
n, alpha, x = self.args
return self.func(n, alpha.conjugate(), x.conjugate())
|
da2ca0f62743cad571923b669ac310119c9abc79b8081aca2b816f0f78fb72ae
|
from sympy import (S, Symbol, symbols, factorial, factorial2, Float, binomial,
rf, ff, gamma, polygamma, EulerGamma, O, pi, nan,
oo, zoo, simplify, expand_func, Product, Mul, Piecewise, Mod,
Eq, sqrt, Poly)
from sympy.functions.combinatorial.factorials import subfactorial
from sympy.functions.special.gamma_functions import uppergamma
from sympy.utilities.pytest import XFAIL, raises, slow
#Solves and Fixes Issue #10388 - This is the updated test for the same solved issue
def test_rf_eval_apply():
x, y = symbols('x,y')
n, k = symbols('n k', integer=True)
m = Symbol('m', integer=True, nonnegative=True)
assert rf(nan, y) == nan
assert rf(x, nan) == nan
assert rf(x, y) == rf(x, y)
assert rf(oo, 0) == 1
assert rf(-oo, 0) == 1
assert rf(oo, 6) == oo
assert rf(-oo, 7) == -oo
assert rf(oo, -6) == oo
assert rf(-oo, -7) == oo
assert rf(x, 0) == 1
assert rf(x, 1) == x
assert rf(x, 2) == x*(x + 1)
assert rf(x, 3) == x*(x + 1)*(x + 2)
assert rf(x, 5) == x*(x + 1)*(x + 2)*(x + 3)*(x + 4)
assert rf(x, -1) == 1/(x - 1)
assert rf(x, -2) == 1/((x - 1)*(x - 2))
assert rf(x, -3) == 1/((x - 1)*(x - 2)*(x - 3))
assert rf(1, 100) == factorial(100)
assert rf(x**2 + 3*x, 2) == (x**2 + 3*x)*(x**2 + 3*x + 1)
assert isinstance(rf(x**2 + 3*x, 2), Mul)
assert rf(x**3 + x, -2) == 1/((x**3 + x - 1)*(x**3 + x - 2))
assert rf(Poly(x**2 + 3*x, x), 2) == Poly(x**4 + 8*x**3 + 19*x**2 + 12*x, x)
assert isinstance(rf(Poly(x**2 + 3*x, x), 2), Poly)
raises(ValueError, lambda: rf(Poly(x**2 + 3*x, x, y), 2))
assert rf(Poly(x**3 + x, x), -2) == 1/(x**6 - 9*x**5 + 35*x**4 - 75*x**3 + 94*x**2 - 66*x + 20)
raises(ValueError, lambda: rf(Poly(x**3 + x, x, y), -2))
assert rf(x, m).is_integer is None
assert rf(n, k).is_integer is None
assert rf(n, m).is_integer is True
assert rf(n, k + pi).is_integer is False
assert rf(n, m + pi).is_integer is False
assert rf(pi, m).is_integer is False
assert rf(x, k).rewrite(ff) == ff(x + k - 1, k)
assert rf(x, k).rewrite(binomial) == factorial(k)*binomial(x + k - 1, k)
assert rf(n, k).rewrite(factorial) == \
factorial(n + k - 1) / factorial(n - 1)
import random
from mpmath import rf as mpmath_rf
for i in range(100):
x = -500 + 500 * random.random()
k = -500 + 500 * random.random()
assert (abs(mpmath_rf(x, k) - rf(x, k)) < 10**(-15))
def test_ff_eval_apply():
x, y = symbols('x,y')
n, k = symbols('n k', integer=True)
m = Symbol('m', integer=True, nonnegative=True)
assert ff(nan, y) == nan
assert ff(x, nan) == nan
assert ff(x, y) == ff(x, y)
assert ff(oo, 0) == 1
assert ff(-oo, 0) == 1
assert ff(oo, 6) == oo
assert ff(-oo, 7) == -oo
assert ff(oo, -6) == oo
assert ff(-oo, -7) == oo
assert ff(x, 0) == 1
assert ff(x, 1) == x
assert ff(x, 2) == x*(x - 1)
assert ff(x, 3) == x*(x - 1)*(x - 2)
assert ff(x, 5) == x*(x - 1)*(x - 2)*(x - 3)*(x - 4)
assert ff(x, -1) == 1/(x + 1)
assert ff(x, -2) == 1/((x + 1)*(x + 2))
assert ff(x, -3) == 1/((x + 1)*(x + 2)*(x + 3))
assert ff(100, 100) == factorial(100)
assert ff(2*x**2 - 5*x, 2) == (2*x**2 - 5*x)*(2*x**2 - 5*x - 1)
assert isinstance(ff(2*x**2 - 5*x, 2), Mul)
assert ff(x**2 + 3*x, -2) == 1/((x**2 + 3*x + 1)*(x**2 + 3*x + 2))
assert ff(Poly(2*x**2 - 5*x, x), 2) == Poly(4*x**4 - 28*x**3 + 59*x**2 - 35*x, x)
assert isinstance(ff(Poly(2*x**2 - 5*x, x), 2), Poly)
raises(ValueError, lambda: ff(Poly(2*x**2 - 5*x, x, y), 2))
assert ff(Poly(x**2 + 3*x, x), -2) == 1/(x**4 + 12*x**3 + 49*x**2 + 78*x + 40)
raises(ValueError, lambda: ff(Poly(x**2 + 3*x, x, y), -2))
assert ff(x, m).is_integer is None
assert ff(n, k).is_integer is None
assert ff(n, m).is_integer is True
assert ff(n, k + pi).is_integer is False
assert ff(n, m + pi).is_integer is False
assert ff(pi, m).is_integer is False
assert isinstance(ff(x, x), ff)
assert ff(n, n) == factorial(n)
assert ff(x, k).rewrite(rf) == rf(x - k + 1, k)
assert ff(x, k).rewrite(gamma) == (-1)**k*gamma(k - x) / gamma(-x)
assert ff(n, k).rewrite(factorial) == factorial(n) / factorial(n - k)
assert ff(x, k).rewrite(binomial) == factorial(k) * binomial(x, k)
import random
from mpmath import ff as mpmath_ff
for i in range(100):
x = -500 + 500 * random.random()
k = -500 + 500 * random.random()
assert (abs(mpmath_ff(x, k) - ff(x, k)) < 10**(-15))
def test_rf_ff_eval_hiprec():
maple = Float('6.9109401292234329956525265438452')
us = ff(18, S(2)/3).evalf(32)
assert abs(us - maple)/us < 1e-31
maple = Float('6.8261540131125511557924466355367')
us = rf(18, S(2)/3).evalf(32)
assert abs(us - maple)/us < 1e-31
maple = Float('34.007346127440197150854651814225')
us = rf(Float('4.4', 32), Float('2.2', 32));
assert abs(us - maple)/us < 1e-31
def test_rf_lambdify_mpmath():
from sympy import lambdify
x, y = symbols('x,y')
f = lambdify((x,y), rf(x, y), 'mpmath')
maple = Float('34.007346127440197')
us = f(4.4, 2.2)
assert abs(us - maple)/us < 1e-15
def test_factorial():
x = Symbol('x')
n = Symbol('n', integer=True)
k = Symbol('k', integer=True, nonnegative=True)
r = Symbol('r', integer=False)
s = Symbol('s', integer=False, negative=True)
t = Symbol('t', nonnegative=True)
u = Symbol('u', noninteger=True)
assert factorial(-2) == zoo
assert factorial(0) == 1
assert factorial(7) == 5040
assert factorial(19) == 121645100408832000
assert factorial(31) == 8222838654177922817725562880000000
assert factorial(n).func == factorial
assert factorial(2*n).func == factorial
assert factorial(x).is_integer is None
assert factorial(n).is_integer is None
assert factorial(k).is_integer
assert factorial(r).is_integer is None
assert factorial(n).is_positive is None
assert factorial(k).is_positive
assert factorial(x).is_real is None
assert factorial(n).is_real is None
assert factorial(k).is_real is True
assert factorial(r).is_real is None
assert factorial(s).is_real is True
assert factorial(t).is_real is True
assert factorial(u).is_real is True
assert factorial(x).is_composite is None
assert factorial(n).is_composite is None
assert factorial(k).is_composite is None
assert factorial(k + 3).is_composite is True
assert factorial(r).is_composite is None
assert factorial(s).is_composite is None
assert factorial(t).is_composite is None
assert factorial(u).is_composite is None
assert factorial(oo) == oo
def test_factorial_Mod():
pr = Symbol('pr', prime=True)
p, q = 10**9 + 9, 10**9 + 33 # prime modulo
r, s = 10**7 + 5, 33333333 # composite modulo
assert Mod(factorial(pr - 1), pr) == pr - 1
assert Mod(factorial(pr - 1), -pr) == -1
assert Mod(factorial(r - 1, evaluate=False), r) == 0
assert Mod(factorial(s - 1, evaluate=False), s) == 0
assert Mod(factorial(p - 1, evaluate=False), p) == p - 1
assert Mod(factorial(q - 1, evaluate=False), q) == q - 1
assert Mod(factorial(p - 50, evaluate=False), p) == 854928834
assert Mod(factorial(q - 1800, evaluate=False), q) == 905504050
assert Mod(factorial(153, evaluate=False), r) == Mod(factorial(153), r)
assert Mod(factorial(255, evaluate=False), s) == Mod(factorial(255), s)
def test_factorial_diff():
n = Symbol('n', integer=True)
assert factorial(n).diff(n) == \
gamma(1 + n)*polygamma(0, 1 + n)
assert factorial(n**2).diff(n) == \
2*n*gamma(1 + n**2)*polygamma(0, 1 + n**2)
def test_factorial_series():
n = Symbol('n', integer=True)
assert factorial(n).series(n, 0, 3) == \
1 - n*EulerGamma + n**2*(EulerGamma**2/2 + pi**2/12) + O(n**3)
def test_factorial_rewrite():
n = Symbol('n', integer=True)
k = Symbol('k', integer=True, nonnegative=True)
assert factorial(n).rewrite(gamma) == gamma(n + 1)
assert str(factorial(k).rewrite(Product)) == 'Product(_i, (_i, 1, k))'
def test_factorial2():
n = Symbol('n', integer=True)
assert factorial2(-1) == 1
assert factorial2(0) == 1
assert factorial2(7) == 105
assert factorial2(8) == 384
# The following is exhaustive
tt = Symbol('tt', integer=True, nonnegative=True)
tte = Symbol('tte', even=True, nonnegative=True)
tpe = Symbol('tpe', even=True, positive=True)
tto = Symbol('tto', odd=True, nonnegative=True)
tf = Symbol('tf', integer=True, nonnegative=False)
tfe = Symbol('tfe', even=True, nonnegative=False)
tfo = Symbol('tfo', odd=True, nonnegative=False)
ft = Symbol('ft', integer=False, nonnegative=True)
ff = Symbol('ff', integer=False, nonnegative=False)
fn = Symbol('fn', integer=False)
nt = Symbol('nt', nonnegative=True)
nf = Symbol('nf', nonnegative=False)
nn = Symbol('nn')
#Solves and Fixes Issue #10388 - This is the updated test for the same solved issue
raises (ValueError, lambda: factorial2(oo))
raises (ValueError, lambda: factorial2(S(5)/2))
assert factorial2(n).is_integer is None
assert factorial2(tt - 1).is_integer
assert factorial2(tte - 1).is_integer
assert factorial2(tpe - 3).is_integer
assert factorial2(tto - 4).is_integer
assert factorial2(tto - 2).is_integer
assert factorial2(tf).is_integer is None
assert factorial2(tfe).is_integer is None
assert factorial2(tfo).is_integer is None
assert factorial2(ft).is_integer is None
assert factorial2(ff).is_integer is None
assert factorial2(fn).is_integer is None
assert factorial2(nt).is_integer is None
assert factorial2(nf).is_integer is None
assert factorial2(nn).is_integer is None
assert factorial2(n).is_positive is None
assert factorial2(tt - 1).is_positive is True
assert factorial2(tte - 1).is_positive is True
assert factorial2(tpe - 3).is_positive is True
assert factorial2(tpe - 1).is_positive is True
assert factorial2(tto - 2).is_positive is True
assert factorial2(tto - 1).is_positive is True
assert factorial2(tf).is_positive is None
assert factorial2(tfe).is_positive is None
assert factorial2(tfo).is_positive is None
assert factorial2(ft).is_positive is None
assert factorial2(ff).is_positive is None
assert factorial2(fn).is_positive is None
assert factorial2(nt).is_positive is None
assert factorial2(nf).is_positive is None
assert factorial2(nn).is_positive is None
assert factorial2(tt).is_even is None
assert factorial2(tt).is_odd is None
assert factorial2(tte).is_even is None
assert factorial2(tte).is_odd is None
assert factorial2(tte + 2).is_even is True
assert factorial2(tpe).is_even is True
assert factorial2(tto).is_odd is True
assert factorial2(tf).is_even is None
assert factorial2(tf).is_odd is None
assert factorial2(tfe).is_even is None
assert factorial2(tfe).is_odd is None
assert factorial2(tfo).is_even is False
assert factorial2(tfo).is_odd is None
def test_factorial2_rewrite():
n = Symbol('n', integer=True)
assert factorial2(n).rewrite(gamma) == \
2**(n/2)*Piecewise((1, Eq(Mod(n, 2), 0)), (sqrt(2)/sqrt(pi), Eq(Mod(n, 2), 1)))*gamma(n/2 + 1)
assert factorial2(2*n).rewrite(gamma) == 2**n*gamma(n + 1)
assert factorial2(2*n + 1).rewrite(gamma) == \
sqrt(2)*2**(n + S(1)/2)*gamma(n + S(3)/2)/sqrt(pi)
def test_binomial():
x = Symbol('x')
n = Symbol('n', integer=True)
nz = Symbol('nz', integer=True, nonzero=True)
k = Symbol('k', integer=True)
kp = Symbol('kp', integer=True, positive=True)
kn = Symbol('kn', integer=True, negative=True)
u = Symbol('u', negative=True)
v = Symbol('v', nonnegative=True)
p = Symbol('p', positive=True)
z = Symbol('z', zero=True)
nt = Symbol('nt', integer=False)
kt = Symbol('kt', integer=False)
a = Symbol('a', integer=True, nonnegative=True)
b = Symbol('b', integer=True, nonnegative=True)
assert binomial(0, 0) == 1
assert binomial(1, 1) == 1
assert binomial(10, 10) == 1
assert binomial(n, z) == 1
assert binomial(1, 2) == 0
assert binomial(-1, 2) == 1
assert binomial(1, -1) == 0
assert binomial(-1, 1) == -1
assert binomial(-1, -1) == 0
assert binomial(S.Half, S.Half) == 1
assert binomial(-10, 1) == -10
assert binomial(-10, 7) == -11440
assert binomial(n, -1) == 0 # holds for all integers (negative, zero, positive)
assert binomial(kp, -1) == 0
assert binomial(nz, 0) == 1
assert expand_func(binomial(n, 1)) == n
assert expand_func(binomial(n, 2)) == n*(n - 1)/2
assert expand_func(binomial(n, n - 2)) == n*(n - 1)/2
assert expand_func(binomial(n, n - 1)) == n
assert binomial(n, 3).func == binomial
assert binomial(n, 3).expand(func=True) == n**3/6 - n**2/2 + n/3
assert expand_func(binomial(n, 3)) == n*(n - 2)*(n - 1)/6
assert binomial(n, n).func == binomial # e.g. (-1, -1) == 0, (2, 2) == 1
assert binomial(n, n + 1).func == binomial # e.g. (-1, 0) == 1
assert binomial(kp, kp + 1) == 0
assert binomial(kn, kn) == 0 # issue #14529
assert binomial(n, u).func == binomial
assert binomial(kp, u).func == binomial
assert binomial(n, p).func == binomial
assert binomial(n, k).func == binomial
assert binomial(n, n + p).func == binomial
assert binomial(kp, kp + p).func == binomial
assert expand_func(binomial(n, n - 3)) == n*(n - 2)*(n - 1)/6
assert binomial(n, k).is_integer
assert binomial(nt, k).is_integer is None
assert binomial(x, nt).is_integer is False
assert binomial(gamma(25), 6) == 79232165267303928292058750056084441948572511312165380965440075720159859792344339983120618959044048198214221915637090855535036339620413440000
assert binomial(1324, 47) == 906266255662694632984994480774946083064699457235920708992926525848438478406790323869952
assert binomial(1735, 43) == 190910140420204130794758005450919715396159959034348676124678207874195064798202216379800
assert binomial(2512, 53) == 213894469313832631145798303740098720367984955243020898718979538096223399813295457822575338958939834177325304000
assert binomial(3383, 52) == 27922807788818096863529701501764372757272890613101645521813434902890007725667814813832027795881839396839287659777235
assert binomial(4321, 51) == 124595639629264868916081001263541480185227731958274383287107643816863897851139048158022599533438936036467601690983780576
assert binomial(a, b).is_nonnegative is True
assert binomial(-1, 2, evaluate=False).is_nonnegative is True
assert binomial(10, 5, evaluate=False).is_nonnegative is True
assert binomial(10, -3, evaluate=False).is_nonnegative is True
assert binomial(-10, -3, evaluate=False).is_nonnegative is True
assert binomial(-10, 2, evaluate=False).is_nonnegative is True
assert binomial(-10, 1, evaluate=False).is_nonnegative is False
assert binomial(-10, 7, evaluate=False).is_nonnegative is False
# issue #14625
for _ in (pi, -pi, nt, v, a):
assert binomial(_, _) == 1
assert binomial(_, _ - 1) == _
assert isinstance(binomial(u, u), binomial)
assert isinstance(binomial(u, u - 1), binomial)
assert isinstance(binomial(x, x), binomial)
assert isinstance(binomial(x, x - 1), binomial)
# issue #13980 and #13981
assert binomial(-7, -5) == 0
assert binomial(-23, -12) == 0
assert binomial(S(13)/2, -10) == 0
assert binomial(-49, -51) == 0
assert binomial(19, S(-7)/2) == S(-68719476736)/(911337863661225*pi)
assert binomial(0, S(3)/2) == S(-2)/(3*pi)
assert binomial(-3, S(-7)/2) == zoo
assert binomial(kn, kt) == zoo
assert binomial(nt, kt).func == binomial
assert binomial(nt, S(15)/6) == 8*gamma(nt + 1)/(15*sqrt(pi)*gamma(nt - S(3)/2))
assert binomial(S(20)/3, S(-10)/8) == gamma(S(23)/3)/(gamma(S(-1)/4)*gamma(S(107)/12))
assert binomial(S(19)/2, S(-7)/2) == S(-1615)/8388608
assert binomial(S(-13)/5, S(-7)/8) == gamma(S(-8)/5)/(gamma(S(-29)/40)*gamma(S(1)/8))
assert binomial(S(-19)/8, S(-13)/5) == gamma(S(-11)/8)/(gamma(S(-8)/5)*gamma(S(49)/40))
# binomial for complexes
from sympy import I
assert binomial(I, S(-89)/8) == gamma(1 + I)/(gamma(S(-81)/8)*gamma(S(97)/8 + I))
assert binomial(I, 2*I) == gamma(1 + I)/(gamma(1 - I)*gamma(1 + 2*I))
assert binomial(-7, I) == zoo
assert binomial(-7/S(6), I) == gamma(-1/S(6))/(gamma(-1/S(6) - I)*gamma(1 + I))
assert binomial((1+2*I), (1+3*I)) == gamma(2 + 2*I)/(gamma(1 - I)*gamma(2 + 3*I))
assert binomial(I, 5) == S(1)/3 - I/S(12)
assert binomial((2*I + 3), 7) == -13*I/S(63)
assert isinstance(binomial(I, n), binomial)
@slow
def test_binomial_Mod():
p, q = 10**5 + 3, 10**9 + 33 # prime modulo
r, s = 10**7 + 5, 33333333 # composite modulo
n, k, m = symbols('n k m')
assert (binomial(n, k) % q).subs({n: s, k: p}) == Mod(binomial(s, p), q)
assert (binomial(n, k) % m).subs({n: 8, k: 5, m: 13}) == 4
assert (binomial(9, k) % 7).subs(k, 2) == 1
# Lucas Theorem
assert Mod(binomial(156675, 4433, evaluate=False), p) == Mod(binomial(156675, 4433), p)
assert Mod(binomial(123456, 43253, evaluate=False), p) == Mod(binomial(123456, 43253), p)
assert Mod(binomial(-178911, 237, evaluate=False), p) == Mod(-binomial(178911 + 237 - 1, 237), p)
assert Mod(binomial(-178911, 238, evaluate=False), p) == Mod(binomial(178911 + 238 - 1, 238), p)
# factorial Mod
assert Mod(binomial(1234, 432, evaluate=False), q) == Mod(binomial(1234, 432), q)
assert Mod(binomial(9734, 451, evaluate=False), q) == Mod(binomial(9734, 451), q)
assert Mod(binomial(-10733, 4459, evaluate=False), q) == Mod(binomial(-10733, 4459), q)
assert Mod(binomial(-15733, 4458, evaluate=False), q) == Mod(binomial(-15733, 4458), q)
# binomial factorize
assert Mod(binomial(253, 113, evaluate=False), r) == Mod(binomial(253, 113), r)
assert Mod(binomial(753, 119, evaluate=False), r) == Mod(binomial(753, 119), r)
assert Mod(binomial(3781, 948, evaluate=False), s) == Mod(binomial(3781, 948), s)
assert Mod(binomial(25773, 1793, evaluate=False), s) == Mod(binomial(25773, 1793), s)
assert Mod(binomial(-753, 118, evaluate=False), r) == Mod(binomial(-753, 118), r)
assert Mod(binomial(-25773, 1793, evaluate=False), s) == Mod(binomial(-25773, 1793), s)
def test_binomial_diff():
n = Symbol('n', integer=True)
k = Symbol('k', integer=True)
assert binomial(n, k).diff(n) == \
(-polygamma(0, 1 + n - k) + polygamma(0, 1 + n))*binomial(n, k)
assert binomial(n**2, k**3).diff(n) == \
2*n*(-polygamma(
0, 1 + n**2 - k**3) + polygamma(0, 1 + n**2))*binomial(n**2, k**3)
assert binomial(n, k).diff(k) == \
(-polygamma(0, 1 + k) + polygamma(0, 1 + n - k))*binomial(n, k)
assert binomial(n**2, k**3).diff(k) == \
3*k**2*(-polygamma(
0, 1 + k**3) + polygamma(0, 1 + n**2 - k**3))*binomial(n**2, k**3)
def test_binomial_rewrite():
n = Symbol('n', integer=True)
k = Symbol('k', integer=True)
assert binomial(n, k).rewrite(
factorial) == factorial(n)/(factorial(k)*factorial(n - k))
assert binomial(
n, k).rewrite(gamma) == gamma(n + 1)/(gamma(k + 1)*gamma(n - k + 1))
assert binomial(n, k).rewrite(ff) == ff(n, k) / factorial(k)
@XFAIL
def test_factorial_simplify_fail():
# simplify(factorial(x + 1).diff(x) - ((x + 1)*factorial(x)).diff(x))) == 0
from sympy.abc import x
assert simplify(x*polygamma(0, x + 1) - x*polygamma(0, x + 2) +
polygamma(0, x + 1) - polygamma(0, x + 2) + 1) == 0
def test_subfactorial():
assert all(subfactorial(i) == ans for i, ans in enumerate(
[1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496]))
assert subfactorial(oo) == oo
assert subfactorial(nan) == nan
x = Symbol('x')
assert subfactorial(x).rewrite(uppergamma) == uppergamma(x + 1, -1)/S.Exp1
tt = Symbol('tt', integer=True, nonnegative=True)
tf = Symbol('tf', integer=True, nonnegative=False)
tn = Symbol('tf', integer=True)
ft = Symbol('ft', integer=False, nonnegative=True)
ff = Symbol('ff', integer=False, nonnegative=False)
fn = Symbol('ff', integer=False)
nt = Symbol('nt', nonnegative=True)
nf = Symbol('nf', nonnegative=False)
nn = Symbol('nf')
te = Symbol('te', even=True, nonnegative=True)
to = Symbol('to', odd=True, nonnegative=True)
assert subfactorial(tt).is_integer
assert subfactorial(tf).is_integer is None
assert subfactorial(tn).is_integer is None
assert subfactorial(ft).is_integer is None
assert subfactorial(ff).is_integer is None
assert subfactorial(fn).is_integer is None
assert subfactorial(nt).is_integer is None
assert subfactorial(nf).is_integer is None
assert subfactorial(nn).is_integer is None
assert subfactorial(tt).is_nonnegative
assert subfactorial(tf).is_nonnegative is None
assert subfactorial(tn).is_nonnegative is None
assert subfactorial(ft).is_nonnegative is None
assert subfactorial(ff).is_nonnegative is None
assert subfactorial(fn).is_nonnegative is None
assert subfactorial(nt).is_nonnegative is None
assert subfactorial(nf).is_nonnegative is None
assert subfactorial(nn).is_nonnegative is None
assert subfactorial(tt).is_even is None
assert subfactorial(tt).is_odd is None
assert subfactorial(te).is_odd is True
assert subfactorial(to).is_even is True
|
807d71188d07f88777235dbc5af7315652b0736c538dcb0ca9d211a48237049b
|
import string
from sympy import (
Symbol, symbols, Dummy, S, Sum, Rational, oo, pi, I,
expand_func, diff, EulerGamma, cancel, re, im, Product, carmichael)
from sympy.functions import (
bernoulli, harmonic, bell, fibonacci, tribonacci, lucas, euler, catalan,
genocchi, partition, binomial, gamma, sqrt, cbrt, hyper, log, digamma,
trigamma, polygamma, factorial, sin, cos, cot, zeta)
from sympy.core.compatibility import range
from sympy.utilities.pytest import XFAIL, raises
from sympy.core.numbers import GoldenRatio
x = Symbol('x')
def test_carmichael():
assert carmichael.find_carmichael_numbers_in_range(0, 561) == []
assert carmichael.find_carmichael_numbers_in_range(561, 562) == [561]
assert carmichael.find_carmichael_numbers_in_range(561, 1105) == carmichael.find_carmichael_numbers_in_range(561,
562)
assert carmichael.find_first_n_carmichaels(5) == [561, 1105, 1729, 2465, 2821]
assert carmichael.is_prime(2821) == False
assert carmichael.is_prime(2465) == False
assert carmichael.is_prime(1729) == False
assert carmichael.is_prime(1105) == False
assert carmichael.is_prime(561) == False
def test_bernoulli():
assert bernoulli(0) == 1
assert bernoulli(1) == Rational(-1, 2)
assert bernoulli(2) == Rational(1, 6)
assert bernoulli(3) == 0
assert bernoulli(4) == Rational(-1, 30)
assert bernoulli(5) == 0
assert bernoulli(6) == Rational(1, 42)
assert bernoulli(7) == 0
assert bernoulli(8) == Rational(-1, 30)
assert bernoulli(10) == Rational(5, 66)
assert bernoulli(1000001) == 0
assert bernoulli(0, x) == 1
assert bernoulli(1, x) == x - Rational(1, 2)
assert bernoulli(2, x) == x**2 - x + Rational(1, 6)
assert bernoulli(3, x) == x**3 - (3*x**2)/2 + x/2
# Should be fast; computed with mpmath
b = bernoulli(1000)
assert b.p % 10**10 == 7950421099
assert b.q == 342999030
b = bernoulli(10**6, evaluate=False).evalf()
assert str(b) == '-2.23799235765713e+4767529'
# Issue #8527
l = Symbol('l', integer=True)
m = Symbol('m', integer=True, nonnegative=True)
n = Symbol('n', integer=True, positive=True)
assert isinstance(bernoulli(2 * l + 1), bernoulli)
assert isinstance(bernoulli(2 * m + 1), bernoulli)
assert bernoulli(2 * n + 1) == 0
def test_fibonacci():
assert [fibonacci(n) for n in range(-3, 5)] == [2, -1, 1, 0, 1, 1, 2, 3]
assert fibonacci(100) == 354224848179261915075
assert [lucas(n) for n in range(-3, 5)] == [-4, 3, -1, 2, 1, 3, 4, 7]
assert lucas(100) == 792070839848372253127
assert fibonacci(1, x) == 1
assert fibonacci(2, x) == x
assert fibonacci(3, x) == x**2 + 1
assert fibonacci(4, x) == x**3 + 2*x
# issue #8800
n = Dummy('n')
assert fibonacci(n).limit(n, S.Infinity) == S.Infinity
assert lucas(n).limit(n, S.Infinity) == S.Infinity
assert fibonacci(n).rewrite(sqrt) == \
2**(-n)*sqrt(5)*((1 + sqrt(5))**n - (-sqrt(5) + 1)**n) / 5
assert fibonacci(n).rewrite(sqrt).subs(n, 10).expand() == fibonacci(10)
assert fibonacci(n).rewrite(GoldenRatio).subs(n,10).evalf() == \
fibonacci(10)
assert lucas(n).rewrite(sqrt) == \
(fibonacci(n-1).rewrite(sqrt) + fibonacci(n+1).rewrite(sqrt)).simplify()
assert lucas(n).rewrite(sqrt).subs(n, 10).expand() == lucas(10)
def test_tribonacci():
assert [tribonacci(n) for n in range(8)] == [0, 1, 1, 2, 4, 7, 13, 24]
assert tribonacci(100) == 98079530178586034536500564
assert tribonacci(0, x) == 0
assert tribonacci(1, x) == 1
assert tribonacci(2, x) == x**2
assert tribonacci(3, x) == x**4 + x
assert tribonacci(4, x) == x**6 + 2*x**3 + 1
assert tribonacci(5, x) == x**8 + 3*x**5 + 3*x**2
n = Dummy('n')
assert tribonacci(n).limit(n, S.Infinity) == S.Infinity
w = (-1 + S.ImaginaryUnit * sqrt(3)) / 2
a = (1 + cbrt(19 + 3*sqrt(33)) + cbrt(19 - 3*sqrt(33))) / 3
b = (1 + w*cbrt(19 + 3*sqrt(33)) + w**2*cbrt(19 - 3*sqrt(33))) / 3
c = (1 + w**2*cbrt(19 + 3*sqrt(33)) + w*cbrt(19 - 3*sqrt(33))) / 3
assert tribonacci(n).rewrite(sqrt) == \
(a**(n + 1)/((a - b)*(a - c))
+ b**(n + 1)/((b - a)*(b - c))
+ c**(n + 1)/((c - a)*(c - b)))
assert tribonacci(n).rewrite(sqrt).subs(n, 4).simplify() == tribonacci(4)
assert tribonacci(n).rewrite(GoldenRatio).subs(n,10).evalf() == \
tribonacci(10)
raises(ValueError, lambda: tribonacci(-1, x))
def test_bell():
assert [bell(n) for n in range(8)] == [1, 1, 2, 5, 15, 52, 203, 877]
assert bell(0, x) == 1
assert bell(1, x) == x
assert bell(2, x) == x**2 + x
assert bell(5, x) == x**5 + 10*x**4 + 25*x**3 + 15*x**2 + x
assert bell(oo) == S.Infinity
raises(ValueError, lambda: bell(oo, x))
raises(ValueError, lambda: bell(-1))
raises(ValueError, lambda: bell(S(1)/2))
X = symbols('x:6')
# X = (x0, x1, .. x5)
# at the same time: X[1] = x1, X[2] = x2 for standard readablity.
# but we must supply zero-based indexed object X[1:] = (x1, .. x5)
assert bell(6, 2, X[1:]) == 6*X[5]*X[1] + 15*X[4]*X[2] + 10*X[3]**2
assert bell(
6, 3, X[1:]) == 15*X[4]*X[1]**2 + 60*X[3]*X[2]*X[1] + 15*X[2]**3
X = (1, 10, 100, 1000, 10000)
assert bell(6, 2, X) == (6 + 15 + 10)*10000
X = (1, 2, 3, 3, 5)
assert bell(6, 2, X) == 6*5 + 15*3*2 + 10*3**2
X = (1, 2, 3, 5)
assert bell(6, 3, X) == 15*5 + 60*3*2 + 15*2**3
# Dobinski's formula
n = Symbol('n', integer=True, nonnegative=True)
# For large numbers, this is too slow
# For nonintegers, there are significant precision errors
for i in [0, 2, 3, 7, 13, 42, 55]:
assert bell(i).evalf() == bell(n).rewrite(Sum).evalf(subs={n: i})
# issue 9184
n = Dummy('n')
assert bell(n).limit(n, S.Infinity) == S.Infinity
def test_harmonic():
n = Symbol("n")
m = Symbol("m")
assert harmonic(n, 0) == n
assert harmonic(n).evalf() == harmonic(n)
assert harmonic(n, 1) == harmonic(n)
assert harmonic(1, n).evalf() == harmonic(1, n)
assert harmonic(0, 1) == 0
assert harmonic(1, 1) == 1
assert harmonic(2, 1) == Rational(3, 2)
assert harmonic(3, 1) == Rational(11, 6)
assert harmonic(4, 1) == Rational(25, 12)
assert harmonic(0, 2) == 0
assert harmonic(1, 2) == 1
assert harmonic(2, 2) == Rational(5, 4)
assert harmonic(3, 2) == Rational(49, 36)
assert harmonic(4, 2) == Rational(205, 144)
assert harmonic(0, 3) == 0
assert harmonic(1, 3) == 1
assert harmonic(2, 3) == Rational(9, 8)
assert harmonic(3, 3) == Rational(251, 216)
assert harmonic(4, 3) == Rational(2035, 1728)
assert harmonic(oo, -1) == S.NaN
assert harmonic(oo, 0) == oo
assert harmonic(oo, S.Half) == oo
assert harmonic(oo, 1) == oo
assert harmonic(oo, 2) == (pi**2)/6
assert harmonic(oo, 3) == zeta(3)
assert harmonic(0, m) == 0
def test_harmonic_rational():
ne = S(6)
no = S(5)
pe = S(8)
po = S(9)
qe = S(10)
qo = S(13)
Heee = harmonic(ne + pe/qe)
Aeee = (-log(10) + 2*(-1/S(4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 - 1/S(4))*log(sqrt(sqrt(5)/8 + 5/S(8)))
+ pi*(1/S(4) + sqrt(5)/4)/(2*sqrt(-sqrt(5)/8 + 5/S(8)))
+ 13944145/S(4720968))
Heeo = harmonic(ne + pe/qo)
Aeeo = (-log(26) + 2*log(sin(3*pi/13))*cos(4*pi/13) + 2*log(sin(2*pi/13))*cos(32*pi/13)
+ 2*log(sin(5*pi/13))*cos(80*pi/13) - 2*log(sin(6*pi/13))*cos(5*pi/13)
- 2*log(sin(4*pi/13))*cos(pi/13) + pi*cot(5*pi/13)/2 - 2*log(sin(pi/13))*cos(3*pi/13)
+ 2422020029/S(702257080))
Heoe = harmonic(ne + po/qe)
Aeoe = (-log(20) + 2*(1/S(4) + sqrt(5)/4)*log(-1/S(4) + sqrt(5)/4)
+ 2*(-1/S(4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 - 1/S(4))*log(sqrt(sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 + 1/S(4))*log(1/S(4) + sqrt(5)/4)
+ 11818877030/S(4286604231) + pi*(sqrt(5)/8 + 5/S(8))/sqrt(-sqrt(5)/8 + 5/S(8)))
Heoo = harmonic(ne + po/qo)
Aeoo = (-log(26) + 2*log(sin(3*pi/13))*cos(54*pi/13) + 2*log(sin(4*pi/13))*cos(6*pi/13)
+ 2*log(sin(6*pi/13))*cos(108*pi/13) - 2*log(sin(5*pi/13))*cos(pi/13)
- 2*log(sin(pi/13))*cos(5*pi/13) + pi*cot(4*pi/13)/2
- 2*log(sin(2*pi/13))*cos(3*pi/13) + 11669332571/S(3628714320))
Hoee = harmonic(no + pe/qe)
Aoee = (-log(10) + 2*(-1/S(4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 - 1/S(4))*log(sqrt(sqrt(5)/8 + 5/S(8)))
+ pi*(1/S(4) + sqrt(5)/4)/(2*sqrt(-sqrt(5)/8 + 5/S(8)))
+ 779405/S(277704))
Hoeo = harmonic(no + pe/qo)
Aoeo = (-log(26) + 2*log(sin(3*pi/13))*cos(4*pi/13) + 2*log(sin(2*pi/13))*cos(32*pi/13)
+ 2*log(sin(5*pi/13))*cos(80*pi/13) - 2*log(sin(6*pi/13))*cos(5*pi/13)
- 2*log(sin(4*pi/13))*cos(pi/13) + pi*cot(5*pi/13)/2
- 2*log(sin(pi/13))*cos(3*pi/13) + 53857323/S(16331560))
Hooe = harmonic(no + po/qe)
Aooe = (-log(20) + 2*(1/S(4) + sqrt(5)/4)*log(-1/S(4) + sqrt(5)/4)
+ 2*(-1/S(4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 - 1/S(4))*log(sqrt(sqrt(5)/8 + 5/S(8)))
+ 2*(-sqrt(5)/4 + 1/S(4))*log(1/S(4) + sqrt(5)/4)
+ 486853480/S(186374097) + pi*(sqrt(5)/8 + 5/S(8))/sqrt(-sqrt(5)/8 + 5/S(8)))
Hooo = harmonic(no + po/qo)
Aooo = (-log(26) + 2*log(sin(3*pi/13))*cos(54*pi/13) + 2*log(sin(4*pi/13))*cos(6*pi/13)
+ 2*log(sin(6*pi/13))*cos(108*pi/13) - 2*log(sin(5*pi/13))*cos(pi/13)
- 2*log(sin(pi/13))*cos(5*pi/13) + pi*cot(4*pi/13)/2
- 2*log(sin(2*pi/13))*cos(3*pi/13) + 383693479/S(125128080))
H = [Heee, Heeo, Heoe, Heoo, Hoee, Hoeo, Hooe, Hooo]
A = [Aeee, Aeeo, Aeoe, Aeoo, Aoee, Aoeo, Aooe, Aooo]
for h, a in zip(H, A):
e = expand_func(h).doit()
assert cancel(e/a) == 1
assert abs(h.n() - a.n()) < 1e-12
def test_harmonic_evalf():
assert str(harmonic(1.5).evalf(n=10)) == '1.280372306'
assert str(harmonic(1.5, 2).evalf(n=10)) == '1.154576311' # issue 7443
def test_harmonic_rewrite_polygamma():
n = Symbol("n")
m = Symbol("m")
assert harmonic(n).rewrite(digamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n).rewrite(trigamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n).rewrite(polygamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n,3).rewrite(polygamma) == polygamma(2, n + 1)/2 - polygamma(2, 1)/2
assert harmonic(n,m).rewrite(polygamma) == (-1)**m*(polygamma(m - 1, 1) - polygamma(m - 1, n + 1))/factorial(m - 1)
assert expand_func(harmonic(n+4)) == harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1)
assert expand_func(harmonic(n-4)) == harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n
assert harmonic(n, m).rewrite("tractable") == harmonic(n, m).rewrite(polygamma)
@XFAIL
def test_harmonic_limit_fail():
n = Symbol("n")
m = Symbol("m")
# For m > 1:
assert limit(harmonic(n, m), n, oo) == zeta(m)
@XFAIL
def test_harmonic_rewrite_sum_fail():
n = Symbol("n")
m = Symbol("m")
_k = Dummy("k")
assert harmonic(n).rewrite(Sum) == Sum(1/_k, (_k, 1, n))
assert harmonic(n, m).rewrite(Sum) == Sum(_k**(-m), (_k, 1, n))
def replace_dummy(expr, sym):
dum = expr.atoms(Dummy)
if not dum:
return expr
assert len(dum) == 1
return expr.xreplace({dum.pop(): sym})
def test_harmonic_rewrite_sum():
n = Symbol("n")
m = Symbol("m")
_k = Dummy("k")
assert replace_dummy(harmonic(n).rewrite(Sum), _k) == Sum(1/_k, (_k, 1, n))
assert replace_dummy(harmonic(n, m).rewrite(Sum), _k) == Sum(_k**(-m), (_k, 1, n))
def test_euler():
assert euler(0) == 1
assert euler(1) == 0
assert euler(2) == -1
assert euler(3) == 0
assert euler(4) == 5
assert euler(6) == -61
assert euler(8) == 1385
assert euler(20, evaluate=False) != 370371188237525
n = Symbol('n', integer=True)
assert euler(n) != -1
assert euler(n).subs(n, 2) == -1
raises(ValueError, lambda: euler(-2))
raises(ValueError, lambda: euler(-3))
raises(ValueError, lambda: euler(2.3))
assert euler(20).evalf() == 370371188237525.0
assert euler(20, evaluate=False).evalf() == 370371188237525.0
assert euler(n).rewrite(Sum) == euler(n)
# XXX: Not sure what the guy who wrote this test was trying to do with the _j and _k stuff
n = Symbol('n', integer=True, nonnegative=True)
assert euler(2*n + 1).rewrite(Sum) == 0
@XFAIL
def test_euler_failing():
# depends on dummy variables being implemented https://github.com/sympy/sympy/issues/5665
assert euler(2*n).rewrite(Sum) == I*Sum(Sum((-1)**_j*2**(-_k)*I**(-_k)*(-2*_j + _k)**(2*n + 1)*binomial(_k, _j)/_k, (_j, 0, _k)), (_k, 1, 2*n + 1))
def test_euler_odd():
n = Symbol('n', odd=True, positive=True)
assert euler(n) == 0
n = Symbol('n', odd=True)
assert euler(n) != 0
def test_euler_polynomials():
assert euler(0, x) == 1
assert euler(1, x) == x - Rational(1, 2)
assert euler(2, x) == x**2 - x
assert euler(3, x) == x**3 - (3*x**2)/2 + Rational(1, 4)
m = Symbol('m')
assert isinstance(euler(m, x), euler)
from sympy import Float
A = Float('-0.46237208575048694923364757452876131e8') # from Maple
B = euler(19, S.Pi.evalf(32))
assert abs((A - B)/A) < 1e-31 # expect low relative error
C = euler(19, S.Pi, evaluate=False).evalf(32)
assert abs((A - C)/A) < 1e-31
def test_euler_polynomial_rewrite():
m = Symbol('m')
A = euler(m, x).rewrite('Sum');
assert A.subs({m:3, x:5}).doit() == euler(3, 5)
def test_catalan():
n = Symbol('n', integer=True)
m = Symbol('m', integer=True, positive=True)
k = Symbol('k', integer=True, nonnegative=True)
p = Symbol('p', nonnegative=True)
catalans = [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786]
for i, c in enumerate(catalans):
assert catalan(i) == c
assert catalan(n).rewrite(factorial).subs(n, i) == c
assert catalan(n).rewrite(Product).subs(n, i).doit() == c
assert catalan(x) == catalan(x)
assert catalan(2*x).rewrite(binomial) == binomial(4*x, 2*x)/(2*x + 1)
assert catalan(Rational(1, 2)).rewrite(gamma) == 8/(3*pi)
assert catalan(Rational(1, 2)).rewrite(factorial).rewrite(gamma) ==\
8 / (3 * pi)
assert catalan(3*x).rewrite(gamma) == 4**(
3*x)*gamma(3*x + Rational(1, 2))/(sqrt(pi)*gamma(3*x + 2))
assert catalan(x).rewrite(hyper) == hyper((-x + 1, -x), (2,), 1)
assert catalan(n).rewrite(factorial) == factorial(2*n) / (factorial(n + 1)
* factorial(n))
assert isinstance(catalan(n).rewrite(Product), catalan)
assert isinstance(catalan(m).rewrite(Product), Product)
assert diff(catalan(x), x) == (polygamma(
0, x + Rational(1, 2)) - polygamma(0, x + 2) + log(4))*catalan(x)
assert catalan(x).evalf() == catalan(x)
c = catalan(S.Half).evalf()
assert str(c) == '0.848826363156775'
c = catalan(I).evalf(3)
assert str((re(c), im(c))) == '(0.398, -0.0209)'
# Assumptions
assert catalan(p).is_positive is True
assert catalan(k).is_integer is True
assert catalan(m+3).is_composite is True
def test_genocchi():
genocchis = [1, -1, 0, 1, 0, -3, 0, 17]
for n, g in enumerate(genocchis):
assert genocchi(n + 1) == g
m = Symbol('m', integer=True)
n = Symbol('n', integer=True, positive=True)
assert genocchi(m) == genocchi(m)
assert genocchi(n).rewrite(bernoulli) == (1 - 2 ** n) * bernoulli(n) * 2
assert genocchi(2 * n).is_odd
assert genocchi(4 * n).is_positive
# these are the only 2 prime Genocchi numbers
assert genocchi(6, evaluate=False).is_prime == S(-3).is_prime
assert genocchi(8, evaluate=False).is_prime
assert genocchi(4 * n + 2).is_negative
assert genocchi(4 * n - 2).is_negative
def test_partition():
partition_nums = [1, 1, 2, 3, 5, 7, 11, 15, 22]
for n, p in enumerate(partition_nums):
assert partition(n) == p
x = Symbol('x')
y = Symbol('y', real=True)
m = Symbol('m', integer=True)
n = Symbol('n', integer=True, negative=True)
p = Symbol('p', integer=True, nonnegative=True)
assert partition(m).is_integer
assert not partition(m).is_negative
assert partition(m).is_nonnegative
assert partition(n).is_zero
assert partition(p).is_positive
assert partition(x).subs(x, 7) == 15
assert partition(y).subs(y, 8) == 22
raises(ValueError, lambda: partition(S(5)/4))
def test_nC_nP_nT():
from sympy.utilities.iterables import (
multiset_permutations, multiset_combinations, multiset_partitions,
partitions, subsets, permutations)
from sympy.functions.combinatorial.numbers import (
nP, nC, nT, stirling, _multiset_histogram, _AOP_product)
from sympy.combinatorics.permutations import Permutation
from sympy.core.numbers import oo
from random import choice
c = string.ascii_lowercase
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(8):
check = nP(s, i)
tot += check
assert len(list(multiset_permutations(s, i))) == check
if u:
assert nP(len(s), i) == check
assert nP(s) == tot
except AssertionError:
print(s, i, 'failed perm test')
raise ValueError()
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(8):
check = nC(s, i)
tot += check
assert len(list(multiset_combinations(s, i))) == check
if u:
assert nC(len(s), i) == check
assert nC(s) == tot
if u:
assert nC(len(s)) == tot
except AssertionError:
print(s, i, 'failed combo test')
raise ValueError()
for i in range(1, 10):
tot = 0
for j in range(1, i + 2):
check = nT(i, j)
tot += check
assert sum(1 for p in partitions(i, j, size=True) if p[0] == j) == check
assert nT(i) == tot
for i in range(1, 10):
tot = 0
for j in range(1, i + 2):
check = nT(range(i), j)
tot += check
assert len(list(multiset_partitions(list(range(i)), j))) == check
assert nT(range(i)) == tot
for i in range(100):
s = ''.join(choice(c) for i in range(7))
u = len(s) == len(set(s))
try:
tot = 0
for i in range(1, 8):
check = nT(s, i)
tot += check
assert len(list(multiset_partitions(s, i))) == check
if u:
assert nT(range(len(s)), i) == check
if u:
assert nT(range(len(s))) == tot
assert nT(s) == tot
except AssertionError:
print(s, i, 'failed partition test')
raise ValueError()
# tests for Stirling numbers of the first kind that are not tested in the
# above
assert [stirling(9, i, kind=1) for i in range(11)] == [
0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1, 0]
perms = list(permutations(range(4)))
assert [sum(1 for p in perms if Permutation(p).cycles == i)
for i in range(5)] == [0, 6, 11, 6, 1] == [
stirling(4, i, kind=1) for i in range(5)]
# http://oeis.org/A008275
assert [stirling(n, k, signed=1)
for n in range(10) for k in range(1, n + 1)] == [
1, -1,
1, 2, -3,
1, -6, 11, -6,
1, 24, -50, 35, -10,
1, -120, 274, -225, 85, -15,
1, 720, -1764, 1624, -735, 175, -21,
1, -5040, 13068, -13132, 6769, -1960, 322, -28,
1, 40320, -109584, 118124, -67284, 22449, -4536, 546, -36, 1]
# https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind
assert [stirling(n, k, kind=1)
for n in range(10) for k in range(n+1)] == [
1,
0, 1,
0, 1, 1,
0, 2, 3, 1,
0, 6, 11, 6, 1,
0, 24, 50, 35, 10, 1,
0, 120, 274, 225, 85, 15, 1,
0, 720, 1764, 1624, 735, 175, 21, 1,
0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1,
0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1]
# https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
assert [stirling(n, k, kind=2)
for n in range(10) for k in range(n+1)] == [
1,
0, 1,
0, 1, 1,
0, 1, 3, 1,
0, 1, 7, 6, 1,
0, 1, 15, 25, 10, 1,
0, 1, 31, 90, 65, 15, 1,
0, 1, 63, 301, 350, 140, 21, 1,
0, 1, 127, 966, 1701, 1050, 266, 28, 1,
0, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1]
assert stirling(3, 4, kind=1) == stirling(3, 4, kind=1) == 0
raises(ValueError, lambda: stirling(-2, 2))
def delta(p):
if len(p) == 1:
return oo
return min(abs(i[0] - i[1]) for i in subsets(p, 2))
parts = multiset_partitions(range(5), 3)
d = 2
assert (sum(1 for p in parts if all(delta(i) >= d for i in p)) ==
stirling(5, 3, d=d) == 7)
# other coverage tests
assert nC('abb', 2) == nC('aab', 2) == 2
assert nP(3, 3, replacement=True) == nP('aabc', 3, replacement=True) == 27
assert nP(3, 4) == 0
assert nP('aabc', 5) == 0
assert nC(4, 2, replacement=True) == nC('abcdd', 2, replacement=True) == \
len(list(multiset_combinations('aabbccdd', 2))) == 10
assert nC('abcdd') == sum(nC('abcdd', i) for i in range(6)) == 24
assert nC(list('abcdd'), 4) == 4
assert nT('aaaa') == nT(4) == len(list(partitions(4))) == 5
assert nT('aaab') == len(list(multiset_partitions('aaab'))) == 7
assert nC('aabb'*3, 3) == 4 # aaa, bbb, abb, baa
assert dict(_AOP_product((4,1,1,1))) == {
0: 1, 1: 4, 2: 7, 3: 8, 4: 8, 5: 7, 6: 4, 7: 1}
# the following was the first t that showed a problem in a previous form of
# the function, so it's not as random as it may appear
t = (3, 9, 4, 6, 6, 5, 5, 2, 10, 4)
assert sum(_AOP_product(t)[i] for i in range(55)) == 58212000
raises(ValueError, lambda: _multiset_histogram({1:'a'}))
def test_PR_14617():
from sympy.functions.combinatorial.numbers import nT
for n in (0, []):
for k in (-1, 0, 1):
if k == 0:
assert nT(n, k) == 1
else:
assert nT(n, k) == 0
def test_issue_8496():
n = Symbol("n")
k = Symbol("k")
raises(TypeError, lambda: catalan(n, k))
def test_issue_8601():
n = Symbol('n', integer=True, negative=True)
assert catalan(n - 1) == S.Zero
assert catalan(-S.Half) == S.ComplexInfinity
assert catalan(-S.One) == -S.Half
c1 = catalan(-5.6).evalf()
assert str(c1) == '6.93334070531408e-5'
c2 = catalan(-35.4).evalf()
assert str(c2) == '-4.14189164517449e-24'
|
a3fe4822fbc93f86a68b80245a1e38dbd873371884c9d61e2723d711a5d4dd48
|
from sympy import (
Abs, adjoint, arg, atan, atan2, conjugate, cos, DiracDelta, E, exp, expand,
Expr, Function, Heaviside, I, im, log, nan, oo, pi, Rational, re, S,
sign, sin, sqrt, Symbol, symbols, transpose, zoo, exp_polar, Piecewise,
Interval, comp, Integral, Matrix, ImmutableMatrix, SparseMatrix,
ImmutableSparseMatrix, MatrixSymbol, FunctionMatrix, Lambda, Derivative)
from sympy.utilities.pytest import XFAIL, raises
from sympy.core.expr import unchanged
def N_equals(a, b):
"""Check whether two complex numbers are numerically close"""
return comp(a.n(), b.n(), 1.e-6)
def test_re():
x, y = symbols('x,y')
a, b = symbols('a,b', real=True)
r = Symbol('r', real=True)
i = Symbol('i', imaginary=True)
assert re(nan) == nan
assert re(oo) == oo
assert re(-oo) == -oo
assert re(0) == 0
assert re(1) == 1
assert re(-1) == -1
assert re(E) == E
assert re(-E) == -E
assert unchanged(re, x)
assert re(x*I) == -im(x)
assert re(r*I) == 0
assert re(r) == r
assert re(i*I) == I * i
assert re(i) == 0
assert re(x + y) == re(x + y)
assert re(x + r) == re(x) + r
assert re(re(x)) == re(x)
assert re(2 + I) == 2
assert re(x + I) == re(x)
assert re(x + y*I) == re(x) - im(y)
assert re(x + r*I) == re(x)
assert re(log(2*I)) == log(2)
assert re((2 + I)**2).expand(complex=True) == 3
assert re(conjugate(x)) == re(x)
assert conjugate(re(x)) == re(x)
assert re(x).as_real_imag() == (re(x), 0)
assert re(i*r*x).diff(r) == re(i*x)
assert re(i*r*x).diff(i) == I*r*im(x)
assert re(
sqrt(a + b*I)) == (a**2 + b**2)**Rational(1, 4)*cos(atan2(b, a)/2)
assert re(a * (2 + b*I)) == 2*a
assert re((1 + sqrt(a + b*I))/2) == \
(a**2 + b**2)**Rational(1, 4)*cos(atan2(b, a)/2)/2 + Rational(1, 2)
assert re(x).rewrite(im) == x - S.ImaginaryUnit*im(x)
assert (x + re(y)).rewrite(re, im) == x + y - S.ImaginaryUnit*im(y)
a = Symbol('a', algebraic=True)
t = Symbol('t', transcendental=True)
x = Symbol('x')
assert re(a).is_algebraic
assert re(x).is_algebraic is None
assert re(t).is_algebraic is False
assert re(S.ComplexInfinity) == S.NaN
n, m, l = symbols('n m l')
A = MatrixSymbol('A',n,m)
assert re(A) == (S(1)/2) * (A + conjugate(A))
A = Matrix([[1 + 4*I,2],[0, -3*I]])
assert re(A) == Matrix([[1, 2],[0, 0]])
A = ImmutableMatrix([[1 + 3*I, 3-2*I],[0, 2*I]])
assert re(A) == ImmutableMatrix([[1, 3],[0, 0]])
X = SparseMatrix([[2*j + i*I for i in range(5)] for j in range(5)])
assert re(X) - Matrix([[0, 0, 0, 0, 0],
[2, 2, 2, 2, 2],
[4, 4, 4, 4, 4],
[6, 6, 6, 6, 6],
[8, 8, 8, 8, 8]]) == Matrix.zeros(5)
assert im(X) - Matrix([[0, 1, 2, 3, 4],
[0, 1, 2, 3, 4],
[0, 1, 2, 3, 4],
[0, 1, 2, 3, 4],
[0, 1, 2, 3, 4]]) == Matrix.zeros(5)
X = FunctionMatrix(3, 3, Lambda((n, m), n + m*I))
assert re(X) == Matrix([[0, 0, 0], [1, 1, 1], [2, 2, 2]])
def test_im():
x, y = symbols('x,y')
a, b = symbols('a,b', real=True)
r = Symbol('r', real=True)
i = Symbol('i', imaginary=True)
assert im(nan) == nan
assert im(oo*I) == oo
assert im(-oo*I) == -oo
assert im(0) == 0
assert im(1) == 0
assert im(-1) == 0
assert im(E*I) == E
assert im(-E*I) == -E
assert unchanged(im, x)
assert im(x*I) == re(x)
assert im(r*I) == r
assert im(r) == 0
assert im(i*I) == 0
assert im(i) == -I * i
assert im(x + y) == im(x + y)
assert im(x + r) == im(x)
assert im(x + r*I) == im(x) + r
assert im(im(x)*I) == im(x)
assert im(2 + I) == 1
assert im(x + I) == im(x) + 1
assert im(x + y*I) == im(x) + re(y)
assert im(x + r*I) == im(x) + r
assert im(log(2*I)) == pi/2
assert im((2 + I)**2).expand(complex=True) == 4
assert im(conjugate(x)) == -im(x)
assert conjugate(im(x)) == im(x)
assert im(x).as_real_imag() == (im(x), 0)
assert im(i*r*x).diff(r) == im(i*x)
assert im(i*r*x).diff(i) == -I * re(r*x)
assert im(
sqrt(a + b*I)) == (a**2 + b**2)**Rational(1, 4)*sin(atan2(b, a)/2)
assert im(a * (2 + b*I)) == a*b
assert im((1 + sqrt(a + b*I))/2) == \
(a**2 + b**2)**Rational(1, 4)*sin(atan2(b, a)/2)/2
assert im(x).rewrite(re) == -S.ImaginaryUnit * (x - re(x))
assert (x + im(y)).rewrite(im, re) == x - S.ImaginaryUnit * (y - re(y))
a = Symbol('a', algebraic=True)
t = Symbol('t', transcendental=True)
x = Symbol('x')
assert re(a).is_algebraic
assert re(x).is_algebraic is None
assert re(t).is_algebraic is False
assert im(S.ComplexInfinity) == S.NaN
n, m, l = symbols('n m l')
A = MatrixSymbol('A',n,m)
assert im(A) == (S(1)/(2*I)) * (A - conjugate(A))
A = Matrix([[1 + 4*I, 2],[0, -3*I]])
assert im(A) == Matrix([[4, 0],[0, -3]])
A = ImmutableMatrix([[1 + 3*I, 3-2*I],[0, 2*I]])
assert im(A) == ImmutableMatrix([[3, -2],[0, 2]])
X = ImmutableSparseMatrix(
[[i*I + i for i in range(5)] for i in range(5)])
Y = SparseMatrix([[i for i in range(5)] for i in range(5)])
assert im(X).as_immutable() == Y
X = FunctionMatrix(3, 3, Lambda((n, m), n + m*I))
assert im(X) == Matrix([[0, 1, 2], [0, 1, 2], [0, 1, 2]])
def test_sign():
assert sign(1.2) == 1
assert sign(-1.2) == -1
assert sign(3*I) == I
assert sign(-3*I) == -I
assert sign(0) == 0
assert sign(nan) == nan
assert sign(2 + 2*I).doit() == sqrt(2)*(2 + 2*I)/4
assert sign(2 + 3*I).simplify() == sign(2 + 3*I)
assert sign(2 + 2*I).simplify() == sign(1 + I)
assert sign(im(sqrt(1 - sqrt(3)))) == 1
assert sign(sqrt(1 - sqrt(3))) == I
x = Symbol('x')
assert sign(x).is_finite is True
assert sign(x).is_complex is True
assert sign(x).is_imaginary is None
assert sign(x).is_integer is None
assert sign(x).is_real is None
assert sign(x).is_zero is None
assert sign(x).doit() == sign(x)
assert sign(1.2*x) == sign(x)
assert sign(2*x) == sign(x)
assert sign(I*x) == I*sign(x)
assert sign(-2*I*x) == -I*sign(x)
assert sign(conjugate(x)) == conjugate(sign(x))
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
m = Symbol('m', negative=True)
assert sign(2*p*x) == sign(x)
assert sign(n*x) == -sign(x)
assert sign(n*m*x) == sign(x)
x = Symbol('x', imaginary=True)
assert sign(x).is_imaginary is True
assert sign(x).is_integer is False
assert sign(x).is_real is False
assert sign(x).is_zero is False
assert sign(x).diff(x) == 2*DiracDelta(-I*x)
assert sign(x).doit() == x / Abs(x)
assert conjugate(sign(x)) == -sign(x)
x = Symbol('x', real=True)
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_real is True
assert sign(x).is_zero is None
assert sign(x).diff(x) == 2*DiracDelta(x)
assert sign(x).doit() == sign(x)
assert conjugate(sign(x)) == sign(x)
x = Symbol('x', nonzero=True)
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_real is True
assert sign(x).is_zero is False
assert sign(x).doit() == x / Abs(x)
assert sign(Abs(x)) == 1
assert Abs(sign(x)) == 1
x = Symbol('x', positive=True)
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_real is True
assert sign(x).is_zero is False
assert sign(x).doit() == x / Abs(x)
assert sign(Abs(x)) == 1
assert Abs(sign(x)) == 1
x = 0
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_real is True
assert sign(x).is_zero is True
assert sign(x).doit() == 0
assert sign(Abs(x)) == 0
assert Abs(sign(x)) == 0
nz = Symbol('nz', nonzero=True, integer=True)
assert sign(nz).is_imaginary is False
assert sign(nz).is_integer is True
assert sign(nz).is_real is True
assert sign(nz).is_zero is False
assert sign(nz)**2 == 1
assert (sign(nz)**3).args == (sign(nz), 3)
assert sign(Symbol('x', nonnegative=True)).is_nonnegative
assert sign(Symbol('x', nonnegative=True)).is_nonpositive is None
assert sign(Symbol('x', nonpositive=True)).is_nonnegative is None
assert sign(Symbol('x', nonpositive=True)).is_nonpositive
assert sign(Symbol('x', real=True)).is_nonnegative is None
assert sign(Symbol('x', real=True)).is_nonpositive is None
assert sign(Symbol('x', real=True, zero=False)).is_nonpositive is None
x, y = Symbol('x', real=True), Symbol('y')
assert sign(x).rewrite(Piecewise) == \
Piecewise((1, x > 0), (-1, x < 0), (0, True))
assert sign(y).rewrite(Piecewise) == sign(y)
assert sign(x).rewrite(Heaviside) == 2*Heaviside(x)-1
assert sign(y).rewrite(Heaviside) == sign(y)
# evaluate what can be evaluated
assert sign(exp_polar(I*pi)*pi) is S.NegativeOne
eq = -sqrt(10 + 6*sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3))
# if there is a fast way to know when and when you cannot prove an
# expression like this is zero then the equality to zero is ok
assert sign(eq).func is sign or sign(eq) == 0
# but sometimes it's hard to do this so it's better not to load
# abs down with tests that will be very slow
q = 1 + sqrt(2) - 2*sqrt(3) + 1331*sqrt(6)
p = expand(q**3)**Rational(1, 3)
d = p - q
assert sign(d).func is sign or sign(d) == 0
def test_as_real_imag():
n = pi**1000
# the special code for working out the real
# and complex parts of a power with Integer exponent
# should not run if there is no imaginary part, hence
# this should not hang
assert n.as_real_imag() == (n, 0)
# issue 6261
x = Symbol('x')
assert sqrt(x).as_real_imag() == \
((re(x)**2 + im(x)**2)**(S(1)/4)*cos(atan2(im(x), re(x))/2),
(re(x)**2 + im(x)**2)**(S(1)/4)*sin(atan2(im(x), re(x))/2))
# issue 3853
a, b = symbols('a,b', real=True)
assert ((1 + sqrt(a + b*I))/2).as_real_imag() == \
(
(a**2 + b**2)**Rational(
1, 4)*cos(atan2(b, a)/2)/2 + Rational(1, 2),
(a**2 + b**2)**Rational(1, 4)*sin(atan2(b, a)/2)/2)
assert sqrt(a**2).as_real_imag() == (sqrt(a**2), 0)
i = symbols('i', imaginary=True)
assert sqrt(i**2).as_real_imag() == (0, abs(i))
assert ((1 + I)/(1 - I)).as_real_imag() == (0, 1)
assert ((1 + I)**3/(1 - I)).as_real_imag() == (-2, 0)
@XFAIL
def test_sign_issue_3068():
n = pi**1000
i = int(n)
assert (n - i).round() == 1 # doesn't hang
assert sign(n - i) == 1
# perhaps it's not possible to get the sign right when
# only 1 digit is being requested for this situation;
# 2 digits works
assert (n - x).n(1, subs={x: i}) > 0
assert (n - x).n(2, subs={x: i}) > 0
def test_Abs():
raises(TypeError, lambda: Abs(Interval(2, 3))) # issue 8717
x, y = symbols('x,y')
assert sign(sign(x)) == sign(x)
assert sign(x*y).func is sign
assert Abs(0) == 0
assert Abs(1) == 1
assert Abs(-1) == 1
assert Abs(I) == 1
assert Abs(-I) == 1
assert Abs(nan) == nan
assert Abs(zoo) == oo
assert Abs(I * pi) == pi
assert Abs(-I * pi) == pi
assert Abs(I * x) == Abs(x)
assert Abs(-I * x) == Abs(x)
assert Abs(-2*x) == 2*Abs(x)
assert Abs(-2.0*x) == 2.0*Abs(x)
assert Abs(2*pi*x*y) == 2*pi*Abs(x*y)
assert Abs(conjugate(x)) == Abs(x)
assert conjugate(Abs(x)) == Abs(x)
assert Abs(x).expand(complex=True) == sqrt(re(x)**2 + im(x)**2)
a = Symbol('a', positive=True)
assert Abs(2*pi*x*a) == 2*pi*a*Abs(x)
assert Abs(2*pi*I*x*a) == 2*pi*a*Abs(x)
x = Symbol('x', real=True)
n = Symbol('n', integer=True)
assert Abs((-1)**n) == 1
assert x**(2*n) == Abs(x)**(2*n)
assert Abs(x).diff(x) == sign(x)
assert abs(x) == Abs(x) # Python built-in
assert Abs(x)**3 == x**2*Abs(x)
assert Abs(x)**4 == x**4
assert (
Abs(x)**(3*n)).args == (Abs(x), 3*n) # leave symbolic odd unchanged
assert (1/Abs(x)).args == (Abs(x), -1)
assert 1/Abs(x)**3 == 1/(x**2*Abs(x))
assert Abs(x)**-3 == Abs(x)/(x**4)
assert Abs(x**3) == x**2*Abs(x)
assert Abs(I**I) == exp(-pi/2)
assert Abs((4 + 5*I)**(6 + 7*I)) == 68921*exp(-7*atan(S(5)/4))
y = Symbol('y', real=True)
assert Abs(I**y) == 1
y = Symbol('y')
assert Abs(I**y) == exp(-pi*im(y)/2)
x = Symbol('x', imaginary=True)
assert Abs(x).diff(x) == -sign(x)
eq = -sqrt(10 + 6*sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3))
# if there is a fast way to know when you can and when you cannot prove an
# expression like this is zero then the equality to zero is ok
assert abs(eq).func is Abs or abs(eq) == 0
# but sometimes it's hard to do this so it's better not to load
# abs down with tests that will be very slow
q = 1 + sqrt(2) - 2*sqrt(3) + 1331*sqrt(6)
p = expand(q**3)**Rational(1, 3)
d = p - q
assert abs(d).func is Abs or abs(d) == 0
assert Abs(4*exp(pi*I/4)) == 4
assert Abs(3**(2 + I)) == 9
assert Abs((-3)**(1 - I)) == 3*exp(pi)
assert Abs(oo) is oo
assert Abs(-oo) is oo
assert Abs(oo + I) is oo
assert Abs(oo + I*oo) is oo
a = Symbol('a', algebraic=True)
t = Symbol('t', transcendental=True)
x = Symbol('x')
assert re(a).is_algebraic
assert re(x).is_algebraic is None
assert re(t).is_algebraic is False
def test_Abs_rewrite():
x = Symbol('x', real=True)
a = Abs(x).rewrite(Heaviside).expand()
assert a == x*Heaviside(x) - x*Heaviside(-x)
for i in [-2, -1, 0, 1, 2]:
assert a.subs(x, i) == abs(i)
y = Symbol('y')
assert Abs(y).rewrite(Heaviside) == Abs(y)
x, y = Symbol('x', real=True), Symbol('y')
assert Abs(x).rewrite(Piecewise) == Piecewise((x, x >= 0), (-x, True))
assert Abs(y).rewrite(Piecewise) == Abs(y)
assert Abs(y).rewrite(sign) == y/sign(y)
def test_Abs_real():
# test some properties of abs that only apply
# to real numbers
x = Symbol('x', complex=True)
assert sqrt(x**2) != Abs(x)
assert Abs(x**2) != x**2
x = Symbol('x', real=True)
assert sqrt(x**2) == Abs(x)
assert Abs(x**2) == x**2
# if the symbol is zero, the following will still apply
nn = Symbol('nn', nonnegative=True, real=True)
np = Symbol('np', nonpositive=True, real=True)
assert Abs(nn) == nn
assert Abs(np) == -np
def test_Abs_properties():
x = Symbol('x')
assert Abs(x).is_real is True
assert Abs(x).is_rational is None
assert Abs(x).is_positive is None
assert Abs(x).is_nonnegative is True
z = Symbol('z', complex=True, zero=False)
assert Abs(z).is_real is True
assert Abs(z).is_rational is None
assert Abs(z).is_positive is True
assert Abs(z).is_zero is False
p = Symbol('p', positive=True)
assert Abs(p).is_real is True
assert Abs(p).is_rational is None
assert Abs(p).is_positive is True
assert Abs(p).is_zero is False
q = Symbol('q', rational=True)
assert Abs(q).is_rational is True
assert Abs(q).is_integer is None
assert Abs(q).is_positive is None
assert Abs(q).is_nonnegative is True
i = Symbol('i', integer=True)
assert Abs(i).is_integer is True
assert Abs(i).is_positive is None
assert Abs(i).is_nonnegative is True
e = Symbol('n', even=True)
ne = Symbol('ne', real=True, even=False)
assert Abs(e).is_even
assert Abs(ne).is_even is False
assert Abs(i).is_even is None
o = Symbol('n', odd=True)
no = Symbol('no', real=True, odd=False)
assert Abs(o).is_odd
assert Abs(no).is_odd is False
assert Abs(i).is_odd is None
def test_abs():
# this tests that abs calls Abs; don't rename to
# test_Abs since that test is already above
a = Symbol('a', positive=True)
assert abs(I*(1 + a)**2) == (1 + a)**2
def test_arg():
assert arg(0) == nan
assert arg(1) == 0
assert arg(-1) == pi
assert arg(I) == pi/2
assert arg(-I) == -pi/2
assert arg(1 + I) == pi/4
assert arg(-1 + I) == 3*pi/4
assert arg(1 - I) == -pi/4
assert arg(exp_polar(4*pi*I)) == 4*pi
assert arg(exp_polar(-7*pi*I)) == -7*pi
assert arg(exp_polar(5 - 3*pi*I/4)) == -3*pi/4
f = Function('f')
assert not arg(f(0) + I*f(1)).atoms(re)
p = Symbol('p', positive=True)
assert arg(p) == 0
n = Symbol('n', negative=True)
assert arg(n) == pi
x = Symbol('x')
assert conjugate(arg(x)) == arg(x)
e = p + I*p**2
assert arg(e) == arg(1 + p*I)
# make sure sign doesn't swap
e = -2*p + 4*I*p**2
assert arg(e) == arg(-1 + 2*p*I)
# make sure sign isn't lost
x = symbols('x', real=True) # could be zero
e = x + I*x
assert arg(e) == arg(x*(1 + I))
assert arg(e/p) == arg(x*(1 + I))
e = p*cos(p) + I*log(p)*exp(p)
assert arg(e).args[0] == e
# keep it simple -- let the user do more advanced cancellation
e = (p + 1) + I*(p**2 - 1)
assert arg(e).args[0] == e
f = Function('f')
e = 2*x*(f(0) - 1) - 2*x*f(0)
assert arg(e) == arg(-2*x)
assert arg(f(0)).func == arg and arg(f(0)).args == (f(0),)
def test_arg_rewrite():
assert arg(1 + I) == atan2(1, 1)
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert arg(x + I*y).rewrite(atan2) == atan2(y, x)
def test_adjoint():
a = Symbol('a', antihermitian=True)
b = Symbol('b', hermitian=True)
assert adjoint(a) == -a
assert adjoint(I*a) == I*a
assert adjoint(b) == b
assert adjoint(I*b) == -I*b
assert adjoint(a*b) == -b*a
assert adjoint(I*a*b) == I*b*a
x, y = symbols('x y')
assert adjoint(adjoint(x)) == x
assert adjoint(x + y) == adjoint(x) + adjoint(y)
assert adjoint(x - y) == adjoint(x) - adjoint(y)
assert adjoint(x * y) == adjoint(x) * adjoint(y)
assert adjoint(x / y) == adjoint(x) / adjoint(y)
assert adjoint(-x) == -adjoint(x)
x, y = symbols('x y', commutative=False)
assert adjoint(adjoint(x)) == x
assert adjoint(x + y) == adjoint(x) + adjoint(y)
assert adjoint(x - y) == adjoint(x) - adjoint(y)
assert adjoint(x * y) == adjoint(y) * adjoint(x)
assert adjoint(x / y) == 1 / adjoint(y) * adjoint(x)
assert adjoint(-x) == -adjoint(x)
def test_conjugate():
a = Symbol('a', real=True)
b = Symbol('b', imaginary=True)
assert conjugate(a) == a
assert conjugate(I*a) == -I*a
assert conjugate(b) == -b
assert conjugate(I*b) == I*b
assert conjugate(a*b) == -a*b
assert conjugate(I*a*b) == I*a*b
x, y = symbols('x y')
assert conjugate(conjugate(x)) == x
assert conjugate(x + y) == conjugate(x) + conjugate(y)
assert conjugate(x - y) == conjugate(x) - conjugate(y)
assert conjugate(x * y) == conjugate(x) * conjugate(y)
assert conjugate(x / y) == conjugate(x) / conjugate(y)
assert conjugate(-x) == -conjugate(x)
a = Symbol('a', algebraic=True)
t = Symbol('t', transcendental=True)
assert re(a).is_algebraic
assert re(x).is_algebraic is None
assert re(t).is_algebraic is False
def test_conjugate_transpose():
x = Symbol('x')
assert conjugate(transpose(x)) == adjoint(x)
assert transpose(conjugate(x)) == adjoint(x)
assert adjoint(transpose(x)) == conjugate(x)
assert transpose(adjoint(x)) == conjugate(x)
assert adjoint(conjugate(x)) == transpose(x)
assert conjugate(adjoint(x)) == transpose(x)
class Symmetric(Expr):
def _eval_adjoint(self):
return None
def _eval_conjugate(self):
return None
def _eval_transpose(self):
return self
x = Symmetric()
assert conjugate(x) == adjoint(x)
assert transpose(x) == x
def test_transpose():
a = Symbol('a', complex=True)
assert transpose(a) == a
assert transpose(I*a) == I*a
x, y = symbols('x y')
assert transpose(transpose(x)) == x
assert transpose(x + y) == transpose(x) + transpose(y)
assert transpose(x - y) == transpose(x) - transpose(y)
assert transpose(x * y) == transpose(x) * transpose(y)
assert transpose(x / y) == transpose(x) / transpose(y)
assert transpose(-x) == -transpose(x)
x, y = symbols('x y', commutative=False)
assert transpose(transpose(x)) == x
assert transpose(x + y) == transpose(x) + transpose(y)
assert transpose(x - y) == transpose(x) - transpose(y)
assert transpose(x * y) == transpose(y) * transpose(x)
assert transpose(x / y) == 1 / transpose(y) * transpose(x)
assert transpose(-x) == -transpose(x)
def test_polarify():
from sympy import polar_lift, polarify
x = Symbol('x')
z = Symbol('z', polar=True)
f = Function('f')
ES = {}
assert polarify(-1) == (polar_lift(-1), ES)
assert polarify(1 + I) == (polar_lift(1 + I), ES)
assert polarify(exp(x), subs=False) == exp(x)
assert polarify(1 + x, subs=False) == 1 + x
assert polarify(f(I) + x, subs=False) == f(polar_lift(I)) + x
assert polarify(x, lift=True) == polar_lift(x)
assert polarify(z, lift=True) == z
assert polarify(f(x), lift=True) == f(polar_lift(x))
assert polarify(1 + x, lift=True) == polar_lift(1 + x)
assert polarify(1 + f(x), lift=True) == polar_lift(1 + f(polar_lift(x)))
newex, subs = polarify(f(x) + z)
assert newex.subs(subs) == f(x) + z
mu = Symbol("mu")
sigma = Symbol("sigma", positive=True)
# Make sure polarify(lift=True) doesn't try to lift the integration
# variable
assert polarify(
Integral(sqrt(2)*x*exp(-(-mu + x)**2/(2*sigma**2))/(2*sqrt(pi)*sigma),
(x, -oo, oo)), lift=True) == Integral(sqrt(2)*(sigma*exp_polar(0))**exp_polar(I*pi)*
exp((sigma*exp_polar(0))**(2*exp_polar(I*pi))*exp_polar(I*pi)*polar_lift(-mu + x)**
(2*exp_polar(0))/2)*exp_polar(0)*polar_lift(x)/(2*sqrt(pi)), (x, -oo, oo))
def test_unpolarify():
from sympy import (exp_polar, polar_lift, exp, unpolarify,
principal_branch)
from sympy import gamma, erf, sin, tanh, uppergamma, Eq, Ne
from sympy.abc import x
p = exp_polar(7*I) + 1
u = exp(7*I) + 1
assert unpolarify(1) == 1
assert unpolarify(p) == u
assert unpolarify(p**2) == u**2
assert unpolarify(p**x) == p**x
assert unpolarify(p*x) == u*x
assert unpolarify(p + x) == u + x
assert unpolarify(sqrt(sin(p))) == sqrt(sin(u))
# Test reduction to principal branch 2*pi.
t = principal_branch(x, 2*pi)
assert unpolarify(t) == x
assert unpolarify(sqrt(t)) == sqrt(t)
# Test exponents_only.
assert unpolarify(p**p, exponents_only=True) == p**u
assert unpolarify(uppergamma(x, p**p)) == uppergamma(x, p**u)
# Test functions.
assert unpolarify(sin(p)) == sin(u)
assert unpolarify(tanh(p)) == tanh(u)
assert unpolarify(gamma(p)) == gamma(u)
assert unpolarify(erf(p)) == erf(u)
assert unpolarify(uppergamma(x, p)) == uppergamma(x, p)
assert unpolarify(uppergamma(sin(p), sin(p + exp_polar(0)))) == \
uppergamma(sin(u), sin(u + 1))
assert unpolarify(uppergamma(polar_lift(0), 2*exp_polar(0))) == \
uppergamma(0, 2)
assert unpolarify(Eq(p, 0)) == Eq(u, 0)
assert unpolarify(Ne(p, 0)) == Ne(u, 0)
assert unpolarify(polar_lift(x) > 0) == (x > 0)
# Test bools
assert unpolarify(True) is True
def test_issue_4035():
x = Symbol('x')
assert Abs(x).expand(trig=True) == Abs(x)
assert sign(x).expand(trig=True) == sign(x)
assert arg(x).expand(trig=True) == arg(x)
def test_issue_3206():
x = Symbol('x')
assert Abs(Abs(x)) == Abs(x)
def test_issue_4754_derivative_conjugate():
x = Symbol('x', real=True)
y = Symbol('y', imaginary=True)
f = Function('f')
assert (f(x).conjugate()).diff(x) == (f(x).diff(x)).conjugate()
assert (f(y).conjugate()).diff(y) == -(f(y).diff(y)).conjugate()
def test_derivatives_issue_4757():
x = Symbol('x', real=True)
y = Symbol('y', imaginary=True)
f = Function('f')
assert re(f(x)).diff(x) == re(f(x).diff(x))
assert im(f(x)).diff(x) == im(f(x).diff(x))
assert re(f(y)).diff(y) == -I*im(f(y).diff(y))
assert im(f(y)).diff(y) == -I*re(f(y).diff(y))
assert Abs(f(x)).diff(x).subs(f(x), 1 + I*x).doit() == x/sqrt(1 + x**2)
assert arg(f(x)).diff(x).subs(f(x), 1 + I*x**2).doit() == 2*x/(1 + x**4)
assert Abs(f(y)).diff(y).subs(f(y), 1 + y).doit() == -y/sqrt(1 - y**2)
assert arg(f(y)).diff(y).subs(f(y), I + y**2).doit() == 2*y/(1 + y**4)
def test_issue_11413():
from sympy import symbols, Matrix, simplify
v0 = Symbol('v0')
v1 = Symbol('v1')
v2 = Symbol('v2')
V = Matrix([[v0],[v1],[v2]])
U = V.normalized()
assert U == Matrix([
[v0/sqrt(Abs(v0)**2 + Abs(v1)**2 + Abs(v2)**2)],
[v1/sqrt(Abs(v0)**2 + Abs(v1)**2 + Abs(v2)**2)],
[v2/sqrt(Abs(v0)**2 + Abs(v1)**2 + Abs(v2)**2)]])
U.norm = sqrt(v0**2/(v0**2 + v1**2 + v2**2) + v1**2/(v0**2 + v1**2 + v2**2) + v2**2/(v0**2 + v1**2 + v2**2))
assert simplify(U.norm) == 1
def test_periodic_argument():
from sympy import (periodic_argument, unbranched_argument, oo,
principal_branch, polar_lift, pi)
x = Symbol('x')
p = Symbol('p', positive=True)
assert unbranched_argument(2 + I) == periodic_argument(2 + I, oo)
assert unbranched_argument(1 + x) == periodic_argument(1 + x, oo)
assert N_equals(unbranched_argument((1 + I)**2), pi/2)
assert N_equals(unbranched_argument((1 - I)**2), -pi/2)
assert N_equals(periodic_argument((1 + I)**2, 3*pi), pi/2)
assert N_equals(periodic_argument((1 - I)**2, 3*pi), -pi/2)
assert unbranched_argument(principal_branch(x, pi)) == \
periodic_argument(x, pi)
assert unbranched_argument(polar_lift(2 + I)) == unbranched_argument(2 + I)
assert periodic_argument(polar_lift(2 + I), 2*pi) == \
periodic_argument(2 + I, 2*pi)
assert periodic_argument(polar_lift(2 + I), 3*pi) == \
periodic_argument(2 + I, 3*pi)
assert periodic_argument(polar_lift(2 + I), pi) == \
periodic_argument(polar_lift(2 + I), pi)
assert unbranched_argument(polar_lift(1 + I)) == pi/4
assert periodic_argument(2*p, p) == periodic_argument(p, p)
assert periodic_argument(pi*p, p) == periodic_argument(p, p)
assert Abs(polar_lift(1 + I)) == Abs(1 + I)
@XFAIL
def test_principal_branch_fail():
# TODO XXX why does abs(x)._eval_evalf() not fall back to global evalf?
assert N_equals(principal_branch((1 + I)**2, pi/2), 0)
def test_principal_branch():
from sympy import principal_branch, polar_lift, exp_polar
p = Symbol('p', positive=True)
x = Symbol('x')
neg = Symbol('x', negative=True)
assert principal_branch(polar_lift(x), p) == principal_branch(x, p)
assert principal_branch(polar_lift(2 + I), p) == principal_branch(2 + I, p)
assert principal_branch(2*x, p) == 2*principal_branch(x, p)
assert principal_branch(1, pi) == exp_polar(0)
assert principal_branch(-1, 2*pi) == exp_polar(I*pi)
assert principal_branch(-1, pi) == exp_polar(0)
assert principal_branch(exp_polar(3*pi*I)*x, 2*pi) == \
principal_branch(exp_polar(I*pi)*x, 2*pi)
assert principal_branch(neg*exp_polar(pi*I), 2*pi) == neg*exp_polar(-I*pi)
# related to issue #14692
assert principal_branch(exp_polar(-I*pi/2)/polar_lift(neg), 2*pi) == \
exp_polar(-I*pi/2)/neg
assert N_equals(principal_branch((1 + I)**2, 2*pi), 2*I)
assert N_equals(principal_branch((1 + I)**2, 3*pi), 2*I)
assert N_equals(principal_branch((1 + I)**2, 1*pi), 2*I)
# test argument sanitization
assert principal_branch(x, I).func is principal_branch
assert principal_branch(x, -4).func is principal_branch
assert principal_branch(x, -oo).func is principal_branch
assert principal_branch(x, zoo).func is principal_branch
@XFAIL
def test_issue_6167_6151():
n = pi**1000
i = int(n)
assert sign(n - i) == 1
assert abs(n - i) == n - i
eps = pi**-1500
big = pi**1000
one = cos(x)**2 + sin(x)**2
e = big*one - big + eps
assert sign(simplify(e)) == 1
for xi in (111, 11, 1, S(1)/10):
assert sign(e.subs(x, xi)) == 1
def test_issue_14216():
from sympy.functions.elementary.complexes import unpolarify
A = MatrixSymbol("A", 2, 2)
assert unpolarify(A[0, 0]) == A[0, 0]
assert unpolarify(A[0, 0]*A[1, 0]) == A[0, 0]*A[1, 0]
def test_issue_14238():
# doesn't cause recursion error
r = Symbol('r', real=True)
assert Abs(r + Piecewise((0, r > 0), (1 - r, True)))
def test_zero_assumptions():
nr = Symbol('nonreal', real=False)
ni = Symbol('nonimaginary', imaginary=False)
# imaginary implies not zero
nzni = Symbol('nonzerononimaginary', zero=False, imaginary=False)
assert re(nr).is_zero is None
assert im(nr).is_zero is False
assert re(ni).is_zero is None
assert im(ni).is_zero is None
assert re(nzni).is_zero is False
assert im(nzni).is_zero is None
def test_issue_15893():
f = Function('f', real=True)
x = Symbol('x', real=True)
eq = Derivative(Abs(f(x)), f(x))
assert eq.doit() == sign(f(x))
|
16e13f605a9bd4d2b993ed3df890bd0860a75ccccd99f0dd7b21da109656fccd
|
from sympy import (
adjoint, And, Basic, conjugate, diff, expand, Eq, Function, I, ITE,
Integral, integrate, Interval, lambdify, log, Max, Min, oo, Or, pi,
Piecewise, piecewise_fold, Rational, solve, symbols, transpose,
cos, sin, exp, Abs, Ne, Not, Symbol, S, sqrt, Tuple, zoo,
factor_terms, DiracDelta, Heaviside, Add, Mul, factorial)
from sympy.printing import srepr
from sympy.utilities.pytest import raises, slow
from sympy.functions.elementary.piecewise import Undefined
a, b, c, d, x, y = symbols('a:d, x, y')
z = symbols('z', nonzero=True)
def test_piecewise():
# Test canonicalization
assert Piecewise((x, x < 1), (0, True)) == Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (0, True), (1, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (0, False), (-1, 1 > 2)) == \
Piecewise((x, x < 1))
assert Piecewise((x, x < 1), (0, x < 1), (0, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (0, x < 2), (0, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (x, x < 2), (0, True)) == \
Piecewise((x, Or(x < 1, x < 2)), (0, True))
assert Piecewise((x, x < 1), (x, x < 2), (x, True)) == x
assert Piecewise((x, True)) == x
# Explicitly constructed empty Piecewise not accepted
raises(TypeError, lambda: Piecewise())
# False condition is never retained
assert Piecewise((2*x, x < 0), (x, False)) == \
Piecewise((2*x, x < 0), (x, False), evaluate=False) == \
Piecewise((2*x, x < 0))
assert Piecewise((x, False)) == Undefined
raises(TypeError, lambda: Piecewise(x))
assert Piecewise((x, 1)) == x # 1 and 0 are accepted as True/False
raises(TypeError, lambda: Piecewise((x, 2)))
raises(TypeError, lambda: Piecewise((x, x**2)))
raises(TypeError, lambda: Piecewise(([1], True)))
assert Piecewise(((1, 2), True)) == Tuple(1, 2)
cond = (Piecewise((1, x < 0), (2, True)) < y)
assert Piecewise((1, cond)
) == Piecewise((1, ITE(x < 0, y > 1, y > 2)))
assert Piecewise((1, x > 0), (2, And(x <= 0, x > -1))
) == Piecewise((1, x > 0), (2, x > -1))
# Test subs
p = Piecewise((-1, x < -1), (x**2, x < 0), (log(x), x >= 0))
p_x2 = Piecewise((-1, x**2 < -1), (x**4, x**2 < 0), (log(x**2), x**2 >= 0))
assert p.subs(x, x**2) == p_x2
assert p.subs(x, -5) == -1
assert p.subs(x, -1) == 1
assert p.subs(x, 1) == log(1)
# More subs tests
p2 = Piecewise((1, x < pi), (-1, x < 2*pi), (0, x > 2*pi))
p3 = Piecewise((1, Eq(x, 0)), (1/x, True))
p4 = Piecewise((1, Eq(x, 0)), (2, 1/x>2))
assert p2.subs(x, 2) == 1
assert p2.subs(x, 4) == -1
assert p2.subs(x, 10) == 0
assert p3.subs(x, 0.0) == 1
assert p4.subs(x, 0.0) == 1
f, g, h = symbols('f,g,h', cls=Function)
pf = Piecewise((f(x), x < -1), (f(x) + h(x) + 2, x <= 1))
pg = Piecewise((g(x), x < -1), (g(x) + h(x) + 2, x <= 1))
assert pg.subs(g, f) == pf
assert Piecewise((1, Eq(x, 0)), (0, True)).subs(x, 0) == 1
assert Piecewise((1, Eq(x, 0)), (0, True)).subs(x, 1) == 0
assert Piecewise((1, Eq(x, y)), (0, True)).subs(x, y) == 1
assert Piecewise((1, Eq(x, z)), (0, True)).subs(x, z) == 1
assert Piecewise((1, Eq(exp(x), cos(z))), (0, True)).subs(x, z) == \
Piecewise((1, Eq(exp(z), cos(z))), (0, True))
p5 = Piecewise( (0, Eq(cos(x) + y, 0)), (1, True))
assert p5.subs(y, 0) == Piecewise( (0, Eq(cos(x), 0)), (1, True))
assert Piecewise((-1, y < 1), (0, x < 0), (1, Eq(x, 0)), (2, True)
).subs(x, 1) == Piecewise((-1, y < 1), (2, True))
assert Piecewise((1, Eq(x**2, -1)), (2, x < 0)).subs(x, I) == 1
p6 = Piecewise((x, x > 0))
n = symbols('n', negative=True)
assert p6.subs(x, n) == Undefined
# Test evalf
assert p.evalf() == p
assert p.evalf(subs={x: -2}) == -1
assert p.evalf(subs={x: -1}) == 1
assert p.evalf(subs={x: 1}) == log(1)
assert p6.evalf(subs={x: -5}) == Undefined
# Test doit
f_int = Piecewise((Integral(x, (x, 0, 1)), x < 1))
assert f_int.doit() == Piecewise( (S(1)/2, x < 1) )
# Test differentiation
f = x
fp = x*p
dp = Piecewise((0, x < -1), (2*x, x < 0), (1/x, x >= 0))
fp_dx = x*dp + p
assert diff(p, x) == dp
assert diff(f*p, x) == fp_dx
# Test simple arithmetic
assert x*p == fp
assert x*p + p == p + x*p
assert p + f == f + p
assert p + dp == dp + p
assert p - dp == -(dp - p)
# Test power
dp2 = Piecewise((0, x < -1), (4*x**2, x < 0), (1/x**2, x >= 0))
assert dp**2 == dp2
# Test _eval_interval
f1 = x*y + 2
f2 = x*y**2 + 3
peval = Piecewise((f1, x < 0), (f2, x > 0))
peval_interval = f1.subs(
x, 0) - f1.subs(x, -1) + f2.subs(x, 1) - f2.subs(x, 0)
assert peval._eval_interval(x, 0, 0) == 0
assert peval._eval_interval(x, -1, 1) == peval_interval
peval2 = Piecewise((f1, x < 0), (f2, True))
assert peval2._eval_interval(x, 0, 0) == 0
assert peval2._eval_interval(x, 1, -1) == -peval_interval
assert peval2._eval_interval(x, -1, -2) == f1.subs(x, -2) - f1.subs(x, -1)
assert peval2._eval_interval(x, -1, 1) == peval_interval
assert peval2._eval_interval(x, None, 0) == peval2.subs(x, 0)
assert peval2._eval_interval(x, -1, None) == -peval2.subs(x, -1)
# Test integration
assert p.integrate() == Piecewise(
(-x, x < -1),
(x**3/3 + S(4)/3, x < 0),
(x*log(x) - x + S(4)/3, True))
p = Piecewise((x, x < 1), (x**2, -1 <= x), (x, 3 < x))
assert integrate(p, (x, -2, 2)) == 5/6.0
assert integrate(p, (x, 2, -2)) == -5/6.0
p = Piecewise((0, x < 0), (1, x < 1), (0, x < 2), (1, x < 3), (0, True))
assert integrate(p, (x, -oo, oo)) == 2
p = Piecewise((x, x < -10), (x**2, x <= -1), (x, 1 < x))
assert integrate(p, (x, -2, 2)) == Undefined
# Test commutativity
assert isinstance(p, Piecewise) and p.is_commutative is True
def test_piecewise_free_symbols():
f = Piecewise((x, a < 0), (y, True))
assert f.free_symbols == {x, y, a}
def test_piecewise_integrate1():
x, y = symbols('x y', real=True, finite=True)
f = Piecewise(((x - 2)**2, x >= 0), (1, True))
assert integrate(f, (x, -2, 2)) == Rational(14, 3)
g = Piecewise(((x - 5)**5, x >= 4), (f, True))
assert integrate(g, (x, -2, 2)) == Rational(14, 3)
assert integrate(g, (x, -2, 5)) == Rational(43, 6)
assert g == Piecewise(((x - 5)**5, x >= 4), (f, x < 4))
g = Piecewise(((x - 5)**5, 2 <= x), (f, x < 2))
assert integrate(g, (x, -2, 2)) == Rational(14, 3)
assert integrate(g, (x, -2, 5)) == -Rational(701, 6)
assert g == Piecewise(((x - 5)**5, 2 <= x), (f, True))
g = Piecewise(((x - 5)**5, 2 <= x), (2*f, True))
assert integrate(g, (x, -2, 2)) == 2 * Rational(14, 3)
assert integrate(g, (x, -2, 5)) == -Rational(673, 6)
def test_piecewise_integrate1b():
g = Piecewise((1, x > 0), (0, Eq(x, 0)), (-1, x < 0))
assert integrate(g, (x, -1, 1)) == 0
g = Piecewise((1, x - y < 0), (0, True))
assert integrate(g, (y, -oo, 0)) == -Min(0, x)
assert g.subs(x, -3).integrate((y, -oo, 0)) == 3
assert integrate(g, (y, 0, -oo)) == Min(0, x)
assert integrate(g, (y, 0, oo)) == -Max(0, x) + oo
assert integrate(g, (y, -oo, 42)) == -Min(42, x) + 42
assert integrate(g, (y, -oo, oo)) == -x + oo
g = Piecewise((0, x < 0), (x, x <= 1), (1, True))
gy1 = g.integrate((x, y, 1))
g1y = g.integrate((x, 1, y))
for yy in (-1, S.Half, 2):
assert g.integrate((x, yy, 1)) == gy1.subs(y, yy)
assert g.integrate((x, 1, yy)) == g1y.subs(y, yy)
assert gy1 == Piecewise(
(-Min(1, Max(0, y))**2/2 + S(1)/2, y < 1),
(-y + 1, True))
assert g1y == Piecewise(
(Min(1, Max(0, y))**2/2 - S(1)/2, y < 1),
(y - 1, True))
@slow
def test_piecewise_integrate1ca():
y = symbols('y', real=True)
g = Piecewise(
(1 - x, Interval(0, 1).contains(x)),
(1 + x, Interval(-1, 0).contains(x)),
(0, True)
)
gy1 = g.integrate((x, y, 1))
g1y = g.integrate((x, 1, y))
assert g.integrate((x, -2, 1)) == gy1.subs(y, -2)
assert g.integrate((x, 1, -2)) == g1y.subs(y, -2)
assert g.integrate((x, 0, 1)) == gy1.subs(y, 0)
assert g.integrate((x, 1, 0)) == g1y.subs(y, 0)
# XXX Make test pass without simplify
assert g.integrate((x, 2, 1)) == gy1.subs(y, 2).simplify()
assert g.integrate((x, 1, 2)) == g1y.subs(y, 2).simplify()
assert piecewise_fold(gy1.rewrite(Piecewise)) == \
Piecewise(
(1, y <= -1),
(-y**2/2 - y + S(1)/2, y <= 0),
(y**2/2 - y + S(1)/2, y < 1),
(0, True))
assert piecewise_fold(g1y.rewrite(Piecewise)) == \
Piecewise(
(-1, y <= -1),
(y**2/2 + y - S(1)/2, y <= 0),
(-y**2/2 + y - S(1)/2, y < 1),
(0, True))
# g1y and gy1 should simplify if the condition that y < 1
# is applied, e.g. Min(1, Max(-1, y)) --> Max(-1, y)
# XXX Make test pass without simplify
assert gy1.simplify() == Piecewise(
(
-Min(1, Max(-1, y))**2/2 - Min(1, Max(-1, y)) +
Min(1, Max(0, y))**2 + S(1)/2, y < 1),
(0, True)
)
assert g1y.simplify() == Piecewise(
(
Min(1, Max(-1, y))**2/2 + Min(1, Max(-1, y)) -
Min(1, Max(0, y))**2 - S(1)/2, y < 1),
(0, True))
@slow
def test_piecewise_integrate1cb():
y = symbols('y', real=True)
g = Piecewise(
(0, Or(x <= -1, x >= 1)),
(1 - x, x > 0),
(1 + x, True)
)
gy1 = g.integrate((x, y, 1))
g1y = g.integrate((x, 1, y))
assert g.integrate((x, -2, 1)) == gy1.subs(y, -2)
assert g.integrate((x, 1, -2)) == g1y.subs(y, -2)
assert g.integrate((x, 0, 1)) == gy1.subs(y, 0)
assert g.integrate((x, 1, 0)) == g1y.subs(y, 0)
assert g.integrate((x, 2, 1)) == gy1.subs(y, 2)
assert g.integrate((x, 1, 2)) == g1y.subs(y, 2)
assert piecewise_fold(gy1.rewrite(Piecewise)) == \
Piecewise(
(1, y <= -1),
(-y**2/2 - y + S(1)/2, y <= 0),
(y**2/2 - y + S(1)/2, y < 1),
(0, True))
assert piecewise_fold(g1y.rewrite(Piecewise)) == \
Piecewise(
(-1, y <= -1),
(y**2/2 + y - S(1)/2, y <= 0),
(-y**2/2 + y - S(1)/2, y < 1),
(0, True))
# g1y and gy1 should simplify if the condition that y < 1
# is applied, e.g. Min(1, Max(-1, y)) --> Max(-1, y)
assert gy1 == Piecewise(
(
-Min(1, Max(-1, y))**2/2 - Min(1, Max(-1, y)) +
Min(1, Max(0, y))**2 + S(1)/2, y < 1),
(0, True)
)
assert g1y == Piecewise(
(
Min(1, Max(-1, y))**2/2 + Min(1, Max(-1, y)) -
Min(1, Max(0, y))**2 - S(1)/2, y < 1),
(0, True))
def test_piecewise_integrate2():
from itertools import permutations
lim = Tuple(x, c, d)
p = Piecewise((1, x < a), (2, x > b), (3, True))
q = p.integrate(lim)
assert q == Piecewise(
(-c + 2*d - 2*Min(d, Max(a, c)) + Min(d, Max(a, b, c)), c < d),
(-2*c + d + 2*Min(c, Max(a, d)) - Min(c, Max(a, b, d)), True))
for v in permutations((1, 2, 3, 4)):
r = dict(zip((a, b, c, d), v))
assert p.subs(r).integrate(lim.subs(r)) == q.subs(r)
def test_meijer_bypass():
# totally bypass meijerg machinery when dealing
# with Piecewise in integrate
assert Piecewise((1, x < 4), (0, True)).integrate((x, oo, 1)) == -3
def test_piecewise_integrate3_inequality_conditions():
from sympy.utilities.iterables import cartes
lim = (x, 0, 5)
# set below includes two pts below range, 2 pts in range,
# 2 pts above range, and the boundaries
N = (-2, -1, 0, 1, 2, 5, 6, 7)
p = Piecewise((1, x > a), (2, x > b), (0, True))
ans = p.integrate(lim)
for i, j in cartes(N, repeat=2):
reps = dict(zip((a, b), (i, j)))
assert ans.subs(reps) == p.subs(reps).integrate(lim)
assert ans.subs(a, 4).subs(b, 1) == 0 + 2*3 + 1
p = Piecewise((1, x > a), (2, x < b), (0, True))
ans = p.integrate(lim)
for i, j in cartes(N, repeat=2):
reps = dict(zip((a, b), (i, j)))
assert ans.subs(reps) == p.subs(reps).integrate(lim)
# delete old tests that involved c1 and c2 since those
# reduce to the above except that a value of 0 was used
# for two expressions whereas the above uses 3 different
# values
@slow
def test_piecewise_integrate4_symbolic_conditions():
a = Symbol('a', real=True, finite=True)
b = Symbol('b', real=True, finite=True)
x = Symbol('x', real=True, finite=True)
y = Symbol('y', real=True, finite=True)
p0 = Piecewise((0, Or(x < a, x > b)), (1, True))
p1 = Piecewise((0, x < a), (0, x > b), (1, True))
p2 = Piecewise((0, x > b), (0, x < a), (1, True))
p3 = Piecewise((0, x < a), (1, x < b), (0, True))
p4 = Piecewise((0, x > b), (1, x > a), (0, True))
p5 = Piecewise((1, And(a < x, x < b)), (0, True))
# check values of a=1, b=3 (and reversed) with values
# of y of 0, 1, 2, 3, 4
lim = Tuple(x, -oo, y)
for p in (p0, p1, p2, p3, p4, p5):
ans = p.integrate(lim)
for i in range(5):
reps = {a:1, b:3, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
reps = {a: 3, b:1, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
lim = Tuple(x, y, oo)
for p in (p0, p1, p2, p3, p4, p5):
ans = p.integrate(lim)
for i in range(5):
reps = {a:1, b:3, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
reps = {a:3, b:1, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
ans = Piecewise(
(0, x <= Min(a, b)),
(x - Min(a, b), x <= b),
(b - Min(a, b), True))
for i in (p0, p1, p2, p4):
assert i.integrate(x) == ans
assert p3.integrate(x) == Piecewise(
(0, x < a),
(-a + x, x <= Max(a, b)),
(-a + Max(a, b), True))
assert p5.integrate(x) == Piecewise(
(0, x <= a),
(-a + x, x <= Max(a, b)),
(-a + Max(a, b), True))
p1 = Piecewise((0, x < a), (0.5, x > b), (1, True))
p2 = Piecewise((0.5, x > b), (0, x < a), (1, True))
p3 = Piecewise((0, x < a), (1, x < b), (0.5, True))
p4 = Piecewise((0.5, x > b), (1, x > a), (0, True))
p5 = Piecewise((1, And(a < x, x < b)), (0.5, x > b), (0, True))
# check values of a=1, b=3 (and reversed) with values
# of y of 0, 1, 2, 3, 4
lim = Tuple(x, -oo, y)
for p in (p1, p2, p3, p4, p5):
ans = p.integrate(lim)
for i in range(5):
reps = {a:1, b:3, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
reps = {a: 3, b:1, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
def test_piecewise_integrate5_independent_conditions():
p = Piecewise((0, Eq(y, 0)), (x*y, True))
assert integrate(p, (x, 1, 3)) == Piecewise((0, Eq(y, 0)), (4*y, True))
def test_piecewise_simplify():
p = Piecewise(((x**2 + 1)/x**2, Eq(x*(1 + x) - x**2, 0)),
((-1)**x*(-1), True))
assert p.simplify() == \
Piecewise((zoo, Eq(x, 0)), ((-1)**(x + 1), True))
# simplify when there are Eq in conditions
assert Piecewise(
(a, And(Eq(a, 0), Eq(a + b, 0))), (1, True)).simplify(
) == Piecewise(
(0, And(Eq(a, 0), Eq(b, 0))), (1, True))
assert Piecewise((2*x*factorial(a)/(factorial(y)*factorial(-y + a)),
Eq(y, 0) & Eq(-y + a, 0)), (2*factorial(a)/(factorial(y)*factorial(-y
+ a)), Eq(y, 0) & Eq(-y + a, 1)), (0, True)).simplify(
) == Piecewise(
(2*x, And(Eq(a, 0), Eq(y, 0))),
(2, And(Eq(a, 1), Eq(y, 0))),
(0, True))
def test_piecewise_solve():
abs2 = Piecewise((-x, x <= 0), (x, x > 0))
f = abs2.subs(x, x - 2)
assert solve(f, x) == [2]
assert solve(f - 1, x) == [1, 3]
f = Piecewise(((x - 2)**2, x >= 0), (1, True))
assert solve(f, x) == [2]
g = Piecewise(((x - 5)**5, x >= 4), (f, True))
assert solve(g, x) == [2, 5]
g = Piecewise(((x - 5)**5, x >= 4), (f, x < 4))
assert solve(g, x) == [2, 5]
g = Piecewise(((x - 5)**5, x >= 2), (f, x < 2))
assert solve(g, x) == [5]
g = Piecewise(((x - 5)**5, x >= 2), (f, True))
assert solve(g, x) == [5]
g = Piecewise(((x - 5)**5, x >= 2), (f, True), (10, False))
assert solve(g, x) == [5]
g = Piecewise(((x - 5)**5, x >= 2),
(-x + 2, x - 2 <= 0), (x - 2, x - 2 > 0))
assert solve(g, x) == [5]
# if no symbol is given the piecewise detection must still work
assert solve(Piecewise((x - 2, x > 2), (2 - x, True)) - 3) == [-1, 5]
f = Piecewise(((x - 2)**2, x >= 0), (0, True))
raises(NotImplementedError, lambda: solve(f, x))
def nona(ans):
return list(filter(lambda x: x is not S.NaN, ans))
p = Piecewise((x**2 - 4, x < y), (x - 2, True))
ans = solve(p, x)
assert nona([i.subs(y, -2) for i in ans]) == [2]
assert nona([i.subs(y, 2) for i in ans]) == [-2, 2]
assert nona([i.subs(y, 3) for i in ans]) == [-2, 2]
assert ans == [
Piecewise((-2, y > -2), (S.NaN, True)),
Piecewise((2, y <= 2), (S.NaN, True)),
Piecewise((2, y > 2), (S.NaN, True))]
# issue 6060
absxm3 = Piecewise(
(x - 3, S(0) <= x - 3),
(3 - x, S(0) > x - 3)
)
assert solve(absxm3 - y, x) == [
Piecewise((-y + 3, -y < 0), (S.NaN, True)),
Piecewise((y + 3, y >= 0), (S.NaN, True))]
p = Symbol('p', positive=True)
assert solve(absxm3 - p, x) == [-p + 3, p + 3]
# issue 6989
f = Function('f')
assert solve(Eq(-f(x), Piecewise((1, x > 0), (0, True))), f(x)) == \
[Piecewise((-1, x > 0), (0, True))]
# issue 8587
f = Piecewise((2*x**2, And(S(0) < x, x < 1)), (2, True))
assert solve(f - 1) == [1/sqrt(2)]
def test_piecewise_fold():
p = Piecewise((x, x < 1), (1, 1 <= x))
assert piecewise_fold(x*p) == Piecewise((x**2, x < 1), (x, 1 <= x))
assert piecewise_fold(p + p) == Piecewise((2*x, x < 1), (2, 1 <= x))
assert piecewise_fold(Piecewise((1, x < 0), (2, True))
+ Piecewise((10, x < 0), (-10, True))) == \
Piecewise((11, x < 0), (-8, True))
p1 = Piecewise((0, x < 0), (x, x <= 1), (0, True))
p2 = Piecewise((0, x < 0), (1 - x, x <= 1), (0, True))
p = 4*p1 + 2*p2
assert integrate(
piecewise_fold(p), (x, -oo, oo)) == integrate(2*x + 2, (x, 0, 1))
assert piecewise_fold(
Piecewise((1, y <= 0), (-Piecewise((2, y >= 0)), True)
)) == Piecewise((1, y <= 0), (-2, y >= 0))
assert piecewise_fold(Piecewise((x, ITE(x > 0, y < 1, y > 1)))
) == Piecewise((x, ((x <= 0) | (y < 1)) & ((x > 0) | (y > 1))))
a, b = (Piecewise((2, Eq(x, 0)), (0, True)),
Piecewise((x, Eq(-x + y, 0)), (1, Eq(-x + y, 1)), (0, True)))
assert piecewise_fold(Mul(a, b, evaluate=False)
) == piecewise_fold(Mul(b, a, evaluate=False))
def test_piecewise_fold_piecewise_in_cond():
p1 = Piecewise((cos(x), x < 0), (0, True))
p2 = Piecewise((0, Eq(p1, 0)), (p1 / Abs(p1), True))
p3 = piecewise_fold(p2)
assert(p2.subs(x, -pi/2) == 0.0)
assert(p2.subs(x, 1) == 0.0)
assert(p2.subs(x, -pi/4) == 1.0)
p4 = Piecewise((0, Eq(p1, 0)), (1,True))
ans = piecewise_fold(p4)
for i in range(-1, 1):
assert ans.subs(x, i) == p4.subs(x, i)
r1 = 1 < Piecewise((1, x < 1), (3, True))
ans = piecewise_fold(r1)
for i in range(2):
assert ans.subs(x, i) == r1.subs(x, i)
p5 = Piecewise((1, x < 0), (3, True))
p6 = Piecewise((1, x < 1), (3, True))
p7 = Piecewise((1, p5 < p6), (0, True))
ans = piecewise_fold(p7)
for i in range(-1, 2):
assert ans.subs(x, i) == p7.subs(x, i)
def test_piecewise_fold_piecewise_in_cond_2():
p1 = Piecewise((cos(x), x < 0), (0, True))
p2 = Piecewise((0, Eq(p1, 0)), (1 / p1, True))
p3 = Piecewise(
(0, (x >= 0) | Eq(cos(x), 0)),
(1/cos(x), x < 0),
(zoo, True)) # redundant b/c all x are already covered
assert(piecewise_fold(p2) == p3)
def test_piecewise_fold_expand():
p1 = Piecewise((1, Interval(0, 1, False, True).contains(x)), (0, True))
p2 = piecewise_fold(expand((1 - x)*p1))
assert p2 == Piecewise((1 - x, (x >= 0) & (x < 1)), (0, True))
assert p2 == expand(piecewise_fold((1 - x)*p1))
def test_piecewise_duplicate():
p = Piecewise((x, x < -10), (x**2, x <= -1), (x, 1 < x))
assert p == Piecewise(*p.args)
def test_doit():
p1 = Piecewise((x, x < 1), (x**2, -1 <= x), (x, 3 < x))
p2 = Piecewise((x, x < 1), (Integral(2 * x), -1 <= x), (x, 3 < x))
assert p2.doit() == p1
assert p2.doit(deep=False) == p2
def test_piecewise_interval():
p1 = Piecewise((x, Interval(0, 1).contains(x)), (0, True))
assert p1.subs(x, -0.5) == 0
assert p1.subs(x, 0.5) == 0.5
assert p1.diff(x) == Piecewise((1, Interval(0, 1).contains(x)), (0, True))
assert integrate(p1, x) == Piecewise(
(0, x <= 0),
(x**2/2, x <= 1),
(S(1)/2, True))
def test_piecewise_collapse():
assert Piecewise((x, True)) == x
a = x < 1
assert Piecewise((x, a), (x + 1, a)) == Piecewise((x, a))
assert Piecewise((x, a), (x + 1, a.reversed)) == Piecewise((x, a))
b = x < 5
def canonical(i):
if isinstance(i, Piecewise):
return Piecewise(*i.args)
return i
for args in [
((1, a), (Piecewise((2, a), (3, b)), b)),
((1, a), (Piecewise((2, a), (3, b.reversed)), b)),
((1, a), (Piecewise((2, a), (3, b)), b), (4, True)),
((1, a), (Piecewise((2, a), (3, b), (4, True)), b)),
((1, a), (Piecewise((2, a), (3, b), (4, True)), b), (5, True))]:
for i in (0, 2, 10):
assert canonical(
Piecewise(*args, evaluate=False).subs(x, i)
) == canonical(Piecewise(*args).subs(x, i))
r1, r2, r3, r4 = symbols('r1:5')
a = x < r1
b = x < r2
c = x < r3
d = x < r4
assert Piecewise((1, a), (Piecewise(
(2, a), (3, b), (4, c)), b), (5, c)
) == Piecewise((1, a), (3, b), (5, c))
assert Piecewise((1, a), (Piecewise(
(2, a), (3, b), (4, c), (6, True)), c), (5, d)
) == Piecewise((1, a), (Piecewise(
(3, b), (4, c)), c), (5, d))
assert Piecewise((1, Or(a, d)), (Piecewise(
(2, d), (3, b), (4, c)), b), (5, c)
) == Piecewise((1, Or(a, d)), (Piecewise(
(2, d), (3, b)), b), (5, c))
assert Piecewise((1, c), (2, ~c), (3, S.true)
) == Piecewise((1, c), (2, S.true))
assert Piecewise((1, c), (2, And(~c, b)), (3,True)
) == Piecewise((1, c), (2, b), (3, True))
assert Piecewise((1, c), (2, Or(~c, b)), (3,True)
).subs(dict(zip((r1, r2, r3, r4, x), (1, 2, 3, 4, 3.5)))) == 2
assert Piecewise((1, c), (2, ~c)) == Piecewise((1, c), (2, True))
def test_piecewise_lambdify():
p = Piecewise(
(x**2, x < 0),
(x, Interval(0, 1, False, True).contains(x)),
(2 - x, x >= 1),
(0, True)
)
f = lambdify(x, p)
assert f(-2.0) == 4.0
assert f(0.0) == 0.0
assert f(0.5) == 0.5
assert f(2.0) == 0.0
def test_piecewise_series():
from sympy import sin, cos, O
p1 = Piecewise((sin(x), x < 0), (cos(x), x > 0))
p2 = Piecewise((x + O(x**2), x < 0), (1 + O(x**2), x > 0))
assert p1.nseries(x, n=2) == p2
def test_piecewise_as_leading_term():
p1 = Piecewise((1/x, x > 1), (0, True))
p2 = Piecewise((x, x > 1), (0, True))
p3 = Piecewise((1/x, x > 1), (x, True))
p4 = Piecewise((x, x > 1), (1/x, True))
p5 = Piecewise((1/x, x > 1), (x, True))
p6 = Piecewise((1/x, x < 1), (x, True))
p7 = Piecewise((x, x < 1), (1/x, True))
p8 = Piecewise((x, x > 1), (1/x, True))
assert p1.as_leading_term(x) == 0
assert p2.as_leading_term(x) == 0
assert p3.as_leading_term(x) == x
assert p4.as_leading_term(x) == 1/x
assert p5.as_leading_term(x) == x
assert p6.as_leading_term(x) == 1/x
assert p7.as_leading_term(x) == x
assert p8.as_leading_term(x) == 1/x
def test_piecewise_complex():
p1 = Piecewise((2, x < 0), (1, 0 <= x))
p2 = Piecewise((2*I, x < 0), (I, 0 <= x))
p3 = Piecewise((I*x, x > 1), (1 + I, True))
p4 = Piecewise((-I*conjugate(x), x > 1), (1 - I, True))
assert conjugate(p1) == p1
assert conjugate(p2) == piecewise_fold(-p2)
assert conjugate(p3) == p4
assert p1.is_imaginary is False
assert p1.is_real is True
assert p2.is_imaginary is True
assert p2.is_real is False
assert p3.is_imaginary is None
assert p3.is_real is None
assert p1.as_real_imag() == (p1, 0)
assert p2.as_real_imag() == (0, -I*p2)
def test_conjugate_transpose():
A, B = symbols("A B", commutative=False)
p = Piecewise((A*B**2, x > 0), (A**2*B, True))
assert p.adjoint() == \
Piecewise((adjoint(A*B**2), x > 0), (adjoint(A**2*B), True))
assert p.conjugate() == \
Piecewise((conjugate(A*B**2), x > 0), (conjugate(A**2*B), True))
assert p.transpose() == \
Piecewise((transpose(A*B**2), x > 0), (transpose(A**2*B), True))
def test_piecewise_evaluate():
assert Piecewise((x, True)) == x
assert Piecewise((x, True), evaluate=True) == x
p = Piecewise((x, True), evaluate=False)
assert p != x
assert p.is_Piecewise
assert all(isinstance(i, Basic) for i in p.args)
assert Piecewise((1, Eq(1, x))).args == ((1, Eq(x, 1)),)
assert Piecewise((1, Eq(1, x)), evaluate=False).args == (
(1, Eq(1, x)),)
def test_as_expr_set_pairs():
assert Piecewise((x, x > 0), (-x, x <= 0)).as_expr_set_pairs() == \
[(x, Interval(0, oo, True, True)), (-x, Interval(-oo, 0))]
assert Piecewise(((x - 2)**2, x >= 0), (0, True)).as_expr_set_pairs() == \
[((x - 2)**2, Interval(0, oo)), (0, Interval(-oo, 0, True, True))]
def test_S_srepr_is_identity():
p = Piecewise((10, Eq(x, 0)), (12, True))
q = S(srepr(p))
assert p == q
def test_issue_12587():
# sort holes into intervals
p = Piecewise((1, x > 4), (2, Not((x <= 3) & (x > -1))), (3, True))
assert p.integrate((x, -5, 5)) == 23
p = Piecewise((1, x > 1), (2, x < y), (3, True))
lim = x, -3, 3
ans = p.integrate(lim)
for i in range(-1, 3):
assert ans.subs(y, i) == p.subs(y, i).integrate(lim)
def test_issue_11045():
assert integrate(1/(x*sqrt(x**2 - 1)), (x, 1, 2)) == pi/3
# handle And with Or arguments
assert Piecewise((1, And(Or(x < 1, x > 3), x < 2)), (0, True)
).integrate((x, 0, 3)) == 1
# hidden false
assert Piecewise((1, x > 1), (2, x > x + 1), (3, True)
).integrate((x, 0, 3)) == 5
# targetcond is Eq
assert Piecewise((1, x > 1), (2, Eq(1, x)), (3, True)
).integrate((x, 0, 4)) == 6
# And has Relational needing to be solved
assert Piecewise((1, And(2*x > x + 1, x < 2)), (0, True)
).integrate((x, 0, 3)) == 1
# Or has Relational needing to be solved
assert Piecewise((1, Or(2*x > x + 2, x < 1)), (0, True)
).integrate((x, 0, 3)) == 2
# ignore hidden false (handled in canonicalization)
assert Piecewise((1, x > 1), (2, x > x + 1), (3, True)
).integrate((x, 0, 3)) == 5
# watch for hidden True Piecewise
assert Piecewise((2, Eq(1 - x, x*(1/x - 1))), (0, True)
).integrate((x, 0, 3)) == 6
# overlapping conditions of targetcond are recognized and ignored;
# the condition x > 3 will be pre-empted by the first condition
assert Piecewise((1, Or(x < 1, x > 2)), (2, x > 3), (3, True)
).integrate((x, 0, 4)) == 6
# convert Ne to Or
assert Piecewise((1, Ne(x, 0)), (2, True)
).integrate((x, -1, 1)) == 2
# no default but well defined
assert Piecewise((x, (x > 1) & (x < 3)), (1, (x < 4))
).integrate((x, 1, 4)) == 5
p = Piecewise((x, (x > 1) & (x < 3)), (1, (x < 4)))
nan = Undefined
i = p.integrate((x, 1, y))
assert i == Piecewise(
(y - 1, y < 1),
(Min(3, y)**2/2 - Min(3, y) + Min(4, y) - S(1)/2,
y <= Min(4, y)),
(nan, True))
assert p.integrate((x, 1, -1)) == i.subs(y, -1)
assert p.integrate((x, 1, 4)) == 5
assert p.integrate((x, 1, 5)) == nan
# handle Not
p = Piecewise((1, x > 1), (2, Not(And(x > 1, x< 3))), (3, True))
assert p.integrate((x, 0, 3)) == 4
# handle updating of int_expr when there is overlap
p = Piecewise(
(1, And(5 > x, x > 1)),
(2, Or(x < 3, x > 7)),
(4, x < 8))
assert p.integrate((x, 0, 10)) == 20
# And with Eq arg handling
assert Piecewise((1, x < 1), (2, And(Eq(x, 3), x > 1))
).integrate((x, 0, 3)) == S.NaN
assert Piecewise((1, x < 1), (2, And(Eq(x, 3), x > 1)), (3, True)
).integrate((x, 0, 3)) == 7
assert Piecewise((1, x < 0), (2, And(Eq(x, 3), x < 1)), (3, True)
).integrate((x, -1, 1)) == 4
# middle condition doesn't matter: it's a zero width interval
assert Piecewise((1, x < 1), (2, Eq(x, 3) & (y < x)), (3, True)
).integrate((x, 0, 3)) == 7
def test_holes():
nan = Undefined
assert Piecewise((1, x < 2)).integrate(x) == Piecewise(
(x, x < 2), (nan, True))
assert Piecewise((1, And(x > 1, x < 2))).integrate(x) == Piecewise(
(nan, x < 1), (x - 1, x < 2), (nan, True))
assert Piecewise((1, And(x > 1, x < 2))).integrate((x, 0, 3)) == nan
assert Piecewise((1, And(x > 0, x < 4))).integrate((x, 1, 3)) == 2
# this also tests that the integrate method is used on non-Piecwise
# arguments in _eval_integral
A, B = symbols("A B")
a, b = symbols('a b', finite=True)
assert Piecewise((A, And(x < 0, a < 1)), (B, Or(x < 1, a > 2))
).integrate(x) == Piecewise(
(B*x, a > 2),
(Piecewise((A*x, x < 0), (B*x, x < 1), (nan, True)), a < 1),
(Piecewise((B*x, x < 1), (nan, True)), True))
def test_issue_11922():
def f(x):
return Piecewise((0, x < -1), (1 - x**2, x < 1), (0, True))
autocorr = lambda k: (
f(x) * f(x + k)).integrate((x, -1, 1))
assert autocorr(1.9) > 0
k = symbols('k')
good_autocorr = lambda k: (
(1 - x**2) * f(x + k)).integrate((x, -1, 1))
a = good_autocorr(k)
assert a.subs(k, 3) == 0
k = symbols('k', positive=True)
a = good_autocorr(k)
assert a.subs(k, 3) == 0
assert Piecewise((0, x < 1), (10, (x >= 1))
).integrate() == Piecewise((0, x < 1), (10*x - 10, True))
def test_issue_5227():
f = 0.0032513612725229*Piecewise((0, x < -80.8461538461539),
(-0.0160799238820171*x + 1.33215984776403, x < 2),
(Piecewise((0.3, x > 123), (0.7, True)) +
Piecewise((0.4, x > 2), (0.6, True)), x <=
123), (-0.00817409766454352*x + 2.10541401273885, x <
380.571428571429), (0, True))
i = integrate(f, (x, -oo, oo))
assert i == Integral(f, (x, -oo, oo)).doit()
assert str(i) == '1.00195081676351'
assert Piecewise((1, x - y < 0), (0, True)
).integrate(y) == Piecewise((0, y <= x), (-x + y, True))
def test_issue_10137():
a = Symbol('a', real=True, finite=True)
b = Symbol('b', real=True, finite=True)
x = Symbol('x', real=True, finite=True)
y = Symbol('y', real=True, finite=True)
p0 = Piecewise((0, Or(x < a, x > b)), (1, True))
p1 = Piecewise((0, Or(a > x, b < x)), (1, True))
assert integrate(p0, (x, y, oo)) == integrate(p1, (x, y, oo))
p3 = Piecewise((1, And(0 < x, x < a)), (0, True))
p4 = Piecewise((1, And(a > x, x > 0)), (0, True))
ip3 = integrate(p3, x)
assert ip3 == Piecewise(
(0, x <= 0),
(x, x <= Max(0, a)),
(Max(0, a), True))
ip4 = integrate(p4, x)
assert ip4 == ip3
assert p3.integrate((x, 2, 4)) == Min(4, Max(2, a)) - 2
assert p4.integrate((x, 2, 4)) == Min(4, Max(2, a)) - 2
def test_stackoverflow_43852159():
f = lambda x: Piecewise((1 , (x >= -1) & (x <= 1)) , (0, True))
Conv = lambda x: integrate(f(x - y)*f(y), (y, -oo, +oo))
cx = Conv(x)
assert cx.subs(x, -1.5) == cx.subs(x, 1.5)
assert cx.subs(x, 3) == 0
assert piecewise_fold(f(x - y)*f(y)) == Piecewise(
(1, (y >= -1) & (y <= 1) & (x - y >= -1) & (x - y <= 1)),
(0, True))
def test_issue_12557():
'''
# 3200 seconds to compute the fourier part of issue
import sympy as sym
x,y,z,t = sym.symbols('x y z t')
k = sym.symbols("k", integer=True)
fourier = sym.fourier_series(sym.cos(k*x)*sym.sqrt(x**2),
(x, -sym.pi, sym.pi))
assert fourier == FourierSeries(
sqrt(x**2)*cos(k*x), (x, -pi, pi), (Piecewise((pi**2,
Eq(k, 0)), (2*(-1)**k/k**2 - 2/k**2, True))/(2*pi),
SeqFormula(Piecewise((pi**2, (Eq(_n, 0) & Eq(k, 0)) | (Eq(_n, 0) &
Eq(_n, k) & Eq(k, 0)) | (Eq(_n, 0) & Eq(k, 0) & Eq(_n, -k)) | (Eq(_n,
0) & Eq(_n, k) & Eq(k, 0) & Eq(_n, -k))), (pi**2/2, Eq(_n, k) | Eq(_n,
-k) | (Eq(_n, 0) & Eq(_n, k)) | (Eq(_n, k) & Eq(k, 0)) | (Eq(_n, 0) &
Eq(_n, -k)) | (Eq(_n, k) & Eq(_n, -k)) | (Eq(k, 0) & Eq(_n, -k)) |
(Eq(_n, 0) & Eq(_n, k) & Eq(_n, -k)) | (Eq(_n, k) & Eq(k, 0) & Eq(_n,
-k))), ((-1)**k*pi**2*_n**3*sin(pi*_n)/(pi*_n**4 - 2*pi*_n**2*k**2 +
pi*k**4) - (-1)**k*pi**2*_n**3*sin(pi*_n)/(-pi*_n**4 + 2*pi*_n**2*k**2
- pi*k**4) + (-1)**k*pi*_n**2*cos(pi*_n)/(pi*_n**4 - 2*pi*_n**2*k**2 +
pi*k**4) - (-1)**k*pi*_n**2*cos(pi*_n)/(-pi*_n**4 + 2*pi*_n**2*k**2 -
pi*k**4) - (-1)**k*pi**2*_n*k**2*sin(pi*_n)/(pi*_n**4 -
2*pi*_n**2*k**2 + pi*k**4) +
(-1)**k*pi**2*_n*k**2*sin(pi*_n)/(-pi*_n**4 + 2*pi*_n**2*k**2 -
pi*k**4) + (-1)**k*pi*k**2*cos(pi*_n)/(pi*_n**4 - 2*pi*_n**2*k**2 +
pi*k**4) - (-1)**k*pi*k**2*cos(pi*_n)/(-pi*_n**4 + 2*pi*_n**2*k**2 -
pi*k**4) - (2*_n**2 + 2*k**2)/(_n**4 - 2*_n**2*k**2 + k**4),
True))*cos(_n*x)/pi, (_n, 1, oo)), SeqFormula(0, (_k, 1, oo))))
'''
x = symbols("x", real=True)
k = symbols('k', integer=True, finite=True)
abs2 = lambda x: Piecewise((-x, x <= 0), (x, x > 0))
assert integrate(abs2(x), (x, -pi, pi)) == pi**2
func = cos(k*x)*sqrt(x**2)
assert integrate(func, (x, -pi, pi)) == Piecewise(
(2*(-1)**k/k**2 - 2/k**2, Ne(k, 0)), (pi**2, True))
def test_issue_6900():
from itertools import permutations
t0, t1, T, t = symbols('t0, t1 T t')
f = Piecewise((0, t < t0), (x, And(t0 <= t, t < t1)), (0, t >= t1))
g = f.integrate(t)
assert g == Piecewise(
(0, t <= t0),
(t*x - t0*x, t <= Max(t0, t1)),
(-t0*x + x*Max(t0, t1), True))
for i in permutations(range(2)):
reps = dict(zip((t0,t1), i))
for tt in range(-1,3):
assert (g.xreplace(reps).subs(t,tt) ==
f.xreplace(reps).integrate(t).subs(t,tt))
lim = Tuple(t, t0, T)
g = f.integrate(lim)
ans = Piecewise(
(-t0*x + x*Min(T, Max(t0, t1)), T > t0),
(0, True))
for i in permutations(range(3)):
reps = dict(zip((t0,t1,T), i))
tru = f.xreplace(reps).integrate(lim.xreplace(reps))
assert tru == ans.xreplace(reps)
assert g == ans
def test_issue_10122():
assert solve(abs(x) + abs(x - 1) - 1 > 0, x
) == Or(And(-oo < x, x < 0), And(S.One < x, x < oo))
def test_issue_4313():
u = Piecewise((0, x <= 0), (1, x >= a), (x/a, True))
e = (u - u.subs(x, y))**2/(x - y)**2
M = Max(0, a)
assert integrate(e, x).expand() == Piecewise(
(Piecewise(
(0, x <= 0),
(-y**2/(a**2*x - a**2*y) + x/a**2 - 2*y*log(-y)/a**2 +
2*y*log(x - y)/a**2 - y/a**2, x <= M),
(-y**2/(-a**2*y + a**2*M) + 1/(-y + M) -
1/(x - y) - 2*y*log(-y)/a**2 + 2*y*log(-y +
M)/a**2 - y/a**2 + M/a**2, True)),
((a <= y) & (y <= 0)) | ((y <= 0) & (y > -oo))),
(Piecewise(
(-1/(x - y), x <= 0),
(-a**2/(a**2*x - a**2*y) + 2*a*y/(a**2*x - a**2*y) -
y**2/(a**2*x - a**2*y) + 2*log(-y)/a - 2*log(x - y)/a +
2/a + x/a**2 - 2*y*log(-y)/a**2 + 2*y*log(x - y)/a**2 -
y/a**2, x <= M),
(-a**2/(-a**2*y + a**2*M) + 2*a*y/(-a**2*y +
a**2*M) - y**2/(-a**2*y + a**2*M) +
2*log(-y)/a - 2*log(-y + M)/a + 2/a -
2*y*log(-y)/a**2 + 2*y*log(-y + M)/a**2 -
y/a**2 + M/a**2, True)),
a <= y),
(Piecewise(
(-y**2/(a**2*x - a**2*y), x <= 0),
(x/a**2 + y/a**2, x <= M),
(a**2/(-a**2*y + a**2*M) -
a**2/(a**2*x - a**2*y) - 2*a*y/(-a**2*y + a**2*M) +
2*a*y/(a**2*x - a**2*y) + y**2/(-a**2*y + a**2*M) -
y**2/(a**2*x - a**2*y) + y/a**2 + M/a**2, True)),
True))
def test__intervals():
assert Piecewise((x + 2, Eq(x, 3)))._intervals(x) == []
assert Piecewise(
(1, x > x + 1),
(Piecewise((1, x < x + 1)), 2*x < 2*x + 1),
(1, True))._intervals(x) == [(-oo, oo, 1, 1)]
assert Piecewise((1, Ne(x, I)), (0, True))._intervals(x) == [
(-oo, oo, 1, 0)]
assert Piecewise((-cos(x), sin(x) >= 0), (cos(x), True)
)._intervals(x) == [(0, pi, -cos(x), 0), (-oo, oo, cos(x), 1)]
# the following tests that duplicates are removed and that non-Eq
# generated zero-width intervals are removed
assert Piecewise((1, Abs(x**(-2)) > 1), (0, True)
)._intervals(x) == [(-1, 0, 1, 0), (0, 1, 1, 0), (-oo, oo, 0, 1)]
def test_containment():
a, b, c, d, e = [1, 2, 3, 4, 5]
p = (Piecewise((d, x > 1), (e, True))*
Piecewise((a, Abs(x - 1) < 1), (b, Abs(x - 2) < 2), (c, True)))
assert p.integrate(x).diff(x) == Piecewise(
(c*e, x <= 0),
(a*e, x <= 1),
(a*d, x < 2), # this is what we want to get right
(b*d, x < 4),
(c*d, True))
def test_piecewise_with_DiracDelta():
d1 = DiracDelta(x - 1)
assert integrate(d1, (x, -oo, oo)) == 1
assert integrate(d1, (x, 0, 2)) == 1
assert Piecewise((d1, Eq(x, 2)), (0, True)).integrate(x) == 0
assert Piecewise((d1, x < 2), (0, True)).integrate(x) == Piecewise(
(Heaviside(x - 1), x < 2), (1, True))
# TODO raise error if function is discontinuous at limit of
# integration, e.g. integrate(d1, (x, -2, 1)) or Piecewise(
# (d1, Eq(x ,1)
def test_issue_10258():
assert Piecewise((0, x < 1), (1, True)).is_zero is None
assert Piecewise((-1, x < 1), (1, True)).is_zero is False
a = Symbol('a', zero=True)
assert Piecewise((0, x < 1), (a, True)).is_zero
assert Piecewise((1, x < 1), (a, x < 3)).is_zero is None
a = Symbol('a')
assert Piecewise((0, x < 1), (a, True)).is_zero is None
assert Piecewise((0, x < 1), (1, True)).is_nonzero is None
assert Piecewise((1, x < 1), (2, True)).is_nonzero
assert Piecewise((0, x < 1), (oo, True)).is_finite is None
assert Piecewise((0, x < 1), (1, True)).is_finite
b = Basic()
assert Piecewise((b, x < 1)).is_finite is None
# 10258
c = Piecewise((1, x < 0), (2, True)) < 3
assert c != True
assert piecewise_fold(c) == True
def test_issue_10087():
a, b = Piecewise((x, x > 1), (2, True)), Piecewise((x, x > 3), (3, True))
m = a*b
f = piecewise_fold(m)
for i in (0, 2, 4):
assert m.subs(x, i) == f.subs(x, i)
m = a + b
f = piecewise_fold(m)
for i in (0, 2, 4):
assert m.subs(x, i) == f.subs(x, i)
def test_issue_8919():
c = symbols('c:5')
x = symbols("x")
f1 = Piecewise((c[1], x < 1), (c[2], True))
f2 = Piecewise((c[3], x < S(1)/3), (c[4], True))
assert integrate(f1*f2, (x, 0, 2)
) == c[1]*c[3]/3 + 2*c[1]*c[4]/3 + c[2]*c[4]
f1 = Piecewise((0, x < 1), (2, True))
f2 = Piecewise((3, x < 2), (0, True))
assert integrate(f1*f2, (x, 0, 3)) == 6
y = symbols("y", positive=True)
a, b, c, x, z = symbols("a,b,c,x,z", real=True)
I = Integral(Piecewise(
(0, (x >= y) | (x < 0) | (b > c)),
(a, True)), (x, 0, z))
ans = I.doit()
assert ans == Piecewise((0, b > c), (a*Min(y, z) - a*Min(0, z), True))
for cond in (True, False):
for yy in range(1, 3):
for zz in range(-yy, 0, yy):
reps = [(b > c, cond), (y, yy), (z, zz)]
assert ans.subs(reps) == I.subs(reps).doit()
def test_unevaluated_integrals():
f = Function('f')
p = Piecewise((1, Eq(f(x) - 1, 0)), (2, x - 10 < 0), (0, True))
assert p.integrate(x) == Integral(p, x)
assert p.integrate((x, 0, 5)) == Integral(p, (x, 0, 5))
# test it by replacing f(x) with x%2 which will not
# affect the answer: the integrand is essentially 2 over
# the domain of integration
assert Integral(p, (x, 0, 5)).subs(f(x), x%2).n() == 10
# this is a test of using _solve_inequality when
# solve_univariate_inequality fails
assert p.integrate(y) == Piecewise(
(y, Eq(f(x), 1) | ((x < 10) & Eq(f(x), 1))),
(2*y, (x >= -oo) & (x < 10)), (0, True))
def test_conditions_as_alternate_booleans():
a, b, c = symbols('a:c')
assert Piecewise((x, Piecewise((y < 1, x > 0), (y > 1, True)))
) == Piecewise((x, ITE(x > 0, y < 1, y > 1)))
def test_Piecewise_rewrite_as_ITE():
a, b, c, d = symbols('a:d')
def _ITE(*args):
return Piecewise(*args).rewrite(ITE)
assert _ITE((a, x < 1), (b, x >= 1)) == ITE(x < 1, a, b)
assert _ITE((a, x < 1), (b, x < oo)) == ITE(x < 1, a, b)
assert _ITE((a, x < 1), (b, Or(y < 1, x < oo)), (c, y > 0)
) == ITE(x < 1, a, b)
assert _ITE((a, x < 1), (b, True)) == ITE(x < 1, a, b)
assert _ITE((a, x < 1), (b, x < 2), (c, True)
) == ITE(x < 1, a, ITE(x < 2, b, c))
assert _ITE((a, x < 1), (b, y < 2), (c, True)
) == ITE(x < 1, a, ITE(y < 2, b, c))
assert _ITE((a, x < 1), (b, x < oo), (c, y < 1)
) == ITE(x < 1, a, b)
assert _ITE((a, x < 1), (c, y < 1), (b, x < oo), (d, True)
) == ITE(x < 1, a, ITE(y < 1, c, b))
assert _ITE((a, x < 0), (b, Or(x < oo, y < 1))
) == ITE(x < 0, a, b)
raises(TypeError, lambda: _ITE((x + 1, x < 1), (x, True)))
# if `a` in the following were replaced with y then the coverage
# is complete but something other than as_set would need to be
# used to detect this
raises(NotImplementedError, lambda: _ITE((x, x < y), (y, x >= a)))
raises(ValueError, lambda: _ITE((a, x < 2), (b, x > 3)))
def test_issue_14052():
assert integrate(abs(sin(x)), (x, 0, 2*pi)) == 4
def test_issue_14240():
assert piecewise_fold(
Piecewise((1, a), (2, b), (4, True)) +
Piecewise((8, a), (16, True))
) == Piecewise((9, a), (18, b), (20, True))
assert piecewise_fold(
Piecewise((2, a), (3, b), (5, True)) *
Piecewise((7, a), (11, True))
) == Piecewise((14, a), (33, b), (55, True))
# these will hang if naive folding is used
assert piecewise_fold(Add(*[
Piecewise((i, a), (0, True)) for i in range(40)])
) == Piecewise((780, a), (0, True))
assert piecewise_fold(Mul(*[
Piecewise((i, a), (0, True)) for i in range(1, 41)])
) == Piecewise((factorial(40), a), (0, True))
def test_issue_14787():
x = Symbol('x')
f = Piecewise((x, x < 1), ((S(58) / 7), True))
assert str(f.evalf()) == "Piecewise((x, x < 1), (8.28571428571429, True))"
def test_issue_8458():
x, y = symbols('x y')
# Original issue
p1 = Piecewise((0, Eq(x, 0)), (sin(x), True))
assert p1.simplify() == sin(x)
# Slightly larger variant
p2 = Piecewise((x, Eq(x, 0)), (4*x + (y-2)**4, Eq(x, 0) & Eq(x+y, 2)), (sin(x), True))
assert p2.simplify() == sin(x)
# Test for problem highlighted during review
p3 = Piecewise((x+1, Eq(x, -1)), (4*x + (y-2)**4, Eq(x, 0) & Eq(x+y, 2)), (sin(x), True))
assert p3.simplify() == Piecewise((0, Eq(x, -1)), (sin(x), True))
|
e52cbdb57426696a48eb58f6d6ee0baa01a71a02d89a0930f183fa2055d0f09b
|
from sympy.functions import bspline_basis_set, interpolating_spline
from sympy.core.compatibility import range
from sympy import Piecewise, Interval, And
from sympy import symbols, Rational, sympify as S
from sympy.utilities.pytest import slow
x, y = symbols('x,y')
def test_basic_degree_0():
d = 0
knots = range(5)
splines = bspline_basis_set(d, knots, x)
for i in range(len(splines)):
assert splines[i] == Piecewise((1, Interval(i, i + 1).contains(x)),
(0, True))
def test_basic_degree_1():
d = 1
knots = range(5)
splines = bspline_basis_set(d, knots, x)
assert splines[0] == Piecewise((x, Interval(0, 1).contains(x)),
(2 - x, Interval(1, 2).contains(x)),
(0, True))
assert splines[1] == Piecewise((-1 + x, Interval(1, 2).contains(x)),
(3 - x, Interval(2, 3).contains(x)),
(0, True))
assert splines[2] == Piecewise((-2 + x, Interval(2, 3).contains(x)),
(4 - x, Interval(3, 4).contains(x)),
(0, True))
def test_basic_degree_2():
d = 2
knots = range(5)
splines = bspline_basis_set(d, knots, x)
b0 = Piecewise((x**2/2, Interval(0, 1).contains(x)),
(Rational(-3, 2) + 3*x - x**2, Interval(1, 2).contains(x)),
(Rational(9, 2) - 3*x + x**2/2, Interval(2, 3).contains(x)),
(0, True))
b1 = Piecewise((Rational(1, 2) - x + x**2/2, Interval(1, 2).contains(x)),
(Rational(-11, 2) + 5*x - x**2, Interval(2, 3).contains(x)),
(8 - 4*x + x**2/2, Interval(3, 4).contains(x)),
(0, True))
assert splines[0] == b0
assert splines[1] == b1
def test_basic_degree_3():
d = 3
knots = range(5)
splines = bspline_basis_set(d, knots, x)
b0 = Piecewise(
(x**3/6, Interval(0, 1).contains(x)),
(Rational(2, 3) - 2*x + 2*x**2 - x**3/2, Interval(1, 2).contains(x)),
(Rational(-22, 3) + 10*x - 4*x**2 + x**3/2, Interval(2, 3).contains(x)),
(Rational(32, 3) - 8*x + 2*x**2 - x**3/6, Interval(3, 4).contains(x)),
(0, True)
)
assert splines[0] == b0
def test_repeated_degree_1():
d = 1
knots = [0, 0, 1, 2, 2, 3, 4, 4]
splines = bspline_basis_set(d, knots, x)
assert splines[0] == Piecewise((1 - x, Interval(0, 1).contains(x)),
(0, True))
assert splines[1] == Piecewise((x, Interval(0, 1).contains(x)),
(2 - x, Interval(1, 2).contains(x)),
(0, True))
assert splines[2] == Piecewise((-1 + x, Interval(1, 2).contains(x)),
(0, True))
assert splines[3] == Piecewise((3 - x, Interval(2, 3).contains(x)),
(0, True))
assert splines[4] == Piecewise((-2 + x, Interval(2, 3).contains(x)),
(4 - x, Interval(3, 4).contains(x)),
(0, True))
assert splines[5] == Piecewise((-3 + x, Interval(3, 4).contains(x)),
(0, True))
def test_repeated_degree_2():
d = 2
knots = [0, 0, 1, 2, 2, 3, 4, 4]
splines = bspline_basis_set(d, knots, x)
assert splines[0] == Piecewise(((-3*x**2/2 + 2*x), And(x <= 1, x >= 0)),
(x**2/2 - 2*x + 2, And(x <= 2, x >= 1)),
(0, True))
assert splines[1] == Piecewise((x**2/2, And(x <= 1, x >= 0)),
(-3*x**2/2 + 4*x - 2, And(x <= 2, x >= 1)),
(0, True))
assert splines[2] == Piecewise((x**2 - 2*x + 1, And(x <= 2, x >= 1)),
(x**2 - 6*x + 9, And(x <= 3, x >= 2)),
(0, True))
assert splines[3] == Piecewise((-3*x**2/2 + 8*x - 10, And(x <= 3, x >= 2)),
(x**2/2 - 4*x + 8, And(x <= 4, x >= 3)),
(0, True))
assert splines[4] == Piecewise((x**2/2 - 2*x + 2, And(x <= 3, x >= 2)),
(-3*x**2/2 + 10*x - 16, And(x <= 4, x >= 3)),
(0, True))
# Tests for interpolating_spline
def test_10_points_degree_1():
d = 1
X = [-5, 2, 3, 4, 7, 9, 10, 30, 31, 34]
Y = [-10, -2, 2, 4, 7, 6, 20, 45, 19, 25]
spline = interpolating_spline(d, x, X, Y)
assert spline == Piecewise((8*x/7 - S(30)/7, (x >= -5) & (x <= 2)), (4*x - 10, (x >= 2) & (x <= 3)),
(2*x - 4, (x >= 3) & (x <= 4)), (x, (x >= 4) & (x <= 7)),
(-x/2 + S(21)/2, (x >= 7) & (x <= 9)), (14*x - 120, (x >= 9) & (x <= 10)),
(5*x/4 + S(15)/2, (x >= 10) & (x <= 30)), (-26*x + 825, (x >= 30) & (x <= 31)),
(2*x - 43, (x >= 31) & (x <= 34)))
def test_3_points_degree_2():
d = 2
X = [-3, 10, 19]
Y = [3, -4, 30]
spline = interpolating_spline(d, x, X, Y)
assert spline == Piecewise((505*x**2/2574 - 4921*x/2574 - S(1931)/429, (x >= -3) & (x <= 19)))
def test_5_points_degree_2():
d = 2
X = [-3, 2, 4, 5, 10]
Y = [-1, 2, 5, 10, 14]
spline = interpolating_spline(d, x, X, Y)
assert spline == Piecewise((4*x**2/329 + 1007*x/1645 + S(1196)/1645, (x >= -3) & (x <= 3)),
(2701*x**2/1645 - 15079*x/1645 + S(5065)/329, (x >= 3) & (x <= S(9)/2)),
(-1319*x**2/1645 + 21101*x/1645 - S(11216)/329, (x >= S(9)/2) & (x <= 10)))
@slow
def test_6_points_degree_3():
d = 3
X = [-1, 0, 2, 3, 9, 12]
Y = [-4, 3, 3, 7, 9, 20]
spline = interpolating_spline(d, x, X, Y)
assert spline == Piecewise((6058*x**3/5301 - 18427*x**2/5301 + 12622*x/5301 + 3, (x >= -1) & (x <= 2)),
(-8327*x**3/5301 + 67883*x**2/5301 - 159998*x/5301 + S(43661)/1767, (x >= 2) & (x <= 3)),
(5414*x**3/47709 - 1386*x**2/589 + 4267*x/279 - S(12232)/589, (x >= 3) & (x <= 12)))
|
cac504da02a289b18f997f8db69e2243f01705ad50cca6b5efdb3b17a969e850
|
from sympy import (
symbols, expand, expand_func, nan, oo, Float, conjugate, diff,
re, im, Abs, O, exp_polar, polar_lift, gruntz, limit,
Symbol, I, integrate, Integral, S,
sqrt, sin, cos, sinc, sinh, cosh, exp, log, pi, EulerGamma,
erf, erfc, erfi, erf2, erfinv, erfcinv, erf2inv,
gamma, uppergamma,
Ei, expint, E1, li, Li, Si, Ci, Shi, Chi,
fresnels, fresnelc,
hyper, meijerg)
from sympy.functions.special.error_functions import _erfs, _eis
from sympy.core.function import ArgumentIndexError
from sympy.utilities.pytest import raises, slow
x, y, z = symbols('x,y,z')
w = Symbol("w", real=True)
n = Symbol("n", integer=True)
def test_erf():
assert erf(nan) == nan
assert erf(oo) == 1
assert erf(-oo) == -1
assert erf(0) == 0
assert erf(I*oo) == oo*I
assert erf(-I*oo) == -oo*I
assert erf(-2) == -erf(2)
assert erf(-x*y) == -erf(x*y)
assert erf(-x - y) == -erf(x + y)
assert erf(erfinv(x)) == x
assert erf(erfcinv(x)) == 1 - x
assert erf(erf2inv(0, x)) == x
assert erf(erf2inv(0, erf(erfcinv(1 - erf(erfinv(x)))))) == x
assert erf(I).is_real is False
assert erf(0).is_real is True
assert conjugate(erf(z)) == erf(conjugate(z))
assert erf(x).as_leading_term(x) == 2*x/sqrt(pi)
assert erf(1/x).as_leading_term(x) == erf(1/x)
assert erf(z).rewrite('uppergamma') == sqrt(z**2)*(1 - erfc(sqrt(z**2)))/z
assert erf(z).rewrite('erfc') == S.One - erfc(z)
assert erf(z).rewrite('erfi') == -I*erfi(I*z)
assert erf(z).rewrite('fresnels') == (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
I*fresnels(z*(1 - I)/sqrt(pi)))
assert erf(z).rewrite('fresnelc') == (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
I*fresnels(z*(1 - I)/sqrt(pi)))
assert erf(z).rewrite('hyper') == 2*z*hyper([S.Half], [3*S.Half], -z**2)/sqrt(pi)
assert erf(z).rewrite('meijerg') == z*meijerg([S.Half], [], [0], [-S.Half], z**2)/sqrt(pi)
assert erf(z).rewrite('expint') == sqrt(z**2)/z - z*expint(S.Half, z**2)/sqrt(S.Pi)
assert limit(exp(x)*exp(x**2)*(erf(x + 1/exp(x)) - erf(x)), x, oo) == \
2/sqrt(pi)
assert limit((1 - erf(z))*exp(z**2)*z, z, oo) == 1/sqrt(pi)
assert limit((1 - erf(x))*exp(x**2)*sqrt(pi)*x, x, oo) == 1
assert limit(((1 - erf(x))*exp(x**2)*sqrt(pi)*x - 1)*2*x**2, x, oo) == -1
assert erf(x).as_real_imag() == \
((erf(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 +
erf(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2,
I*(erf(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) -
erf(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) *
re(x)*Abs(im(x))/(2*im(x)*Abs(re(x)))))
raises(ArgumentIndexError, lambda: erf(x).fdiff(2))
def test_erf_series():
assert erf(x).series(x, 0, 7) == 2*x/sqrt(pi) - \
2*x**3/3/sqrt(pi) + x**5/5/sqrt(pi) + O(x**7)
def test_erf_evalf():
assert abs( erf(Float(2.0)) - 0.995322265 ) < 1E-8 # XXX
def test__erfs():
assert _erfs(z).diff(z) == -2/sqrt(S.Pi) + 2*z*_erfs(z)
assert _erfs(1/z).series(z) == \
z/sqrt(pi) - z**3/(2*sqrt(pi)) + 3*z**5/(4*sqrt(pi)) + O(z**6)
assert expand(erf(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== erf(z).diff(z)
assert _erfs(z).rewrite("intractable") == (-erf(z) + 1)*exp(z**2)
def test_erfc():
assert erfc(nan) == nan
assert erfc(oo) == 0
assert erfc(-oo) == 2
assert erfc(0) == 1
assert erfc(I*oo) == -oo*I
assert erfc(-I*oo) == oo*I
assert erfc(-x) == S(2) - erfc(x)
assert erfc(erfcinv(x)) == x
assert erfc(I).is_real is False
assert erfc(0).is_real is True
assert conjugate(erfc(z)) == erfc(conjugate(z))
assert erfc(x).as_leading_term(x) == S.One
assert erfc(1/x).as_leading_term(x) == erfc(1/x)
assert erfc(z).rewrite('erf') == 1 - erf(z)
assert erfc(z).rewrite('erfi') == 1 + I*erfi(I*z)
assert erfc(z).rewrite('fresnels') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
I*fresnels(z*(1 - I)/sqrt(pi)))
assert erfc(z).rewrite('fresnelc') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) -
I*fresnels(z*(1 - I)/sqrt(pi)))
assert erfc(z).rewrite('hyper') == 1 - 2*z*hyper([S.Half], [3*S.Half], -z**2)/sqrt(pi)
assert erfc(z).rewrite('meijerg') == 1 - z*meijerg([S.Half], [], [0], [-S.Half], z**2)/sqrt(pi)
assert erfc(z).rewrite('uppergamma') == 1 - sqrt(z**2)*(1 - erfc(sqrt(z**2)))/z
assert erfc(z).rewrite('expint') == S.One - sqrt(z**2)/z + z*expint(S.Half, z**2)/sqrt(S.Pi)
assert expand_func(erf(x) + erfc(x)) == S.One
assert erfc(x).as_real_imag() == \
((erfc(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 +
erfc(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2,
I*(erfc(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) -
erfc(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) *
re(x)*Abs(im(x))/(2*im(x)*Abs(re(x)))))
raises(ArgumentIndexError, lambda: erfc(x).fdiff(2))
def test_erfc_series():
assert erfc(x).series(x, 0, 7) == 1 - 2*x/sqrt(pi) + \
2*x**3/3/sqrt(pi) - x**5/5/sqrt(pi) + O(x**7)
def test_erfc_evalf():
assert abs( erfc(Float(2.0)) - 0.00467773 ) < 1E-8 # XXX
def test_erfi():
assert erfi(nan) == nan
assert erfi(oo) == S.Infinity
assert erfi(-oo) == S.NegativeInfinity
assert erfi(0) == S.Zero
assert erfi(I*oo) == I
assert erfi(-I*oo) == -I
assert erfi(-x) == -erfi(x)
assert erfi(I*erfinv(x)) == I*x
assert erfi(I*erfcinv(x)) == I*(1 - x)
assert erfi(I*erf2inv(0, x)) == I*x
assert erfi(I).is_real is False
assert erfi(0).is_real is True
assert conjugate(erfi(z)) == erfi(conjugate(z))
assert erfi(z).rewrite('erf') == -I*erf(I*z)
assert erfi(z).rewrite('erfc') == I*erfc(I*z) - I
assert erfi(z).rewrite('fresnels') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) -
I*fresnels(z*(1 + I)/sqrt(pi)))
assert erfi(z).rewrite('fresnelc') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) -
I*fresnels(z*(1 + I)/sqrt(pi)))
assert erfi(z).rewrite('hyper') == 2*z*hyper([S.Half], [3*S.Half], z**2)/sqrt(pi)
assert erfi(z).rewrite('meijerg') == z*meijerg([S.Half], [], [0], [-S.Half], -z**2)/sqrt(pi)
assert erfi(z).rewrite('uppergamma') == (sqrt(-z**2)/z*(uppergamma(S.Half,
-z**2)/sqrt(S.Pi) - S.One))
assert erfi(z).rewrite('expint') == sqrt(-z**2)/z - z*expint(S.Half, -z**2)/sqrt(S.Pi)
assert expand_func(erfi(I*z)) == I*erf(z)
assert erfi(x).as_real_imag() == \
((erfi(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 +
erfi(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2,
I*(erfi(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) -
erfi(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) *
re(x)*Abs(im(x))/(2*im(x)*Abs(re(x)))))
raises(ArgumentIndexError, lambda: erfi(x).fdiff(2))
def test_erfi_series():
assert erfi(x).series(x, 0, 7) == 2*x/sqrt(pi) + \
2*x**3/3/sqrt(pi) + x**5/5/sqrt(pi) + O(x**7)
def test_erfi_evalf():
assert abs( erfi(Float(2.0)) - 18.5648024145756 ) < 1E-13 # XXX
def test_erf2():
assert erf2(0, 0) == S.Zero
assert erf2(x, x) == S.Zero
assert erf2(nan, 0) == nan
assert erf2(-oo, y) == erf(y) + 1
assert erf2( oo, y) == erf(y) - 1
assert erf2( x, oo) == 1 - erf(x)
assert erf2( x,-oo) == -1 - erf(x)
assert erf2(x, erf2inv(x, y)) == y
assert erf2(-x, -y) == -erf2(x,y)
assert erf2(-x, y) == erf(y) + erf(x)
assert erf2( x, -y) == -erf(y) - erf(x)
assert erf2(x, y).rewrite('fresnels') == erf(y).rewrite(fresnels)-erf(x).rewrite(fresnels)
assert erf2(x, y).rewrite('fresnelc') == erf(y).rewrite(fresnelc)-erf(x).rewrite(fresnelc)
assert erf2(x, y).rewrite('hyper') == erf(y).rewrite(hyper)-erf(x).rewrite(hyper)
assert erf2(x, y).rewrite('meijerg') == erf(y).rewrite(meijerg)-erf(x).rewrite(meijerg)
assert erf2(x, y).rewrite('uppergamma') == erf(y).rewrite(uppergamma) - erf(x).rewrite(uppergamma)
assert erf2(x, y).rewrite('expint') == erf(y).rewrite(expint)-erf(x).rewrite(expint)
assert erf2(I, 0).is_real is False
assert erf2(0, 0).is_real is True
assert expand_func(erf(x) + erf2(x, y)) == erf(y)
assert conjugate(erf2(x, y)) == erf2(conjugate(x), conjugate(y))
assert erf2(x, y).rewrite('erf') == erf(y) - erf(x)
assert erf2(x, y).rewrite('erfc') == erfc(x) - erfc(y)
assert erf2(x, y).rewrite('erfi') == I*(erfi(I*x) - erfi(I*y))
raises(ArgumentIndexError, lambda: erfi(x).fdiff(3))
def test_erfinv():
assert erfinv(0) == 0
assert erfinv(1) == S.Infinity
assert erfinv(nan) == S.NaN
assert erfinv(erf(w)) == w
assert erfinv(erf(-w)) == -w
assert erfinv(x).diff() == sqrt(pi)*exp(erfinv(x)**2)/2
assert erfinv(z).rewrite('erfcinv') == erfcinv(1-z)
def test_erfinv_evalf():
assert abs( erfinv(Float(0.2)) - 0.179143454621292 ) < 1E-13
def test_erfcinv():
assert erfcinv(1) == 0
assert erfcinv(0) == S.Infinity
assert erfcinv(nan) == S.NaN
assert erfcinv(x).diff() == -sqrt(pi)*exp(erfcinv(x)**2)/2
assert erfcinv(z).rewrite('erfinv') == erfinv(1-z)
def test_erf2inv():
assert erf2inv(0, 0) == S.Zero
assert erf2inv(0, 1) == S.Infinity
assert erf2inv(1, 0) == S.One
assert erf2inv(0, y) == erfinv(y)
assert erf2inv(oo,y) == erfcinv(-y)
assert erf2inv(x, y).diff(x) == exp(-x**2 + erf2inv(x, y)**2)
assert erf2inv(x, y).diff(y) == sqrt(pi)*exp(erf2inv(x, y)**2)/2
# NOTE we multiply by exp_polar(I*pi) and need this to be on the principal
# branch, hence take x in the lower half plane (d=0).
def mytn(expr1, expr2, expr3, x, d=0):
from sympy.utilities.randtest import verify_numerically, random_complex_number
subs = {}
for a in expr1.free_symbols:
if a != x:
subs[a] = random_complex_number()
return expr2 == expr3 and verify_numerically(expr1.subs(subs),
expr2.subs(subs), x, d=d)
def mytd(expr1, expr2, x):
from sympy.utilities.randtest import test_derivative_numerically, \
random_complex_number
subs = {}
for a in expr1.free_symbols:
if a != x:
subs[a] = random_complex_number()
return expr1.diff(x) == expr2 and test_derivative_numerically(expr1.subs(subs), x)
def tn_branch(func, s=None):
from sympy import I, pi, exp_polar
from random import uniform
def fn(x):
if s is None:
return func(x)
return func(s, x)
c = uniform(1, 5)
expr = fn(c*exp_polar(I*pi)) - fn(c*exp_polar(-I*pi))
eps = 1e-15
expr2 = fn(-c + eps*I) - fn(-c - eps*I)
return abs(expr.n() - expr2.n()).n() < 1e-10
def test_ei():
assert tn_branch(Ei)
assert mytd(Ei(x), exp(x)/x, x)
assert mytn(Ei(x), Ei(x).rewrite(uppergamma),
-uppergamma(0, x*polar_lift(-1)) - I*pi, x)
assert mytn(Ei(x), Ei(x).rewrite(expint),
-expint(1, x*polar_lift(-1)) - I*pi, x)
assert Ei(x).rewrite(expint).rewrite(Ei) == Ei(x)
assert Ei(x*exp_polar(2*I*pi)) == Ei(x) + 2*I*pi
assert Ei(x*exp_polar(-2*I*pi)) == Ei(x) - 2*I*pi
assert mytn(Ei(x), Ei(x).rewrite(Shi), Chi(x) + Shi(x), x)
assert mytn(Ei(x*polar_lift(I)), Ei(x*polar_lift(I)).rewrite(Si),
Ci(x) + I*Si(x) + I*pi/2, x)
assert Ei(log(x)).rewrite(li) == li(x)
assert Ei(2*log(x)).rewrite(li) == li(x**2)
assert gruntz(Ei(x+exp(-x))*exp(-x)*x, x, oo) == 1
assert Ei(x).series(x) == EulerGamma + log(x) + x + x**2/4 + \
x**3/18 + x**4/96 + x**5/600 + O(x**6)
assert str(Ei(cos(2)).evalf(n=10)) == '-0.6760647401'
def test_expint():
assert mytn(expint(x, y), expint(x, y).rewrite(uppergamma),
y**(x - 1)*uppergamma(1 - x, y), x)
assert mytd(
expint(x, y), -y**(x - 1)*meijerg([], [1, 1], [0, 0, 1 - x], [], y), x)
assert mytd(expint(x, y), -expint(x - 1, y), y)
assert mytn(expint(1, x), expint(1, x).rewrite(Ei),
-Ei(x*polar_lift(-1)) + I*pi, x)
assert expint(-4, x) == exp(-x)/x + 4*exp(-x)/x**2 + 12*exp(-x)/x**3 \
+ 24*exp(-x)/x**4 + 24*exp(-x)/x**5
assert expint(-S(3)/2, x) == \
exp(-x)/x + 3*exp(-x)/(2*x**2) + 3*sqrt(pi)*erfc(sqrt(x))/(4*x**S('5/2'))
assert tn_branch(expint, 1)
assert tn_branch(expint, 2)
assert tn_branch(expint, 3)
assert tn_branch(expint, 1.7)
assert tn_branch(expint, pi)
assert expint(y, x*exp_polar(2*I*pi)) == \
x**(y - 1)*(exp(2*I*pi*y) - 1)*gamma(-y + 1) + expint(y, x)
assert expint(y, x*exp_polar(-2*I*pi)) == \
x**(y - 1)*(exp(-2*I*pi*y) - 1)*gamma(-y + 1) + expint(y, x)
assert expint(2, x*exp_polar(2*I*pi)) == 2*I*pi*x + expint(2, x)
assert expint(2, x*exp_polar(-2*I*pi)) == -2*I*pi*x + expint(2, x)
assert expint(1, x).rewrite(Ei).rewrite(expint) == expint(1, x)
assert mytn(E1(x), E1(x).rewrite(Shi), Shi(x) - Chi(x), x)
assert mytn(E1(polar_lift(I)*x), E1(polar_lift(I)*x).rewrite(Si),
-Ci(x) + I*Si(x) - I*pi/2, x)
assert mytn(expint(2, x), expint(2, x).rewrite(Ei).rewrite(expint),
-x*E1(x) + exp(-x), x)
assert mytn(expint(3, x), expint(3, x).rewrite(Ei).rewrite(expint),
x**2*E1(x)/2 + (1 - x)*exp(-x)/2, x)
assert expint(S(3)/2, z).nseries(z) == \
2 + 2*z - z**2/3 + z**3/15 - z**4/84 + z**5/540 - \
2*sqrt(pi)*sqrt(z) + O(z**6)
assert E1(z).series(z) == -EulerGamma - log(z) + z - \
z**2/4 + z**3/18 - z**4/96 + z**5/600 + O(z**6)
assert expint(4, z).series(z) == S(1)/3 - z/2 + z**2/2 + \
z**3*(log(z)/6 - S(11)/36 + EulerGamma/6) - z**4/24 + \
z**5/240 + O(z**6)
def test__eis():
assert _eis(z).diff(z) == -_eis(z) + 1/z
assert _eis(1/z).series(z) == \
z + z**2 + 2*z**3 + 6*z**4 + 24*z**5 + O(z**6)
assert Ei(z).rewrite('tractable') == exp(z)*_eis(z)
assert li(z).rewrite('tractable') == z*_eis(log(z))
assert _eis(z).rewrite('intractable') == exp(-z)*Ei(z)
assert expand(li(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== li(z).diff(z)
assert expand(Ei(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== Ei(z).diff(z)
assert _eis(z).series(z, n=3) == EulerGamma + log(z) + z*(-log(z) - \
EulerGamma + 1) + z**2*(log(z)/2 - S(3)/4 + EulerGamma/2) + O(z**3*log(z))
def tn_arg(func):
def test(arg, e1, e2):
from random import uniform
v = uniform(1, 5)
v1 = func(arg*x).subs(x, v).n()
v2 = func(e1*v + e2*1e-15).n()
return abs(v1 - v2).n() < 1e-10
return test(exp_polar(I*pi/2), I, 1) and \
test(exp_polar(-I*pi/2), -I, 1) and \
test(exp_polar(I*pi), -1, I) and \
test(exp_polar(-I*pi), -1, -I)
def test_li():
z = Symbol("z")
zr = Symbol("z", real=True)
zp = Symbol("z", positive=True)
zn = Symbol("z", negative=True)
assert li(0) == 0
assert li(1) == -oo
assert li(oo) == oo
assert isinstance(li(z), li)
assert diff(li(z), z) == 1/log(z)
assert conjugate(li(z)) == li(conjugate(z))
assert conjugate(li(-zr)) == li(-zr)
assert conjugate(li(-zp)) == conjugate(li(-zp))
assert conjugate(li(zn)) == conjugate(li(zn))
assert li(z).rewrite(Li) == Li(z) + li(2)
assert li(z).rewrite(Ei) == Ei(log(z))
assert li(z).rewrite(uppergamma) == (-log(1/log(z))/2 - log(-log(z)) +
log(log(z))/2 - expint(1, -log(z)))
assert li(z).rewrite(Si) == (-log(I*log(z)) - log(1/log(z))/2 +
log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)))
assert li(z).rewrite(Ci) == (-log(I*log(z)) - log(1/log(z))/2 +
log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)))
assert li(z).rewrite(Shi) == (-log(1/log(z))/2 + log(log(z))/2 +
Chi(log(z)) - Shi(log(z)))
assert li(z).rewrite(Chi) == (-log(1/log(z))/2 + log(log(z))/2 +
Chi(log(z)) - Shi(log(z)))
assert li(z).rewrite(hyper) ==(log(z)*hyper((1, 1), (2, 2), log(z)) -
log(1/log(z))/2 + log(log(z))/2 + EulerGamma)
assert li(z).rewrite(meijerg) == (-log(1/log(z))/2 - log(-log(z)) + log(log(z))/2 -
meijerg(((), (1,)), ((0, 0), ()), -log(z)))
assert gruntz(1/li(z), z, oo) == 0
def test_Li():
assert Li(2) == 0
assert Li(oo) == oo
assert isinstance(Li(z), Li)
assert diff(Li(z), z) == 1/log(z)
assert gruntz(1/Li(z), z, oo) == 0
assert Li(z).rewrite(li) == li(z) - li(2)
def test_si():
assert Si(I*x) == I*Shi(x)
assert Shi(I*x) == I*Si(x)
assert Si(-I*x) == -I*Shi(x)
assert Shi(-I*x) == -I*Si(x)
assert Si(-x) == -Si(x)
assert Shi(-x) == -Shi(x)
assert Si(exp_polar(2*pi*I)*x) == Si(x)
assert Si(exp_polar(-2*pi*I)*x) == Si(x)
assert Shi(exp_polar(2*pi*I)*x) == Shi(x)
assert Shi(exp_polar(-2*pi*I)*x) == Shi(x)
assert Si(oo) == pi/2
assert Si(-oo) == -pi/2
assert Shi(oo) == oo
assert Shi(-oo) == -oo
assert mytd(Si(x), sin(x)/x, x)
assert mytd(Shi(x), sinh(x)/x, x)
assert mytn(Si(x), Si(x).rewrite(Ei),
-I*(-Ei(x*exp_polar(-I*pi/2))/2
+ Ei(x*exp_polar(I*pi/2))/2 - I*pi) + pi/2, x)
assert mytn(Si(x), Si(x).rewrite(expint),
-I*(-expint(1, x*exp_polar(-I*pi/2))/2 +
expint(1, x*exp_polar(I*pi/2))/2) + pi/2, x)
assert mytn(Shi(x), Shi(x).rewrite(Ei),
Ei(x)/2 - Ei(x*exp_polar(I*pi))/2 + I*pi/2, x)
assert mytn(Shi(x), Shi(x).rewrite(expint),
expint(1, x)/2 - expint(1, x*exp_polar(I*pi))/2 - I*pi/2, x)
assert tn_arg(Si)
assert tn_arg(Shi)
assert Si(x).nseries(x, n=8) == \
x - x**3/18 + x**5/600 - x**7/35280 + O(x**9)
assert Shi(x).nseries(x, n=8) == \
x + x**3/18 + x**5/600 + x**7/35280 + O(x**9)
assert Si(sin(x)).nseries(x, n=5) == x - 2*x**3/9 + 17*x**5/450 + O(x**6)
assert Si(x).nseries(x, 1, n=3) == \
Si(1) + (x - 1)*sin(1) + (x - 1)**2*(-sin(1)/2 + cos(1)/2) + O((x - 1)**3, (x, 1))
t = Symbol('t', Dummy=True)
assert Si(x).rewrite(sinc) == Integral(sinc(t), (t, 0, x))
def test_ci():
m1 = exp_polar(I*pi)
m1_ = exp_polar(-I*pi)
pI = exp_polar(I*pi/2)
mI = exp_polar(-I*pi/2)
assert Ci(m1*x) == Ci(x) + I*pi
assert Ci(m1_*x) == Ci(x) - I*pi
assert Ci(pI*x) == Chi(x) + I*pi/2
assert Ci(mI*x) == Chi(x) - I*pi/2
assert Chi(m1*x) == Chi(x) + I*pi
assert Chi(m1_*x) == Chi(x) - I*pi
assert Chi(pI*x) == Ci(x) + I*pi/2
assert Chi(mI*x) == Ci(x) - I*pi/2
assert Ci(exp_polar(2*I*pi)*x) == Ci(x) + 2*I*pi
assert Chi(exp_polar(-2*I*pi)*x) == Chi(x) - 2*I*pi
assert Chi(exp_polar(2*I*pi)*x) == Chi(x) + 2*I*pi
assert Ci(exp_polar(-2*I*pi)*x) == Ci(x) - 2*I*pi
assert Ci(oo) == 0
assert Ci(-oo) == I*pi
assert Chi(oo) == oo
assert Chi(-oo) == oo
assert mytd(Ci(x), cos(x)/x, x)
assert mytd(Chi(x), cosh(x)/x, x)
assert mytn(Ci(x), Ci(x).rewrite(Ei),
Ei(x*exp_polar(-I*pi/2))/2 + Ei(x*exp_polar(I*pi/2))/2, x)
assert mytn(Chi(x), Chi(x).rewrite(Ei),
Ei(x)/2 + Ei(x*exp_polar(I*pi))/2 - I*pi/2, x)
assert tn_arg(Ci)
assert tn_arg(Chi)
from sympy import O, EulerGamma, log, limit
assert Ci(x).nseries(x, n=4) == \
EulerGamma + log(x) - x**2/4 + x**4/96 + O(x**5)
assert Chi(x).nseries(x, n=4) == \
EulerGamma + log(x) + x**2/4 + x**4/96 + O(x**5)
assert limit(log(x) - Ci(2*x), x, 0) == -log(2) - EulerGamma
@slow
def test_fresnel():
assert fresnels(0) == 0
assert fresnels(oo) == S.Half
assert fresnels(-oo) == -S.Half
assert fresnels(z) == fresnels(z)
assert fresnels(-z) == -fresnels(z)
assert fresnels(I*z) == -I*fresnels(z)
assert fresnels(-I*z) == I*fresnels(z)
assert conjugate(fresnels(z)) == fresnels(conjugate(z))
assert fresnels(z).diff(z) == sin(pi*z**2/2)
assert fresnels(z).rewrite(erf) == (S.One + I)/4 * (
erf((S.One + I)/2*sqrt(pi)*z) - I*erf((S.One - I)/2*sqrt(pi)*z))
assert fresnels(z).rewrite(hyper) == \
pi*z**3/6 * hyper([S(3)/4], [S(3)/2, S(7)/4], -pi**2*z**4/16)
assert fresnels(z).series(z, n=15) == \
pi*z**3/6 - pi**3*z**7/336 + pi**5*z**11/42240 + O(z**15)
assert fresnels(w).is_real is True
assert fresnels(z).as_real_imag() == \
((fresnels(re(z) - I*re(z)*Abs(im(z))/Abs(re(z)))/2 +
fresnels(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))/2,
I*(fresnels(re(z) - I*re(z)*Abs(im(z))/Abs(re(z))) -
fresnels(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))) *
re(z)*Abs(im(z))/(2*im(z)*Abs(re(z)))))
assert fresnels(2 + 3*I).as_real_imag() == (
fresnels(2 + 3*I)/2 + fresnels(2 - 3*I)/2,
I*(fresnels(2 - 3*I) - fresnels(2 + 3*I))/2
)
assert expand_func(integrate(fresnels(z), z)) == \
z*fresnels(z) + cos(pi*z**2/2)/pi
assert fresnels(z).rewrite(meijerg) == sqrt(2)*pi*z**(S(9)/4) * \
meijerg(((), (1,)), ((S(3)/4,),
(S(1)/4, 0)), -pi**2*z**4/16)/(2*(-z)**(S(3)/4)*(z**2)**(S(3)/4))
assert fresnelc(0) == 0
assert fresnelc(oo) == S.Half
assert fresnelc(-oo) == -S.Half
assert fresnelc(z) == fresnelc(z)
assert fresnelc(-z) == -fresnelc(z)
assert fresnelc(I*z) == I*fresnelc(z)
assert fresnelc(-I*z) == -I*fresnelc(z)
assert conjugate(fresnelc(z)) == fresnelc(conjugate(z))
assert fresnelc(z).diff(z) == cos(pi*z**2/2)
assert fresnelc(z).rewrite(erf) == (S.One - I)/4 * (
erf((S.One + I)/2*sqrt(pi)*z) + I*erf((S.One - I)/2*sqrt(pi)*z))
assert fresnelc(z).rewrite(hyper) == \
z * hyper([S.One/4], [S.One/2, S(5)/4], -pi**2*z**4/16)
assert fresnelc(z).series(z, n=15) == \
z - pi**2*z**5/40 + pi**4*z**9/3456 - pi**6*z**13/599040 + O(z**15)
# issues 6510, 10102
fs = (S.Half - sin(pi*z**2/2)/(pi**2*z**3)
+ (-1/(pi*z) + 3/(pi**3*z**5))*cos(pi*z**2/2))
fc = (S.Half - cos(pi*z**2/2)/(pi**2*z**3)
+ (1/(pi*z) - 3/(pi**3*z**5))*sin(pi*z**2/2))
assert fresnels(z).series(z, oo) == fs + O(z**(-6), (z, oo))
assert fresnelc(z).series(z, oo) == fc + O(z**(-6), (z, oo))
assert (fresnels(z).series(z, -oo) + fs.subs(z, -z)).expand().is_Order
assert (fresnelc(z).series(z, -oo) + fc.subs(z, -z)).expand().is_Order
assert (fresnels(1/z).series(z) - fs.subs(z, 1/z)).expand().is_Order
assert (fresnelc(1/z).series(z) - fc.subs(z, 1/z)).expand().is_Order
assert ((2*fresnels(3*z)).series(z, oo) - 2*fs.subs(z, 3*z)).expand().is_Order
assert ((3*fresnelc(2*z)).series(z, oo) - 3*fc.subs(z, 2*z)).expand().is_Order
assert fresnelc(w).is_real is True
assert fresnelc(z).as_real_imag() == \
((fresnelc(re(z) - I*re(z)*Abs(im(z))/Abs(re(z)))/2 +
fresnelc(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))/2,
I*(fresnelc(re(z) - I*re(z)*Abs(im(z))/Abs(re(z))) -
fresnelc(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))) *
re(z)*Abs(im(z))/(2*im(z)*Abs(re(z)))))
assert fresnelc(2 + 3*I).as_real_imag() == (
fresnelc(2 - 3*I)/2 + fresnelc(2 + 3*I)/2,
I*(fresnelc(2 - 3*I) - fresnelc(2 + 3*I))/2
)
assert expand_func(integrate(fresnelc(z), z)) == \
z*fresnelc(z) - sin(pi*z**2/2)/pi
assert fresnelc(z).rewrite(meijerg) == sqrt(2)*pi*z**(S(3)/4) * \
meijerg(((), (1,)), ((S(1)/4,),
(S(3)/4, 0)), -pi**2*z**4/16)/(2*(-z)**(S(1)/4)*(z**2)**(S(1)/4))
from sympy.utilities.randtest import verify_numerically
verify_numerically(re(fresnels(z)), fresnels(z).as_real_imag()[0], z)
verify_numerically(im(fresnels(z)), fresnels(z).as_real_imag()[1], z)
verify_numerically(fresnels(z), fresnels(z).rewrite(hyper), z)
verify_numerically(fresnels(z), fresnels(z).rewrite(meijerg), z)
verify_numerically(re(fresnelc(z)), fresnelc(z).as_real_imag()[0], z)
verify_numerically(im(fresnelc(z)), fresnelc(z).as_real_imag()[1], z)
verify_numerically(fresnelc(z), fresnelc(z).rewrite(hyper), z)
verify_numerically(fresnelc(z), fresnelc(z).rewrite(meijerg), z)
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6be46306867cf89f375e0b88a68cfd3374a9744d603571b1e79881090eeca5e6
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from sympy.core.containers import Tuple
from sympy.core.function import (Function, Lambda, nfloat)
from sympy.core.mod import Mod
from sympy.core.numbers import (E, I, Rational, oo, pi)
from sympy.core.relational import (Eq, Gt,
Ne)
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Symbol, symbols)
from sympy.functions.elementary.complexes import (Abs, arg, im, re, sign)
from sympy.functions.elementary.exponential import (LambertW, exp, log)
from sympy.functions.elementary.hyperbolic import (HyperbolicFunction,
atanh, sinh, tanh)
from sympy.functions.elementary.miscellaneous import sqrt, Min, Max
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (
TrigonometricFunction, acos, acot, acsc, asec, asin, atan, atan2,
cos, cot, csc, sec, sin, tan)
from sympy.functions.special.error_functions import (erf, erfc,
erfcinv, erfinv)
from sympy.logic.boolalg import And
from sympy.matrices.dense import MutableDenseMatrix as Matrix
from sympy.polys.polytools import Poly
from sympy.polys.rootoftools import CRootOf
from sympy.sets.contains import Contains
from sympy.sets.conditionset import ConditionSet
from sympy.sets.fancysets import ImageSet
from sympy.sets.sets import (Complement, EmptySet, FiniteSet,
Intersection, Interval, Union, imageset)
from sympy.tensor.indexed import Indexed
from sympy.utilities.iterables import numbered_symbols
from sympy.utilities.pytest import XFAIL, raises, skip, slow, SKIP
from sympy.utilities.randtest import verify_numerically as tn
from sympy.physics.units import cm
from sympy.core.containers import Dict
from sympy.solvers.solveset import (
solveset_real, domain_check, solveset_complex, linear_eq_to_matrix,
linsolve, _is_function_class_equation, invert_real, invert_complex,
solveset, solve_decomposition, substitution, nonlinsolve, solvify,
_is_finite_with_finite_vars, _transolve, _is_exponential,
_solve_exponential, _is_logarithmic,
_solve_logarithm, _term_factors)
a = Symbol('a', real=True)
b = Symbol('b', real=True)
c = Symbol('c', real=True)
x = Symbol('x', real=True)
y = Symbol('y', real=True)
z = Symbol('z', real=True)
q = Symbol('q', real=True)
m = Symbol('m', real=True)
n = Symbol('n', real=True)
def test_invert_real():
x = Symbol('x', real=True)
y = Symbol('y')
n = Symbol('n')
def ireal(x, s=S.Reals):
return Intersection(s, x)
# issue 14223
assert invert_real(x, 0, x, Interval(1, 2)) == (x, S.EmptySet)
assert invert_real(exp(x), y, x) == (x, ireal(FiniteSet(log(y))))
y = Symbol('y', positive=True)
n = Symbol('n', real=True)
assert invert_real(x + 3, y, x) == (x, FiniteSet(y - 3))
assert invert_real(x*3, y, x) == (x, FiniteSet(y / 3))
assert invert_real(exp(x), y, x) == (x, FiniteSet(log(y)))
assert invert_real(exp(3*x), y, x) == (x, FiniteSet(log(y) / 3))
assert invert_real(exp(x + 3), y, x) == (x, FiniteSet(log(y) - 3))
assert invert_real(exp(x) + 3, y, x) == (x, ireal(FiniteSet(log(y - 3))))
assert invert_real(exp(x)*3, y, x) == (x, FiniteSet(log(y / 3)))
assert invert_real(log(x), y, x) == (x, FiniteSet(exp(y)))
assert invert_real(log(3*x), y, x) == (x, FiniteSet(exp(y) / 3))
assert invert_real(log(x + 3), y, x) == (x, FiniteSet(exp(y) - 3))
assert invert_real(Abs(x), y, x) == (x, FiniteSet(y, -y))
assert invert_real(2**x, y, x) == (x, FiniteSet(log(y)/log(2)))
assert invert_real(2**exp(x), y, x) == (x, ireal(FiniteSet(log(log(y)/log(2)))))
assert invert_real(x**2, y, x) == (x, FiniteSet(sqrt(y), -sqrt(y)))
assert invert_real(x**Rational(1, 2), y, x) == (x, FiniteSet(y**2))
raises(ValueError, lambda: invert_real(x, x, x))
raises(ValueError, lambda: invert_real(x**pi, y, x))
raises(ValueError, lambda: invert_real(S.One, y, x))
assert invert_real(x**31 + x, y, x) == (x**31 + x, FiniteSet(y))
lhs = x**31 + x
conditions = Contains(y, Interval(0, oo), evaluate=False)
base_values = FiniteSet(y - 1, -y - 1)
assert invert_real(Abs(x**31 + x + 1), y, x) == (lhs, base_values)
assert invert_real(sin(x), y, x) == \
(x, imageset(Lambda(n, n*pi + (-1)**n*asin(y)), S.Integers))
assert invert_real(sin(exp(x)), y, x) == \
(x, imageset(Lambda(n, log((-1)**n*asin(y) + n*pi)), S.Integers))
assert invert_real(csc(x), y, x) == \
(x, imageset(Lambda(n, n*pi + (-1)**n*acsc(y)), S.Integers))
assert invert_real(csc(exp(x)), y, x) == \
(x, imageset(Lambda(n, log((-1)**n*acsc(y) + n*pi)), S.Integers))
assert invert_real(cos(x), y, x) == \
(x, Union(imageset(Lambda(n, 2*n*pi + acos(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - acos(y)), S.Integers)))
assert invert_real(cos(exp(x)), y, x) == \
(x, Union(imageset(Lambda(n, log(2*n*pi + Mod(acos(y), 2*pi))), S.Integers), \
imageset(Lambda(n, log(2*n*pi + Mod(-acos(y), 2*pi))), S.Integers)))
assert invert_real(sec(x), y, x) == \
(x, Union(imageset(Lambda(n, 2*n*pi + asec(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - asec(y)), S.Integers)))
assert invert_real(sec(exp(x)), y, x) == \
(x, Union(imageset(Lambda(n, log(2*n*pi + Mod(asec(y), 2*pi))), S.Integers), \
imageset(Lambda(n, log(2*n*pi + Mod(-asec(y), 2*pi))), S.Integers)))
assert invert_real(tan(x), y, x) == \
(x, imageset(Lambda(n, n*pi + atan(y) % pi), S.Integers))
assert invert_real(tan(exp(x)), y, x) == \
(x, imageset(Lambda(n, log(n*pi + atan(y) % pi)), S.Integers))
assert invert_real(cot(x), y, x) == \
(x, imageset(Lambda(n, n*pi + acot(y) % pi), S.Integers))
assert invert_real(cot(exp(x)), y, x) == \
(x, imageset(Lambda(n, log(n*pi + acot(y) % pi)), S.Integers))
assert invert_real(tan(tan(x)), y, x) == \
(tan(x), imageset(Lambda(n, n*pi + atan(y) % pi), S.Integers))
x = Symbol('x', positive=True)
assert invert_real(x**pi, y, x) == (x, FiniteSet(y**(1/pi)))
def test_invert_complex():
assert invert_complex(x + 3, y, x) == (x, FiniteSet(y - 3))
assert invert_complex(x*3, y, x) == (x, FiniteSet(y / 3))
assert invert_complex(exp(x), y, x) == \
(x, imageset(Lambda(n, I*(2*pi*n + arg(y)) + log(Abs(y))), S.Integers))
assert invert_complex(log(x), y, x) == (x, FiniteSet(exp(y)))
raises(ValueError, lambda: invert_real(1, y, x))
raises(ValueError, lambda: invert_complex(x, x, x))
raises(ValueError, lambda: invert_complex(x, x, 1))
# https://github.com/skirpichev/omg/issues/16
assert invert_complex(sinh(x), 0, x) != (x, FiniteSet(0))
def test_domain_check():
assert domain_check(1/(1 + (1/(x+1))**2), x, -1) is False
assert domain_check(x**2, x, 0) is True
assert domain_check(x, x, oo) is False
assert domain_check(0, x, oo) is False
def test_issue_11536():
assert solveset(0**x - 100, x, S.Reals) == S.EmptySet
assert solveset(0**x - 1, x, S.Reals) == FiniteSet(0)
def test_is_function_class_equation():
from sympy.abc import x, a
assert _is_function_class_equation(TrigonometricFunction,
tan(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) - 1, x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) + sin(x) - a, x) is True
assert _is_function_class_equation(TrigonometricFunction,
sin(x)*tan(x) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
sin(x)*tan(x + a) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
sin(x)*tan(x*a) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
a*tan(x) - 1, x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x)**2 + sin(x) - 1, x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) + x, x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(x**2), x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(x**2) + sin(x), x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(x)**sin(x), x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(sin(x)) + sin(x), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) - 1, x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) + sinh(x) - a, x) is True
assert _is_function_class_equation(HyperbolicFunction,
sinh(x)*tanh(x) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
sinh(x)*tanh(x + a) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
sinh(x)*tanh(x*a) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
a*tanh(x) - 1, x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x)**2 + sinh(x) - 1, x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) + x, x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x**2), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x**2) + sinh(x), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x)**sinh(x), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(sinh(x)) + sinh(x), x) is False
def test_garbage_input():
raises(ValueError, lambda: solveset_real([x], x))
assert solveset_real(x, 1) == S.EmptySet
assert solveset_real(x - 1, 1) == FiniteSet(x)
assert solveset_real(x, pi) == S.EmptySet
assert solveset_real(x, x**2) == S.EmptySet
raises(ValueError, lambda: solveset_complex([x], x))
assert solveset_complex(x, pi) == S.EmptySet
raises(ValueError, lambda: solveset((x, y), x))
raises(ValueError, lambda: solveset(x + 1, S.Reals))
raises(ValueError, lambda: solveset(x + 1, x, 2))
def test_solve_mul():
assert solveset_real((a*x + b)*(exp(x) - 3), x) == \
FiniteSet(-b/a, log(3))
assert solveset_real((2*x + 8)*(8 + exp(x)), x) == FiniteSet(S(-4))
assert solveset_real(x/log(x), x) == EmptySet()
def test_solve_invert():
assert solveset_real(exp(x) - 3, x) == FiniteSet(log(3))
assert solveset_real(log(x) - 3, x) == FiniteSet(exp(3))
assert solveset_real(3**(x + 2), x) == FiniteSet()
assert solveset_real(3**(2 - x), x) == FiniteSet()
assert solveset_real(y - b*exp(a/x), x) == Intersection(
S.Reals, FiniteSet(a/log(y/b)))
# issue 4504
assert solveset_real(2**x - 10, x) == FiniteSet(1 + log(5)/log(2))
def test_errorinverses():
assert solveset_real(erf(x) - S.One/2, x) == \
FiniteSet(erfinv(S.One/2))
assert solveset_real(erfinv(x) - 2, x) == \
FiniteSet(erf(2))
assert solveset_real(erfc(x) - S.One, x) == \
FiniteSet(erfcinv(S.One))
assert solveset_real(erfcinv(x) - 2, x) == FiniteSet(erfc(2))
def test_solve_polynomial():
assert solveset_real(3*x - 2, x) == FiniteSet(Rational(2, 3))
assert solveset_real(x**2 - 1, x) == FiniteSet(-S(1), S(1))
assert solveset_real(x - y**3, x) == FiniteSet(y ** 3)
a11, a12, a21, a22, b1, b2 = symbols('a11, a12, a21, a22, b1, b2')
assert solveset_real(x**3 - 15*x - 4, x) == FiniteSet(
-2 + 3 ** Rational(1, 2),
S(4),
-2 - 3 ** Rational(1, 2))
assert solveset_real(sqrt(x) - 1, x) == FiniteSet(1)
assert solveset_real(sqrt(x) - 2, x) == FiniteSet(4)
assert solveset_real(x**Rational(1, 4) - 2, x) == FiniteSet(16)
assert solveset_real(x**Rational(1, 3) - 3, x) == FiniteSet(27)
assert len(solveset_real(x**5 + x**3 + 1, x)) == 1
assert len(solveset_real(-2*x**3 + 4*x**2 - 2*x + 6, x)) > 0
assert solveset_real(x**6 + x**4 + I, x) == ConditionSet(x,
Eq(x**6 + x**4 + I, 0), S.Reals)
def test_return_root_of():
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = list(solveset_complex(f, x))
for root in s:
assert root.func == CRootOf
# if one uses solve to get the roots of a polynomial that has a CRootOf
# solution, make sure that the use of nfloat during the solve process
# doesn't fail. Note: if you want numerical solutions to a polynomial
# it is *much* faster to use nroots to get them than to solve the
# equation only to get CRootOf solutions which are then numerically
# evaluated. So for eq = x**5 + 3*x + 7 do Poly(eq).nroots() rather
# than [i.n() for i in solve(eq)] to get the numerical roots of eq.
assert nfloat(list(solveset_complex(x**5 + 3*x**3 + 7, x))[0],
exponent=False) == CRootOf(x**5 + 3*x**3 + 7, 0).n()
sol = list(solveset_complex(x**6 - 2*x + 2, x))
assert all(isinstance(i, CRootOf) for i in sol) and len(sol) == 6
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = list(solveset_complex(f, x))
for root in s:
assert root.func == CRootOf
s = x**5 + 4*x**3 + 3*x**2 + S(7)/4
assert solveset_complex(s, x) == \
FiniteSet(*Poly(s*4, domain='ZZ').all_roots())
# Refer issue #7876
eq = x*(x - 1)**2*(x + 1)*(x**6 - x + 1)
assert solveset_complex(eq, x) == \
FiniteSet(-1, 0, 1, CRootOf(x**6 - x + 1, 0),
CRootOf(x**6 - x + 1, 1),
CRootOf(x**6 - x + 1, 2),
CRootOf(x**6 - x + 1, 3),
CRootOf(x**6 - x + 1, 4),
CRootOf(x**6 - x + 1, 5))
def test__has_rational_power():
from sympy.solvers.solveset import _has_rational_power
assert _has_rational_power(sqrt(2), x)[0] is False
assert _has_rational_power(x*sqrt(2), x)[0] is False
assert _has_rational_power(x**2*sqrt(x), x) == (True, 2)
assert _has_rational_power(sqrt(2)*x**(S(1)/3), x) == (True, 3)
assert _has_rational_power(sqrt(x)*x**(S(1)/3), x) == (True, 6)
def test_solveset_sqrt_1():
assert solveset_real(sqrt(5*x + 6) - 2 - x, x) == \
FiniteSet(-S(1), S(2))
assert solveset_real(sqrt(x - 1) - x + 7, x) == FiniteSet(10)
assert solveset_real(sqrt(x - 2) - 5, x) == FiniteSet(27)
assert solveset_real(sqrt(x) - 2 - 5, x) == FiniteSet(49)
assert solveset_real(sqrt(x**3), x) == FiniteSet(0)
assert solveset_real(sqrt(x - 1), x) == FiniteSet(1)
def test_solveset_sqrt_2():
# http://tutorial.math.lamar.edu/Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a
assert solveset_real(sqrt(2*x - 1) - sqrt(x - 4) - 2, x) == \
FiniteSet(S(5), S(13))
assert solveset_real(sqrt(x + 7) + 2 - sqrt(3 - x), x) == \
FiniteSet(-6)
# http://www.purplemath.com/modules/solverad.htm
assert solveset_real(sqrt(17*x - sqrt(x**2 - 5)) - 7, x) == \
FiniteSet(3)
eq = x + 1 - (x**4 + 4*x**3 - x)**Rational(1, 4)
assert solveset_real(eq, x) == FiniteSet(-S(1)/2, -S(1)/3)
eq = sqrt(2*x + 9) - sqrt(x + 1) - sqrt(x + 4)
assert solveset_real(eq, x) == FiniteSet(0)
eq = sqrt(x + 4) + sqrt(2*x - 1) - 3*sqrt(x - 1)
assert solveset_real(eq, x) == FiniteSet(5)
eq = sqrt(x)*sqrt(x - 7) - 12
assert solveset_real(eq, x) == FiniteSet(16)
eq = sqrt(x - 3) + sqrt(x) - 3
assert solveset_real(eq, x) == FiniteSet(4)
eq = sqrt(2*x**2 - 7) - (3 - x)
assert solveset_real(eq, x) == FiniteSet(-S(8), S(2))
# others
eq = sqrt(9*x**2 + 4) - (3*x + 2)
assert solveset_real(eq, x) == FiniteSet(0)
assert solveset_real(sqrt(x - 3) - sqrt(x) - 3, x) == FiniteSet()
eq = (2*x - 5)**Rational(1, 3) - 3
assert solveset_real(eq, x) == FiniteSet(16)
assert solveset_real(sqrt(x) + sqrt(sqrt(x)) - 4, x) == \
FiniteSet((-S.Half + sqrt(17)/2)**4)
eq = sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))
assert solveset_real(eq, x) == FiniteSet()
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
ans = solveset_real(eq, x)
ra = S('''-1484/375 - 4*(-1/2 + sqrt(3)*I/2)*(-12459439/52734375 +
114*sqrt(12657)/78125)**(1/3) - 172564/(140625*(-1/2 +
sqrt(3)*I/2)*(-12459439/52734375 + 114*sqrt(12657)/78125)**(1/3))''')
rb = S(4)/5
assert all(abs(eq.subs(x, i).n()) < 1e-10 for i in (ra, rb)) and \
len(ans) == 2 and \
set([i.n(chop=True) for i in ans]) == \
set([i.n(chop=True) for i in (ra, rb)])
assert solveset_real(sqrt(x) + x**Rational(1, 3) +
x**Rational(1, 4), x) == FiniteSet(0)
assert solveset_real(x/sqrt(x**2 + 1), x) == FiniteSet(0)
eq = (x - y**3)/((y**2)*sqrt(1 - y**2))
assert solveset_real(eq, x) == FiniteSet(y**3)
# issue 4497
assert solveset_real(1/(5 + x)**(S(1)/5) - 9, x) == \
FiniteSet(-295244/S(59049))
@XFAIL
def test_solve_sqrt_fail():
# this only works if we check real_root(eq.subs(x, S(1)/3))
# but checksol doesn't work like that
eq = (x**3 - 3*x**2)**Rational(1, 3) + 1 - x
assert solveset_real(eq, x) == FiniteSet(S(1)/3)
@slow
def test_solve_sqrt_3():
R = Symbol('R')
eq = sqrt(2)*R*sqrt(1/(R + 1)) + (R + 1)*(sqrt(2)*sqrt(1/(R + 1)) - 1)
sol = solveset_complex(eq, R)
fset = [S(5)/3 + 4*sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3,
-sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3 +
40*re(1/((-S(1)/2 - sqrt(3)*I/2)*(S(251)/27 + sqrt(111)*I/9)**(S(1)/3)))/9 +
sqrt(30)*sin(atan(3*sqrt(111)/251)/3)/3 + S(5)/3 +
I*(-sqrt(30)*cos(atan(3*sqrt(111)/251)/3)/3 -
sqrt(10)*sin(atan(3*sqrt(111)/251)/3)/3 +
40*im(1/((-S(1)/2 - sqrt(3)*I/2)*(S(251)/27 + sqrt(111)*I/9)**(S(1)/3)))/9)]
cset = [40*re(1/((-S(1)/2 + sqrt(3)*I/2)*(S(251)/27 + sqrt(111)*I/9)**(S(1)/3)))/9 -
sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3 - sqrt(30)*sin(atan(3*sqrt(111)/251)/3)/3 +
S(5)/3 +
I*(40*im(1/((-S(1)/2 + sqrt(3)*I/2)*(S(251)/27 + sqrt(111)*I/9)**(S(1)/3)))/9 -
sqrt(10)*sin(atan(3*sqrt(111)/251)/3)/3 +
sqrt(30)*cos(atan(3*sqrt(111)/251)/3)/3)]
assert sol._args[0] == FiniteSet(*fset)
assert sol._args[1] == ConditionSet(
R,
Eq(sqrt(2)*R*sqrt(1/(R + 1)) + (R + 1)*(sqrt(2)*sqrt(1/(R + 1)) - 1), 0),
FiniteSet(*cset))
# the number of real roots will depend on the value of m: for m=1 there are 4
# and for m=-1 there are none.
eq = -sqrt((m - q)**2 + (-m/(2*q) + S(1)/2)**2) + sqrt((-m**2/2 - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - S(1)/4)**2 + (m**2/2 - m - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - S(1)/4)**2)
unsolved_object = ConditionSet(q, Eq(sqrt((m - q)**2 + (-m/(2*q) + S(1)/2)**2) -
sqrt((-m**2/2 - sqrt(4*m**4 - 4*m**2 + 8*m + 1)/4 - S(1)/4)**2 + (m**2/2 - m -
sqrt(4*m**4 - 4*m**2 + 8*m + 1)/4 - S(1)/4)**2), 0), S.Reals)
assert solveset_real(eq, q) == unsolved_object
def test_solve_polynomial_symbolic_param():
assert solveset_complex((x**2 - 1)**2 - a, x) == \
FiniteSet(sqrt(1 + sqrt(a)), -sqrt(1 + sqrt(a)),
sqrt(1 - sqrt(a)), -sqrt(1 - sqrt(a)))
# issue 4507
assert solveset_complex(y - b/(1 + a*x), x) == \
FiniteSet((b/y - 1)/a) - FiniteSet(-1/a)
# issue 4508
assert solveset_complex(y - b*x/(a + x), x) == \
FiniteSet(-a*y/(y - b)) - FiniteSet(-a)
def test_solve_rational():
assert solveset_real(1/x + 1, x) == FiniteSet(-S.One)
assert solveset_real(1/exp(x) - 1, x) == FiniteSet(0)
assert solveset_real(x*(1 - 5/x), x) == FiniteSet(5)
assert solveset_real(2*x/(x + 2) - 1, x) == FiniteSet(2)
assert solveset_real((x**2/(7 - x)).diff(x), x) == \
FiniteSet(S(0), S(14))
def test_solveset_real_gen_is_pow():
assert solveset_real(sqrt(1) + 1, x) == EmptySet()
def test_no_sol():
assert solveset(1 - oo*x) == EmptySet()
assert solveset(oo*x, x) == EmptySet()
assert solveset(oo*x - oo, x) == EmptySet()
assert solveset_real(4, x) == EmptySet()
assert solveset_real(exp(x), x) == EmptySet()
assert solveset_real(x**2 + 1, x) == EmptySet()
assert solveset_real(-3*a/sqrt(x), x) == EmptySet()
assert solveset_real(1/x, x) == EmptySet()
assert solveset_real(-(1 + x)/(2 + x)**2 + 1/(2 + x), x) == \
EmptySet()
def test_sol_zero_real():
assert solveset_real(0, x) == S.Reals
assert solveset(0, x, Interval(1, 2)) == Interval(1, 2)
assert solveset_real(-x**2 - 2*x + (x + 1)**2 - 1, x) == S.Reals
def test_no_sol_rational_extragenous():
assert solveset_real((x/(x + 1) + 3)**(-2), x) == EmptySet()
assert solveset_real((x - 1)/(1 + 1/(x - 1)), x) == EmptySet()
def test_solve_polynomial_cv_1a():
"""
Test for solving on equations that can be converted to
a polynomial equation using the change of variable y -> x**Rational(p, q)
"""
assert solveset_real(sqrt(x) - 1, x) == FiniteSet(1)
assert solveset_real(sqrt(x) - 2, x) == FiniteSet(4)
assert solveset_real(x**Rational(1, 4) - 2, x) == FiniteSet(16)
assert solveset_real(x**Rational(1, 3) - 3, x) == FiniteSet(27)
assert solveset_real(x*(x**(S(1) / 3) - 3), x) == \
FiniteSet(S(0), S(27))
def test_solveset_real_rational():
"""Test solveset_real for rational functions"""
assert solveset_real((x - y**3) / ((y**2)*sqrt(1 - y**2)), x) \
== FiniteSet(y**3)
# issue 4486
assert solveset_real(2*x/(x + 2) - 1, x) == FiniteSet(2)
def test_solveset_real_log():
assert solveset_real(log((x-1)*(x+1)), x) == \
FiniteSet(sqrt(2), -sqrt(2))
def test_poly_gens():
assert solveset_real(4**(2*(x**2) + 2*x) - 8, x) == \
FiniteSet(-Rational(3, 2), S.Half)
def test_solve_abs():
x = Symbol('x')
n = Dummy('n')
raises(ValueError, lambda: solveset(Abs(x) - 1, x))
assert solveset(Abs(x) - n, x, S.Reals) == ConditionSet(x, Contains(n, Interval(0, oo)), {-n, n})
assert solveset_real(Abs(x) - 2, x) == FiniteSet(-2, 2)
assert solveset_real(Abs(x) + 2, x) is S.EmptySet
assert solveset_real(Abs(x + 3) - 2*Abs(x - 3), x) == \
FiniteSet(1, 9)
assert solveset_real(2*Abs(x) - Abs(x - 1), x) == \
FiniteSet(-1, Rational(1, 3))
sol = ConditionSet(
x,
And(
Contains(b, Interval(0, oo)),
Contains(a + b, Interval(0, oo)),
Contains(a - b, Interval(0, oo))),
FiniteSet(-a - b - 3, -a + b - 3, a - b - 3, a + b - 3))
eq = Abs(Abs(x + 3) - a) - b
assert invert_real(eq, 0, x)[1] == sol
reps = {a: 3, b: 1}
eqab = eq.subs(reps)
for i in sol.subs(reps):
assert not eqab.subs(x, i)
assert solveset(Eq(sin(Abs(x)), 1), x, domain=S.Reals) == Union(
Intersection(Interval(0, oo),
ImageSet(Lambda(n, (-1)**n*pi/2 + n*pi), S.Integers)),
Intersection(Interval(-oo, 0),
ImageSet(Lambda(n, n*pi - (-1)**(-n)*pi/2), S.Integers)))
def test_issue_9565():
assert solveset_real(Abs((x - 1)/(x - 5)) <= S(1)/3, x) == Interval(-1, 2)
def test_issue_10069():
eq = abs(1/(x - 1)) - 1 > 0
u = Union(Interval.open(0, 1), Interval.open(1, 2))
assert solveset_real(eq, x) == u
@XFAIL
def test_rewrite_trigh():
# if this import passes then the test below should also pass
from sympy import sech
assert solveset_real(sinh(x) + sech(x), x) == FiniteSet(
2*atanh(-S.Half + sqrt(5)/2 - sqrt(-2*sqrt(5) + 2)/2),
2*atanh(-S.Half + sqrt(5)/2 + sqrt(-2*sqrt(5) + 2)/2),
2*atanh(-sqrt(5)/2 - S.Half + sqrt(2 + 2*sqrt(5))/2),
2*atanh(-sqrt(2 + 2*sqrt(5))/2 - sqrt(5)/2 - S.Half))
def test_real_imag_splitting():
a, b = symbols('a b', real=True, finite=True)
assert solveset_real(sqrt(a**2 - b**2) - 3, a) == \
FiniteSet(-sqrt(b**2 + 9), sqrt(b**2 + 9))
assert solveset_real(sqrt(a**2 + b**2) - 3, a) != \
S.EmptySet
def test_units():
assert solveset_real(1/x - 1/(2*cm), x) == FiniteSet(2*cm)
def test_solve_only_exp_1():
y = Symbol('y', positive=True, finite=True)
assert solveset_real(exp(x) - y, x) == FiniteSet(log(y))
assert solveset_real(exp(x) + exp(-x) - 4, x) == \
FiniteSet(log(-sqrt(3) + 2), log(sqrt(3) + 2))
assert solveset_real(exp(x) + exp(-x) - y, x) != S.EmptySet
def test_atan2():
# The .inverse() method on atan2 works only if x.is_real is True and the
# second argument is a real constant
assert solveset_real(atan2(x, 2) - pi/3, x) == FiniteSet(2*sqrt(3))
def test_piecewise_solveset():
eq = Piecewise((x - 2, Gt(x, 2)), (2 - x, True)) - 3
assert set(solveset_real(eq, x)) == set(FiniteSet(-1, 5))
absxm3 = Piecewise(
(x - 3, S(0) <= x - 3),
(3 - x, S(0) > x - 3))
y = Symbol('y', positive=True)
assert solveset_real(absxm3 - y, x) == FiniteSet(-y + 3, y + 3)
f = Piecewise(((x - 2)**2, x >= 0), (0, True))
assert solveset(f, x, domain=S.Reals) == Union(FiniteSet(2), Interval(-oo, 0, True, True))
assert solveset(
Piecewise((x + 1, x > 0), (I, True)) - I, x, S.Reals
) == Interval(-oo, 0)
assert solveset(Piecewise((x - 1, Ne(x, I)), (x, True)), x) == FiniteSet(1)
def test_solveset_complex_polynomial():
from sympy.abc import x, a, b, c
assert solveset_complex(a*x**2 + b*x + c, x) == \
FiniteSet(-b/(2*a) - sqrt(-4*a*c + b**2)/(2*a),
-b/(2*a) + sqrt(-4*a*c + b**2)/(2*a))
assert solveset_complex(x - y**3, y) == FiniteSet(
(-x**Rational(1, 3))/2 + I*sqrt(3)*x**Rational(1, 3)/2,
x**Rational(1, 3),
(-x**Rational(1, 3))/2 - I*sqrt(3)*x**Rational(1, 3)/2)
assert solveset_complex(x + 1/x - 1, x) == \
FiniteSet(Rational(1, 2) + I*sqrt(3)/2, Rational(1, 2) - I*sqrt(3)/2)
def test_sol_zero_complex():
assert solveset_complex(0, x) == S.Complexes
def test_solveset_complex_rational():
assert solveset_complex((x - 1)*(x - I)/(x - 3), x) == \
FiniteSet(1, I)
assert solveset_complex((x - y**3)/((y**2)*sqrt(1 - y**2)), x) == \
FiniteSet(y**3)
assert solveset_complex(-x**2 - I, x) == \
FiniteSet(-sqrt(2)/2 + sqrt(2)*I/2, sqrt(2)/2 - sqrt(2)*I/2)
def test_solve_quintics():
skip("This test is too slow")
f = x**5 - 110*x**3 - 55*x**2 + 2310*x + 979
s = solveset_complex(f, x)
for root in s:
res = f.subs(x, root.n()).n()
assert tn(res, 0)
f = x**5 + 15*x + 12
s = solveset_complex(f, x)
for root in s:
res = f.subs(x, root.n()).n()
assert tn(res, 0)
def test_solveset_complex_exp():
from sympy.abc import x, n
assert solveset_complex(exp(x) - 1, x) == \
imageset(Lambda(n, I*2*n*pi), S.Integers)
assert solveset_complex(exp(x) - I, x) == \
imageset(Lambda(n, I*(2*n*pi + pi/2)), S.Integers)
assert solveset_complex(1/exp(x), x) == S.EmptySet
assert solveset_complex(sinh(x).rewrite(exp), x) == \
imageset(Lambda(n, n*pi*I), S.Integers)
def test_solveset_real_exp():
from sympy.abc import x, y
assert solveset(Eq((-2)**x, 4), x, S.Reals) == FiniteSet(2)
assert solveset(Eq(-2**x, 4), x, S.Reals) == S.EmptySet
assert solveset(Eq((-3)**x, 27), x, S.Reals) == S.EmptySet
assert solveset(Eq((-5)**(x+1), 625), x, S.Reals) == FiniteSet(3)
assert solveset(Eq(2**(x-3), -16), x, S.Reals) == S.EmptySet
assert solveset(Eq((-3)**(x - 3), -3**39), x, S.Reals) == FiniteSet(42)
assert solveset(Eq(2**x, y), x, S.Reals) == Intersection(S.Reals, FiniteSet(log(y)/log(2)))
assert invert_real((-2)**(2*x) - 16, 0, x) == (x, FiniteSet(2))
def test_solve_complex_log():
assert solveset_complex(log(x), x) == FiniteSet(1)
assert solveset_complex(1 - log(a + 4*x**2), x) == \
FiniteSet(-sqrt(-a + E)/2, sqrt(-a + E)/2)
def test_solve_complex_sqrt():
assert solveset_complex(sqrt(5*x + 6) - 2 - x, x) == \
FiniteSet(-S(1), S(2))
assert solveset_complex(sqrt(5*x + 6) - (2 + 2*I) - x, x) == \
FiniteSet(-S(2), 3 - 4*I)
assert solveset_complex(4*x*(1 - a * sqrt(x)), x) == \
FiniteSet(S(0), 1 / a ** 2)
def test_solveset_complex_tan():
s = solveset_complex(tan(x).rewrite(exp), x)
assert s == imageset(Lambda(n, pi*n), S.Integers) - \
imageset(Lambda(n, pi*n + pi/2), S.Integers)
def test_solve_trig():
from sympy.abc import n
assert solveset_real(sin(x), x) == \
Union(imageset(Lambda(n, 2*pi*n), S.Integers),
imageset(Lambda(n, 2*pi*n + pi), S.Integers))
assert solveset_real(sin(x) - 1, x) == \
imageset(Lambda(n, 2*pi*n + pi/2), S.Integers)
assert solveset_real(cos(x), x) == \
Union(imageset(Lambda(n, 2*pi*n + pi/2), S.Integers),
imageset(Lambda(n, 2*pi*n + 3*pi/2), S.Integers))
assert solveset_real(sin(x) + cos(x), x) == \
Union(imageset(Lambda(n, 2*n*pi + 3*pi/4), S.Integers),
imageset(Lambda(n, 2*n*pi + 7*pi/4), S.Integers))
assert solveset_real(sin(x)**2 + cos(x)**2, x) == S.EmptySet
assert solveset_complex(cos(x) - S.Half, x) == \
Union(imageset(Lambda(n, 2*n*pi + 5*pi/3), S.Integers),
imageset(Lambda(n, 2*n*pi + pi/3), S.Integers))
y, a = symbols('y,a')
assert solveset(sin(y + a) - sin(y), a, domain=S.Reals) == \
imageset(Lambda(n, 2*n*pi), S.Integers)
assert solveset_real(sin(2*x)*cos(x) + cos(2*x)*sin(x)-1, x) == \
ImageSet(Lambda(n, 2*n*pi/3 + pi/6), S.Integers)
# Tests for _solve_trig2() function
assert solveset_real(2*cos(x)*cos(2*x) - 1, x) == \
Union(ImageSet(Lambda(n, 2*n*pi + 2*atan(sqrt(-2*2**(S(1)/3)*(67 +
9*sqrt(57))**(S(2)/3) + 8*2**(S(2)/3) + 11*(67 +
9*sqrt(57))**(S(1)/3))/(3*(67 + 9*sqrt(57))**(S(1)/6)))), S.Integers),
ImageSet(Lambda(n, 2*n*pi - 2*atan(sqrt(-2*2**(S(1)/3)*(67 +
9*sqrt(57))**(S(2)/3) + 8*2**(S(2)/3) + 11*(67 +
9*sqrt(57))**(S(1)/3))/(3*(67 + 9*sqrt(57))**(S(1)/6))) +
2*pi), S.Integers))
assert solveset_real(2*tan(x)*sin(x) + 1, x) == Union(
ImageSet(Lambda(n, 2*n*pi + atan(sqrt(2)*sqrt(-1 + sqrt(17))/
(-sqrt(17) + 1)) + pi), S.Integers),
ImageSet(Lambda(n, 2*n*pi - atan(sqrt(2)*sqrt(-1 + sqrt(17))/
(-sqrt(17) + 1)) + pi), S.Integers))
assert solveset_real(cos(2*x)*cos(4*x) - 1, x) == \
ImageSet(Lambda(n, n*pi), S.Integers)
def test_solve_invalid_sol():
assert 0 not in solveset_real(sin(x)/x, x)
assert 0 not in solveset_complex((exp(x) - 1)/x, x)
@XFAIL
def test_solve_trig_simplified():
from sympy.abc import n
assert solveset_real(sin(x), x) == \
imageset(Lambda(n, n*pi), S.Integers)
assert solveset_real(cos(x), x) == \
imageset(Lambda(n, n*pi + pi/2), S.Integers)
assert solveset_real(cos(x) + sin(x), x) == \
imageset(Lambda(n, n*pi - pi/4), S.Integers)
@XFAIL
def test_solve_lambert():
assert solveset_real(x*exp(x) - 1, x) == FiniteSet(LambertW(1))
assert solveset_real(exp(x) + x, x) == FiniteSet(-LambertW(1))
assert solveset_real(x + 2**x, x) == \
FiniteSet(-LambertW(log(2))/log(2))
# issue 4739
ans = solveset_real(3*x + 5 + 2**(-5*x + 3), x)
assert ans == FiniteSet(-Rational(5, 3) +
LambertW(-10240*2**(S(1)/3)*log(2)/3)/(5*log(2)))
eq = 2*(3*x + 4)**5 - 6*7**(3*x + 9)
result = solveset_real(eq, x)
ans = FiniteSet((log(2401) +
5*LambertW(-log(7**(7*3**Rational(1, 5)/5))))/(3*log(7))/-1)
assert result == ans
assert solveset_real(eq.expand(), x) == result
assert solveset_real(5*x - 1 + 3*exp(2 - 7*x), x) == \
FiniteSet(Rational(1, 5) + LambertW(-21*exp(Rational(3, 5))/5)/7)
assert solveset_real(2*x + 5 + log(3*x - 2), x) == \
FiniteSet(Rational(2, 3) + LambertW(2*exp(-Rational(19, 3))/3)/2)
assert solveset_real(3*x + log(4*x), x) == \
FiniteSet(LambertW(Rational(3, 4))/3)
assert solveset_real(x**x - 2) == FiniteSet(exp(LambertW(log(2))))
a = Symbol('a')
assert solveset_real(-a*x + 2*x*log(x), x) == FiniteSet(exp(a/2))
a = Symbol('a', real=True)
assert solveset_real(a/x + exp(x/2), x) == \
FiniteSet(2*LambertW(-a/2))
assert solveset_real((a/x + exp(x/2)).diff(x), x) == \
FiniteSet(4*LambertW(sqrt(2)*sqrt(a)/4))
# coverage test
assert solveset_real(tanh(x + 3)*tanh(x - 3) - 1, x) == EmptySet()
assert solveset_real((x**2 - 2*x + 1).subs(x, log(x) + 3*x), x) == \
FiniteSet(LambertW(3*S.Exp1)/3)
assert solveset_real((x**2 - 2*x + 1).subs(x, (log(x) + 3*x)**2 - 1), x) == \
FiniteSet(LambertW(3*exp(-sqrt(2)))/3, LambertW(3*exp(sqrt(2)))/3)
assert solveset_real((x**2 - 2*x - 2).subs(x, log(x) + 3*x), x) == \
FiniteSet(LambertW(3*exp(1 + sqrt(3)))/3, LambertW(3*exp(-sqrt(3) + 1))/3)
assert solveset_real(x*log(x) + 3*x + 1, x) == \
FiniteSet(exp(-3 + LambertW(-exp(3))))
eq = (x*exp(x) - 3).subs(x, x*exp(x))
assert solveset_real(eq, x) == \
FiniteSet(LambertW(3*exp(-LambertW(3))))
assert solveset_real(3*log(a**(3*x + 5)) + a**(3*x + 5), x) == \
FiniteSet(-((log(a**5) + LambertW(S(1)/3))/(3*log(a))))
p = symbols('p', positive=True)
assert solveset_real(3*log(p**(3*x + 5)) + p**(3*x + 5), x) == \
FiniteSet(
log((-3**(S(1)/3) - 3**(S(5)/6)*I)*LambertW(S(1)/3)**(S(1)/3)/(2*p**(S(5)/3)))/log(p),
log((-3**(S(1)/3) + 3**(S(5)/6)*I)*LambertW(S(1)/3)**(S(1)/3)/(2*p**(S(5)/3)))/log(p),
log((3*LambertW(S(1)/3)/p**5)**(1/(3*log(p)))),) # checked numerically
# check collection
b = Symbol('b')
eq = 3*log(a**(3*x + 5)) + b*log(a**(3*x + 5)) + a**(3*x + 5)
assert solveset_real(eq, x) == FiniteSet(
-((log(a**5) + LambertW(1/(b + 3)))/(3*log(a))))
# issue 4271
assert solveset_real((a/x + exp(x/2)).diff(x, 2), x) == FiniteSet(
6*LambertW((-1)**(S(1)/3)*a**(S(1)/3)/3))
assert solveset_real(x**3 - 3**x, x) == \
FiniteSet(-3/log(3)*LambertW(-log(3)/3))
assert solveset_real(3**cos(x) - cos(x)**3) == FiniteSet(
acos(-3*LambertW(-log(3)/3)/log(3)))
assert solveset_real(x**2 - 2**x, x) == \
solveset_real(-x**2 + 2**x, x)
assert solveset_real(3*log(x) - x*log(3)) == FiniteSet(
-3*LambertW(-log(3)/3)/log(3),
-3*LambertW(-log(3)/3, -1)/log(3))
assert solveset_real(LambertW(2*x) - y) == FiniteSet(
y*exp(y)/2)
@XFAIL
def test_other_lambert():
a = S(6)/5
assert solveset_real(x**a - a**x, x) == FiniteSet(
a, -a*LambertW(-log(a)/a)/log(a))
def test_solveset():
x = Symbol('x')
f = Function('f')
raises(ValueError, lambda: solveset(x + y))
assert solveset(x, 1) == S.EmptySet
assert solveset(f(1)**2 + y + 1, f(1)
) == FiniteSet(-sqrt(-y - 1), sqrt(-y - 1))
assert solveset(f(1)**2 - 1, f(1), S.Reals) == FiniteSet(-1, 1)
assert solveset(f(1)**2 + 1, f(1)) == FiniteSet(-I, I)
assert solveset(x - 1, 1) == FiniteSet(x)
assert solveset(sin(x) - cos(x), sin(x)) == FiniteSet(cos(x))
assert solveset(0, domain=S.Reals) == S.Reals
assert solveset(1) == S.EmptySet
assert solveset(True, domain=S.Reals) == S.Reals # issue 10197
assert solveset(False, domain=S.Reals) == S.EmptySet
assert solveset(exp(x) - 1, domain=S.Reals) == FiniteSet(0)
assert solveset(exp(x) - 1, x, S.Reals) == FiniteSet(0)
assert solveset(Eq(exp(x), 1), x, S.Reals) == FiniteSet(0)
assert solveset(exp(x) - 1, exp(x), S.Reals) == FiniteSet(1)
A = Indexed('A', x)
assert solveset(A - 1, A, S.Reals) == FiniteSet(1)
assert solveset(x - 1 >= 0, x, S.Reals) == Interval(1, oo)
assert solveset(exp(x) - 1 >= 0, x, S.Reals) == Interval(0, oo)
assert solveset(exp(x) - 1, x) == imageset(Lambda(n, 2*I*pi*n), S.Integers)
assert solveset(Eq(exp(x), 1), x) == imageset(Lambda(n, 2*I*pi*n),
S.Integers)
# issue 13825
assert solveset(x**2 + f(0) + 1, x) == {-sqrt(-f(0) - 1), sqrt(-f(0) - 1)}
def test_conditionset():
assert solveset(Eq(sin(x)**2 + cos(x)**2, 1), x, domain=S.Reals) == \
ConditionSet(x, True, S.Reals)
assert solveset(Eq(x**2 + x*sin(x), 1), x, domain=S.Reals
) == ConditionSet(x, Eq(x**2 + x*sin(x) - 1, 0), S.Reals)
assert solveset(Eq(-I*(exp(I*x) - exp(-I*x))/2, 1), x
) == imageset(Lambda(n, 2*n*pi + pi/2), S.Integers)
assert solveset(x + sin(x) > 1, x, domain=S.Reals
) == ConditionSet(x, x + sin(x) > 1, S.Reals)
assert solveset(Eq(sin(Abs(x)), x), x, domain=S.Reals
) == ConditionSet(x, Eq(-x + sin(Abs(x)), 0), S.Reals)
assert solveset(y**x-z, x, S.Reals) == \
ConditionSet(x, Eq(y**x - z, 0), S.Reals)
@XFAIL
def test_conditionset_equality():
''' Checking equality of different representations of ConditionSet'''
assert solveset(Eq(tan(x), y), x) == ConditionSet(x, Eq(tan(x), y), S.Complexes)
def test_solveset_domain():
x = Symbol('x')
assert solveset(x**2 - x - 6, x, Interval(0, oo)) == FiniteSet(3)
assert solveset(x**2 - 1, x, Interval(0, oo)) == FiniteSet(1)
assert solveset(x**4 - 16, x, Interval(0, 10)) == FiniteSet(2)
def test_improve_coverage():
from sympy.solvers.solveset import _has_rational_power
x = Symbol('x')
solution = solveset(exp(x) + sin(x), x, S.Reals)
unsolved_object = ConditionSet(x, Eq(exp(x) + sin(x), 0), S.Reals)
assert solution == unsolved_object
assert _has_rational_power(sin(x)*exp(x) + 1, x) == (False, S.One)
assert _has_rational_power((sin(x)**2)*(exp(x) + 1)**3, x) == (False, S.One)
def test_issue_9522():
x = Symbol('x')
expr1 = Eq(1/(x**2 - 4) + x, 1/(x**2 - 4) + 2)
expr2 = Eq(1/x + x, 1/x)
assert solveset(expr1, x, S.Reals) == EmptySet()
assert solveset(expr2, x, S.Reals) == EmptySet()
def test_solvify():
x = Symbol('x')
assert solvify(x**2 + 10, x, S.Reals) == []
assert solvify(x**3 + 1, x, S.Complexes) == [-1, S(1)/2 - sqrt(3)*I/2,
S(1)/2 + sqrt(3)*I/2]
assert solvify(log(x), x, S.Reals) == [1]
assert solvify(cos(x), x, S.Reals) == [pi/2, 3*pi/2]
assert solvify(sin(x) + 1, x, S.Reals) == [3*pi/2]
raises(NotImplementedError, lambda: solvify(sin(exp(x)), x, S.Complexes))
def test_abs_invert_solvify():
assert solvify(sin(Abs(x)), x, S.Reals) is None
def test_linear_eq_to_matrix():
x, y, z = symbols('x, y, z')
a, b, c, d, e, f, g, h, i, j, k, l = symbols('a:l')
eqns1 = [2*x + y - 2*z - 3, x - y - z, x + y + 3*z - 12]
eqns2 = [Eq(3*x + 2*y - z, 1), Eq(2*x - 2*y + 4*z, -2), -2*x + y - 2*z]
A, B = linear_eq_to_matrix(eqns1, x, y, z)
assert A == Matrix([[2, 1, -2], [1, -1, -1], [1, 1, 3]])
assert B == Matrix([[3], [0], [12]])
A, B = linear_eq_to_matrix(eqns2, x, y, z)
assert A == Matrix([[3, 2, -1], [2, -2, 4], [-2, 1, -2]])
assert B == Matrix([[1], [-2], [0]])
# Pure symbolic coefficients
eqns3 = [a*b*x + b*y + c*z - d, e*x + d*x + f*y + g*z - h, i*x + j*y + k*z - l]
A, B = linear_eq_to_matrix(eqns3, x, y, z)
assert A == Matrix([[a*b, b, c], [d + e, f, g], [i, j, k]])
assert B == Matrix([[d], [h], [l]])
# raise ValueError if
# 1) no symbols are given
raises(ValueError, lambda: linear_eq_to_matrix(eqns3))
# 2) there are duplicates
raises(ValueError, lambda: linear_eq_to_matrix(eqns3, [x, x, y]))
# 3) there are non-symbols
raises(ValueError, lambda: linear_eq_to_matrix(eqns3, [x, 1/a, y]))
# 4) a nonlinear term is detected in the original expression
raises(ValueError, lambda: linear_eq_to_matrix(Eq(1/x + x, 1/x)))
assert linear_eq_to_matrix(1, x) == (Matrix([[0]]), Matrix([[-1]]))
# issue 15195
assert linear_eq_to_matrix(x + y*(z*(3*x + 2) + 3), x) == (
Matrix([[3*y*z + 1]]), Matrix([[-y*(2*z + 3)]]))
assert linear_eq_to_matrix(Matrix(
[[a*x + b*y - 7], [5*x + 6*y - c]]), x, y) == (
Matrix([[a, b], [5, 6]]), Matrix([[7], [c]]))
# issue 15312
assert linear_eq_to_matrix(Eq(x + 2, 1), x) == (
Matrix([[1]]), Matrix([[-1]]))
def test_linsolve():
x, y, z, u, v, w = symbols("x, y, z, u, v, w")
x1, x2, x3, x4 = symbols('x1, x2, x3, x4')
# Test for different input forms
M = Matrix([[1, 2, 1, 1, 7], [1, 2, 2, -1, 12], [2, 4, 0, 6, 4]])
system1 = A, b = M[:, :-1], M[:, -1]
Eqns = [x1 + 2*x2 + x3 + x4 - 7, x1 + 2*x2 + 2*x3 - x4 - 12,
2*x1 + 4*x2 + 6*x4 - 4]
sol = FiniteSet((-2*x2 - 3*x4 + 2, x2, 2*x4 + 5, x4))
assert linsolve(Eqns, (x1, x2, x3, x4)) == sol
assert linsolve(Eqns, *(x1, x2, x3, x4)) == sol
assert linsolve(system1, (x1, x2, x3, x4)) == sol
assert linsolve(system1, *(x1, x2, x3, x4)) == sol
# issue 9667 - symbols can be Dummy symbols
x1, x2, x3, x4 = symbols('x:4', cls=Dummy)
assert linsolve(system1, x1, x2, x3, x4) == FiniteSet(
(-2*x2 - 3*x4 + 2, x2, 2*x4 + 5, x4))
# raise ValueError for garbage value
raises(ValueError, lambda: linsolve(Eqns))
raises(ValueError, lambda: linsolve(x1))
raises(ValueError, lambda: linsolve(x1, x2))
raises(ValueError, lambda: linsolve((A,), x1, x2))
raises(ValueError, lambda: linsolve(A, b, x1, x2))
#raise ValueError if equations are non-linear in given variables
raises(ValueError, lambda: linsolve([x + y - 1, x ** 2 + y - 3], [x, y]))
raises(ValueError, lambda: linsolve([cos(x) + y, x + y], [x, y]))
assert linsolve([x + z - 1, x ** 2 + y - 3], [z, y]) == {(-x + 1, -x**2 + 3)}
# Fully symbolic test
a, b, c, d, e, f = symbols('a, b, c, d, e, f')
A = Matrix([[a, b], [c, d]])
B = Matrix([[e], [f]])
system2 = (A, B)
sol = FiniteSet(((-b*f + d*e)/(a*d - b*c), (a*f - c*e)/(a*d - b*c)))
assert linsolve(system2, [x, y]) == sol
# No solution
A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
b = Matrix([0, 0, 1])
assert linsolve((A, b), (x, y, z)) == EmptySet()
# Issue #10056
A, B, J1, J2 = symbols('A B J1 J2')
Augmatrix = Matrix([
[2*I*J1, 2*I*J2, -2/J1],
[-2*I*J2, -2*I*J1, 2/J2],
[0, 2, 2*I/(J1*J2)],
[2, 0, 0],
])
assert linsolve(Augmatrix, A, B) == FiniteSet((0, I/(J1*J2)))
# Issue #10121 - Assignment of free variables
a, b, c, d, e = symbols('a, b, c, d, e')
Augmatrix = Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]])
assert linsolve(Augmatrix, a, b, c, d, e) == FiniteSet((a, 0, c, 0, e))
raises(IndexError, lambda: linsolve(Augmatrix, a, b, c))
x0, x1, x2, _x0 = symbols('tau0 tau1 tau2 _tau0')
assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
) == FiniteSet((x0, 0, x1, _x0, x2))
x0, x1, x2, _x0 = symbols('_tau0 _tau1 _tau2 tau0')
assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
) == FiniteSet((x0, 0, x1, _x0, x2))
x0, x1, x2, _x0 = symbols('_tau0 _tau1 _tau2 tau1')
assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
) == FiniteSet((x0, 0, x1, _x0, x2))
# symbols can be given as generators
x0, x2, x4 = symbols('x0, x2, x4')
assert linsolve(Augmatrix, numbered_symbols('x')
) == FiniteSet((x0, 0, x2, 0, x4))
Augmatrix[-1, -1] = x0
# use Dummy to avoid clash; the names may clash but the symbols
# will not
Augmatrix[-1, -1] = symbols('_x0')
assert len(linsolve(
Augmatrix, numbered_symbols('x', cls=Dummy)).free_symbols) == 4
# Issue #12604
f = Function('f')
assert linsolve([f(x) - 5], f(x)) == FiniteSet((5,))
# Issue #14860
from sympy.physics.units import meter, newton, kilo
Eqns = [8*kilo*newton + x + y, 28*kilo*newton*meter + 3*x*meter]
assert linsolve(Eqns, x, y) == {(-28000*newton/3, 4000*newton/3)}
# linsolve fully expands expressions, so removable singularities
# and other nonlinearity does not raise an error
assert linsolve([Eq(x, x + y)], [x, y]) == {(x, 0)}
assert linsolve([Eq(1/x, 1/x + y)], [x, y]) == {(x, 0)}
assert linsolve([Eq(y/x, y/x + y)], [x, y]) == {(x, 0)}
assert linsolve([Eq(x*(x + 1), x**2 + y)], [x, y]) == {(y, y)}
def test_solve_decomposition():
x = Symbol('x')
n = Dummy('n')
f1 = exp(3*x) - 6*exp(2*x) + 11*exp(x) - 6
f2 = sin(x)**2 - 2*sin(x) + 1
f3 = sin(x)**2 - sin(x)
f4 = sin(x + 1)
f5 = exp(x + 2) - 1
f6 = 1/log(x)
f7 = 1/x
s1 = ImageSet(Lambda(n, 2*n*pi), S.Integers)
s2 = ImageSet(Lambda(n, 2*n*pi + pi), S.Integers)
s3 = ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers)
s4 = ImageSet(Lambda(n, 2*n*pi - 1), S.Integers)
s5 = ImageSet(Lambda(n, 2*n*pi - 1 + pi), S.Integers)
assert solve_decomposition(f1, x, S.Reals) == FiniteSet(0, log(2), log(3))
assert solve_decomposition(f2, x, S.Reals) == s3
assert solve_decomposition(f3, x, S.Reals) == Union(s1, s2, s3)
assert solve_decomposition(f4, x, S.Reals) == Union(s4, s5)
assert solve_decomposition(f5, x, S.Reals) == FiniteSet(-2)
assert solve_decomposition(f6, x, S.Reals) == S.EmptySet
assert solve_decomposition(f7, x, S.Reals) == S.EmptySet
assert solve_decomposition(x, x, Interval(1, 2)) == S.EmptySet
# nonlinsolve testcases
def test_nonlinsolve_basic():
assert nonlinsolve([],[]) == S.EmptySet
assert nonlinsolve([],[x, y]) == S.EmptySet
system = [x, y - x - 5]
assert nonlinsolve([x],[x, y]) == FiniteSet((0, y))
assert nonlinsolve(system, [y]) == FiniteSet((x + 5,))
soln = (ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers),)
assert nonlinsolve([sin(x) - 1], [x]) == FiniteSet(tuple(soln))
assert nonlinsolve([x**2 - 1], [x]) == FiniteSet((-1,), (1,))
soln = FiniteSet((y, y))
assert nonlinsolve([x - y, 0], x, y) == soln
assert nonlinsolve([0, x - y], x, y) == soln
assert nonlinsolve([x - y, x - y], x, y) == soln
assert nonlinsolve([x, 0], x, y) == FiniteSet((0, y))
f = Function('f')
assert nonlinsolve([f(x), 0], f(x), y) == FiniteSet((0, y))
assert nonlinsolve([f(x), 0], f(x), f(y)) == FiniteSet((0, f(y)))
A = Indexed('A', x)
assert nonlinsolve([A, 0], A, y) == FiniteSet((0, y))
assert nonlinsolve([x**2 -1], [sin(x)]) == FiniteSet((S.EmptySet,))
assert nonlinsolve([x**2 -1], sin(x)) == FiniteSet((S.EmptySet,))
assert nonlinsolve([x**2 -1], 1) == FiniteSet((x**2,))
assert nonlinsolve([x**2 -1], x + y) == FiniteSet((S.EmptySet,))
def test_nonlinsolve_abs():
soln = FiniteSet((x, Abs(x)))
assert nonlinsolve([Abs(x) - y], x, y) == soln
def test_raise_exception_nonlinsolve():
raises(IndexError, lambda: nonlinsolve([x**2 -1], []))
raises(ValueError, lambda: nonlinsolve([x**2 -1]))
raises(NotImplementedError, lambda: nonlinsolve([(x+y)**2 - 9, x**2 - y**2 - 0.75], (x, y)))
def test_trig_system():
# TODO: add more simple testcases when solveset returns
# simplified soln for Trig eq
assert nonlinsolve([sin(x) - 1, cos(x) -1 ], x) == S.EmptySet
soln1 = (ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers),)
soln = FiniteSet(soln1)
assert nonlinsolve([sin(x) - 1, cos(x)], x) == soln
@XFAIL
def test_trig_system_fail():
# fails because solveset trig solver is not much smart.
sys = [x + y - pi/2, sin(x) + sin(y) - 1]
# solveset returns conditonset for sin(x) + sin(y) - 1
soln_1 = (ImageSet(Lambda(n, n*pi + pi/2), S.Integers),
ImageSet(Lambda(n, n*pi)), S.Integers)
soln_1 = FiniteSet(soln_1)
soln_2 = (ImageSet(Lambda(n, n*pi), S.Integers),
ImageSet(Lambda(n, n*pi+ pi/2), S.Integers))
soln_2 = FiniteSet(soln_2)
soln = soln_1 + soln_2
assert nonlinsolve(sys, [x, y]) == soln
# Add more cases from here
# http://www.vitutor.com/geometry/trigonometry/equations_systems.html#uno
sys = [sin(x) + sin(y) - (sqrt(3)+1)/2, sin(x) - sin(y) - (sqrt(3) - 1)/2]
soln_x = Union(ImageSet(Lambda(n, 2*n*pi + pi/3), S.Integers),
ImageSet(Lambda(n, 2*n*pi + 2*pi/3), S.Integers))
soln_y = Union(ImageSet(Lambda(n, 2*n*pi + pi/6), S.Integers),
ImageSet(Lambda(n, 2*n*pi + 5*pi/6), S.Integers))
assert nonlinsolve(sys, [x, y]) ==FiniteSet((soln_x, soln_y))
def test_nonlinsolve_positive_dimensional():
x, y, z, a, b, c, d = symbols('x, y, z, a, b, c, d', real = True)
assert nonlinsolve([x*y, x*y - x], [x, y]) == FiniteSet((0, y))
system = [a**2 + a*c, a - b]
assert nonlinsolve(system, [a, b]) == FiniteSet((0, 0), (-c, -c))
# here (a= 0, b = 0) is independent soln so both is printed.
# if symbols = [a, b, c] then only {a : -c ,b : -c}
eq1 = a + b + c + d
eq2 = a*b + b*c + c*d + d*a
eq3 = a*b*c + b*c*d + c*d*a + d*a*b
eq4 = a*b*c*d - 1
system = [eq1, eq2, eq3, eq4]
sol1 = (-1/d, -d, 1/d, FiniteSet(d) - FiniteSet(0))
sol2 = (1/d, -d, -1/d, FiniteSet(d) - FiniteSet(0))
soln = FiniteSet(sol1, sol2)
assert nonlinsolve(system, [a, b, c, d]) == soln
def test_nonlinsolve_polysys():
x, y, z = symbols('x, y, z', real = True)
assert nonlinsolve([x**2 + y - 2, x**2 + y], [x, y]) == S.EmptySet
s = (-y + 2, y)
assert nonlinsolve([(x + y)**2 - 4, x + y - 2], [x, y]) == FiniteSet(s)
system = [x**2 - y**2]
soln_real = FiniteSet((-y, y), (y, y))
soln_complex = FiniteSet((-Abs(y), y), (Abs(y), y))
soln =soln_real + soln_complex
assert nonlinsolve(system, [x, y]) == soln
system = [x**2 - y**2]
soln_real= FiniteSet((y, -y), (y, y))
soln_complex = FiniteSet((y, -Abs(y)), (y, Abs(y)))
soln = soln_real + soln_complex
assert nonlinsolve(system, [y, x]) == soln
system = [x**2 + y - 3, x - y - 4]
assert nonlinsolve(system, (x, y)) != nonlinsolve(system, (y, x))
def test_nonlinsolve_using_substitution():
x, y, z, n = symbols('x, y, z, n', real = True)
system = [(x + y)*n - y**2 + 2]
s_x = (n*y - y**2 + 2)/n
soln = (-s_x, y)
assert nonlinsolve(system, [x, y]) == FiniteSet(soln)
system = [z**2*x**2 - z**2*y**2/exp(x)]
soln_real_1 = (y, x, 0)
soln_real_2 = (-exp(x/2)*Abs(x), x, z)
soln_real_3 = (exp(x/2)*Abs(x), x, z)
soln_complex_1 = (-x*exp(x/2), x, z)
soln_complex_2 = (x*exp(x/2), x, z)
syms = [y, x, z]
soln = FiniteSet(soln_real_1, soln_complex_1, soln_complex_2,\
soln_real_2, soln_real_3)
assert nonlinsolve(system,syms) == soln
def test_nonlinsolve_complex():
x, y, z = symbols('x, y, z')
n = Dummy('n')
real_soln = (log(sin(S(1)/3)), S(1)/3)
img_lamda = Lambda(n, 2*n*I*pi + Mod(log(sin(S(1)/3)), 2*I*pi))
complex_soln = (ImageSet(img_lamda, S.Integers), S(1)/3)
soln = FiniteSet(real_soln, complex_soln)
assert nonlinsolve([exp(x) - sin(y), 1/y - 3], [x, y]) == soln
system = [exp(x) - sin(y), 1/exp(y) - 3]
soln_x = ImageSet(Lambda(n, I*(2*n*pi + pi) + log(sin(log(3)))), S.Integers)
soln_real = FiniteSet((soln_x, -log(S(3))))
# Mod(-log(3), 2*I*pi) is equal to -log(3).
expr_x = I*(2*n*pi + arg(sin(2*n*I*pi + Mod(-log(3), 2*I*pi)))) + \
log(Abs(sin(2*n*I*pi + Mod(-log(3), 2*I*pi))))
soln_x = ImageSet(Lambda(n, expr_x), S.Integers)
expr_y = 2*n*I*pi + Mod(-log(3), 2*I*pi)
soln_y = ImageSet(Lambda(n, expr_y), S.Integers)
soln_complex = FiniteSet((soln_x, soln_y))
soln = soln_real + soln_complex
assert nonlinsolve(system, [x, y]) == soln
system = [exp(x) - sin(y), y**2 - 4]
s1 = (log(sin(2)), 2)
s2 = (ImageSet(Lambda(n, I*(2*n*pi + pi) + log(sin(2))), S.Integers), -2 )
img = ImageSet(Lambda(n, 2*n*I*pi + Mod(log(sin(2)), 2*I*pi)), S.Integers)
s3 = (img, 2)
assert nonlinsolve(system, [x, y]) == FiniteSet(s1, s2, s3)
@XFAIL
def test_solve_nonlinear_trans():
# After the transcendental equation solver these will work
x, y, z = symbols('x, y, z', real=True)
soln1 = FiniteSet((2*LambertW(y/2), y))
soln2 = FiniteSet((-x*sqrt(exp(x)), y), (x*sqrt(exp(x)), y))
soln3 = FiniteSet((x*exp(x/2), x))
soln4 = FiniteSet(2*LambertW(y/2), y)
assert nonlinsolve([x**2 - y**2/exp(x)], [x, y]) == soln1
assert nonlinsolve([x**2 - y**2/exp(x)], [y, x]) == soln2
assert nonlinsolve([x**2 - y**2/exp(x)], [y, x]) == soln3
assert nonlinsolve([x**2 - y**2/exp(x)], [x, y]) == soln4
def test_issue_5132_1():
system = [sqrt(x**2 + y**2) - sqrt(10), x + y - 4]
assert nonlinsolve(system, [x, y]) == FiniteSet((1, 3), (3, 1))
n = Dummy('n')
eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3]
s_real_y = -log(3)
s_real_z = sqrt(-exp(2*x) - sin(log(3)))
soln_real = FiniteSet((s_real_y, s_real_z), (s_real_y, -s_real_z))
lam = Lambda(n, 2*n*I*pi + Mod(-log(3), 2*I*pi))
s_complex_y = ImageSet(lam, S.Integers)
lam = Lambda(n, sqrt(-exp(2*x) + sin(2*n*I*pi + Mod(-log(3), 2*I*pi))))
s_complex_z_1 = ImageSet(lam, S.Integers)
lam = Lambda(n, -sqrt(-exp(2*x) + sin(2*n*I*pi + Mod(-log(3), 2*I*pi))))
s_complex_z_2 = ImageSet(lam, S.Integers)
soln_complex = FiniteSet(
(s_complex_y, s_complex_z_1),
(s_complex_y, s_complex_z_2)
)
soln = soln_real + soln_complex
assert nonlinsolve(eqs, [y, z]) == soln
def test_issue_5132_2():
x, y = symbols('x, y', real=True)
eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3]
n = Dummy('n')
soln_real = (log(-z**2 + sin(y))/2, z)
lam = Lambda( n, I*(2*n*pi + arg(-z**2 + sin(y)))/2 + log(Abs(z**2 - sin(y)))/2)
img = ImageSet(lam, S.Integers)
# not sure about the complex soln. But it looks correct.
soln_complex = (img, z)
soln = FiniteSet(soln_real, soln_complex)
assert nonlinsolve(eqs, [x, z]) == soln
r, t = symbols('r, t')
system = [r - x**2 - y**2, tan(t) - y/x]
s_x = sqrt(r/(tan(t)**2 + 1))
s_y = sqrt(r/(tan(t)**2 + 1))*tan(t)
soln = FiniteSet((s_x, s_y), (-s_x, -s_y))
assert nonlinsolve(system, [x, y]) == soln
def test_issue_6752():
a,b,c,d = symbols('a, b, c, d', real=True)
assert nonlinsolve([a**2 + a, a - b], [a, b]) == {(-1, -1), (0, 0)}
@SKIP("slow")
def test_issue_5114_solveset():
# slow testcase
a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r = symbols('a:r')
# there is no 'a' in the equation set but this is how the
# problem was originally posed
syms = [a, b, c, f, h, k, n]
eqs = [b + r/d - c/d,
c*(1/d + 1/e + 1/g) - f/g - r/d,
f*(1/g + 1/i + 1/j) - c/g - h/i,
h*(1/i + 1/l + 1/m) - f/i - k/m,
k*(1/m + 1/o + 1/p) - h/m - n/p,
n*(1/p + 1/q) - k/p]
assert len(nonlinsolve(eqs, syms)) == 1
@SKIP("Hangs")
def _test_issue_5335():
# Not able to check zero dimensional system.
# is_zero_dimensional Hangs
lam, a0, conc = symbols('lam a0 conc')
eqs = [lam + 2*y - a0*(1 - x/2)*x - 0.005*x/2*x,
a0*(1 - x/2)*x - 1*y - 0.743436700916726*y,
x + y - conc]
sym = [x, y, a0]
# there are 4 solutions but only two are valid
assert len(nonlinsolve(eqs, sym)) == 2
# float
lam, a0, conc = symbols('lam a0 conc')
eqs = [lam + 2*y - a0*(1 - x/2)*x - 0.005*x/2*x,
a0*(1 - x/2)*x - 1*y - 0.743436700916726*y,
x + y - conc]
sym = [x, y, a0]
assert len(nonlinsolve(eqs, sym)) == 2
def test_issue_2777():
# the equations represent two circles
x, y = symbols('x y', real=True)
e1, e2 = sqrt(x**2 + y**2) - 10, sqrt(y**2 + (-x + 10)**2) - 3
a, b = 191/S(20), 3*sqrt(391)/20
ans = {(a, -b), (a, b)}
assert nonlinsolve((e1, e2), (x, y)) == ans
assert nonlinsolve((e1, e2/(x - a)), (x, y)) == S.EmptySet
# make the 2nd circle's radius be -3
e2 += 6
assert nonlinsolve((e1, e2), (x, y)) == S.EmptySet
def test_issue_8828():
x1 = 0
y1 = -620
r1 = 920
x2 = 126
y2 = 276
x3 = 51
y3 = 205
r3 = 104
v = [x, y, z]
f1 = (x - x1)**2 + (y - y1)**2 - (r1 - z)**2
f2 = (x2 - x)**2 + (y2 - y)**2 - z**2
f3 = (x - x3)**2 + (y - y3)**2 - (r3 - z)**2
F = [f1, f2, f3]
g1 = sqrt((x - x1)**2 + (y - y1)**2) + z - r1
g2 = f2
g3 = sqrt((x - x3)**2 + (y - y3)**2) + z - r3
G = [g1, g2, g3]
# both soln same
A = nonlinsolve(F, v)
B = nonlinsolve(G, v)
assert A == B
def test_nonlinsolve_conditionset():
# when solveset failed to solve all the eq
# return conditionset
f = Function('f')
f1 = f(x) - pi/2
f2 = f(y) - 3*pi/2
intermediate_system = FiniteSet(2*f(x) - pi, 2*f(y) - 3*pi)
symbols = Tuple(x, y)
soln = ConditionSet(
symbols,
intermediate_system,
S.Complexes)
assert nonlinsolve([f1, f2], [x, y]) == soln
def test_substitution_basic():
assert substitution([], [x, y]) == S.EmptySet
assert substitution([], []) == S.EmptySet
system = [2*x**2 + 3*y**2 - 30, 3*x**2 - 2*y**2 - 19]
soln = FiniteSet((-3, -2), (-3, 2), (3, -2), (3, 2))
assert substitution(system, [x, y]) == soln
soln = FiniteSet((-1, 1))
assert substitution([x + y], [x], [{y: 1}], [y], set([]), [x, y]) == soln
assert substitution(
[x + y], [x], [{y: 1}], [y],
set([x + 1]), [y, x]) == S.EmptySet
def test_issue_5132_substitution():
x, y, z, r, t = symbols('x, y, z, r, t', real=True)
system = [r - x**2 - y**2, tan(t) - y/x]
s_x_1 = Complement(FiniteSet(-sqrt(r/(tan(t)**2 + 1))), FiniteSet(0))
s_x_2 = Complement(FiniteSet(sqrt(r/(tan(t)**2 + 1))), FiniteSet(0))
s_y = sqrt(r/(tan(t)**2 + 1))*tan(t)
soln = FiniteSet((s_x_2, s_y)) + FiniteSet((s_x_1, -s_y))
assert substitution(system, [x, y]) == soln
n = Dummy('n')
eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3]
s_real_y = -log(3)
s_real_z = sqrt(-exp(2*x) - sin(log(3)))
soln_real = FiniteSet((s_real_y, s_real_z), (s_real_y, -s_real_z))
lam = Lambda(n, 2*n*I*pi + Mod(-log(3), 2*I*pi))
s_complex_y = ImageSet(lam, S.Integers)
lam = Lambda(n, sqrt(-exp(2*x) + sin(2*n*I*pi + Mod(-log(3), 2*I*pi))))
s_complex_z_1 = ImageSet(lam, S.Integers)
lam = Lambda(n, -sqrt(-exp(2*x) + sin(2*n*I*pi + Mod(-log(3), 2*I*pi))))
s_complex_z_2 = ImageSet(lam, S.Integers)
soln_complex = FiniteSet(
(s_complex_y, s_complex_z_1),
(s_complex_y, s_complex_z_2)
)
soln = soln_real + soln_complex
assert substitution(eqs, [y, z]) == soln
def test_raises_substitution():
raises(ValueError, lambda: substitution([x**2 -1], []))
raises(TypeError, lambda: substitution([x**2 -1]))
raises(ValueError, lambda: substitution([x**2 -1], [sin(x)]))
raises(TypeError, lambda: substitution([x**2 -1], x))
raises(TypeError, lambda: substitution([x**2 -1], 1))
# end of tests for nonlinsolve
def test_issue_9556():
x = Symbol('x')
b = Symbol('b', positive=True)
assert solveset(Abs(x) + 1, x, S.Reals) == EmptySet()
assert solveset(Abs(x) + b, x, S.Reals) == EmptySet()
assert solveset(Eq(b, -1), b, S.Reals) == EmptySet()
def test_issue_9611():
x = Symbol('x')
a = Symbol('a')
y = Symbol('y')
assert solveset(Eq(x - x + a, a), x, S.Reals) == S.Reals
assert solveset(Eq(y - y + a, a), y) == S.Complexes
def test_issue_9557():
x = Symbol('x')
a = Symbol('a')
assert solveset(x**2 + a, x, S.Reals) == Intersection(S.Reals,
FiniteSet(-sqrt(-a), sqrt(-a)))
def test_issue_9778():
assert solveset(x**3 + 1, x, S.Reals) == FiniteSet(-1)
assert solveset(x**(S(3)/5) + 1, x, S.Reals) == S.EmptySet
assert solveset(x**3 + y, x, S.Reals) == \
FiniteSet(-Abs(y)**(S(1)/3)*sign(y))
def test_issue_10214():
assert solveset(x**(S(3)/2) + 4, x, S.Reals) == S.EmptySet
assert solveset(x**(S(-3)/2) + 4, x, S.Reals) == S.EmptySet
ans = FiniteSet(-2**(S(2)/3))
assert solveset(x**(S(3)) + 4, x, S.Reals) == ans
assert (x**(S(3)) + 4).subs(x,list(ans)[0]) == 0 # substituting ans and verifying the result.
assert (x**(S(3)) + 4).subs(x,-(-2)**(2/S(3))) == 0
def test_issue_9849():
assert solveset(Abs(sin(x)) + 1, x, S.Reals) == S.EmptySet
def test_issue_9953():
assert linsolve([ ], x) == S.EmptySet
def test_issue_9913():
assert solveset(2*x + 1/(x - 10)**2, x, S.Reals) == \
FiniteSet(-(3*sqrt(24081)/4 + S(4027)/4)**(S(1)/3)/3 - 100/
(3*(3*sqrt(24081)/4 + S(4027)/4)**(S(1)/3)) + S(20)/3)
def test_issue_10397():
assert solveset(sqrt(x), x, S.Complexes) == FiniteSet(0)
def test_issue_14987():
raises(ValueError, lambda: linear_eq_to_matrix(
[x**2], x))
raises(ValueError, lambda: linear_eq_to_matrix(
[x*(-3/x + 1) + 2*y - a], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[(x**2 - 3*x)/(x - 3) - 3], x))
raises(ValueError, lambda: linear_eq_to_matrix(
[(x + 1)**3 - x**3 - 3*x**2 + 7], x))
raises(ValueError, lambda: linear_eq_to_matrix(
[x*(1/x + 1) + y], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[(x + 1)*y], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[Eq(1/x, 1/x + y)], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[Eq(y/x, y/x + y)], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[Eq(x*(x + 1), x**2 + y)], [x, y]))
def test_simplification():
eq = x + (a - b)/(-2*a + 2*b)
assert solveset(eq, x) == FiniteSet(S.Half)
assert solveset(eq, x, S.Reals) == FiniteSet(S.Half)
def test_issue_10555():
f = Function('f')
g = Function('g')
assert solveset(f(x) - pi/2, x, S.Reals) == \
ConditionSet(x, Eq(f(x) - pi/2, 0), S.Reals)
assert solveset(f(g(x)) - pi/2, g(x), S.Reals) == \
ConditionSet(g(x), Eq(f(g(x)) - pi/2, 0), S.Reals)
def test_issue_8715():
eq = x + 1/x > -2 + 1/x
assert solveset(eq, x, S.Reals) == \
(Interval.open(-2, oo) - FiniteSet(0))
assert solveset(eq.subs(x,log(x)), x, S.Reals) == \
Interval.open(exp(-2), oo) - FiniteSet(1)
def test_issue_11174():
r, t = symbols('r t')
eq = z**2 + exp(2*x) - sin(y)
soln = Intersection(S.Reals, FiniteSet(log(-z**2 + sin(y))/2))
assert solveset(eq, x, S.Reals) == soln
eq = sqrt(r)*Abs(tan(t))/sqrt(tan(t)**2 + 1) + x*tan(t)
s = -sqrt(r)*Abs(tan(t))/(sqrt(tan(t)**2 + 1)*tan(t))
soln = Intersection(S.Reals, FiniteSet(s))
assert solveset(eq, x, S.Reals) == soln
def test_issue_11534():
# eq and eq2 should give the same solution as a Complement
eq = -y + x/sqrt(-x**2 + 1)
eq2 = -y**2 + x**2/(-x**2 + 1)
soln = Complement(FiniteSet(-y/sqrt(y**2 + 1), y/sqrt(y**2 + 1)), FiniteSet(-1, 1))
assert solveset(eq, x, S.Reals) == soln
assert solveset(eq2, x, S.Reals) == soln
def test_issue_10477():
assert solveset((x**2 + 4*x - 3)/x < 2, x, S.Reals) == \
Union(Interval.open(-oo, -3), Interval.open(0, 1))
def test_issue_10671():
assert solveset(sin(y), y, Interval(0, pi)) == FiniteSet(0, pi)
i = Interval(1, 10)
assert solveset((1/x).diff(x) < 0, x, i) == i
def test_issue_11064():
eq = x + sqrt(x**2 - 5)
assert solveset(eq > 0, x, S.Reals) == \
Interval(sqrt(5), oo)
assert solveset(eq < 0, x, S.Reals) == \
Interval(-oo, -sqrt(5))
assert solveset(eq > sqrt(5), x, S.Reals) == \
Interval.Lopen(sqrt(5), oo)
def test_issue_12478():
eq = sqrt(x - 2) + 2
soln = solveset_real(eq, x)
assert soln is S.EmptySet
assert solveset(eq < 0, x, S.Reals) is S.EmptySet
assert solveset(eq > 0, x, S.Reals) == Interval(2, oo)
def test_issue_12429():
eq = solveset(log(x)/x <= 0, x, S.Reals)
sol = Interval.Lopen(0, 1)
assert eq == sol
def test_solveset_arg():
assert solveset(arg(x), x, S.Reals) == Interval.open(0, oo)
assert solveset(arg(4*x -3), x) == Interval.open(S(3)/4, oo)
def test__is_finite_with_finite_vars():
f = _is_finite_with_finite_vars
# issue 12482
assert all(f(1/x) is None for x in (
Dummy(), Dummy(real=True), Dummy(complex=True)))
assert f(1/Dummy(real=False)) is True # b/c it's finite but not 0
def test_issue_13550():
assert solveset(x**2 - 2*x - 15, symbol = x, domain = Interval(-oo, 0)) == FiniteSet(-3)
def test_issue_13849():
t = symbols('t')
assert nonlinsolve((t*(sqrt(5) + sqrt(2)) - sqrt(2), t), t) == EmptySet()
def test_issue_14223():
x = Symbol('x')
assert solveset((Abs(x + Min(x, 2)) - 2).rewrite(Piecewise), x,
S.Reals) == FiniteSet(-1, 1)
assert solveset((Abs(x + Min(x, 2)) - 2).rewrite(Piecewise), x,
Interval(0, 2)) == FiniteSet(1)
def test_issue_10158():
x = Symbol('x')
dom = S.Reals
assert solveset(x*Max(x, 15) - 10, x, dom) == FiniteSet(2/S(3))
assert solveset(x*Min(x, 15) - 10, x, dom) == FiniteSet(-sqrt(10), sqrt(10))
assert solveset(Max(Abs(x - 3) - 1, x + 2) - 3, x, dom) == FiniteSet(-1, 1)
assert solveset(Abs(x - 1) - Abs(y), x, dom) == FiniteSet(-Abs(y) + 1, Abs(y) + 1)
assert solveset(Abs(x + 4*Abs(x + 1)), x, dom) == FiniteSet(-4/S(3), -4/S(5))
assert solveset(2*Abs(x + Abs(x + Max(3, x))) - 2, x, S.Reals) == FiniteSet(-1, -2)
dom = S.Complexes
raises(ValueError, lambda: solveset(x*Max(x, 15) - 10, x, dom))
raises(ValueError, lambda: solveset(x*Min(x, 15) - 10, x, dom))
raises(ValueError, lambda: solveset(Max(Abs(x - 3) - 1, x + 2) - 3, x, dom))
raises(ValueError, lambda: solveset(Abs(x - 1) - Abs(y), x, dom))
raises(ValueError, lambda: solveset(Abs(x + 4*Abs(x + 1)), x, dom))
def test_issue_14300():
x, y, n = symbols('x y n')
f = 1 - exp(-18000000*x) - y
a1 = FiniteSet(-log(-y + 1)/18000000)
assert solveset(f, x, S.Reals) == \
Intersection(S.Reals, a1)
assert solveset(f, x) == \
ImageSet(Lambda(n, -I*(2*n*pi + arg(-y + 1))/18000000 -
log(Abs(y - 1))/18000000), S.Integers)
def test_issue_14454():
x = Symbol('x')
number = CRootOf(x**4 + x - 1, 2)
raises(ValueError, lambda: invert_real(number, 0, x, S.Reals))
assert invert_real(x**2, number, x, S.Reals) # no error
def test_term_factors():
assert list(_term_factors(3**x - 2)) == [-2, 3**x]
expr = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3)
assert set(_term_factors(expr)) == set([
3**(x + 2), 4**(x + 2), 3**(x + 3), 4**(x - 1), -1, 4**(x + 1)])
#################### tests for transolve and its helpers ###############
def test_transolve():
assert _transolve(3**x, x, S.Reals) == S.EmptySet
assert _transolve(3**x - 9**(x + 5), x, S.Reals) == FiniteSet(-10)
# exponential tests
def test_exponential_real():
from sympy.abc import x, y, z
e1 = 3**(2*x) - 2**(x + 3)
e2 = 4**(5 - 9*x) - 8**(2 - x)
e3 = 2**x + 4**x
e4 = exp(log(5)*x) - 2**x
e5 = exp(x/y)*exp(-z/y) - 2
e6 = 5**(x/2) - 2**(x/3)
e7 = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3)
e8 = -9*exp(-2*x + 5) + 4*exp(3*x + 1)
e9 = 2**x + 4**x + 8**x - 84
assert solveset(e1, x, S.Reals) == FiniteSet(
-3*log(2)/(-2*log(3) + log(2)))
assert solveset(e2, x, S.Reals) == FiniteSet(4/S(15))
assert solveset(e3, x, S.Reals) == S.EmptySet
assert solveset(e4, x, S.Reals) == FiniteSet(0)
assert solveset(e5, x, S.Reals) == Intersection(
S.Reals, FiniteSet(y*log(2*exp(z/y))))
assert solveset(e6, x, S.Reals) == FiniteSet(0)
assert solveset(e7, x, S.Reals) == FiniteSet(2)
assert solveset(e8, x, S.Reals) == FiniteSet(-2*log(2)/5 + 2*log(3)/5 + S(4)/5)
assert solveset(e9, x, S.Reals) == FiniteSet(2)
assert solveset_real(-9*exp(-2*x + 5) + 2**(x + 1), x) == FiniteSet(
-((-5 - 2*log(3) + log(2))/(log(2) + 2)))
assert solveset_real(4**(x/2) - 2**(x/3), x) == FiniteSet(0)
b = sqrt(6)*sqrt(log(2))/sqrt(log(5))
assert solveset_real(5**(x/2) - 2**(3/x), x) == FiniteSet(-b, b)
# coverage test
C1, C2 = symbols('C1 C2')
f = Function('f')
assert solveset_real(C1 + C2/x**2 - exp(-f(x)), f(x)) == Intersection(
S.Reals, FiniteSet(-log(C1 + C2/x**2)))
y = symbols('y', positive=True)
assert solveset_real(x**2 - y**2/exp(x), y) == Intersection(
S.Reals, FiniteSet(-sqrt(x**2*exp(x)), sqrt(x**2*exp(x))))
p = Symbol('p', positive=True)
assert solveset_real((1/p + 1)**(p + 1), p) == EmptySet()
@XFAIL
def test_exponential_complex():
from sympy.abc import x
from sympy import Dummy
n = Dummy('n')
assert solveset_complex(2**x + 4**x, x) == imageset(
Lambda(n, I*(2*n*pi + pi)/log(2)), S.Integers)
assert solveset_complex(x**z*y**z - 2, z) == FiniteSet(
log(2)/(log(x) + log(y)))
assert solveset_complex(4**(x/2) - 2**(x/3), x) == imageset(
Lambda(n, 3*n*I*pi/log(2)), S.Integers)
assert solveset(2**x + 32, x) == imageset(
Lambda(n, (I*(2*n*pi + pi) + 5*log(2))/log(2)), S.Integers)
eq = (2**exp(y**2/x) + 2)/(x**2 + 15)
a = sqrt(x)*sqrt(-log(log(2)) + log(log(2) + 2*n*I*pi))
assert solveset_complex(eq, y) == FiniteSet(-a, a)
union1 = imageset(Lambda(n, I*(2*n*pi - 2*pi/3)/log(2)), S.Integers)
union2 = imageset(Lambda(n, I*(2*n*pi + 2*pi/3)/log(2)), S.Integers)
assert solveset(2**x + 4**x + 8**x, x) == Union(union1, union2)
eq = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3)
res = solveset(eq, x)
num = 2*n*I*pi - 4*log(2) + 2*log(3)
den = -2*log(2) + log(3)
ans = imageset(Lambda(n, num/den), S.Integers)
assert res == ans
def test_expo_conditionset():
from sympy.abc import x, y
f1 = (exp(x) + 1)**x - 2
f2 = (x + 2)**y*x - 3
f3 = 2**x - exp(x) - 3
f4 = log(x) - exp(x)
f5 = 2**x + 3**x - 5**x
assert solveset(f1, x, S.Reals) == ConditionSet(
x, Eq((exp(x) + 1)**x - 2, 0), S.Reals)
assert solveset(f2, x, S.Reals) == ConditionSet(
x, Eq(x*(x + 2)**y - 3, 0), S.Reals)
assert solveset(f3, x, S.Reals) == ConditionSet(
x, Eq(2**x - exp(x) - 3, 0), S.Reals)
assert solveset(f4, x, S.Reals) == ConditionSet(
x, Eq(-exp(x) + log(x), 0), S.Reals)
assert solveset(f5, x, S.Reals) == ConditionSet(
x, Eq(2**x + 3**x - 5**x, 0), S.Reals)
def test_exponential_symbols():
x, y, z = symbols('x y z', positive=True)
from sympy import simplify
assert solveset(z**x - y, x, S.Reals) == Intersection(
S.Reals, FiniteSet(log(y)/log(z)))
w = symbols('w')
f1 = 2*x**w - 4*y**w
f2 = (x/y)**w - 2
ans1 = solveset(f1, w, S.Reals)
ans2 = solveset(f2, w, S.Reals)
assert ans1 == simplify(ans2)
assert solveset(x**x, x, S.Reals) == S.EmptySet
assert solveset(x**y - 1, y, S.Reals) == FiniteSet(0)
assert solveset(exp(x/y)*exp(-z/y) - 2, y, S.Reals) == FiniteSet(
(x - z)/log(2)) - FiniteSet(0)
a, b, x, y = symbols('a b x y')
assert solveset_real(a**x - b**x, x) == ConditionSet(
x, (a > 0) & (b > 0), FiniteSet(0))
assert solveset(a**x - b**x, x) == ConditionSet(
x, Ne(a, 0) & Ne(b, 0), FiniteSet(0))
@XFAIL
def test_issue_10864():
assert solveset(x**(y*z) - x, x, S.Reals) == FiniteSet(1)
@XFAIL
def test_solve_only_exp_2():
assert solveset_real(sqrt(exp(x)) + sqrt(exp(-x)) - 4, x) == \
FiniteSet(2*log(-sqrt(3) + 2), 2*log(sqrt(3) + 2))
def test_is_exponential():
x, y, z = symbols('x y z')
assert _is_exponential(y, x) is False
assert _is_exponential(3**x - 2, x) is True
assert _is_exponential(5**x - 7**(2 - x), x) is True
assert _is_exponential(sin(2**x) - 4*x, x) is False
assert _is_exponential(x**y - z, y) is True
assert _is_exponential(x**y - z, x) is False
assert _is_exponential(2**x + 4**x - 1, x) is True
assert _is_exponential(x**(y*z) - x, x) is False
assert _is_exponential(x**(2*x) - 3**x, x) is False
assert _is_exponential(x**y - y*z, y) is False
assert _is_exponential(x**y - x*z, y) is True
def test_solve_exponential():
assert _solve_exponential(3**(2*x) - 2**(x + 3), 0, x, S.Reals) == \
FiniteSet(-3*log(2)/(-2*log(3) + log(2)))
assert _solve_exponential(2**y + 4**y, 1, y, S.Reals) == \
FiniteSet(log(-S(1)/2 + sqrt(5)/2)/log(2))
assert _solve_exponential(2**y + 4**y, 0, y, S.Reals) == \
S.EmptySet
assert _solve_exponential(2**x + 3**x - 5**x, 0, x, S.Reals) == \
ConditionSet(x, Eq(2**x + 3**x - 5**x, 0), S.Reals)
# end of exponential tests
# logarithmic tests
def test_logarithmic():
assert solveset_real(log(x - 3) + log(x + 3), x) == FiniteSet(
-sqrt(10), sqrt(10))
assert solveset_real(log(x + 1) - log(2*x - 1), x) == FiniteSet(2)
assert solveset_real(log(x + 3) + log(1 + 3/x) - 3, x) == FiniteSet(
-3 + sqrt(-12 + exp(3))*exp(S(3)/2)/2 + exp(3)/2,
-sqrt(-12 + exp(3))*exp(S(3)/2)/2 - 3 + exp(3)/2)
eq = z - log(x) + log(y/(x*(-1 + y**2/x**2)))
assert solveset_real(eq, x) == \
Intersection(S.Reals, FiniteSet(-sqrt(y**2 - y*exp(z)),
sqrt(y**2 - y*exp(z)))) - \
Intersection(S.Reals, FiniteSet(-sqrt(y**2), sqrt(y**2)))
assert solveset_real(
log(3*x) - log(-x + 1) - log(4*x + 1), x) == FiniteSet(-S(1)/2, S(1)/2)
assert solveset(log(x**y) - y*log(x), x, S.Reals) == S.Reals
@XFAIL
def test_uselogcombine_2():
eq = log(exp(2*x) + 1) + log(-tanh(x) + 1) - log(2)
assert solveset_real(eq, x) == EmptySet()
eq = log(8*x) - log(sqrt(x) + 1) - 2
assert solveset_real(eq, x) == EmptySet()
def test_is_logarithmic():
assert _is_logarithmic(y, x) is False
assert _is_logarithmic(log(x), x) is True
assert _is_logarithmic(log(x) - 3, x) is True
assert _is_logarithmic(log(x)*log(y), x) is True
assert _is_logarithmic(log(x)**2, x) is False
assert _is_logarithmic(log(x - 3) + log(x + 3), x) is True
assert _is_logarithmic(log(x**y) - y*log(x), x) is True
assert _is_logarithmic(sin(log(x)), x) is False
assert _is_logarithmic(x + y, x) is False
assert _is_logarithmic(log(3*x) - log(1 - x) + 4, x) is True
assert _is_logarithmic(log(x) + log(y) + x, x) is False
assert _is_logarithmic(log(log(x - 3)) + log(x - 3), x) is True
assert _is_logarithmic(log(log(3) + x) + log(x), x) is True
assert _is_logarithmic(log(x)*(y + 3) + log(x), y) is False
def test_solve_logarithm():
y = Symbol('y')
assert _solve_logarithm(log(x**y) - y*log(x), 0, x, S.Reals) == S.Reals
y = Symbol('y', positive=True)
assert _solve_logarithm(log(x)*log(y), 0, x, S.Reals) == FiniteSet(1)
# end of logarithmic tests
def test_linear_coeffs():
from sympy.solvers.solveset import linear_coeffs
assert linear_coeffs(0, x) == [0, 0]
assert all(i is S.Zero for i in linear_coeffs(0, x))
assert linear_coeffs(x + 2*y + 3, x, y) == [1, 2, 3]
assert linear_coeffs(x + 2*y + 3, y, x) == [2, 1, 3]
assert linear_coeffs(x + 2*x**2 + 3, x, x**2) == [1, 2, 3]
raises(ValueError, lambda:
linear_coeffs(x + 2*x**2 + x**3, x, x**2))
raises(ValueError, lambda:
linear_coeffs(1/x*(x - 1) + 1/x, x))
|
3aa2259c314a82e127dd768899b6189bd10fe1cbd17eae0950dce7c3212206f0
|
from sympy import (
Abs, And, Derivative, Dummy, Eq, Float, Function, Gt, I, Integral,
LambertW, Lt, Matrix, Or, Poly, Q, Rational, S, Symbol, Ne,
Wild, acos, asin, atan, atanh, cos, cosh, diff, erf, erfinv, erfc,
erfcinv, exp, im, log, pi, re, sec, sin,
sinh, solve, solve_linear, sqrt, sstr, symbols, sympify, tan, tanh,
root, simplify, atan2, arg, Mul, SparseMatrix, ask, Tuple, nsolve, oo,
E, cbrt, denom, Add)
from sympy.core.compatibility import range
from sympy.core.function import nfloat
from sympy.solvers import solve_linear_system, solve_linear_system_LU, \
solve_undetermined_coeffs
from sympy.solvers.solvers import _invert, unrad, checksol, posify, _ispow, \
det_quick, det_perm, det_minor, _simple_dens, check_assumptions, denoms, \
failing_assumptions
from sympy.physics.units import cm
from sympy.polys.rootoftools import CRootOf
from sympy.utilities.pytest import slow, XFAIL, SKIP, raises, skip, ON_TRAVIS
from sympy.utilities.randtest import verify_numerically as tn
from sympy.abc import a, b, c, d, k, h, p, x, y, z, t, q, m
def NS(e, n=15, **options):
return sstr(sympify(e).evalf(n, **options), full_prec=True)
def test_swap_back():
f, g = map(Function, 'fg')
fx, gx = f(x), g(x)
assert solve([fx + y - 2, fx - gx - 5], fx, y, gx) == \
{fx: gx + 5, y: -gx - 3}
assert solve(fx + gx*x - 2, [fx, gx], dict=True)[0] == {fx: 2, gx: 0}
assert solve(fx + gx**2*x - y, [fx, gx], dict=True) == [{fx: y - gx**2*x}]
assert solve([f(1) - 2, x + 2], dict=True) == [{x: -2, f(1): 2}]
def guess_solve_strategy(eq, symbol):
try:
solve(eq, symbol)
return True
except (TypeError, NotImplementedError):
return False
def test_guess_poly():
# polynomial equations
assert guess_solve_strategy( S(4), x ) # == GS_POLY
assert guess_solve_strategy( x, x ) # == GS_POLY
assert guess_solve_strategy( x + a, x ) # == GS_POLY
assert guess_solve_strategy( 2*x, x ) # == GS_POLY
assert guess_solve_strategy( x + sqrt(2), x) # == GS_POLY
assert guess_solve_strategy( x + 2**Rational(1, 4), x) # == GS_POLY
assert guess_solve_strategy( x**2 + 1, x ) # == GS_POLY
assert guess_solve_strategy( x**2 - 1, x ) # == GS_POLY
assert guess_solve_strategy( x*y + y, x ) # == GS_POLY
assert guess_solve_strategy( x*exp(y) + y, x) # == GS_POLY
assert guess_solve_strategy(
(x - y**3)/(y**2*sqrt(1 - y**2)), x) # == GS_POLY
def test_guess_poly_cv():
# polynomial equations via a change of variable
assert guess_solve_strategy( sqrt(x) + 1, x ) # == GS_POLY_CV_1
assert guess_solve_strategy(
x**Rational(1, 3) + sqrt(x) + 1, x ) # == GS_POLY_CV_1
assert guess_solve_strategy( 4*x*(1 - sqrt(x)), x ) # == GS_POLY_CV_1
# polynomial equation multiplying both sides by x**n
assert guess_solve_strategy( x + 1/x + y, x ) # == GS_POLY_CV_2
def test_guess_rational_cv():
# rational functions
assert guess_solve_strategy( (x + 1)/(x**2 + 2), x) # == GS_RATIONAL
assert guess_solve_strategy(
(x - y**3)/(y**2*sqrt(1 - y**2)), y) # == GS_RATIONAL_CV_1
# rational functions via the change of variable y -> x**n
assert guess_solve_strategy( (sqrt(x) + 1)/(x**Rational(1, 3) + sqrt(x) + 1), x ) \
#== GS_RATIONAL_CV_1
def test_guess_transcendental():
#transcendental functions
assert guess_solve_strategy( exp(x) + 1, x ) # == GS_TRANSCENDENTAL
assert guess_solve_strategy( 2*cos(x) - y, x ) # == GS_TRANSCENDENTAL
assert guess_solve_strategy(
exp(x) + exp(-x) - y, x ) # == GS_TRANSCENDENTAL
assert guess_solve_strategy(3**x - 10, x) # == GS_TRANSCENDENTAL
assert guess_solve_strategy(-3**x + 10, x) # == GS_TRANSCENDENTAL
assert guess_solve_strategy(a*x**b - y, x) # == GS_TRANSCENDENTAL
def test_solve_args():
# equation container, issue 5113
ans = {x: -3, y: 1}
eqs = (x + 5*y - 2, -3*x + 6*y - 15)
assert all(solve(container(eqs), x, y) == ans for container in
(tuple, list, set, frozenset))
assert solve(Tuple(*eqs), x, y) == ans
# implicit symbol to solve for
assert set(solve(x**2 - 4)) == set([S(2), -S(2)])
assert solve([x + y - 3, x - y - 5]) == {x: 4, y: -1}
assert solve(x - exp(x), x, implicit=True) == [exp(x)]
# no symbol to solve for
assert solve(42) == solve(42, x) == []
assert solve([1, 2]) == []
# duplicate symbols removed
assert solve((x - 3, y + 2), x, y, x) == {x: 3, y: -2}
# unordered symbols
# only 1
assert solve(y - 3, set([y])) == [3]
# more than 1
assert solve(y - 3, set([x, y])) == [{y: 3}]
# multiple symbols: take the first linear solution+
# - return as tuple with values for all requested symbols
assert solve(x + y - 3, [x, y]) == [(3 - y, y)]
# - unless dict is True
assert solve(x + y - 3, [x, y], dict=True) == [{x: 3 - y}]
# - or no symbols are given
assert solve(x + y - 3) == [{x: 3 - y}]
# multiple symbols might represent an undetermined coefficients system
assert solve(a + b*x - 2, [a, b]) == {a: 2, b: 0}
args = (a + b)*x - b**2 + 2, a, b
assert solve(*args) == \
[(-sqrt(2), sqrt(2)), (sqrt(2), -sqrt(2))]
assert solve(*args, set=True) == \
([a, b], set([(-sqrt(2), sqrt(2)), (sqrt(2), -sqrt(2))]))
assert solve(*args, dict=True) == \
[{b: sqrt(2), a: -sqrt(2)}, {b: -sqrt(2), a: sqrt(2)}]
eq = a*x**2 + b*x + c - ((x - h)**2 + 4*p*k)/4/p
flags = dict(dict=True)
assert solve(eq, [h, p, k], exclude=[a, b, c], **flags) == \
[{k: c - b**2/(4*a), h: -b/(2*a), p: 1/(4*a)}]
flags.update(dict(simplify=False))
assert solve(eq, [h, p, k], exclude=[a, b, c], **flags) == \
[{k: (4*a*c - b**2)/(4*a), h: -b/(2*a), p: 1/(4*a)}]
# failing undetermined system
assert solve(a*x + b**2/(x + 4) - 3*x - 4/x, a, b, dict=True) == \
[{a: (-b**2*x + 3*x**3 + 12*x**2 + 4*x + 16)/(x**2*(x + 4))}]
# failed single equation
assert solve(1/(1/x - y + exp(y))) == []
raises(
NotImplementedError, lambda: solve(exp(x) + sin(x) + exp(y) + sin(y)))
# failed system
# -- when no symbols given, 1 fails
assert solve([y, exp(x) + x]) == [{x: -LambertW(1), y: 0}]
# both fail
assert solve(
(exp(x) - x, exp(y) - y)) == [{x: -LambertW(-1), y: -LambertW(-1)}]
# -- when symbols given
solve([y, exp(x) + x], x, y) == [(-LambertW(1), 0)]
# symbol is a number
assert solve(x**2 - pi, pi) == [x**2]
# no equations
assert solve([], [x]) == []
# overdetermined system
# - nonlinear
assert solve([(x + y)**2 - 4, x + y - 2]) == [{x: -y + 2}]
# - linear
assert solve((x + y - 2, 2*x + 2*y - 4)) == {x: -y + 2}
# When one or more args are Boolean
assert solve([True, Eq(x, 0)], [x], dict=True) == [{x: 0}]
assert solve([Eq(x, x), Eq(x, 0), Eq(x, x+1)], [x], dict=True) == []
assert not solve([Eq(x, x+1), x < 2], x)
assert solve([Eq(x, 0), x+1<2]) == Eq(x, 0)
assert solve([Eq(x, x), Eq(x, x+1)], x) == []
assert solve(True, x) == []
assert solve([x-1, False], [x], set=True) == ([], set())
def test_solve_polynomial1():
assert solve(3*x - 2, x) == [Rational(2, 3)]
assert solve(Eq(3*x, 2), x) == [Rational(2, 3)]
assert set(solve(x**2 - 1, x)) == set([-S(1), S(1)])
assert set(solve(Eq(x**2, 1), x)) == set([-S(1), S(1)])
assert solve(x - y**3, x) == [y**3]
rx = root(x, 3)
assert solve(x - y**3, y) == [
rx, -rx/2 - sqrt(3)*I*rx/2, -rx/2 + sqrt(3)*I*rx/2]
a11, a12, a21, a22, b1, b2 = symbols('a11,a12,a21,a22,b1,b2')
assert solve([a11*x + a12*y - b1, a21*x + a22*y - b2], x, y) == \
{
x: (a22*b1 - a12*b2)/(a11*a22 - a12*a21),
y: (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
}
solution = {y: S.Zero, x: S.Zero}
assert solve((x - y, x + y), x, y ) == solution
assert solve((x - y, x + y), (x, y)) == solution
assert solve((x - y, x + y), [x, y]) == solution
assert set(solve(x**3 - 15*x - 4, x)) == set([
-2 + 3**Rational(1, 2),
S(4),
-2 - 3**Rational(1, 2)
])
assert set(solve((x**2 - 1)**2 - a, x)) == \
set([sqrt(1 + sqrt(a)), -sqrt(1 + sqrt(a)),
sqrt(1 - sqrt(a)), -sqrt(1 - sqrt(a))])
def test_solve_polynomial2():
assert solve(4, x) == []
def test_solve_polynomial_cv_1a():
"""
Test for solving on equations that can be converted to a polynomial equation
using the change of variable y -> x**Rational(p, q)
"""
assert solve( sqrt(x) - 1, x) == [1]
assert solve( sqrt(x) - 2, x) == [4]
assert solve( x**Rational(1, 4) - 2, x) == [16]
assert solve( x**Rational(1, 3) - 3, x) == [27]
assert solve(sqrt(x) + x**Rational(1, 3) + x**Rational(1, 4), x) == [0]
def test_solve_polynomial_cv_1b():
assert set(solve(4*x*(1 - a*sqrt(x)), x)) == set([S(0), 1/a**2])
assert set(solve(x*(root(x, 3) - 3), x)) == set([S(0), S(27)])
def test_solve_polynomial_cv_2():
"""
Test for solving on equations that can be converted to a polynomial equation
multiplying both sides of the equation by x**m
"""
assert solve(x + 1/x - 1, x) in \
[[ Rational(1, 2) + I*sqrt(3)/2, Rational(1, 2) - I*sqrt(3)/2],
[ Rational(1, 2) - I*sqrt(3)/2, Rational(1, 2) + I*sqrt(3)/2]]
def test_quintics_1():
f = x**5 - 110*x**3 - 55*x**2 + 2310*x + 979
s = solve(f, check=False)
for root in s:
res = f.subs(x, root.n()).n()
assert tn(res, 0)
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = solve(f)
for root in s:
assert root.func == CRootOf
# if one uses solve to get the roots of a polynomial that has a CRootOf
# solution, make sure that the use of nfloat during the solve process
# doesn't fail. Note: if you want numerical solutions to a polynomial
# it is *much* faster to use nroots to get them than to solve the
# equation only to get RootOf solutions which are then numerically
# evaluated. So for eq = x**5 + 3*x + 7 do Poly(eq).nroots() rather
# than [i.n() for i in solve(eq)] to get the numerical roots of eq.
assert nfloat(solve(x**5 + 3*x**3 + 7)[0], exponent=False) == \
CRootOf(x**5 + 3*x**3 + 7, 0).n()
def test_highorder_poly():
# just testing that the uniq generator is unpacked
sol = solve(x**6 - 2*x + 2)
assert all(isinstance(i, CRootOf) for i in sol) and len(sol) == 6
def test_quintics_2():
f = x**5 + 15*x + 12
s = solve(f, check=False)
for root in s:
res = f.subs(x, root.n()).n()
assert tn(res, 0)
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = solve(f)
for root in s:
assert root.func == CRootOf
def test_solve_rational():
"""Test solve for rational functions"""
assert solve( ( x - y**3 )/( (y**2)*sqrt(1 - y**2) ), x) == [y**3]
def test_solve_nonlinear():
assert solve(x**2 - y**2, x, y, dict=True) == [{x: -y}, {x: y}]
assert solve(x**2 - y**2/exp(x), x, y, dict=True) == [{x: 2*LambertW(y/2)}]
assert solve(x**2 - y**2/exp(x), y, x, dict=True) == [{y: -x*sqrt(exp(x))},
{y: x*sqrt(exp(x))}]
def test_issue_8666():
x = symbols('x')
assert solve(Eq(x**2 - 1/(x**2 - 4), 4 - 1/(x**2 - 4)), x) == []
assert solve(Eq(x + 1/x, 1/x), x) == []
def test_issue_7228():
assert solve(4**(2*(x**2) + 2*x) - 8, x) == [-Rational(3, 2), S.Half]
def test_issue_7190():
assert solve(log(x-3) + log(x+3), x) == [sqrt(10)]
def test_linear_system():
x, y, z, t, n = symbols('x, y, z, t, n')
assert solve([x - 1, x - y, x - 2*y, y - 1], [x, y]) == []
assert solve([x - 1, x - y, x - 2*y, x - 1], [x, y]) == []
assert solve([x - 1, x - 1, x - y, x - 2*y], [x, y]) == []
assert solve([x + 5*y - 2, -3*x + 6*y - 15], x, y) == {x: -3, y: 1}
M = Matrix([[0, 0, n*(n + 1), (n + 1)**2, 0],
[n + 1, n + 1, -2*n - 1, -(n + 1), 0],
[-1, 0, 1, 0, 0]])
assert solve_linear_system(M, x, y, z, t) == \
{x: -t - t/n, z: -t - t/n, y: 0}
assert solve([x + y + z + t, -z - t], x, y, z, t) == {x: -y, z: -t}
def test_linear_system_function():
a = Function('a')
assert solve([a(0, 0) + a(0, 1) + a(1, 0) + a(1, 1), -a(1, 0) - a(1, 1)],
a(0, 0), a(0, 1), a(1, 0), a(1, 1)) == {a(1, 0): -a(1, 1), a(0, 0): -a(0, 1)}
def test_linear_systemLU():
n = Symbol('n')
M = Matrix([[1, 2, 0, 1], [1, 3, 2*n, 1], [4, -1, n**2, 1]])
assert solve_linear_system_LU(M, [x, y, z]) == {z: -3/(n**2 + 18*n),
x: 1 - 12*n/(n**2 + 18*n),
y: 6*n/(n**2 + 18*n)}
# Note: multiple solutions exist for some of these equations, so the tests
# should be expected to break if the implementation of the solver changes
# in such a way that a different branch is chosen
@slow
def test_solve_transcendental():
from sympy.abc import a, b
assert solve(exp(x) - 3, x) == [log(3)]
assert set(solve((a*x + b)*(exp(x) - 3), x)) == set([-b/a, log(3)])
assert solve(cos(x) - y, x) == [-acos(y) + 2*pi, acos(y)]
assert solve(2*cos(x) - y, x) == [-acos(y/2) + 2*pi, acos(y/2)]
assert solve(Eq(cos(x), sin(x)), x) == [-3*pi/4, pi/4]
assert set(solve(exp(x) + exp(-x) - y, x)) in [set([
log(y/2 - sqrt(y**2 - 4)/2),
log(y/2 + sqrt(y**2 - 4)/2),
]), set([
log(y - sqrt(y**2 - 4)) - log(2),
log(y + sqrt(y**2 - 4)) - log(2)]),
set([
log(y/2 - sqrt((y - 2)*(y + 2))/2),
log(y/2 + sqrt((y - 2)*(y + 2))/2)])]
assert solve(exp(x) - 3, x) == [log(3)]
assert solve(Eq(exp(x), 3), x) == [log(3)]
assert solve(log(x) - 3, x) == [exp(3)]
assert solve(sqrt(3*x) - 4, x) == [Rational(16, 3)]
assert solve(3**(x + 2), x) == []
assert solve(3**(2 - x), x) == []
assert solve(x + 2**x, x) == [-LambertW(log(2))/log(2)]
ans = solve(3*x + 5 + 2**(-5*x + 3), x)
assert len(ans) == 1 and ans[0].expand() == \
-Rational(5, 3) + LambertW(-10240*root(2, 3)*log(2)/3)/(5*log(2))
assert solve(5*x - 1 + 3*exp(2 - 7*x), x) == \
[Rational(1, 5) + LambertW(-21*exp(Rational(3, 5))/5)/7]
assert solve(2*x + 5 + log(3*x - 2), x) == \
[Rational(2, 3) + LambertW(2*exp(-Rational(19, 3))/3)/2]
assert solve(3*x + log(4*x), x) == [LambertW(Rational(3, 4))/3]
assert set(solve((2*x + 8)*(8 + exp(x)), x)) == set([S(-4), log(8) + pi*I])
eq = 2*exp(3*x + 4) - 3
ans = solve(eq, x) # this generated a failure in flatten
assert len(ans) == 3 and all(eq.subs(x, a).n(chop=True) == 0 for a in ans)
assert solve(2*log(3*x + 4) - 3, x) == [(exp(Rational(3, 2)) - 4)/3]
assert solve(exp(x) + 1, x) == [pi*I]
eq = 2*(3*x + 4)**5 - 6*7**(3*x + 9)
result = solve(eq, x)
ans = [(log(2401) + 5*LambertW(-log(7**(7*3**Rational(1, 5)/5))))/(3*log(7))/-1]
assert result == ans
# it works if expanded, too
assert solve(eq.expand(), x) == result
assert solve(z*cos(x) - y, x) == [-acos(y/z) + 2*pi, acos(y/z)]
assert solve(z*cos(2*x) - y, x) == [-acos(y/z)/2 + pi, acos(y/z)/2]
assert solve(z*cos(sin(x)) - y, x) == [
pi - asin(acos(y/z)), asin(acos(y/z) - 2*pi) + pi,
-asin(acos(y/z) - 2*pi), asin(acos(y/z))]
assert solve(z*cos(x), x) == [pi/2, 3*pi/2]
# issue 4508
assert solve(y - b*x/(a + x), x) in [[-a*y/(y - b)], [a*y/(b - y)]]
assert solve(y - b*exp(a/x), x) == [a/log(y/b)]
# issue 4507
assert solve(y - b/(1 + a*x), x) in [[(b - y)/(a*y)], [-((y - b)/(a*y))]]
# issue 4506
assert solve(y - a*x**b, x) == [(y/a)**(1/b)]
# issue 4505
assert solve(z**x - y, x) == [log(y)/log(z)]
# issue 4504
assert solve(2**x - 10, x) == [log(10)/log(2)]
# issue 6744
assert solve(x*y) == [{x: 0}, {y: 0}]
assert solve([x*y]) == [{x: 0}, {y: 0}]
assert solve(x**y - 1) == [{x: 1}, {y: 0}]
assert solve([x**y - 1]) == [{x: 1}, {y: 0}]
assert solve(x*y*(x**2 - y**2)) == [{x: 0}, {x: -y}, {x: y}, {y: 0}]
assert solve([x*y*(x**2 - y**2)]) == [{x: 0}, {x: -y}, {x: y}, {y: 0}]
# issue 4739
assert solve(exp(log(5)*x) - 2**x, x) == [0]
# issue 14791
assert solve(exp(log(5)*x) - exp(log(2)*x), x) == [0]
f = Function('f')
assert solve(y*f(log(5)*x) - y*f(log(2)*x), x) == [0]
assert solve(f(x) - f(0), x) == [0]
assert solve(f(x) - f(2 - x), x) == [1]
raises(NotImplementedError, lambda: solve(f(x, y) - f(1, 2), x))
raises(NotImplementedError, lambda: solve(f(x, y) - f(2 - x, 2), x))
raises(ValueError, lambda: solve(f(x, y) - f(1 - x), x))
raises(ValueError, lambda: solve(f(x, y) - f(1), x))
# misc
# make sure that the right variables is picked up in tsolve
# shouldn't generate a GeneratorsNeeded error in _tsolve when the NaN is generated
# for eq_down. Actual answers, as determined numerically are approx. +/- 0.83
raises(NotImplementedError, lambda:
solve(sinh(x)*sinh(sinh(x)) + cosh(x)*cosh(sinh(x)) - 3))
# watch out for recursive loop in tsolve
raises(NotImplementedError, lambda: solve((x + 2)**y*x - 3, x))
# issue 7245
assert solve(sin(sqrt(x))) == [0, pi**2]
# issue 7602
a, b = symbols('a, b', real=True, negative=False)
assert str(solve(Eq(a, 0.5 - cos(pi*b)/2), b)) == \
'[2.0 - 0.318309886183791*acos(1.0 - 2.0*a), 0.318309886183791*acos(1.0 - 2.0*a)]'
# issue 15325
assert solve(y**(1/x) - z, x) == [log(y)/log(z)]
def test_solve_for_functions_derivatives():
t = Symbol('t')
x = Function('x')(t)
y = Function('y')(t)
a11, a12, a21, a22, b1, b2 = symbols('a11,a12,a21,a22,b1,b2')
soln = solve([a11*x + a12*y - b1, a21*x + a22*y - b2], x, y)
assert soln == {
x: (a22*b1 - a12*b2)/(a11*a22 - a12*a21),
y: (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
}
assert solve(x - 1, x) == [1]
assert solve(3*x - 2, x) == [Rational(2, 3)]
soln = solve([a11*x.diff(t) + a12*y.diff(t) - b1, a21*x.diff(t) +
a22*y.diff(t) - b2], x.diff(t), y.diff(t))
assert soln == { y.diff(t): (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
x.diff(t): (a22*b1 - a12*b2)/(a11*a22 - a12*a21) }
assert solve(x.diff(t) - 1, x.diff(t)) == [1]
assert solve(3*x.diff(t) - 2, x.diff(t)) == [Rational(2, 3)]
eqns = set((3*x - 1, 2*y - 4))
assert solve(eqns, set((x, y))) == { x: Rational(1, 3), y: 2 }
x = Symbol('x')
f = Function('f')
F = x**2 + f(x)**2 - 4*x - 1
assert solve(F.diff(x), diff(f(x), x)) == [(-x + 2)/f(x)]
# Mixed cased with a Symbol and a Function
x = Symbol('x')
y = Function('y')(t)
soln = solve([a11*x + a12*y.diff(t) - b1, a21*x +
a22*y.diff(t) - b2], x, y.diff(t))
assert soln == { y.diff(t): (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
x: (a22*b1 - a12*b2)/(a11*a22 - a12*a21) }
def test_issue_3725():
f = Function('f')
F = x**2 + f(x)**2 - 4*x - 1
e = F.diff(x)
assert solve(e, f(x).diff(x)) in [[(2 - x)/f(x)], [-((x - 2)/f(x))]]
def test_issue_3870():
a, b, c, d = symbols('a b c d')
A = Matrix(2, 2, [a, b, c, d])
B = Matrix(2, 2, [0, 2, -3, 0])
C = Matrix(2, 2, [1, 2, 3, 4])
assert solve(A*B - C, [a, b, c, d]) == {a: 1, b: -S(1)/3, c: 2, d: -1}
assert solve([A*B - C], [a, b, c, d]) == {a: 1, b: -S(1)/3, c: 2, d: -1}
assert solve(Eq(A*B, C), [a, b, c, d]) == {a: 1, b: -S(1)/3, c: 2, d: -1}
assert solve([A*B - B*A], [a, b, c, d]) == {a: d, b: -S(2)/3*c}
assert solve([A*C - C*A], [a, b, c, d]) == {a: d - c, b: S(2)/3*c}
assert solve([A*B - B*A, A*C - C*A], [a, b, c, d]) == {a: d, b: 0, c: 0}
assert solve([Eq(A*B, B*A)], [a, b, c, d]) == {a: d, b: -S(2)/3*c}
assert solve([Eq(A*C, C*A)], [a, b, c, d]) == {a: d - c, b: S(2)/3*c}
assert solve([Eq(A*B, B*A), Eq(A*C, C*A)], [a, b, c, d]) == {a: d, b: 0, c: 0}
def test_solve_linear():
w = Wild('w')
assert solve_linear(x, x) == (0, 1)
assert solve_linear(x, exclude=[x]) == (0, 1)
assert solve_linear(x, symbols=[w]) == (0, 1)
assert solve_linear(x, y - 2*x) in [(x, y/3), (y, 3*x)]
assert solve_linear(x, y - 2*x, exclude=[x]) == (y, 3*x)
assert solve_linear(3*x - y, 0) in [(x, y/3), (y, 3*x)]
assert solve_linear(3*x - y, 0, [x]) == (x, y/3)
assert solve_linear(3*x - y, 0, [y]) == (y, 3*x)
assert solve_linear(x**2/y, 1) == (y, x**2)
assert solve_linear(w, x) in [(w, x), (x, w)]
assert solve_linear(cos(x)**2 + sin(x)**2 + 2 + y) == \
(y, -2 - cos(x)**2 - sin(x)**2)
assert solve_linear(cos(x)**2 + sin(x)**2 + 2 + y, symbols=[x]) == (0, 1)
assert solve_linear(Eq(x, 3)) == (x, 3)
assert solve_linear(1/(1/x - 2)) == (0, 0)
assert solve_linear((x + 1)*exp(-x), symbols=[x]) == (x, -1)
assert solve_linear((x + 1)*exp(x), symbols=[x]) == ((x + 1)*exp(x), 1)
assert solve_linear(x*exp(-x**2), symbols=[x]) == (x, 0)
assert solve_linear(0**x - 1) == (0**x - 1, 1)
assert solve_linear(1 + 1/(x - 1)) == (x, 0)
eq = y*cos(x)**2 + y*sin(x)**2 - y # = y*(1 - 1) = 0
assert solve_linear(eq) == (0, 1)
eq = cos(x)**2 + sin(x)**2 # = 1
assert solve_linear(eq) == (0, 1)
raises(ValueError, lambda: solve_linear(Eq(x, 3), 3))
def test_solve_undetermined_coeffs():
assert solve_undetermined_coeffs(a*x**2 + b*x**2 + b*x + 2*c*x + c + 1, [a, b, c], x) == \
{a: -2, b: 2, c: -1}
# Test that rational functions work
assert solve_undetermined_coeffs(a/x + b/(x + 1) - (2*x + 1)/(x**2 + x), [a, b], x) == \
{a: 1, b: 1}
# Test cancellation in rational functions
assert solve_undetermined_coeffs(((c + 1)*a*x**2 + (c + 1)*b*x**2 +
(c + 1)*b*x + (c + 1)*2*c*x + (c + 1)**2)/(c + 1), [a, b, c], x) == \
{a: -2, b: 2, c: -1}
def test_solve_inequalities():
x = Symbol('x')
sol = And(S(0) < x, x < oo)
assert solve(x + 1 > 1) == sol
assert solve([x + 1 > 1]) == sol
assert solve([x + 1 > 1], x) == sol
assert solve([x + 1 > 1], [x]) == sol
system = [Lt(x**2 - 2, 0), Gt(x**2 - 1, 0)]
assert solve(system) == \
And(Or(And(Lt(-sqrt(2), x), Lt(x, -1)),
And(Lt(1, x), Lt(x, sqrt(2)))), Eq(0, 0))
x = Symbol('x', real=True)
system = [Lt(x**2 - 2, 0), Gt(x**2 - 1, 0)]
assert solve(system) == \
Or(And(Lt(-sqrt(2), x), Lt(x, -1)), And(Lt(1, x), Lt(x, sqrt(2))))
# issues 6627, 3448
assert solve((x - 3)/(x - 2) < 0, x) == And(Lt(2, x), Lt(x, 3))
assert solve(x/(x + 1) > 1, x) == And(Lt(-oo, x), Lt(x, -1))
assert solve(sin(x) > S.Half) == And(pi/6 < x, x < 5*pi/6)
assert solve(Eq(False, x < 1)) == (S(1) <= x) & (x < oo)
assert solve(Eq(True, x < 1)) == (-oo < x) & (x < 1)
assert solve(Eq(x < 1, False)) == (S(1) <= x) & (x < oo)
assert solve(Eq(x < 1, True)) == (-oo < x) & (x < 1)
assert solve(Eq(False, x)) == False
assert solve(Eq(True, x)) == True
assert solve(Eq(False, ~x)) == True
assert solve(Eq(True, ~x)) == False
assert solve(Ne(True, x)) == False
def test_issue_4793():
assert solve(1/x) == []
assert solve(x*(1 - 5/x)) == [5]
assert solve(x + sqrt(x) - 2) == [1]
assert solve(-(1 + x)/(2 + x)**2 + 1/(2 + x)) == []
assert solve(-x**2 - 2*x + (x + 1)**2 - 1) == []
assert solve((x/(x + 1) + 3)**(-2)) == []
assert solve(x/sqrt(x**2 + 1), x) == [0]
assert solve(exp(x) - y, x) == [log(y)]
assert solve(exp(x)) == []
assert solve(x**2 + x + sin(y)**2 + cos(y)**2 - 1, x) in [[0, -1], [-1, 0]]
eq = 4*3**(5*x + 2) - 7
ans = solve(eq, x)
assert len(ans) == 5 and all(eq.subs(x, a).n(chop=True) == 0 for a in ans)
assert solve(log(x**2) - y**2/exp(x), x, y, set=True) == (
[x, y],
{(x, sqrt(exp(x) * log(x ** 2))), (x, -sqrt(exp(x) * log(x ** 2)))})
assert solve(x**2*z**2 - z**2*y**2) == [{x: -y}, {x: y}, {z: 0}]
assert solve((x - 1)/(1 + 1/(x - 1))) == []
assert solve(x**(y*z) - x, x) == [1]
raises(NotImplementedError, lambda: solve(log(x) - exp(x), x))
raises(NotImplementedError, lambda: solve(2**x - exp(x) - 3))
def test_PR1964():
# issue 5171
assert solve(sqrt(x)) == solve(sqrt(x**3)) == [0]
assert solve(sqrt(x - 1)) == [1]
# issue 4462
a = Symbol('a')
assert solve(-3*a/sqrt(x), x) == []
# issue 4486
assert solve(2*x/(x + 2) - 1, x) == [2]
# issue 4496
assert set(solve((x**2/(7 - x)).diff(x))) == set([S(0), S(14)])
# issue 4695
f = Function('f')
assert solve((3 - 5*x/f(x))*f(x), f(x)) == [5*x/3]
# issue 4497
assert solve(1/root(5 + x, 5) - 9, x) == [-295244/S(59049)]
assert solve(sqrt(x) + sqrt(sqrt(x)) - 4) == [(-S.Half + sqrt(17)/2)**4]
assert set(solve(Poly(sqrt(exp(x)) + sqrt(exp(-x)) - 4))) in \
[
set([log((-sqrt(3) + 2)**2), log((sqrt(3) + 2)**2)]),
set([2*log(-sqrt(3) + 2), 2*log(sqrt(3) + 2)]),
set([log(-4*sqrt(3) + 7), log(4*sqrt(3) + 7)]),
]
assert set(solve(Poly(exp(x) + exp(-x) - 4))) == \
set([log(-sqrt(3) + 2), log(sqrt(3) + 2)])
assert set(solve(x**y + x**(2*y) - 1, x)) == \
set([(-S.Half + sqrt(5)/2)**(1/y), (-S.Half - sqrt(5)/2)**(1/y)])
assert solve(exp(x/y)*exp(-z/y) - 2, y) == [(x - z)/log(2)]
assert solve(
x**z*y**z - 2, z) in [[log(2)/(log(x) + log(y))], [log(2)/(log(x*y))]]
# if you do inversion too soon then multiple roots (as for the following)
# will be missed, e.g. if exp(3*x) = exp(3) -> 3*x = 3
E = S.Exp1
assert solve(exp(3*x) - exp(3), x) in [
[1, log(E*(-S.Half - sqrt(3)*I/2)), log(E*(-S.Half + sqrt(3)*I/2))],
[1, log(-E/2 - sqrt(3)*E*I/2), log(-E/2 + sqrt(3)*E*I/2)],
]
# coverage test
p = Symbol('p', positive=True)
assert solve((1/p + 1)**(p + 1)) == []
def test_issue_5197():
x = Symbol('x', real=True)
assert solve(x**2 + 1, x) == []
n = Symbol('n', integer=True, positive=True)
assert solve((n - 1)*(n + 2)*(2*n - 1), n) == [1]
x = Symbol('x', positive=True)
y = Symbol('y')
assert solve([x + 5*y - 2, -3*x + 6*y - 15], x, y) == []
# not {x: -3, y: 1} b/c x is positive
# The solution following should not contain (-sqrt(2), sqrt(2))
assert solve((x + y)*n - y**2 + 2, x, y) == [(sqrt(2), -sqrt(2))]
y = Symbol('y', positive=True)
# The solution following should not contain {y: -x*exp(x/2)}
assert solve(x**2 - y**2/exp(x), y, x, dict=True) == [{y: x*exp(x/2)}]
assert solve(x**2 - y**2/exp(x), x, y, dict=True) == [{x: 2*LambertW(y/2)}]
x, y, z = symbols('x y z', positive=True)
assert solve(z**2*x**2 - z**2*y**2/exp(x), y, x, z, dict=True) == [{y: x*exp(x/2)}]
def test_checking():
assert set(
solve(x*(x - y/x), x, check=False)) == set([sqrt(y), S(0), -sqrt(y)])
assert set(solve(x*(x - y/x), x, check=True)) == set([sqrt(y), -sqrt(y)])
# {x: 0, y: 4} sets denominator to 0 in the following so system should return None
assert solve((1/(1/x + 2), 1/(y - 3) - 1)) == []
# 0 sets denominator of 1/x to zero so None is returned
assert solve(1/(1/x + 2)) == []
def test_issue_4671_4463_4467():
assert solve((sqrt(x**2 - 1) - 2)) in ([sqrt(5), -sqrt(5)],
[-sqrt(5), sqrt(5)])
assert solve((2**exp(y**2/x) + 2)/(x**2 + 15), y) == [
-sqrt(x*log(1 + I*pi/log(2))), sqrt(x*log(1 + I*pi/log(2)))]
C1, C2 = symbols('C1 C2')
f = Function('f')
assert solve(C1 + C2/x**2 - exp(-f(x)), f(x)) == [log(x**2/(C1*x**2 + C2))]
a = Symbol('a')
E = S.Exp1
assert solve(1 - log(a + 4*x**2), x) in (
[-sqrt(-a + E)/2, sqrt(-a + E)/2],
[sqrt(-a + E)/2, -sqrt(-a + E)/2]
)
assert solve(log(a**(-3) - x**2)/a, x) in (
[-sqrt(-1 + a**(-3)), sqrt(-1 + a**(-3))],
[sqrt(-1 + a**(-3)), -sqrt(-1 + a**(-3))],)
assert solve(1 - log(a + 4*x**2), x) in (
[-sqrt(-a + E)/2, sqrt(-a + E)/2],
[sqrt(-a + E)/2, -sqrt(-a + E)/2],)
assert set(solve((
a**2 + 1) * (sin(a*x) + cos(a*x)), x)) == set([-pi/(4*a), 3*pi/(4*a)])
assert solve(3 - (sinh(a*x) + cosh(a*x)), x) == [log(3)/a]
assert set(solve(3 - (sinh(a*x) + cosh(a*x)**2), x)) == \
set([log(-2 + sqrt(5))/a, log(-sqrt(2) + 1)/a,
log(-sqrt(5) - 2)/a, log(1 + sqrt(2))/a])
assert solve(atan(x) - 1) == [tan(1)]
def test_issue_5132():
r, t = symbols('r,t')
assert set(solve([r - x**2 - y**2, tan(t) - y/x], [x, y])) == \
set([(
-sqrt(r*cos(t)**2), -1*sqrt(r*cos(t)**2)*tan(t)),
(sqrt(r*cos(t)**2), sqrt(r*cos(t)**2)*tan(t))])
assert solve([exp(x) - sin(y), 1/y - 3], [x, y]) == \
[(log(sin(S(1)/3)), S(1)/3)]
assert solve([exp(x) - sin(y), 1/exp(y) - 3], [x, y]) == \
[(log(-sin(log(3))), -log(3))]
assert set(solve([exp(x) - sin(y), y**2 - 4], [x, y])) == \
set([(log(-sin(2)), -S(2)), (log(sin(2)), S(2))])
eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3]
assert solve(eqs, set=True) == \
([x, y], set([
(log(-sqrt(-z**2 - sin(log(3)))), -log(3)),
(log(-z**2 - sin(log(3)))/2, -log(3))]))
assert solve(eqs, x, z, set=True) == (
[x, z],
{(log(-z**2 + sin(y))/2, z), (log(-sqrt(-z**2 + sin(y))), z)})
assert set(solve(eqs, x, y)) == \
set([
(log(-sqrt(-z**2 - sin(log(3)))), -log(3)),
(log(-z**2 - sin(log(3)))/2, -log(3))])
assert set(solve(eqs, y, z)) == \
set([
(-log(3), -sqrt(-exp(2*x) - sin(log(3)))),
(-log(3), sqrt(-exp(2*x) - sin(log(3))))])
eqs = [exp(x)**2 - sin(y) + z, 1/exp(y) - 3]
assert solve(eqs, set=True) == ([x, y], set(
[
(log(-sqrt(-z - sin(log(3)))), -log(3)),
(log(-z - sin(log(3)))/2, -log(3))]))
assert solve(eqs, x, z, set=True) == (
[x, z],
{(log(-sqrt(-z + sin(y))), z), (log(-z + sin(y))/2, z)})
assert set(solve(eqs, x, y)) == set(
[
(log(-sqrt(-z - sin(log(3)))), -log(3)),
(log(-z - sin(log(3)))/2, -log(3))])
assert solve(eqs, z, y) == \
[(-exp(2*x) - sin(log(3)), -log(3))]
assert solve((sqrt(x**2 + y**2) - sqrt(10), x + y - 4), set=True) == (
[x, y], set([(S(1), S(3)), (S(3), S(1))]))
assert set(solve((sqrt(x**2 + y**2) - sqrt(10), x + y - 4), x, y)) == \
set([(S(1), S(3)), (S(3), S(1))])
def test_issue_5335():
lam, a0, conc = symbols('lam a0 conc')
a = 0.005
b = 0.743436700916726
eqs = [lam + 2*y - a0*(1 - x/2)*x - a*x/2*x,
a0*(1 - x/2)*x - 1*y - b*y,
x + y - conc]
sym = [x, y, a0]
# there are 4 solutions obtained manually but only two are valid
assert len(solve(eqs, sym, manual=True, minimal=True)) == 2
assert len(solve(eqs, sym)) == 2 # cf below with rational=False
@SKIP("Hangs")
def _test_issue_5335_float():
# gives ZeroDivisionError: polynomial division
lam, a0, conc = symbols('lam a0 conc')
a = 0.005
b = 0.743436700916726
eqs = [lam + 2*y - a0*(1 - x/2)*x - a*x/2*x,
a0*(1 - x/2)*x - 1*y - b*y,
x + y - conc]
sym = [x, y, a0]
assert len(solve(eqs, sym, rational=False)) == 2
def test_issue_5767():
assert set(solve([x**2 + y + 4], [x])) == \
set([(-sqrt(-y - 4),), (sqrt(-y - 4),)])
def test_polysys():
assert set(solve([x**2 + 2/y - 2, x + y - 3], [x, y])) == \
set([(S(1), S(2)), (1 + sqrt(5), 2 - sqrt(5)),
(1 - sqrt(5), 2 + sqrt(5))])
assert solve([x**2 + y - 2, x**2 + y]) == []
# the ordering should be whatever the user requested
assert solve([x**2 + y - 3, x - y - 4], (x, y)) != solve([x**2 +
y - 3, x - y - 4], (y, x))
@slow
def test_unrad1():
raises(NotImplementedError, lambda:
unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x)) + 3))
raises(NotImplementedError, lambda:
unrad(sqrt(x) + (x + 1)**Rational(1, 3) + 2*sqrt(y)))
s = symbols('s', cls=Dummy)
# checkers to deal with possibility of answer coming
# back with a sign change (cf issue 5203)
def check(rv, ans):
assert bool(rv[1]) == bool(ans[1])
if ans[1]:
return s_check(rv, ans)
e = rv[0].expand()
a = ans[0].expand()
return e in [a, -a] and rv[1] == ans[1]
def s_check(rv, ans):
# get the dummy
rv = list(rv)
d = rv[0].atoms(Dummy)
reps = list(zip(d, [s]*len(d)))
# replace s with this dummy
rv = (rv[0].subs(reps).expand(), [rv[1][0].subs(reps), rv[1][1].subs(reps)])
ans = (ans[0].subs(reps).expand(), [ans[1][0].subs(reps), ans[1][1].subs(reps)])
return str(rv[0]) in [str(ans[0]), str(-ans[0])] and \
str(rv[1]) == str(ans[1])
assert check(unrad(sqrt(x)),
(x, []))
assert check(unrad(sqrt(x) + 1),
(x - 1, []))
assert check(unrad(sqrt(x) + root(x, 3) + 2),
(s**3 + s**2 + 2, [s, s**6 - x]))
assert check(unrad(sqrt(x)*root(x, 3) + 2),
(x**5 - 64, []))
assert check(unrad(sqrt(x) + (x + 1)**Rational(1, 3)),
(x**3 - (x + 1)**2, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(2*x)),
(-2*sqrt(2)*x - 2*x + 1, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + 2),
(16*x - 9, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - x)),
(5*x**2 - 4*x, []))
assert check(unrad(a*sqrt(x) + b*sqrt(x) + c*sqrt(y) + d*sqrt(y)),
((a*sqrt(x) + b*sqrt(x))**2 - (c*sqrt(y) + d*sqrt(y))**2, []))
assert check(unrad(sqrt(x) + sqrt(1 - x)),
(2*x - 1, []))
assert check(unrad(sqrt(x) + sqrt(1 - x) - 3),
(x**2 - x + 16, []))
assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x)),
(5*x**2 - 2*x + 1, []))
assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - 3) in [
(25*x**4 + 376*x**3 + 1256*x**2 - 2272*x + 784, []),
(25*x**8 - 476*x**6 + 2534*x**4 - 1468*x**2 + 169, [])]
assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - sqrt(1 - 2*x)) == \
(41*x**4 + 40*x**3 + 232*x**2 - 160*x + 16, []) # orig root at 0.487
assert check(unrad(sqrt(x) + sqrt(x + 1)), (S(1), []))
eq = sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x))
assert check(unrad(eq),
(16*x**2 - 9*x, []))
assert set(solve(eq, check=False)) == set([S(0), S(9)/16])
assert solve(eq) == []
# but this one really does have those solutions
assert set(solve(sqrt(x) - sqrt(x + 1) + sqrt(1 - sqrt(x)))) == \
set([S.Zero, S(9)/16])
assert check(unrad(sqrt(x) + root(x + 1, 3) + 2*sqrt(y), y),
(S('2*sqrt(x)*(x + 1)**(1/3) + x - 4*y + (x + 1)**(2/3)'), []))
assert check(unrad(sqrt(x/(1 - x)) + (x + 1)**Rational(1, 3)),
(x**5 - x**4 - x**3 + 2*x**2 + x - 1, []))
assert check(unrad(sqrt(x/(1 - x)) + 2*sqrt(y), y),
(4*x*y + x - 4*y, []))
assert check(unrad(sqrt(x)*sqrt(1 - x) + 2, x),
(x**2 - x + 4, []))
# http://tutorial.math.lamar.edu/
# Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a
assert solve(Eq(x, sqrt(x + 6))) == [3]
assert solve(Eq(x + sqrt(x - 4), 4)) == [4]
assert solve(Eq(1, x + sqrt(2*x - 3))) == []
assert set(solve(Eq(sqrt(5*x + 6) - 2, x))) == set([-S(1), S(2)])
assert set(solve(Eq(sqrt(2*x - 1) - sqrt(x - 4), 2))) == set([S(5), S(13)])
assert solve(Eq(sqrt(x + 7) + 2, sqrt(3 - x))) == [-6]
# http://www.purplemath.com/modules/solverad.htm
assert solve((2*x - 5)**Rational(1, 3) - 3) == [16]
assert set(solve(x + 1 - root(x**4 + 4*x**3 - x, 4))) == \
set([-S(1)/2, -S(1)/3])
assert set(solve(sqrt(2*x**2 - 7) - (3 - x))) == set([-S(8), S(2)])
assert solve(sqrt(2*x + 9) - sqrt(x + 1) - sqrt(x + 4)) == [0]
assert solve(sqrt(x + 4) + sqrt(2*x - 1) - 3*sqrt(x - 1)) == [5]
assert solve(sqrt(x)*sqrt(x - 7) - 12) == [16]
assert solve(sqrt(x - 3) + sqrt(x) - 3) == [4]
assert solve(sqrt(9*x**2 + 4) - (3*x + 2)) == [0]
assert solve(sqrt(x) - 2 - 5) == [49]
assert solve(sqrt(x - 3) - sqrt(x) - 3) == []
assert solve(sqrt(x - 1) - x + 7) == [10]
assert solve(sqrt(x - 2) - 5) == [27]
assert solve(sqrt(17*x - sqrt(x**2 - 5)) - 7) == [3]
assert solve(sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))) == []
# don't posify the expression in unrad and do use _mexpand
z = sqrt(2*x + 1)/sqrt(x) - sqrt(2 + 1/x)
p = posify(z)[0]
assert solve(p) == []
assert solve(z) == []
assert solve(z + 6*I) == [-S(1)/11]
assert solve(p + 6*I) == []
# issue 8622
assert unrad((root(x + 1, 5) - root(x, 3))) == (
x**5 - x**3 - 3*x**2 - 3*x - 1, [])
# issue #8679
assert check(unrad(x + root(x, 3) + root(x, 3)**2 + sqrt(y), x),
(s**3 + s**2 + s + sqrt(y), [s, s**3 - x]))
# for coverage
assert check(unrad(sqrt(x) + root(x, 3) + y),
(s**3 + s**2 + y, [s, s**6 - x]))
assert solve(sqrt(x) + root(x, 3) - 2) == [1]
raises(NotImplementedError, lambda:
solve(sqrt(x) + root(x, 3) + root(x + 1, 5) - 2))
# fails through a different code path
raises(NotImplementedError, lambda: solve(-sqrt(2) + cosh(x)/x))
# unrad some
assert solve(sqrt(x + root(x, 3))+root(x - y, 5), y) == [
x + (x**(S(1)/3) + x)**(S(5)/2)]
assert check(unrad(sqrt(x) - root(x + 1, 3)*sqrt(x + 2) + 2),
(s**10 + 8*s**8 + 24*s**6 - 12*s**5 - 22*s**4 - 160*s**3 - 212*s**2 -
192*s - 56, [s, s**2 - x]))
e = root(x + 1, 3) + root(x, 3)
assert unrad(e) == (2*x + 1, [])
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
assert check(unrad(eq),
(15625*x**4 + 173000*x**3 + 355600*x**2 - 817920*x + 331776, []))
assert check(unrad(root(x, 4) + root(x, 4)**3 - 1),
(s**3 + s - 1, [s, s**4 - x]))
assert check(unrad(root(x, 2) + root(x, 2)**3 - 1),
(x**3 + 2*x**2 + x - 1, []))
assert unrad(x**0.5) is None
assert check(unrad(t + root(x + y, 5) + root(x + y, 5)**3),
(s**3 + s + t, [s, s**5 - x - y]))
assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, y),
(s**3 + s + x, [s, s**5 - x - y]))
assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, x),
(s**5 + s**3 + s - y, [s, s**5 - x - y]))
assert check(unrad(root(x - 1, 3) + root(x + 1, 5) + root(2, 5)),
(s**5 + 5*2**(S(1)/5)*s**4 + s**3 + 10*2**(S(2)/5)*s**3 +
10*2**(S(3)/5)*s**2 + 5*2**(S(4)/5)*s + 4, [s, s**3 - x + 1]))
raises(NotImplementedError, lambda:
unrad((root(x, 2) + root(x, 3) + root(x, 4)).subs(x, x**5 - x + 1)))
# the simplify flag should be reset to False for unrad results;
# if it's not then this next test will take a long time
assert solve(root(x, 3) + root(x, 5) - 2) == [1]
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
assert check(unrad(eq),
((5*x - 4)*(3125*x**3 + 37100*x**2 + 100800*x - 82944), []))
ans = S('''
[4/5, -1484/375 + 172564/(140625*(114*sqrt(12657)/78125 +
12459439/52734375)**(1/3)) +
4*(114*sqrt(12657)/78125 + 12459439/52734375)**(1/3)]''')
assert solve(eq) == ans
# duplicate radical handling
assert check(unrad(sqrt(x + root(x + 1, 3)) - root(x + 1, 3) - 2),
(s**3 - s**2 - 3*s - 5, [s, s**3 - x - 1]))
# cov post-processing
e = root(x**2 + 1, 3) - root(x**2 - 1, 5) - 2
assert check(unrad(e),
(s**5 - 10*s**4 + 39*s**3 - 80*s**2 + 80*s - 30,
[s, s**3 - x**2 - 1]))
e = sqrt(x + root(x + 1, 2)) - root(x + 1, 3) - 2
assert check(unrad(e),
(s**6 - 2*s**5 - 7*s**4 - 3*s**3 + 26*s**2 + 40*s + 25,
[s, s**3 - x - 1]))
assert check(unrad(e, _reverse=True),
(s**6 - 14*s**5 + 73*s**4 - 187*s**3 + 276*s**2 - 228*s + 89,
[s, s**2 - x - sqrt(x + 1)]))
# this one needs r0, r1 reversal to work
assert check(unrad(sqrt(x + sqrt(root(x, 3) - 1)) - root(x, 6) - 2),
(s**12 - 2*s**8 - 8*s**7 - 8*s**6 + s**4 + 8*s**3 + 23*s**2 +
32*s + 17, [s, s**6 - x]))
# is this needed?
#assert unrad(root(cosh(x), 3)/x*root(x + 1, 5) - 1) == (
# x**15 - x**3*cosh(x)**5 - 3*x**2*cosh(x)**5 - 3*x*cosh(x)**5 - cosh(x)**5, [])
raises(NotImplementedError, lambda:
unrad(sqrt(cosh(x)/x) + root(x + 1,3)*sqrt(x) - 1))
assert unrad(S('(x+y)**(2*y/3) + (x+y)**(1/3) + 1')) is None
assert check(unrad(S('(x+y)**(2*y/3) + (x+y)**(1/3) + 1'), x),
(s**(2*y) + s + 1, [s, s**3 - x - y]))
# This tests two things: that if full unrad is attempted and fails
# the solution should still be found; also it tests that the use of
# composite
assert len(solve(sqrt(y)*x + x**3 - 1, x)) == 3
assert len(solve(-512*y**3 + 1344*(x + 2)**(S(1)/3)*y**2 -
1176*(x + 2)**(S(2)/3)*y - 169*x + 686, y, _unrad=False)) == 3
# watch out for when the cov doesn't involve the symbol of interest
eq = S('-x + (7*y/8 - (27*x/2 + 27*sqrt(x**2)/2)**(1/3)/3)**3 - 1')
assert solve(eq, y) == [
4*2**(S(2)/3)*(27*x + 27*sqrt(x**2))**(S(1)/3)/21 - (-S(1)/2 -
sqrt(3)*I/2)*(-6912*x/343 + sqrt((-13824*x/343 - S(13824)/343)**2)/2 -
S(6912)/343)**(S(1)/3)/3, 4*2**(S(2)/3)*(27*x + 27*sqrt(x**2))**(S(1)/3)/21 -
(-S(1)/2 + sqrt(3)*I/2)*(-6912*x/343 + sqrt((-13824*x/343 -
S(13824)/343)**2)/2 - S(6912)/343)**(S(1)/3)/3, 4*2**(S(2)/3)*(27*x +
27*sqrt(x**2))**(S(1)/3)/21 - (-6912*x/343 + sqrt((-13824*x/343 -
S(13824)/343)**2)/2 - S(6912)/343)**(S(1)/3)/3]
eq = root(x + 1, 3) - (root(x, 3) + root(x, 5))
assert check(unrad(eq),
(3*s**13 + 3*s**11 + s**9 - 1, [s, s**15 - x]))
assert check(unrad(eq - 2),
(3*s**13 + 3*s**11 + 6*s**10 + s**9 + 12*s**8 + 6*s**6 + 12*s**5 +
12*s**3 + 7, [s, s**15 - x]))
assert check(unrad(root(x, 3) - root(x + 1, 4)/2 + root(x + 2, 3)),
(4096*s**13 + 960*s**12 + 48*s**11 - s**10 - 1728*s**4,
[s, s**4 - x - 1])) # orig expr has two real roots: -1, -.389
assert check(unrad(root(x, 3) + root(x + 1, 4) - root(x + 2, 3)/2),
(343*s**13 + 2904*s**12 + 1344*s**11 + 512*s**10 - 1323*s**9 -
3024*s**8 - 1728*s**7 + 1701*s**5 + 216*s**4 - 729*s, [s, s**4 - x -
1])) # orig expr has one real root: -0.048
assert check(unrad(root(x, 3)/2 - root(x + 1, 4) + root(x + 2, 3)),
(729*s**13 - 216*s**12 + 1728*s**11 - 512*s**10 + 1701*s**9 -
3024*s**8 + 1344*s**7 + 1323*s**5 - 2904*s**4 + 343*s, [s, s**4 - x -
1])) # orig expr has 2 real roots: -0.91, -0.15
assert check(unrad(root(x, 3)/2 - root(x + 1, 4) + root(x + 2, 3) - 2),
(729*s**13 + 1242*s**12 + 18496*s**10 + 129701*s**9 + 388602*s**8 +
453312*s**7 - 612864*s**6 - 3337173*s**5 - 6332418*s**4 - 7134912*s**3
- 5064768*s**2 - 2111913*s - 398034, [s, s**4 - x - 1]))
# orig expr has 1 real root: 19.53
ans = solve(sqrt(x) + sqrt(x + 1) -
sqrt(1 - x) - sqrt(2 + x))
assert len(ans) == 1 and NS(ans[0])[:4] == '0.73'
# the fence optimization problem
# https://github.com/sympy/sympy/issues/4793#issuecomment-36994519
F = Symbol('F')
eq = F - (2*x + 2*y + sqrt(x**2 + y**2))
ans = 2*F/7 - sqrt(2)*F/14
X = solve(eq, x, check=False)
for xi in reversed(X): # reverse since currently, ans is the 2nd one
Y = solve((x*y).subs(x, xi).diff(y), y, simplify=False, check=False)
if any((a - ans).expand().is_zero for a in Y):
break
else:
assert None # no answer was found
assert solve(sqrt(x + 1) + root(x, 3) - 2) == S('''
[(-11/(9*(47/54 + sqrt(93)/6)**(1/3)) + 1/3 + (47/54 +
sqrt(93)/6)**(1/3))**3]''')
assert solve(sqrt(sqrt(x + 1)) + x**Rational(1, 3) - 2) == S('''
[(-sqrt(-2*(-1/16 + sqrt(6913)/16)**(1/3) + 6/(-1/16 +
sqrt(6913)/16)**(1/3) + 17/2 + 121/(4*sqrt(-6/(-1/16 +
sqrt(6913)/16)**(1/3) + 2*(-1/16 + sqrt(6913)/16)**(1/3) + 17/4)))/2 +
sqrt(-6/(-1/16 + sqrt(6913)/16)**(1/3) + 2*(-1/16 +
sqrt(6913)/16)**(1/3) + 17/4)/2 + 9/4)**3]''')
assert solve(sqrt(x) + root(sqrt(x) + 1, 3) - 2) == S('''
[(-(81/2 + 3*sqrt(741)/2)**(1/3)/3 + (81/2 + 3*sqrt(741)/2)**(-1/3) +
2)**2]''')
eq = S('''
-x + (1/2 - sqrt(3)*I/2)*(3*x**3/2 - x*(3*x**2 - 34)/2 + sqrt((-3*x**3
+ x*(3*x**2 - 34) + 90)**2/4 - 39304/27) - 45)**(1/3) + 34/(3*(1/2 -
sqrt(3)*I/2)*(3*x**3/2 - x*(3*x**2 - 34)/2 + sqrt((-3*x**3 + x*(3*x**2
- 34) + 90)**2/4 - 39304/27) - 45)**(1/3))''')
assert check(unrad(eq),
(-s*(-s**6 + sqrt(3)*s**6*I - 153*2**(S(2)/3)*3**(S(1)/3)*s**4 +
51*12**(S(1)/3)*s**4 - 102*2**(S(2)/3)*3**(S(5)/6)*s**4*I - 1620*s**3 +
1620*sqrt(3)*s**3*I + 13872*18**(S(1)/3)*s**2 - 471648 +
471648*sqrt(3)*I), [s, s**3 - 306*x - sqrt(3)*sqrt(31212*x**2 -
165240*x + 61484) + 810]))
assert solve(eq) == [] # not other code errors
@slow
def test_unrad_slow():
# this has roots with multiplicity > 1; there should be no
# repeats in roots obtained, however
eq = (sqrt(1 + sqrt(1 - 4*x**2)) - x*((1 + sqrt(1 + 2*sqrt(1 - 4*x**2)))))
assert solve(eq) == [S.Half]
@XFAIL
def test_unrad_fail():
# this only works if we check real_root(eq.subs(x, S(1)/3))
# but checksol doesn't work like that
assert solve(root(x**3 - 3*x**2, 3) + 1 - x) == [S(1)/3]
assert solve(root(x + 1, 3) + root(x**2 - 2, 5) + 1) == [
-1, -1 + CRootOf(x**5 + x**4 + 5*x**3 + 8*x**2 + 10*x + 5, 0)**3]
def test_checksol():
x, y, r, t = symbols('x, y, r, t')
eq = r - x**2 - y**2
dict_var_soln = {y: - sqrt(r) / sqrt(tan(t)**2 + 1),
x: -sqrt(r)*tan(t)/sqrt(tan(t)**2 + 1)}
assert checksol(eq, dict_var_soln) == True
assert checksol(Eq(x, False), {x: False}) is True
assert checksol(Ne(x, False), {x: False}) is False
assert checksol(Eq(x < 1, True), {x: 0}) is True
assert checksol(Eq(x < 1, True), {x: 1}) is False
assert checksol(Eq(x < 1, False), {x: 1}) is True
assert checksol(Eq(x < 1, False), {x: 0}) is False
assert checksol(Eq(x + 1, x**2 + 1), {x: 1}) is True
assert checksol([x - 1, x**2 - 1], x, 1) is True
assert checksol([x - 1, x**2 - 2], x, 1) is False
assert checksol(Poly(x**2 - 1), x, 1) is True
raises(ValueError, lambda: checksol(x, 1))
raises(ValueError, lambda: checksol([], x, 1))
def test__invert():
assert _invert(x - 2) == (2, x)
assert _invert(2) == (2, 0)
assert _invert(exp(1/x) - 3, x) == (1/log(3), x)
assert _invert(exp(1/x + a/x) - 3, x) == ((a + 1)/log(3), x)
assert _invert(a, x) == (a, 0)
def test_issue_4463():
assert solve(-a*x + 2*x*log(x), x) == [exp(a/2)]
assert solve(a/x + exp(x/2), x) == [2*LambertW(-a/2)]
assert solve(x**x) == []
assert solve(x**x - 2) == [exp(LambertW(log(2)))]
assert solve(((x - 3)*(x - 2))**((x - 3)*(x - 4))) == [2]
assert solve(
(a/x + exp(x/2)).diff(x), x) == [4*LambertW(sqrt(2)*sqrt(a)/4)]
@slow
def test_issue_5114_solvers():
a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r = symbols('a:r')
# there is no 'a' in the equation set but this is how the
# problem was originally posed
syms = a, b, c, f, h, k, n
eqs = [b + r/d - c/d,
c*(1/d + 1/e + 1/g) - f/g - r/d,
f*(1/g + 1/i + 1/j) - c/g - h/i,
h*(1/i + 1/l + 1/m) - f/i - k/m,
k*(1/m + 1/o + 1/p) - h/m - n/p,
n*(1/p + 1/q) - k/p]
assert len(solve(eqs, syms, manual=True, check=False, simplify=False)) == 1
def test_issue_5849():
I1, I2, I3, I4, I5, I6 = symbols('I1:7')
dI1, dI4, dQ2, dQ4, Q2, Q4 = symbols('dI1,dI4,dQ2,dQ4,Q2,Q4')
e = (
I1 - I2 - I3,
I3 - I4 - I5,
I4 + I5 - I6,
-I1 + I2 + I6,
-2*I1 - 2*I3 - 2*I5 - 3*I6 - dI1/2 + 12,
-I4 + dQ4,
-I2 + dQ2,
2*I3 + 2*I5 + 3*I6 - Q2,
I4 - 2*I5 + 2*Q4 + dI4
)
ans = [{
dQ4: I3 - I5,
dI1: -4*I2 - 8*I3 - 4*I5 - 6*I6 + 24,
I4: I3 - I5,
dQ2: I2,
Q2: 2*I3 + 2*I5 + 3*I6,
I1: I2 + I3,
Q4: -I3/2 + 3*I5/2 - dI4/2}]
v = I1, I4, Q2, Q4, dI1, dI4, dQ2, dQ4
assert solve(e, *v, manual=True, check=False, dict=True) == ans
assert solve(e, *v, manual=True) == []
# the matrix solver (tested below) doesn't like this because it produces
# a zero row in the matrix. Is this related to issue 4551?
assert [ei.subs(
ans[0]) for ei in e] == [0, 0, I3 - I6, -I3 + I6, 0, 0, 0, 0, 0]
def test_issue_5849_matrix():
'''Same as test_2750 but solved with the matrix solver.'''
I1, I2, I3, I4, I5, I6 = symbols('I1:7')
dI1, dI4, dQ2, dQ4, Q2, Q4 = symbols('dI1,dI4,dQ2,dQ4,Q2,Q4')
e = (
I1 - I2 - I3,
I3 - I4 - I5,
I4 + I5 - I6,
-I1 + I2 + I6,
-2*I1 - 2*I3 - 2*I5 - 3*I6 - dI1/2 + 12,
-I4 + dQ4,
-I2 + dQ2,
2*I3 + 2*I5 + 3*I6 - Q2,
I4 - 2*I5 + 2*Q4 + dI4
)
assert solve(e, I1, I4, Q2, Q4, dI1, dI4, dQ2, dQ4) == {
dI4: -I3 + 3*I5 - 2*Q4,
dI1: -4*I2 - 8*I3 - 4*I5 - 6*I6 + 24,
dQ2: I2,
I1: I2 + I3,
Q2: 2*I3 + 2*I5 + 3*I6,
dQ4: I3 - I5,
I4: I3 - I5}
def test_issue_5901():
f, g, h = map(Function, 'fgh')
a = Symbol('a')
D = Derivative(f(x), x)
G = Derivative(g(a), a)
assert solve(f(x) + f(x).diff(x), f(x)) == \
[-D]
assert solve(f(x) - 3, f(x)) == \
[3]
assert solve(f(x) - 3*f(x).diff(x), f(x)) == \
[3*D]
assert solve([f(x) - 3*f(x).diff(x)], f(x)) == \
{f(x): 3*D}
assert solve([f(x) - 3*f(x).diff(x), f(x)**2 - y + 4], f(x), y) == \
[{f(x): 3*D, y: 9*D**2 + 4}]
assert solve(-f(a)**2*g(a)**2 + f(a)**2*h(a)**2 + g(a).diff(a),
h(a), g(a), set=True) == \
([g(a)], set([
(-sqrt(h(a)**2*f(a)**2 + G)/f(a),),
(sqrt(h(a)**2*f(a)**2+ G)/f(a),)]))
args = [f(x).diff(x, 2)*(f(x) + g(x)) - g(x)**2 + 2, f(x), g(x)]
assert set(solve(*args)) == \
set([(-sqrt(2), sqrt(2)), (sqrt(2), -sqrt(2))])
eqs = [f(x)**2 + g(x) - 2*f(x).diff(x), g(x)**2 - 4]
assert solve(eqs, f(x), g(x), set=True) == \
([f(x), g(x)], set([
(-sqrt(2*D - 2), S(2)),
(sqrt(2*D - 2), S(2)),
(-sqrt(2*D + 2), -S(2)),
(sqrt(2*D + 2), -S(2))]))
# the underlying problem was in solve_linear that was not masking off
# anything but a Mul or Add; it now raises an error if it gets anything
# but a symbol and solve handles the substitutions necessary so solve_linear
# won't make this error
raises(
ValueError, lambda: solve_linear(f(x) + f(x).diff(x), symbols=[f(x)]))
assert solve_linear(f(x) + f(x).diff(x), symbols=[x]) == \
(f(x) + Derivative(f(x), x), 1)
assert solve_linear(f(x) + Integral(x, (x, y)), symbols=[x]) == \
(f(x) + Integral(x, (x, y)), 1)
assert solve_linear(f(x) + Integral(x, (x, y)) + x, symbols=[x]) == \
(x + f(x) + Integral(x, (x, y)), 1)
assert solve_linear(f(y) + Integral(x, (x, y)) + x, symbols=[x]) == \
(x, -f(y) - Integral(x, (x, y)))
assert solve_linear(x - f(x)/a + (f(x) - 1)/a, symbols=[x]) == \
(x, 1/a)
assert solve_linear(x + Derivative(2*x, x)) == \
(x, -2)
assert solve_linear(x + Integral(x, y), symbols=[x]) == \
(x, 0)
assert solve_linear(x + Integral(x, y) - 2, symbols=[x]) == \
(x, 2/(y + 1))
assert set(solve(x + exp(x)**2, exp(x))) == \
set([-sqrt(-x), sqrt(-x)])
assert solve(x + exp(x), x, implicit=True) == \
[-exp(x)]
assert solve(cos(x) - sin(x), x, implicit=True) == []
assert solve(x - sin(x), x, implicit=True) == \
[sin(x)]
assert solve(x**2 + x - 3, x, implicit=True) == \
[-x**2 + 3]
assert solve(x**2 + x - 3, x**2, implicit=True) == \
[-x + 3]
def test_issue_5912():
assert set(solve(x**2 - x - 0.1, rational=True)) == \
set([S(1)/2 + sqrt(35)/10, -sqrt(35)/10 + S(1)/2])
ans = solve(x**2 - x - 0.1, rational=False)
assert len(ans) == 2 and all(a.is_Number for a in ans)
ans = solve(x**2 - x - 0.1)
assert len(ans) == 2 and all(a.is_Number for a in ans)
def test_float_handling():
def test(e1, e2):
return len(e1.atoms(Float)) == len(e2.atoms(Float))
assert solve(x - 0.5, rational=True)[0].is_Rational
assert solve(x - 0.5, rational=False)[0].is_Float
assert solve(x - S.Half, rational=False)[0].is_Rational
assert solve(x - 0.5, rational=None)[0].is_Float
assert solve(x - S.Half, rational=None)[0].is_Rational
assert test(nfloat(1 + 2*x), 1.0 + 2.0*x)
for contain in [list, tuple, set]:
ans = nfloat(contain([1 + 2*x]))
assert type(ans) is contain and test(list(ans)[0], 1.0 + 2.0*x)
k, v = list(nfloat({2*x: [1 + 2*x]}).items())[0]
assert test(k, 2*x) and test(v[0], 1.0 + 2.0*x)
assert test(nfloat(cos(2*x)), cos(2.0*x))
assert test(nfloat(3*x**2), 3.0*x**2)
assert test(nfloat(3*x**2, exponent=True), 3.0*x**2.0)
assert test(nfloat(exp(2*x)), exp(2.0*x))
assert test(nfloat(x/3), x/3.0)
assert test(nfloat(x**4 + 2*x + cos(S(1)/3) + 1),
x**4 + 2.0*x + 1.94495694631474)
# don't call nfloat if there is no solution
tot = 100 + c + z + t
assert solve(((.7 + c)/tot - .6, (.2 + z)/tot - .3, t/tot - .1)) == []
def test_check_assumptions():
x = symbols('x', positive=True)
assert solve(x**2 - 1) == [1]
assert check_assumptions(1, x) == True
raises(AssertionError, lambda: check_assumptions(2*x, x, positive=True))
raises(TypeError, lambda: check_assumptions(1, 1))
def test_failing_assumptions():
x = Symbol('x', real=True, positive=True)
y = Symbol('y')
assert failing_assumptions(6*x + y, **x.assumptions0) == \
{'real': None, 'imaginary': None, 'complex': None, 'hermitian': None,
'positive': None, 'nonpositive': None, 'nonnegative': None, 'nonzero': None,
'negative': None, 'zero': None}
def test_issue_6056():
assert solve(tanh(x + 3)*tanh(x - 3) - 1) == []
assert set([simplify(w) for w in solve(tanh(x - 1)*tanh(x + 1) + 1)]) == set([
-log(2)/2 + log(1 - I),
-log(2)/2 + log(-1 - I),
-log(2)/2 + log(1 + I),
-log(2)/2 + log(-1 + I),])
assert set([simplify(w) for w in solve((tanh(x + 3)*tanh(x - 3) + 1)**2)]) == set([
-log(2)/2 + log(1 - I),
-log(2)/2 + log(-1 - I),
-log(2)/2 + log(1 + I),
-log(2)/2 + log(-1 + I),])
def test_issue_5673():
eq = -x + exp(exp(LambertW(log(x)))*LambertW(log(x)))
assert checksol(eq, x, 2) is True
assert checksol(eq, x, 2, numerical=False) is None
def test_exclude():
R, C, Ri, Vout, V1, Vminus, Vplus, s = \
symbols('R, C, Ri, Vout, V1, Vminus, Vplus, s')
Rf = symbols('Rf', positive=True) # to eliminate Rf = 0 soln
eqs = [C*V1*s + Vplus*(-2*C*s - 1/R),
Vminus*(-1/Ri - 1/Rf) + Vout/Rf,
C*Vplus*s + V1*(-C*s - 1/R) + Vout/R,
-Vminus + Vplus]
assert solve(eqs, exclude=s*C*R) == [
{
Rf: Ri*(C*R*s + 1)**2/(C*R*s),
Vminus: Vplus,
V1: 2*Vplus + Vplus/(C*R*s),
Vout: C*R*Vplus*s + 3*Vplus + Vplus/(C*R*s)},
{
Vplus: 0,
Vminus: 0,
V1: 0,
Vout: 0},
]
# TODO: Investigate why currently solution [0] is preferred over [1].
assert solve(eqs, exclude=[Vplus, s, C]) in [[{
Vminus: Vplus,
V1: Vout/2 + Vplus/2 + sqrt((Vout - 5*Vplus)*(Vout - Vplus))/2,
R: (Vout - 3*Vplus - sqrt(Vout**2 - 6*Vout*Vplus + 5*Vplus**2))/(2*C*Vplus*s),
Rf: Ri*(Vout - Vplus)/Vplus,
}, {
Vminus: Vplus,
V1: Vout/2 + Vplus/2 - sqrt((Vout - 5*Vplus)*(Vout - Vplus))/2,
R: (Vout - 3*Vplus + sqrt(Vout**2 - 6*Vout*Vplus + 5*Vplus**2))/(2*C*Vplus*s),
Rf: Ri*(Vout - Vplus)/Vplus,
}], [{
Vminus: Vplus,
Vout: (V1**2 - V1*Vplus - Vplus**2)/(V1 - 2*Vplus),
Rf: Ri*(V1 - Vplus)**2/(Vplus*(V1 - 2*Vplus)),
R: Vplus/(C*s*(V1 - 2*Vplus)),
}]]
def test_high_order_roots():
s = x**5 + 4*x**3 + 3*x**2 + S(7)/4
assert set(solve(s)) == set(Poly(s*4, domain='ZZ').all_roots())
def test_minsolve_linear_system():
def count(dic):
return len([x for x in dic.values() if x == 0])
assert count(solve([x + y + z, y + z + a + t], particular=True, quick=True)) \
== 3
assert count(solve([x + y + z, y + z + a + t], particular=True, quick=False)) \
== 3
assert count(solve([x + y + z, y + z + a], particular=True, quick=True)) == 1
assert count(solve([x + y + z, y + z + a], particular=True, quick=False)) == 2
def test_real_roots():
# cf. issue 6650
x = Symbol('x', real=True)
assert len(solve(x**5 + x**3 + 1)) == 1
def test_issue_6528():
eqs = [
327600995*x**2 - 37869137*x + 1809975124*y**2 - 9998905626,
895613949*x**2 - 273830224*x*y + 530506983*y**2 - 10000000000]
# two expressions encountered are > 1400 ops long so if this hangs
# it is likely because simplification is being done
assert len(solve(eqs, y, x, check=False)) == 4
def test_overdetermined():
x = symbols('x', real=True)
eqs = [Abs(4*x - 7) - 5, Abs(3 - 8*x) - 1]
assert solve(eqs, x) == [(S.Half,)]
assert solve(eqs, x, manual=True) == [(S.Half,)]
assert solve(eqs, x, manual=True, check=False) == [(S.Half,), (S(3),)]
def test_issue_6605():
x = symbols('x')
assert solve(4**(x/2) - 2**(x/3)) == [0, 3*I*pi/log(2)]
# while the first one passed, this one failed
x = symbols('x', real=True)
assert solve(5**(x/2) - 2**(x/3)) == [0]
b = sqrt(6)*sqrt(log(2))/sqrt(log(5))
assert solve(5**(x/2) - 2**(3/x)) == [-b, b]
def test__ispow():
assert _ispow(x**2)
assert not _ispow(x)
assert not _ispow(True)
def test_issue_6644():
eq = -sqrt((m - q)**2 + (-m/(2*q) + S(1)/2)**2) + sqrt((-m**2/2 - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - S(1)/4)**2 + (m**2/2 - m - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - S(1)/4)**2)
sol = solve(eq, q, simplify=False, check=False)
assert len(sol) == 5
def test_issue_6752():
assert solve([a**2 + a, a - b], [a, b]) == [(-1, -1), (0, 0)]
assert solve([a**2 + a*c, a - b], [a, b]) == [(0, 0), (-c, -c)]
def test_issue_6792():
assert solve(x*(x - 1)**2*(x + 1)*(x**6 - x + 1)) == [
-1, 0, 1, CRootOf(x**6 - x + 1, 0), CRootOf(x**6 - x + 1, 1),
CRootOf(x**6 - x + 1, 2), CRootOf(x**6 - x + 1, 3),
CRootOf(x**6 - x + 1, 4), CRootOf(x**6 - x + 1, 5)]
def test_issues_6819_6820_6821_6248_8692():
# issue 6821
x, y = symbols('x y', real=True)
assert solve(abs(x + 3) - 2*abs(x - 3)) == [1, 9]
assert solve([abs(x) - 2, arg(x) - pi], x) == [(-2,), (2,)]
assert set(solve(abs(x - 7) - 8)) == set([-S(1), S(15)])
# issue 8692
assert solve(Eq(Abs(x + 1) + Abs(x**2 - 7), 9), x) == [
-S(1)/2 + sqrt(61)/2, -sqrt(69)/2 + S(1)/2]
# issue 7145
assert solve(2*abs(x) - abs(x - 1)) == [-1, Rational(1, 3)]
x = symbols('x')
assert solve([re(x) - 1, im(x) - 2], x) == [
{re(x): 1, x: 1 + 2*I, im(x): 2}]
# check for 'dict' handling of solution
eq = sqrt(re(x)**2 + im(x)**2) - 3
assert solve(eq) == solve(eq, x)
i = symbols('i', imaginary=True)
assert solve(abs(i) - 3) == [-3*I, 3*I]
raises(NotImplementedError, lambda: solve(abs(x) - 3))
w = symbols('w', integer=True)
assert solve(2*x**w - 4*y**w, w) == solve((x/y)**w - 2, w)
x, y = symbols('x y', real=True)
assert solve(x + y*I + 3) == {y: 0, x: -3}
# issue 2642
assert solve(x*(1 + I)) == [0]
x, y = symbols('x y', imaginary=True)
assert solve(x + y*I + 3 + 2*I) == {x: -2*I, y: 3*I}
x = symbols('x', real=True)
assert solve(x + y + 3 + 2*I) == {x: -3, y: -2*I}
# issue 6248
f = Function('f')
assert solve(f(x + 1) - f(2*x - 1)) == [2]
assert solve(log(x + 1) - log(2*x - 1)) == [2]
x = symbols('x')
assert solve(2**x + 4**x) == [I*pi/log(2)]
def test_issue_14607():
# issue 14607
s, tau_c, tau_1, tau_2, phi, K = symbols(
's, tau_c, tau_1, tau_2, phi, K')
target = (s**2*tau_1*tau_2 + s*tau_1 + s*tau_2 + 1)/(K*s*(-phi + tau_c))
K_C, tau_I, tau_D = symbols('K_C, tau_I, tau_D',
positive=True, nonzero=True)
PID = K_C*(1 + 1/(tau_I*s) + tau_D*s)
eq = (target - PID).together()
eq *= denom(eq).simplify()
eq = Poly(eq, s)
c = eq.coeffs()
vars = [K_C, tau_I, tau_D]
s = solve(c, vars, dict=True)
assert len(s) == 1
knownsolution = {K_C: -(tau_1 + tau_2)/(K*(phi - tau_c)),
tau_I: tau_1 + tau_2,
tau_D: tau_1*tau_2/(tau_1 + tau_2)}
for var in vars:
assert s[0][var].simplify() == knownsolution[var].simplify()
@slow
def test_lambert_multivariate():
from sympy.abc import a, x, y
from sympy.solvers.bivariate import _filtered_gens, _lambert, _solve_lambert
assert _filtered_gens(Poly(x + 1/x + exp(x) + y), x) == set([x, exp(x)])
assert _lambert(x, x) == []
assert solve((x**2 - 2*x + 1).subs(x, log(x) + 3*x)) == [LambertW(3*S.Exp1)/3]
assert solve((x**2 - 2*x + 1).subs(x, (log(x) + 3*x)**2 - 1)) == \
[LambertW(3*exp(-sqrt(2)))/3, LambertW(3*exp(sqrt(2)))/3]
assert solve((x**2 - 2*x - 2).subs(x, log(x) + 3*x)) == \
[LambertW(3*exp(1 - sqrt(3)))/3, LambertW(3*exp(1 + sqrt(3)))/3]
assert solve(x*log(x) + 3*x + 1, x) == [exp(-3 + LambertW(-exp(3)))]
eq = (x*exp(x) - 3).subs(x, x*exp(x))
assert solve(eq) == [LambertW(3*exp(-LambertW(3)))]
# coverage test
raises(NotImplementedError, lambda: solve(x - sin(x)*log(y - x), x))
x0 = 1/log(a)
x1 = LambertW(S(1)/3)
x2 = a**(-5)
x3 = 3**(S(1)/3)
x4 = 3**(S(5)/6)*I
x5 = x1**(S(1)/3)*x2**(S(1)/3)/2
ans = solve(3*log(a**(3*x + 5)) + a**(3*x + 5), x)
assert ans == [
x0*log(3*x1*x2)/3, x0*log(-x5*(x3 - x4)), x0*log(-x5*(x3 + x4))]
# check collection
K = ((b + 3)*LambertW(1/(b + 3))/a**5)**(S(1)/3)
assert solve(
3*log(a**(3*x + 5)) + b*log(a**(3*x + 5)) + a**(3*x + 5),
x) == [
log(K*(1 - sqrt(3)*I)/-2)/log(a),
log(K*(1 + sqrt(3)*I)/-2)/log(a),
log((b + 3)*LambertW(1/(b + 3))/a**5)/(3*log(a))]
p = symbols('p', positive=True)
eq = 4*2**(2*p + 3) - 2*p - 3
assert _solve_lambert(eq, p, _filtered_gens(Poly(eq), p)) == [
-S(3)/2 - LambertW(-4*log(2))/(2*log(2))]
# issue 4271
assert solve((a/x + exp(x/2)).diff(x, 2), x) == [
6*LambertW(root(-1, 3)*root(a, 3)/3)]
assert solve((log(x) + x).subs(x, x**2 + 1)) == [
-I*sqrt(-LambertW(1) + 1), sqrt(-1 + LambertW(1))]
assert solve(x**3 - 3**x, x) == [3, -3*LambertW(-log(3)/3)/log(3)]
assert solve(x**2 - 2**x, x) == [2, 4]
assert solve(-x**2 + 2**x, x) == [2, 4]
assert solve(3**cos(x) - cos(x)**3) == [acos(3), acos(-3*LambertW(-log(3)/3)/log(3))]
assert set(solve(3*log(x) - x*log(3))) == set( # 2.478... and 3
[3, -3*LambertW(-log(3)/3)/log(3)])
assert solve(LambertW(2*x) - y, x) == [y*exp(y)/2]
@XFAIL
def test_other_lambert():
from sympy.abc import x
assert solve(3*sin(x) - x*sin(3), x) == [3]
a = S(6)/5
assert set(solve(x**a - a**x)) == set(
[a, -a*LambertW(-log(a)/a)/log(a)])
assert set(solve(3**cos(x) - cos(x)**3)) == set(
[acos(3), acos(-3*LambertW(-log(3)/3)/log(3))])
def test_rewrite_trig():
assert solve(sin(x) + tan(x)) == [0, -pi, pi, 2*pi]
assert solve(sin(x) + sec(x)) == [
-2*atan(-S.Half + sqrt(2)*sqrt(1 - sqrt(3)*I)/2 + sqrt(3)*I/2),
2*atan(S.Half - sqrt(2)*sqrt(1 + sqrt(3)*I)/2 + sqrt(3)*I/2), 2*atan(S.Half
+ sqrt(2)*sqrt(1 + sqrt(3)*I)/2 + sqrt(3)*I/2), 2*atan(S.Half -
sqrt(3)*I/2 + sqrt(2)*sqrt(1 - sqrt(3)*I)/2)]
assert solve(sinh(x) + tanh(x)) == [0, I*pi]
# issue 6157
assert solve(2*sin(x) - cos(x), x) == [-2*atan(2 - sqrt(5)),
-2*atan(2 + sqrt(5))]
@XFAIL
def test_rewrite_trigh():
# if this import passes then the test below should also pass
from sympy import sech
assert solve(sinh(x) + sech(x)) == [
2*atanh(-S.Half + sqrt(5)/2 - sqrt(-2*sqrt(5) + 2)/2),
2*atanh(-S.Half + sqrt(5)/2 + sqrt(-2*sqrt(5) + 2)/2),
2*atanh(-sqrt(5)/2 - S.Half + sqrt(2 + 2*sqrt(5))/2),
2*atanh(-sqrt(2 + 2*sqrt(5))/2 - sqrt(5)/2 - S.Half)]
def test_uselogcombine():
eq = z - log(x) + log(y/(x*(-1 + y**2/x**2)))
assert solve(eq, x, force=True) == [-sqrt(y*(y - exp(z))), sqrt(y*(y - exp(z)))]
assert solve(log(x + 3) + log(1 + 3/x) - 3) in [
[-3 + sqrt(-12 + exp(3))*exp(S(3)/2)/2 + exp(3)/2,
-sqrt(-12 + exp(3))*exp(S(3)/2)/2 - 3 + exp(3)/2],
[-3 + sqrt(-36 + (-exp(3) + 6)**2)/2 + exp(3)/2,
-3 - sqrt(-36 + (-exp(3) + 6)**2)/2 + exp(3)/2],
]
assert solve(log(exp(2*x) + 1) + log(-tanh(x) + 1) - log(2)) == []
def test_atan2():
assert solve(atan2(x, 2) - pi/3, x) == [2*sqrt(3)]
def test_errorinverses():
assert solve(erf(x) - y, x) == [erfinv(y)]
assert solve(erfinv(x) - y, x) == [erf(y)]
assert solve(erfc(x) - y, x) == [erfcinv(y)]
assert solve(erfcinv(x) - y, x) == [erfc(y)]
def test_issue_2725():
R = Symbol('R')
eq = sqrt(2)*R*sqrt(1/(R + 1)) + (R + 1)*(sqrt(2)*sqrt(1/(R + 1)) - 1)
sol = solve(eq, R, set=True)[1]
assert sol == set([(S(5)/3 + (-S(1)/2 - sqrt(3)*I/2)*(S(251)/27 +
sqrt(111)*I/9)**(S(1)/3) + 40/(9*((-S(1)/2 - sqrt(3)*I/2)*(S(251)/27 +
sqrt(111)*I/9)**(S(1)/3))),), (S(5)/3 + 40/(9*(S(251)/27 +
sqrt(111)*I/9)**(S(1)/3)) + (S(251)/27 + sqrt(111)*I/9)**(S(1)/3),)])
def test_issue_5114_6611():
# See that it doesn't hang; this solves in about 2 seconds.
# Also check that the solution is relatively small.
# Note: the system in issue 6611 solves in about 5 seconds and has
# an op-count of 138336 (with simplify=False).
b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r = symbols('b:r')
eqs = Matrix([
[b - c/d + r/d], [c*(1/g + 1/e + 1/d) - f/g - r/d],
[-c/g + f*(1/j + 1/i + 1/g) - h/i], [-f/i + h*(1/m + 1/l + 1/i) - k/m],
[-h/m + k*(1/p + 1/o + 1/m) - n/p], [-k/p + n*(1/q + 1/p)]])
v = Matrix([f, h, k, n, b, c])
ans = solve(list(eqs), list(v), simplify=False)
# If time is taken to simplify then then 2617 below becomes
# 1168 and the time is about 50 seconds instead of 2.
assert sum([s.count_ops() for s in ans.values()]) <= 2617
def test_det_quick():
m = Matrix(3, 3, symbols('a:9'))
assert m.det() == det_quick(m) # calls det_perm
m[0, 0] = 1
assert m.det() == det_quick(m) # calls det_minor
m = Matrix(3, 3, list(range(9)))
assert m.det() == det_quick(m) # defaults to .det()
# make sure they work with Sparse
s = SparseMatrix(2, 2, (1, 2, 1, 4))
assert det_perm(s) == det_minor(s) == s.det()
def test_real_imag_splitting():
a, b = symbols('a b', real=True)
assert solve(sqrt(a**2 + b**2) - 3, a) == \
[-sqrt(-b**2 + 9), sqrt(-b**2 + 9)]
a, b = symbols('a b', imaginary=True)
assert solve(sqrt(a**2 + b**2) - 3, a) == []
def test_issue_7110():
y = -2*x**3 + 4*x**2 - 2*x + 5
assert any(ask(Q.real(i)) for i in solve(y))
def test_units():
assert solve(1/x - 1/(2*cm)) == [2*cm]
def test_issue_7547():
A, B, V = symbols('A,B,V')
eq1 = Eq(630.26*(V - 39.0)*V*(V + 39) - A + B, 0)
eq2 = Eq(B, 1.36*10**8*(V - 39))
eq3 = Eq(A, 5.75*10**5*V*(V + 39.0))
sol = Matrix(nsolve(Tuple(eq1, eq2, eq3), [A, B, V], (0, 0, 0)))
assert str(sol) == str(Matrix(
[['4442890172.68209'],
['4289299466.1432'],
['70.5389666628177']]))
def test_issue_7895():
r = symbols('r', real=True)
assert solve(sqrt(r) - 2) == [4]
def test_issue_2777():
# the equations represent two circles
x, y = symbols('x y', real=True)
e1, e2 = sqrt(x**2 + y**2) - 10, sqrt(y**2 + (-x + 10)**2) - 3
a, b = 191/S(20), 3*sqrt(391)/20
ans = [(a, -b), (a, b)]
assert solve((e1, e2), (x, y)) == ans
assert solve((e1, e2/(x - a)), (x, y)) == []
# make the 2nd circle's radius be -3
e2 += 6
assert solve((e1, e2), (x, y)) == []
assert solve((e1, e2), (x, y), check=False) == ans
def test_issue_7322():
number = 5.62527e-35
assert solve(x - number, x)[0] == number
def test_nsolve():
raises(ValueError, lambda: nsolve(x, (-1, 1), method='bisect'))
raises(TypeError, lambda: nsolve((x - y + 3,x + y,z - y),(x,y,z),(-50,50)))
raises(TypeError, lambda: nsolve((x + y, x - y), (0, 1)))
@slow
def test_high_order_multivariate():
assert len(solve(a*x**3 - x + 1, x)) == 3
assert len(solve(a*x**4 - x + 1, x)) == 4
assert solve(a*x**5 - x + 1, x) == [] # incomplete solution allowed
raises(NotImplementedError, lambda:
solve(a*x**5 - x + 1, x, incomplete=False))
# result checking must always consider the denominator and CRootOf
# must be checked, too
d = x**5 - x + 1
assert solve(d*(1 + 1/d)) == [CRootOf(d + 1, i) for i in range(5)]
d = x - 1
assert solve(d*(2 + 1/d)) == [S.Half]
def test_base_0_exp_0():
assert solve(0**x - 1) == [0]
assert solve(0**(x - 2) - 1) == [2]
assert solve(S('x*(1/x**0 - x)', evaluate=False)) == \
[0, 1]
def test__simple_dens():
assert _simple_dens(1/x**0, [x]) == set()
assert _simple_dens(1/x**y, [x]) == set([x**y])
assert _simple_dens(1/root(x, 3), [x]) == set([x])
def test_issue_8755():
# This tests two things: that if full unrad is attempted and fails
# the solution should still be found; also it tests the use of
# keyword `composite`.
assert len(solve(sqrt(y)*x + x**3 - 1, x)) == 3
assert len(solve(-512*y**3 + 1344*(x + 2)**(S(1)/3)*y**2 -
1176*(x + 2)**(S(2)/3)*y - 169*x + 686, y, _unrad=False)) == 3
@slow
def test_issue_8828():
x1 = 0
y1 = -620
r1 = 920
x2 = 126
y2 = 276
x3 = 51
y3 = 205
r3 = 104
v = x, y, z
f1 = (x - x1)**2 + (y - y1)**2 - (r1 - z)**2
f2 = (x2 - x)**2 + (y2 - y)**2 - z**2
f3 = (x - x3)**2 + (y - y3)**2 - (r3 - z)**2
F = f1,f2,f3
g1 = sqrt((x - x1)**2 + (y - y1)**2) + z - r1
g2 = f2
g3 = sqrt((x - x3)**2 + (y - y3)**2) + z - r3
G = g1,g2,g3
A = solve(F, v)
B = solve(G, v)
C = solve(G, v, manual=True)
p, q, r = [set([tuple(i.evalf(2) for i in j) for j in R]) for R in [A, B, C]]
assert p == q == r
@slow
def test_issue_2840_8155():
assert solve(sin(3*x) + sin(6*x)) == [
0, -pi, pi, 14*pi/9, 16*pi/9, 2*pi, 2*I*(log(2) - log(-1 - sqrt(3)*I)),
2*I*(log(2) - log(-1 + sqrt(3)*I)), 2*I*(log(2) - log(1 - sqrt(3)*I)),
2*I*(log(2) - log(1 + sqrt(3)*I)), 2*I*(log(2) - log(-sqrt(3) - I)),
2*I*(log(2) - log(-sqrt(3) + I)), 2*I*(log(2) - log(sqrt(3) - I)),
2*I*(log(2) - log(sqrt(3) + I)), -2*I*log(-(-1)**(S(1)/9)), -2*I*log(
-(-1)**(S(2)/9)), -2*I*log(-sin(pi/18) - I*cos(pi/18)), -2*I*log(-sin(
pi/18) + I*cos(pi/18)), -2*I*log(sin(pi/18) - I*cos(pi/18)), -2*I*log(
sin(pi/18) + I*cos(pi/18)), -2*I*log(exp(-2*I*pi/9)), -2*I*log(exp(
-I*pi/9)), -2*I*log(exp(I*pi/9)), -2*I*log(exp(2*I*pi/9))]
assert solve(2*sin(x) - 2*sin(2*x)) == [
0, -pi, pi, 2*I*(log(2) - log(-sqrt(3) - I)), 2*I*(log(2) -
log(-sqrt(3) + I)), 2*I*(log(2) - log(sqrt(3) - I)), 2*I*(log(2) -
log(sqrt(3) + I))]
def test_issue_9567():
assert solve(1 + 1/(x - 1)) == [0]
def test_issue_11538():
assert solve(x + E) == [-E]
assert solve(x**2 + E) == [-I*sqrt(E), I*sqrt(E)]
assert solve(x**3 + 2*E) == [
-cbrt(2 * E),
cbrt(2)*cbrt(E)/2 - cbrt(2)*sqrt(3)*I*cbrt(E)/2,
cbrt(2)*cbrt(E)/2 + cbrt(2)*sqrt(3)*I*cbrt(E)/2]
assert solve([x + 4, y + E], x, y) == {x: -4, y: -E}
assert solve([x**2 + 4, y + E], x, y) == [
(-2*I, -E), (2*I, -E)]
e1 = x - y**3 + 4
e2 = x + y + 4 + 4 * E
assert len(solve([e1, e2], x, y)) == 3
@slow
def test_issue_12114():
a, b, c, d, e, f, g = symbols('a,b,c,d,e,f,g')
terms = [1 + a*b + d*e, 1 + a*c + d*f, 1 + b*c + e*f,
g - a**2 - d**2, g - b**2 - e**2, g - c**2 - f**2]
s = solve(terms, [a, b, c, d, e, f, g], dict=True)
assert s == [{a: -sqrt(-f**2 - 1), b: -sqrt(-f**2 - 1),
c: -sqrt(-f**2 - 1), d: f, e: f, g: -1},
{a: sqrt(-f**2 - 1), b: sqrt(-f**2 - 1),
c: sqrt(-f**2 - 1), d: f, e: f, g: -1},
{a: -sqrt(3)*f/2 - sqrt(-f**2 + 2)/2,
b: sqrt(3)*f/2 - sqrt(-f**2 + 2)/2, c: sqrt(-f**2 + 2),
d: -f/2 + sqrt(-3*f**2 + 6)/2,
e: -f/2 - sqrt(3)*sqrt(-f**2 + 2)/2, g: 2},
{a: -sqrt(3)*f/2 + sqrt(-f**2 + 2)/2,
b: sqrt(3)*f/2 + sqrt(-f**2 + 2)/2, c: -sqrt(-f**2 + 2),
d: -f/2 - sqrt(-3*f**2 + 6)/2,
e: -f/2 + sqrt(3)*sqrt(-f**2 + 2)/2, g: 2},
{a: sqrt(3)*f/2 - sqrt(-f**2 + 2)/2,
b: -sqrt(3)*f/2 - sqrt(-f**2 + 2)/2, c: sqrt(-f**2 + 2),
d: -f/2 - sqrt(-3*f**2 + 6)/2,
e: -f/2 + sqrt(3)*sqrt(-f**2 + 2)/2, g: 2},
{a: sqrt(3)*f/2 + sqrt(-f**2 + 2)/2,
b: -sqrt(3)*f/2 + sqrt(-f**2 + 2)/2, c: -sqrt(-f**2 + 2),
d: -f/2 + sqrt(-3*f**2 + 6)/2,
e: -f/2 - sqrt(3)*sqrt(-f**2 + 2)/2, g: 2}]
def test_inf():
assert solve(1 - oo*x) == []
assert solve(oo*x, x) == []
assert solve(oo*x - oo, x) == []
def test_issue_12448():
f = Function('f')
fun = [f(i) for i in range(15)]
sym = symbols('x:15')
reps = dict(zip(fun, sym))
(x, y, z), c = sym[:3], sym[3:]
ssym = solve([c[4*i]*x + c[4*i + 1]*y + c[4*i + 2]*z + c[4*i + 3]
for i in range(3)], (x, y, z))
(x, y, z), c = fun[:3], fun[3:]
sfun = solve([c[4*i]*x + c[4*i + 1]*y + c[4*i + 2]*z + c[4*i + 3]
for i in range(3)], (x, y, z))
assert sfun[fun[0]].xreplace(reps).count_ops() == \
ssym[sym[0]].count_ops()
def test_denoms():
assert denoms(x/2 + 1/y) == set([2, y])
assert denoms(x/2 + 1/y, y) == set([y])
assert denoms(x/2 + 1/y, [y]) == set([y])
assert denoms(1/x + 1/y + 1/z, [x, y]) == set([x, y])
assert denoms(1/x + 1/y + 1/z, x, y) == set([x, y])
assert denoms(1/x + 1/y + 1/z, set([x, y])) == set([x, y])
def test_issue_12476():
x0, x1, x2, x3, x4, x5 = symbols('x0 x1 x2 x3 x4 x5')
eqns = [x0**2 - x0, x0*x1 - x1, x0*x2 - x2, x0*x3 - x3, x0*x4 - x4, x0*x5 - x5,
x0*x1 - x1, -x0/3 + x1**2 - 2*x2/3, x1*x2 - x1/3 - x2/3 - x3/3,
x1*x3 - x2/3 - x3/3 - x4/3, x1*x4 - 2*x3/3 - x5/3, x1*x5 - x4, x0*x2 - x2,
x1*x2 - x1/3 - x2/3 - x3/3, -x0/6 - x1/6 + x2**2 - x2/6 - x3/3 - x4/6,
-x1/6 + x2*x3 - x2/3 - x3/6 - x4/6 - x5/6, x2*x4 - x2/3 - x3/3 - x4/3,
x2*x5 - x3, x0*x3 - x3, x1*x3 - x2/3 - x3/3 - x4/3,
-x1/6 + x2*x3 - x2/3 - x3/6 - x4/6 - x5/6,
-x0/6 - x1/6 - x2/6 + x3**2 - x3/3 - x4/6, -x1/3 - x2/3 + x3*x4 - x3/3,
-x2 + x3*x5, x0*x4 - x4, x1*x4 - 2*x3/3 - x5/3, x2*x4 - x2/3 - x3/3 - x4/3,
-x1/3 - x2/3 + x3*x4 - x3/3, -x0/3 - 2*x2/3 + x4**2, -x1 + x4*x5, x0*x5 - x5,
x1*x5 - x4, x2*x5 - x3, -x2 + x3*x5, -x1 + x4*x5, -x0 + x5**2, x0 - 1]
sols = [{x0: 1, x3: S(1)/6, x2: S(1)/6, x4: -S(2)/3, x1: -S(2)/3, x5: 1},
{x0: 1, x3: S(1)/2, x2: -S(1)/2, x4: 0, x1: 0, x5: -1},
{x0: 1, x3: -S(1)/3, x2: -S(1)/3, x4: S(1)/3, x1: S(1)/3, x5: 1},
{x0: 1, x3: 1, x2: 1, x4: 1, x1: 1, x5: 1},
{x0: 1, x3: -S(1)/3, x2: S(1)/3, x4: sqrt(5)/3, x1: -sqrt(5)/3, x5: -1},
{x0: 1, x3: -S(1)/3, x2: S(1)/3, x4: -sqrt(5)/3, x1: sqrt(5)/3, x5: -1}]
assert solve(eqns) == sols
def test_issue_13849():
t = symbols('t')
assert solve((t*(sqrt(5) + sqrt(2)) - sqrt(2), t), t) == []
def test_issue_14860():
from sympy.physics.units import newton, kilo
assert solve(8*kilo*newton + x + y, x) == [-8000*newton - y]
def test_issue_14721():
k, h, a, b = symbols(':4')
assert solve([
-1 + (-k + 1)**2/b**2 + (-h - 1)**2/a**2,
-1 + (-k + 1)**2/b**2 + (-h + 1)**2/a**2,
h, k + 2], h, k, a, b) == [
(0, -2, -b*sqrt(1/(b**2 - 9)), b),
(0, -2, b*sqrt(1/(b**2 - 9)), b)]
assert solve([
h, h/a + 1/b**2 - 2, -h/2 + 1/b**2 - 2], a, h, b) == [
(a, 0, -sqrt(2)/2), (a, 0, sqrt(2)/2)]
assert solve((a + b**2 - 1, a + b**2 - 2)) == []
def test_issue_14779():
x = symbols('x', real=True)
assert solve(sqrt(x**4 - 130*x**2 + 1089) + sqrt(x**4 - 130*x**2
+ 3969) - 96*Abs(x)/x,x) == [sqrt(130)]
def test_issue_15307():
assert solve((y - 2, Mul(x + 3,x - 2, evaluate=False))) == \
[{x: -3, y: 2}, {x: 2, y: 2}]
assert solve((y - 2, Mul(3, x - 2, evaluate=False))) == \
{x: 2, y: 2}
assert solve((y - 2, Add(x + 4, x - 2, evaluate=False))) == \
{x: -1, y: 2}
eq1 = Eq(12513*x + 2*y - 219093, -5726*x - y)
eq2 = Eq(-2*x + 8, 2*x - 40)
assert solve([eq1, eq2]) == {x:12, y:75}
def test_issue_15415():
assert solve(x - 3, x) == [3]
assert solve([x - 3], x) == {x:3}
assert solve(Eq(y + 3*x**2/2, y + 3*x), y) == []
assert solve([Eq(y + 3*x**2/2, y + 3*x)], y) == []
assert solve([Eq(y + 3*x**2/2, y + 3*x), Eq(x, 1)], y) == []
@slow
def test_issue_15731():
# f(x)**g(x)=c
assert solve(Eq((x**2 - 7*x + 11)**(x**2 - 13*x + 42), 1)) == [2, 3, 4, 5, 6, 7]
assert solve((x)**(x + 4) - 4) == [-2]
assert solve((-x)**(-x + 4) - 4) == [2]
assert solve((x**2 - 6)**(x**2 - 2) - 4) == [-2, 2]
assert solve((x**2 - 2*x - 1)**(x**2 - 3) - 1/(1 - 2*sqrt(2))) == [sqrt(2)]
assert solve(x**(x + S.Half) - 4*sqrt(2)) == [S(2)]
assert solve((x**2 + 1)**x - 25) == [2]
assert solve(x**(2/x) - 2) == [2, 4]
assert solve((x/2)**(2/x) - sqrt(2)) == [4, 8]
assert solve(x**(x + S.Half) - S(9)/4) == [S(3)/2]
# a**g(x)=c
assert solve((-sqrt(sqrt(2)))**x - 2) == [4, log(2)/(log(2**(S(1)/4)) + I*pi)]
assert solve((sqrt(2))**x - sqrt(sqrt(2))) == [S(1)/2]
assert solve((-sqrt(2))**x + 2*(sqrt(2))) == [3,
(3*log(2)**2 + 4*pi**2 - 4*I*pi*log(2))/(log(2)**2 + 4*pi**2)]
assert solve((sqrt(2))**x - 2*(sqrt(2))) == [3]
assert solve(I**x + 1) == [2]
assert solve((1 + I)**x - 2*I) == [2]
assert solve((sqrt(2) + sqrt(3))**x - (2*sqrt(6) + 5)**(S(1)/3)) == [S(2)/3]
# bases of both sides are equal
b = Symbol('b')
assert solve(b**x - b**2, x) == [2]
assert solve(b**x - 1/b, x) == [-1]
assert solve(b**x - b, x) == [1]
b = Symbol('b', positive=True)
assert solve(b**x - b**2, x) == [2]
assert solve(b**x - 1/b, x) == [-1]
|
bff4206fb3da7edd001a2e24f9e0162812c4dd097753e926312ca0d3a90bd0e3
|
"""Tests for solvers of systems of polynomial equations. """
from sympy import (flatten, I, Integer, Poly, QQ, Rational, S, sqrt,
solve, symbols)
from sympy.abc import x, y, z
from sympy.polys import PolynomialError
from sympy.solvers.polysys import (solve_poly_system,
solve_triangulated, solve_biquadratic, SolveFailed)
from sympy.polys.polytools import parallel_poly_from_expr
from sympy.utilities.pytest import raises
def test_solve_poly_system():
assert solve_poly_system([x - 1], x) == [(S.One,)]
assert solve_poly_system([y - x, y - x - 1], x, y) is None
assert solve_poly_system([y - x**2, y + x**2], x, y) == [(S.Zero, S.Zero)]
assert solve_poly_system([2*x - 3, 3*y/2 - 2*x, z - 5*y], x, y, z) == \
[(Rational(3, 2), Integer(2), Integer(10))]
assert solve_poly_system([x*y - 2*y, 2*y**2 - x**2], x, y) == \
[(0, 0), (2, -sqrt(2)), (2, sqrt(2))]
assert solve_poly_system([y - x**2, y + x**2 + 1], x, y) == \
[(-I*sqrt(S.Half), -S.Half), (I*sqrt(S.Half), -S.Half)]
f_1 = x**2 + y + z - 1
f_2 = x + y**2 + z - 1
f_3 = x + y + z**2 - 1
a, b = sqrt(2) - 1, -sqrt(2) - 1
assert solve_poly_system([f_1, f_2, f_3], x, y, z) == \
[(0, 0, 1), (0, 1, 0), (1, 0, 0), (a, a, a), (b, b, b)]
solution = [(1, -1), (1, 1)]
assert solve_poly_system([Poly(x**2 - y**2), Poly(x - 1)]) == solution
assert solve_poly_system([x**2 - y**2, x - 1], x, y) == solution
assert solve_poly_system([x**2 - y**2, x - 1]) == solution
assert solve_poly_system(
[x + x*y - 3, y + x*y - 4], x, y) == [(-3, -2), (1, 2)]
raises(NotImplementedError, lambda: solve_poly_system([x**3 - y**3], x, y))
raises(NotImplementedError, lambda: solve_poly_system(
[z, -2*x*y**2 + x + y**2*z, y**2*(-z - 4) + 2]))
raises(PolynomialError, lambda: solve_poly_system([1/x], x))
def test_solve_biquadratic():
x0, y0, x1, y1, r = symbols('x0 y0 x1 y1 r')
f_1 = (x - 1)**2 + (y - 1)**2 - r**2
f_2 = (x - 2)**2 + (y - 2)**2 - r**2
s = sqrt(2*r**2 - 1)
a = (3 - s)/2
b = (3 + s)/2
assert solve_poly_system([f_1, f_2], x, y) == [(a, b), (b, a)]
f_1 = (x - 1)**2 + (y - 2)**2 - r**2
f_2 = (x - 1)**2 + (y - 1)**2 - r**2
assert solve_poly_system([f_1, f_2], x, y) == \
[(1 - sqrt(((2*r - 1)*(2*r + 1)))/2, S(3)/2),
(1 + sqrt(((2*r - 1)*(2*r + 1)))/2, S(3)/2)]
query = lambda expr: expr.is_Pow and expr.exp is S.Half
f_1 = (x - 1 )**2 + (y - 2)**2 - r**2
f_2 = (x - x1)**2 + (y - 1)**2 - r**2
result = solve_poly_system([f_1, f_2], x, y)
assert len(result) == 2 and all(len(r) == 2 for r in result)
assert all(r.count(query) == 1 for r in flatten(result))
f_1 = (x - x0)**2 + (y - y0)**2 - r**2
f_2 = (x - x1)**2 + (y - y1)**2 - r**2
result = solve_poly_system([f_1, f_2], x, y)
assert len(result) == 2 and all(len(r) == 2 for r in result)
assert all(len(r.find(query)) == 1 for r in flatten(result))
s1 = (x*y - y, x**2 - x)
assert solve(s1) == [{x: 1}, {x: 0, y: 0}]
s2 = (x*y - x, y**2 - y)
assert solve(s2) == [{y: 1}, {x: 0, y: 0}]
gens = (x, y)
for seq in (s1, s2):
(f, g), opt = parallel_poly_from_expr(seq, *gens)
raises(SolveFailed, lambda: solve_biquadratic(f, g, opt))
seq = (x**2 + y**2 - 2, y**2 - 1)
(f, g), opt = parallel_poly_from_expr(seq, *gens)
assert solve_biquadratic(f, g, opt) == [
(-1, -1), (-1, 1), (1, -1), (1, 1)]
ans = [(0, -1), (0, 1)]
seq = (x**2 + y**2 - 1, y**2 - 1)
(f, g), opt = parallel_poly_from_expr(seq, *gens)
assert solve_biquadratic(f, g, opt) == ans
seq = (x**2 + y**2 - 1, x**2 - x + y**2 - 1)
(f, g), opt = parallel_poly_from_expr(seq, *gens)
assert solve_biquadratic(f, g, opt) == ans
def test_solve_triangulated():
f_1 = x**2 + y + z - 1
f_2 = x + y**2 + z - 1
f_3 = x + y + z**2 - 1
a, b = sqrt(2) - 1, -sqrt(2) - 1
assert solve_triangulated([f_1, f_2, f_3], x, y, z) == \
[(0, 0, 1), (0, 1, 0), (1, 0, 0)]
dom = QQ.algebraic_field(sqrt(2))
assert solve_triangulated([f_1, f_2, f_3], x, y, z, domain=dom) == \
[(0, 0, 1), (0, 1, 0), (1, 0, 0), (a, a, a), (b, b, b)]
def test_solve_issue_3686():
roots = solve_poly_system([((x - 5)**2/250000 + (y - S(5)/10)**2/250000) - 1, x], x, y)
assert roots == [(0, S(1)/2 - 15*sqrt(1111)), (0, S(1)/2 + 15*sqrt(1111))]
roots = solve_poly_system([((x - 5)**2/250000 + (y - 5.0/10)**2/250000) - 1, x], x, y)
# TODO: does this really have to be so complicated?!
assert len(roots) == 2
assert roots[0][0] == 0
assert roots[0][1].epsilon_eq(-499.474999374969, 1e12)
assert roots[1][0] == 0
assert roots[1][1].epsilon_eq(500.474999374969, 1e12)
|
3b0dbc6421360cdc2a20361bf642df2b6980d39c0060b0d3d70e8d3e0b9b3795
|
from sympy import (Add, factor_list, igcd, Matrix, Mul, S, simplify,
Symbol, symbols, Eq, pi, factorint, oo, powsimp)
from sympy.core.function import _mexpand
from sympy.core.compatibility import range
from sympy.functions.elementary.trigonometric import sin
from sympy.solvers.diophantine import (descent, diop_bf_DN, diop_DN,
diop_solve, diophantine, divisible, equivalent, find_DN, ldescent, length,
reconstruct, partition, power_representation,
prime_as_sum_of_two_squares, square_factor, sum_of_four_squares,
sum_of_three_squares, transformation_to_DN, transformation_to_normal,
classify_diop, base_solution_linear, cornacchia, sqf_normal,
diop_ternary_quadratic_normal, _diop_ternary_quadratic_normal,
gaussian_reduce, holzer,diop_general_pythagorean,
_diop_general_sum_of_squares, _nint_or_floor, _odd, _even,
_remove_gcd, check_param, parametrize_ternary_quadratic,
diop_ternary_quadratic, diop_linear, diop_quadratic,
diop_general_sum_of_squares, sum_of_powers, sum_of_squares,
diop_general_sum_of_even_powers, _can_do_sum_of_squares)
from sympy.utilities import default_sort_key
from sympy.utilities.pytest import slow, raises, XFAIL
from sympy.utilities.iterables import (
permute_signs,
signed_permutations)
a, b, c, d, p, q, x, y, z, w, t, u, v, X, Y, Z = symbols(
"a, b, c, d, p, q, x, y, z, w, t, u, v, X, Y, Z", integer=True)
t_0, t_1, t_2, t_3, t_4, t_5, t_6 = symbols("t_:7", integer=True)
m1, m2, m3 = symbols('m1:4', integer=True)
n1 = symbols('n1', integer=True)
def diop_simplify(eq):
return _mexpand(powsimp(_mexpand(eq)))
def test_input_format():
raises(TypeError, lambda: diophantine(sin(x)))
raises(TypeError, lambda: diophantine(3))
raises(TypeError, lambda: diophantine(x/pi - 3))
def test_univariate():
assert diop_solve((x - 1)*(x - 2)**2) == set([(1,), (2,)])
assert diop_solve((x - 1)*(x - 2)) == set([(1,), (2,)])
def test_classify_diop():
raises(TypeError, lambda: classify_diop(x**2/3 - 1))
raises(ValueError, lambda: classify_diop(1))
raises(NotImplementedError, lambda: classify_diop(w*x*y*z - 1))
raises(NotImplementedError, lambda: classify_diop(x**3 + y**3 + z**4 - 90))
assert classify_diop(14*x**2 + 15*x - 42) == (
[x], {1: -42, x: 15, x**2: 14}, 'univariate')
assert classify_diop(x*y + z) == (
[x, y, z], {x*y: 1, z: 1}, 'inhomogeneous_ternary_quadratic')
assert classify_diop(x*y + z + w + x**2) == (
[w, x, y, z], {x*y: 1, w: 1, x**2: 1, z: 1}, 'inhomogeneous_general_quadratic')
assert classify_diop(x*y + x*z + x**2 + 1) == (
[x, y, z], {x*y: 1, x*z: 1, x**2: 1, 1: 1}, 'inhomogeneous_general_quadratic')
assert classify_diop(x*y + z + w + 42) == (
[w, x, y, z], {x*y: 1, w: 1, 1: 42, z: 1}, 'inhomogeneous_general_quadratic')
assert classify_diop(x*y + z*w) == (
[w, x, y, z], {x*y: 1, w*z: 1}, 'homogeneous_general_quadratic')
assert classify_diop(x*y**2 + 1) == (
[x, y], {x*y**2: 1, 1: 1}, 'cubic_thue')
assert classify_diop(x**4 + y**4 + z**4 - (1 + 16 + 81)) == (
[x, y, z], {1: -98, x**4: 1, z**4: 1, y**4: 1}, 'general_sum_of_even_powers')
def test_linear():
assert diop_solve(x) == (0,)
assert diop_solve(1*x) == (0,)
assert diop_solve(3*x) == (0,)
assert diop_solve(x + 1) == (-1,)
assert diop_solve(2*x + 1) == (None,)
assert diop_solve(2*x + 4) == (-2,)
assert diop_solve(y + x) == (t_0, -t_0)
assert diop_solve(y + x + 0) == (t_0, -t_0)
assert diop_solve(y + x - 0) == (t_0, -t_0)
assert diop_solve(0*x - y - 5) == (-5,)
assert diop_solve(3*y + 2*x - 5) == (3*t_0 - 5, -2*t_0 + 5)
assert diop_solve(2*x - 3*y - 5) == (3*t_0 - 5, 2*t_0 - 5)
assert diop_solve(-2*x - 3*y - 5) == (3*t_0 + 5, -2*t_0 - 5)
assert diop_solve(7*x + 5*y) == (5*t_0, -7*t_0)
assert diop_solve(2*x + 4*y) == (2*t_0, -t_0)
assert diop_solve(4*x + 6*y - 4) == (3*t_0 - 2, -2*t_0 + 2)
assert diop_solve(4*x + 6*y - 3) == (None, None)
assert diop_solve(0*x + 3*y - 4*z + 5) == (4*t_0 + 5, 3*t_0 + 5)
assert diop_solve(4*x + 3*y - 4*z + 5) == (t_0, 8*t_0 + 4*t_1 + 5, 7*t_0 + 3*t_1 + 5)
assert diop_solve(4*x + 3*y - 4*z + 5, None) == (0, 5, 5)
assert diop_solve(4*x + 2*y + 8*z - 5) == (None, None, None)
assert diop_solve(5*x + 7*y - 2*z - 6) == (t_0, -3*t_0 + 2*t_1 + 6, -8*t_0 + 7*t_1 + 18)
assert diop_solve(3*x - 6*y + 12*z - 9) == (2*t_0 + 3, t_0 + 2*t_1, t_1)
assert diop_solve(6*w + 9*x + 20*y - z) == (t_0, t_1, t_1 + t_2, 6*t_0 + 29*t_1 + 20*t_2)
# to ignore constant factors, use diophantine
raises(TypeError, lambda: diop_solve(x/2))
def test_quadratic_simple_hyperbolic_case():
# Simple Hyperbolic case: A = C = 0 and B != 0
assert diop_solve(3*x*y + 34*x - 12*y + 1) == \
set([(-133, -11), (5, -57)])
assert diop_solve(6*x*y + 2*x + 3*y + 1) == set([])
assert diop_solve(-13*x*y + 2*x - 4*y - 54) == set([(27, 0)])
assert diop_solve(-27*x*y - 30*x - 12*y - 54) == set([(-14, -1)])
assert diop_solve(2*x*y + 5*x + 56*y + 7) == set([(-161, -3),\
(-47,-6), (-35, -12), (-29, -69),\
(-27, 64), (-21, 7),(-9, 1),\
(105, -2)])
assert diop_solve(6*x*y + 9*x + 2*y + 3) == set([])
assert diop_solve(x*y + x + y + 1) == set([(-1, t), (t, -1)])
assert diophantine(48*x*y)
def test_quadratic_elliptical_case():
# Elliptical case: B**2 - 4AC < 0
# Two test cases highlighted require lot of memory due to quadratic_congruence() method.
# This above method should be replaced by Pernici's square_mod() method when his PR gets merged.
#assert diop_solve(42*x**2 + 8*x*y + 15*y**2 + 23*x + 17*y - 4915) == set([(-11, -1)])
assert diop_solve(4*x**2 + 3*y**2 + 5*x - 11*y + 12) == set([])
assert diop_solve(x**2 + y**2 + 2*x + 2*y + 2) == set([(-1, -1)])
#assert diop_solve(15*x**2 - 9*x*y + 14*y**2 - 23*x - 14*y - 4950) == set([(-15, 6)])
assert diop_solve(10*x**2 + 12*x*y + 12*y**2 - 34) == \
set([(-1, -1), (-1, 2), (1, -2), (1, 1)])
def test_quadratic_parabolic_case():
# Parabolic case: B**2 - 4AC = 0
assert check_solutions(8*x**2 - 24*x*y + 18*y**2 + 5*x + 7*y + 16)
assert check_solutions(8*x**2 - 24*x*y + 18*y**2 + 6*x + 12*y - 6)
assert check_solutions(8*x**2 + 24*x*y + 18*y**2 + 4*x + 6*y - 7)
assert check_solutions(-4*x**2 + 4*x*y - y**2 + 2*x - 3)
assert check_solutions(x**2 + 2*x*y + y**2 + 2*x + 2*y + 1)
assert check_solutions(x**2 - 2*x*y + y**2 + 2*x + 2*y + 1)
assert check_solutions(y**2 - 41*x + 40)
def test_quadratic_perfect_square():
# B**2 - 4*A*C > 0
# B**2 - 4*A*C is a perfect square
assert check_solutions(48*x*y)
assert check_solutions(4*x**2 - 5*x*y + y**2 + 2)
assert check_solutions(-2*x**2 - 3*x*y + 2*y**2 -2*x - 17*y + 25)
assert check_solutions(12*x**2 + 13*x*y + 3*y**2 - 2*x + 3*y - 12)
assert check_solutions(8*x**2 + 10*x*y + 2*y**2 - 32*x - 13*y - 23)
assert check_solutions(4*x**2 - 4*x*y - 3*y- 8*x - 3)
assert check_solutions(- 4*x*y - 4*y**2 - 3*y- 5*x - 10)
assert check_solutions(x**2 - y**2 - 2*x - 2*y)
assert check_solutions(x**2 - 9*y**2 - 2*x - 6*y)
assert check_solutions(4*x**2 - 9*y**2 - 4*x - 12*y - 3)
def test_quadratic_non_perfect_square():
# B**2 - 4*A*C is not a perfect square
# Used check_solutions() since the solutions are complex expressions involving
# square roots and exponents
assert check_solutions(x**2 - 2*x - 5*y**2)
assert check_solutions(3*x**2 - 2*y**2 - 2*x - 2*y)
assert check_solutions(x**2 - x*y - y**2 - 3*y)
assert check_solutions(x**2 - 9*y**2 - 2*x - 6*y)
def test_issue_9106():
eq = -48 - 2*x*(3*x - 1) + y*(3*y - 1)
v = (x, y)
for sol in diophantine(eq):
assert not diop_simplify(eq.xreplace(dict(zip(v, sol))))
@slow
def test_quadratic_non_perfect_slow():
assert check_solutions(8*x**2 + 10*x*y - 2*y**2 - 32*x - 13*y - 23)
# This leads to very large numbers.
# assert check_solutions(5*x**2 - 13*x*y + y**2 - 4*x - 4*y - 15)
assert check_solutions(-3*x**2 - 2*x*y + 7*y**2 - 5*x - 7)
assert check_solutions(-4 - x + 4*x**2 - y - 3*x*y - 4*y**2)
assert check_solutions(1 + 2*x + 2*x**2 + 2*y + x*y - 2*y**2)
def test_DN():
# Most of the test cases were adapted from,
# Solving the generalized Pell equation x**2 - D*y**2 = N, John P. Robertson, July 31, 2004.
# http://www.jpr2718.org/pell.pdf
# others are verified using Wolfram Alpha.
# Covers cases where D <= 0 or D > 0 and D is a square or N = 0
# Solutions are straightforward in these cases.
assert diop_DN(3, 0) == [(0, 0)]
assert diop_DN(-17, -5) == []
assert diop_DN(-19, 23) == [(2, 1)]
assert diop_DN(-13, 17) == [(2, 1)]
assert diop_DN(-15, 13) == []
assert diop_DN(0, 5) == []
assert diop_DN(0, 9) == [(3, t)]
assert diop_DN(9, 0) == [(3*t, t)]
assert diop_DN(16, 24) == []
assert diop_DN(9, 180) == [(18, 4)]
assert diop_DN(9, -180) == [(12, 6)]
assert diop_DN(7, 0) == [(0, 0)]
# When equation is x**2 + y**2 = N
# Solutions are interchangeable
assert diop_DN(-1, 5) == [(2, 1), (1, 2)]
assert diop_DN(-1, 169) == [(12, 5), (5, 12), (13, 0), (0, 13)]
# D > 0 and D is not a square
# N = 1
assert diop_DN(13, 1) == [(649, 180)]
assert diop_DN(980, 1) == [(51841, 1656)]
assert diop_DN(981, 1) == [(158070671986249, 5046808151700)]
assert diop_DN(986, 1) == [(49299, 1570)]
assert diop_DN(991, 1) == [(379516400906811930638014896080, 12055735790331359447442538767)]
assert diop_DN(17, 1) == [(33, 8)]
assert diop_DN(19, 1) == [(170, 39)]
# N = -1
assert diop_DN(13, -1) == [(18, 5)]
assert diop_DN(991, -1) == []
assert diop_DN(41, -1) == [(32, 5)]
assert diop_DN(290, -1) == [(17, 1)]
assert diop_DN(21257, -1) == [(13913102721304, 95427381109)]
assert diop_DN(32, -1) == []
# |N| > 1
# Some tests were created using calculator at
# http://www.numbertheory.org/php/patz.html
assert diop_DN(13, -4) == [(3, 1), (393, 109), (36, 10)]
# Source I referred returned (3, 1), (393, 109) and (-3, 1) as fundamental solutions
# So (-3, 1) and (393, 109) should be in the same equivalent class
assert equivalent(-3, 1, 393, 109, 13, -4) == True
assert diop_DN(13, 27) == [(220, 61), (40, 11), (768, 213), (12, 3)]
assert set(diop_DN(157, 12)) == \
set([(13, 1), (10663, 851), (579160, 46222), \
(483790960,38610722), (26277068347, 2097138361), (21950079635497, 1751807067011)])
assert diop_DN(13, 25) == [(3245, 900)]
assert diop_DN(192, 18) == []
assert diop_DN(23, 13) == [(-6, 1), (6, 1)]
assert diop_DN(167, 2) == [(13, 1)]
assert diop_DN(167, -2) == []
assert diop_DN(123, -2) == [(11, 1)]
# One calculator returned [(11, 1), (-11, 1)] but both of these are in
# the same equivalence class
assert equivalent(11, 1, -11, 1, 123, -2)
assert diop_DN(123, -23) == [(-10, 1), (10, 1)]
assert diop_DN(0, 0, t) == [(0, t)]
assert diop_DN(0, -1, t) == []
def test_bf_pell():
assert diop_bf_DN(13, -4) == [(3, 1), (-3, 1), (36, 10)]
assert diop_bf_DN(13, 27) == [(12, 3), (-12, 3), (40, 11), (-40, 11)]
assert diop_bf_DN(167, -2) == []
assert diop_bf_DN(1729, 1) == [(44611924489705, 1072885712316)]
assert diop_bf_DN(89, -8) == [(9, 1), (-9, 1)]
assert diop_bf_DN(21257, -1) == [(13913102721304, 95427381109)]
assert diop_bf_DN(340, -4) == [(756, 41)]
assert diop_bf_DN(-1, 0, t) == [(0, 0)]
assert diop_bf_DN(0, 0, t) == [(0, t)]
assert diop_bf_DN(4, 0, t) == [(2*t, t), (-2*t, t)]
assert diop_bf_DN(3, 0, t) == [(0, 0)]
assert diop_bf_DN(1, -2, t) == []
def test_length():
assert length(2, 1, 0) == 1
assert length(-2, 4, 5) == 3
assert length(-5, 4, 17) == 5
assert length(0, 4, 13) == 6
assert length(-31, 8, 613) == 69
assert length(7, 13, 11) == 23
assert length(-40, 5, 23) == 4
assert length(1, 6, 4) == 2
def is_pell_transformation_ok(eq):
"""
Test whether X*Y, X, or Y terms are present in the equation
after transforming the equation using the transformation returned
by transformation_to_pell(). If they are not present we are good.
Moreover, coefficient of X**2 should be a divisor of coefficient of
Y**2 and the constant term.
"""
A, B = transformation_to_DN(eq)
u = (A*Matrix([X, Y]) + B)[0]
v = (A*Matrix([X, Y]) + B)[1]
simplified = diop_simplify(eq.subs(zip((x, y), (u, v))))
coeff = dict([reversed(t.as_independent(*[X, Y])) for t in simplified.args])
for term in [X*Y, X, Y]:
if term in coeff.keys():
return False
for term in [X**2, Y**2, 1]:
if term not in coeff.keys():
coeff[term] = 0
if coeff[X**2] != 0:
return divisible(coeff[Y**2], coeff[X**2]) and \
divisible(coeff[1], coeff[X**2])
return True
def test_transformation_to_pell():
assert is_pell_transformation_ok(-13*x**2 - 7*x*y + y**2 + 2*x - 2*y - 14)
assert is_pell_transformation_ok(-17*x**2 + 19*x*y - 7*y**2 - 5*x - 13*y - 23)
assert is_pell_transformation_ok(x**2 - y**2 + 17)
assert is_pell_transformation_ok(-x**2 + 7*y**2 - 23)
assert is_pell_transformation_ok(25*x**2 - 45*x*y + 5*y**2 - 5*x - 10*y + 5)
assert is_pell_transformation_ok(190*x**2 + 30*x*y + y**2 - 3*y - 170*x - 130)
assert is_pell_transformation_ok(x**2 - 2*x*y -190*y**2 - 7*y - 23*x - 89)
assert is_pell_transformation_ok(15*x**2 - 9*x*y + 14*y**2 - 23*x - 14*y - 4950)
def test_find_DN():
assert find_DN(x**2 - 2*x - y**2) == (1, 1)
assert find_DN(x**2 - 3*y**2 - 5) == (3, 5)
assert find_DN(x**2 - 2*x*y - 4*y**2 - 7) == (5, 7)
assert find_DN(4*x**2 - 8*x*y - y**2 - 9) == (20, 36)
assert find_DN(7*x**2 - 2*x*y - y**2 - 12) == (8, 84)
assert find_DN(-3*x**2 + 4*x*y -y**2) == (1, 0)
assert find_DN(-13*x**2 - 7*x*y + y**2 + 2*x - 2*y -14) == (101, -7825480)
def test_ldescent():
# Equations which have solutions
u = ([(13, 23), (3, -11), (41, -113), (4, -7), (-7, 4), (91, -3), (1, 1), (1, -1),
(4, 32), (17, 13), (123689, 1), (19, -570)])
for a, b in u:
w, x, y = ldescent(a, b)
assert a*x**2 + b*y**2 == w**2
assert ldescent(-1, -1) is None
def test_diop_ternary_quadratic_normal():
assert check_solutions(234*x**2 - 65601*y**2 - z**2)
assert check_solutions(23*x**2 + 616*y**2 - z**2)
assert check_solutions(5*x**2 + 4*y**2 - z**2)
assert check_solutions(3*x**2 + 6*y**2 - 3*z**2)
assert check_solutions(x**2 + 3*y**2 - z**2)
assert check_solutions(4*x**2 + 5*y**2 - z**2)
assert check_solutions(x**2 + y**2 - z**2)
assert check_solutions(16*x**2 + y**2 - 25*z**2)
assert check_solutions(6*x**2 - y**2 + 10*z**2)
assert check_solutions(213*x**2 + 12*y**2 - 9*z**2)
assert check_solutions(34*x**2 - 3*y**2 - 301*z**2)
assert check_solutions(124*x**2 - 30*y**2 - 7729*z**2)
def is_normal_transformation_ok(eq):
A = transformation_to_normal(eq)
X, Y, Z = A*Matrix([x, y, z])
simplified = diop_simplify(eq.subs(zip((x, y, z), (X, Y, Z))))
coeff = dict([reversed(t.as_independent(*[X, Y, Z])) for t in simplified.args])
for term in [X*Y, Y*Z, X*Z]:
if term in coeff.keys():
return False
return True
def test_transformation_to_normal():
assert is_normal_transformation_ok(x**2 + 3*y**2 + z**2 - 13*x*y - 16*y*z + 12*x*z)
assert is_normal_transformation_ok(x**2 + 3*y**2 - 100*z**2)
assert is_normal_transformation_ok(x**2 + 23*y*z)
assert is_normal_transformation_ok(3*y**2 - 100*z**2 - 12*x*y)
assert is_normal_transformation_ok(x**2 + 23*x*y - 34*y*z + 12*x*z)
assert is_normal_transformation_ok(z**2 + 34*x*y - 23*y*z + x*z)
assert is_normal_transformation_ok(x**2 + y**2 + z**2 - x*y - y*z - x*z)
assert is_normal_transformation_ok(x**2 + 2*y*z + 3*z**2)
assert is_normal_transformation_ok(x*y + 2*x*z + 3*y*z)
assert is_normal_transformation_ok(2*x*z + 3*y*z)
def test_diop_ternary_quadratic():
assert check_solutions(2*x**2 + z**2 + y**2 - 4*x*y)
assert check_solutions(x**2 - y**2 - z**2 - x*y - y*z)
assert check_solutions(3*x**2 - x*y - y*z - x*z)
assert check_solutions(x**2 - y*z - x*z)
assert check_solutions(5*x**2 - 3*x*y - x*z)
assert check_solutions(4*x**2 - 5*y**2 - x*z)
assert check_solutions(3*x**2 + 2*y**2 - z**2 - 2*x*y + 5*y*z - 7*y*z)
assert check_solutions(8*x**2 - 12*y*z)
assert check_solutions(45*x**2 - 7*y**2 - 8*x*y - z**2)
assert check_solutions(x**2 - 49*y**2 - z**2 + 13*z*y -8*x*y)
assert check_solutions(90*x**2 + 3*y**2 + 5*x*y + 2*z*y + 5*x*z)
assert check_solutions(x**2 + 3*y**2 + z**2 - x*y - 17*y*z)
assert check_solutions(x**2 + 3*y**2 + z**2 - x*y - 16*y*z + 12*x*z)
assert check_solutions(x**2 + 3*y**2 + z**2 - 13*x*y - 16*y*z + 12*x*z)
assert check_solutions(x*y - 7*y*z + 13*x*z)
assert diop_ternary_quadratic_normal(x**2 + y**2 + z**2) == (None, None, None)
assert diop_ternary_quadratic_normal(x**2 + y**2) is None
raises(ValueError, lambda:
_diop_ternary_quadratic_normal((x, y, z),
{x*y: 1, x**2: 2, y**2: 3, z**2: 0}))
eq = -2*x*y - 6*x*z + 7*y**2 - 3*y*z + 4*z**2
assert diop_ternary_quadratic(eq) == (7, 2, 0)
assert diop_ternary_quadratic_normal(4*x**2 + 5*y**2 - z**2) == \
(1, 0, 2)
assert diop_ternary_quadratic(x*y + 2*y*z) == \
(-2, 0, n1)
eq = -5*x*y - 8*x*z - 3*y*z + 8*z**2
assert parametrize_ternary_quadratic(eq) == \
(64*p**2 - 24*p*q, -64*p*q + 64*q**2, 40*p*q)
# this cannot be tested with diophantine because it will
# factor into a product
assert diop_solve(x*y + 2*y*z) == (-4*p*q, -2*n1*p**2 + 2*p**2, 2*p*q)
def test_square_factor():
assert square_factor(1) == square_factor(-1) == 1
assert square_factor(0) == 1
assert square_factor(5) == square_factor(-5) == 1
assert square_factor(4) == square_factor(-4) == 2
assert square_factor(12) == square_factor(-12) == 2
assert square_factor(6) == 1
assert square_factor(18) == 3
assert square_factor(52) == 2
assert square_factor(49) == 7
assert square_factor(392) == 14
assert square_factor(factorint(-12)) == 2
def test_parametrize_ternary_quadratic():
assert check_solutions(x**2 + y**2 - z**2)
assert check_solutions(x**2 + 2*x*y + z**2)
assert check_solutions(234*x**2 - 65601*y**2 - z**2)
assert check_solutions(3*x**2 + 2*y**2 - z**2 - 2*x*y + 5*y*z - 7*y*z)
assert check_solutions(x**2 - y**2 - z**2)
assert check_solutions(x**2 - 49*y**2 - z**2 + 13*z*y - 8*x*y)
assert check_solutions(8*x*y + z**2)
assert check_solutions(124*x**2 - 30*y**2 - 7729*z**2)
assert check_solutions(236*x**2 - 225*y**2 - 11*x*y - 13*y*z - 17*x*z)
assert check_solutions(90*x**2 + 3*y**2 + 5*x*y + 2*z*y + 5*x*z)
assert check_solutions(124*x**2 - 30*y**2 - 7729*z**2)
def test_no_square_ternary_quadratic():
assert check_solutions(2*x*y + y*z - 3*x*z)
assert check_solutions(189*x*y - 345*y*z - 12*x*z)
assert check_solutions(23*x*y + 34*y*z)
assert check_solutions(x*y + y*z + z*x)
assert check_solutions(23*x*y + 23*y*z + 23*x*z)
def test_descent():
u = ([(13, 23), (3, -11), (41, -113), (91, -3), (1, 1), (1, -1), (17, 13), (123689, 1), (19, -570)])
for a, b in u:
w, x, y = descent(a, b)
assert a*x**2 + b*y**2 == w**2
# the docstring warns against bad input, so these are expected results
# - can't both be negative
raises(TypeError, lambda: descent(-1, -3))
# A can't be zero unless B != 1
raises(ZeroDivisionError, lambda: descent(0, 3))
# supposed to be square-free
raises(TypeError, lambda: descent(4, 3))
def test_diophantine():
assert check_solutions((x - y)*(y - z)*(z - x))
assert check_solutions((x - y)*(x**2 + y**2 - z**2))
assert check_solutions((x - 3*y + 7*z)*(x**2 + y**2 - z**2))
assert check_solutions((x**2 - 3*y**2 - 1))
assert check_solutions(y**2 + 7*x*y)
assert check_solutions(x**2 - 3*x*y + y**2)
assert check_solutions(z*(x**2 - y**2 - 15))
assert check_solutions(x*(2*y - 2*z + 5))
assert check_solutions((x**2 - 3*y**2 - 1)*(x**2 - y**2 - 15))
assert check_solutions((x**2 - 3*y**2 - 1)*(y - 7*z))
assert check_solutions((x**2 + y**2 - z**2)*(x - 7*y - 3*z + 4*w))
# Following test case caused problems in parametric representation
# But this can be solved by factroing out y.
# No need to use methods for ternary quadratic equations.
assert check_solutions(y**2 - 7*x*y + 4*y*z)
assert check_solutions(x**2 - 2*x + 1)
assert diophantine(x - y) == diophantine(Eq(x, y))
assert diophantine(3*x*pi - 2*y*pi) == set([(2*t_0, 3*t_0)])
eq = x**2 + y**2 + z**2 - 14
base_sol = set([(1, 2, 3)])
assert diophantine(eq) == base_sol
complete_soln = set(signed_permutations(base_sol.pop()))
assert diophantine(eq, permute=True) == complete_soln
assert diophantine(x**2 + 15*x/14 - 3) == set()
# test issue 11049
eq = 92*x**2 - 99*y**2 - z**2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \
(9, 7, 51)
assert diophantine(eq) == set([(
891*p**2 + 9*q**2, -693*p**2 - 102*p*q + 7*q**2,
5049*p**2 - 1386*p*q - 51*q**2)])
eq = 2*x**2 + 2*y**2 - z**2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \
(1, 1, 2)
assert diophantine(eq) == set([(
2*p**2 - q**2, -2*p**2 + 4*p*q - q**2,
4*p**2 - 4*p*q + 2*q**2)])
eq = 411*x**2+57*y**2-221*z**2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \
(2021, 2645, 3066)
assert diophantine(eq) == \
set([(115197*p**2 - 446641*q**2, -150765*p**2 + 1355172*p*q -
584545*q**2, 174762*p**2 - 301530*p*q + 677586*q**2)])
eq = 573*x**2+267*y**2-984*z**2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \
(49, 233, 127)
assert diophantine(eq) == \
set([(4361*p**2 - 16072*q**2, -20737*p**2 + 83312*p*q - 76424*q**2,
11303*p**2 - 41474*p*q + 41656*q**2)])
# this produces factors during reconstruction
eq = x**2 + 3*y**2 - 12*z**2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \
(0, 2, 1)
assert diophantine(eq) == \
set([(24*p*q, 2*p**2 - 24*q**2, p**2 + 12*q**2)])
# solvers have not been written for every type
raises(NotImplementedError, lambda: diophantine(x*y**2 + 1))
# rational expressions
assert diophantine(1/x) == set()
assert diophantine(1/x + 1/y - S.Half)
set([(6, 3), (-2, 1), (4, 4), (1, -2), (3, 6)])
assert diophantine(x**2 + y**2 +3*x- 5, permute=True) == \
set([(-1, 1), (-4, -1), (1, -1), (1, 1), (-4, 1), (-1, -1), (4, 1), (4, -1)])
def test_general_pythagorean():
from sympy.abc import a, b, c, d, e
assert check_solutions(a**2 + b**2 + c**2 - d**2)
assert check_solutions(a**2 + 4*b**2 + 4*c**2 - d**2)
assert check_solutions(9*a**2 + 4*b**2 + 4*c**2 - d**2)
assert check_solutions(9*a**2 + 4*b**2 - 25*d**2 + 4*c**2 )
assert check_solutions(9*a**2 - 16*d**2 + 4*b**2 + 4*c**2)
assert check_solutions(-e**2 + 9*a**2 + 4*b**2 + 4*c**2 + 25*d**2)
assert check_solutions(16*a**2 - b**2 + 9*c**2 + d**2 + 25*e**2)
def test_diop_general_sum_of_squares_quick():
for i in range(3, 10):
assert check_solutions(sum(i**2 for i in symbols(':%i' % i)) - i)
raises(ValueError, lambda: _diop_general_sum_of_squares((x, y), 2))
assert _diop_general_sum_of_squares((x, y, z), -2) == set()
eq = x**2 + y**2 + z**2 - (1 + 4 + 9)
assert diop_general_sum_of_squares(eq) == \
set([(1, 2, 3)])
eq = u**2 + v**2 + x**2 + y**2 + z**2 - 1313
assert len(diop_general_sum_of_squares(eq, 3)) == 3
# issue 11016
var = symbols(':5') + (symbols('6', negative=True),)
eq = Add(*[i**2 for i in var]) - 112
base_soln = set(
[(0, 1, 1, 5, 6, -7), (1, 1, 1, 3, 6, -8), (2, 3, 3, 4, 5, -7),
(0, 1, 1, 1, 3, -10), (0, 0, 4, 4, 4, -8), (1, 2, 3, 3, 5, -8),
(0, 1, 2, 3, 7, -7), (2, 2, 4, 4, 6, -6), (1, 1, 3, 4, 6, -7),
(0, 2, 3, 3, 3, -9), (0, 0, 2, 2, 2, -10), (1, 1, 2, 3, 4, -9),
(0, 1, 1, 2, 5, -9), (0, 0, 2, 6, 6, -6), (1, 3, 4, 5, 5, -6),
(0, 2, 2, 2, 6, -8), (0, 3, 3, 3, 6, -7), (0, 2, 3, 5, 5, -7),
(0, 1, 5, 5, 5, -6)])
assert diophantine(eq) == base_soln
assert len(diophantine(eq, permute=True)) == 196800
# handle negated squares with signsimp
assert diophantine(12 - x**2 - y**2 - z**2) == set([(2, 2, 2)])
# diophantine handles simplification, so classify_diop should
# not have to look for additional patterns that are removed
# by diophantine
eq = a**2 + b**2 + c**2 + d**2 - 4
raises(NotImplementedError, lambda: classify_diop(-eq))
def test_diop_partition():
for n in [8, 10]:
for k in range(1, 8):
for p in partition(n, k):
assert len(p) == k
assert [p for p in partition(3, 5)] == []
assert [list(p) for p in partition(3, 5, 1)] == [
[0, 0, 0, 0, 3], [0, 0, 0, 1, 2], [0, 0, 1, 1, 1]]
assert list(partition(0)) == [()]
assert list(partition(1, 0)) == [()]
assert [list(i) for i in partition(3)] == [[1, 1, 1], [1, 2], [3]]
def test_prime_as_sum_of_two_squares():
for i in [5, 13, 17, 29, 37, 41, 2341, 3557, 34841, 64601]:
a, b = prime_as_sum_of_two_squares(i)
assert a**2 + b**2 == i
assert prime_as_sum_of_two_squares(7) is None
ans = prime_as_sum_of_two_squares(800029)
assert ans == (450, 773) and type(ans[0]) is int
def test_sum_of_three_squares():
for i in [0, 1, 2, 34, 123, 34304595905, 34304595905394941, 343045959052344,
800, 801, 802, 803, 804, 805, 806]:
a, b, c = sum_of_three_squares(i)
assert a**2 + b**2 + c**2 == i
assert sum_of_three_squares(7) is None
assert sum_of_three_squares((4**5)*15) is None
assert sum_of_three_squares(25) == (5, 0, 0)
assert sum_of_three_squares(4) == (0, 0, 2)
def test_sum_of_four_squares():
from random import randint
# this should never fail
n = randint(1, 100000000000000)
assert sum(i**2 for i in sum_of_four_squares(n)) == n
assert sum_of_four_squares(0) == (0, 0, 0, 0)
assert sum_of_four_squares(14) == (0, 1, 2, 3)
assert sum_of_four_squares(15) == (1, 1, 2, 3)
assert sum_of_four_squares(18) == (1, 2, 2, 3)
assert sum_of_four_squares(19) == (0, 1, 3, 3)
assert sum_of_four_squares(48) == (0, 4, 4, 4)
def test_power_representation():
tests = [(1729, 3, 2), (234, 2, 4), (2, 1, 2), (3, 1, 3), (5, 2, 2), (12352, 2, 4),
(32760, 2, 3)]
for test in tests:
n, p, k = test
f = power_representation(n, p, k)
while True:
try:
l = next(f)
assert len(l) == k
chk_sum = 0
for l_i in l:
chk_sum = chk_sum + l_i**p
assert chk_sum == n
except StopIteration:
break
assert list(power_representation(20, 2, 4, True)) == \
[(1, 1, 3, 3), (0, 0, 2, 4)]
raises(ValueError, lambda: list(power_representation(1.2, 2, 2)))
raises(ValueError, lambda: list(power_representation(2, 0, 2)))
raises(ValueError, lambda: list(power_representation(2, 2, 0)))
assert list(power_representation(-1, 2, 2)) == []
assert list(power_representation(1, 1, 1)) == [(1,)]
assert list(power_representation(3, 2, 1)) == []
assert list(power_representation(4, 2, 1)) == [(2,)]
assert list(power_representation(3**4, 4, 6, zeros=True)) == \
[(1, 2, 2, 2, 2, 2), (0, 0, 0, 0, 0, 3)]
assert list(power_representation(3**4, 4, 5, zeros=False)) == []
assert list(power_representation(-2, 3, 2)) == [(-1, -1)]
assert list(power_representation(-2, 4, 2)) == []
assert list(power_representation(0, 3, 2, True)) == [(0, 0)]
assert list(power_representation(0, 3, 2, False)) == []
# when we are dealing with squares, do feasibility checks
assert len(list(power_representation(4**10*(8*10 + 7), 2, 3))) == 0
# there will be a recursion error if these aren't recognized
big = 2**30
for i in [13, 10, 7, 5, 4, 2, 1]:
assert list(sum_of_powers(big, 2, big - i)) == []
def test_assumptions():
"""
Test whether diophantine respects the assumptions.
"""
#Test case taken from the below so question regarding assumptions in diophantine module
#https://stackoverflow.com/questions/23301941/how-can-i-declare-natural-symbols-with-sympy
m, n = symbols('m n', integer=True, positive=True)
diof = diophantine(n ** 2 + m * n - 500)
assert diof == set([(5, 20), (40, 10), (95, 5), (121, 4), (248, 2), (499, 1)])
a, b = symbols('a b', integer=True, positive=False)
diof = diophantine(a*b + 2*a + 3*b - 6)
assert diof == set([(-15, -3), (-9, -4), (-7, -5), (-6, -6), (-5, -8), (-4, -14)])
def check_solutions(eq):
"""
Determines whether solutions returned by diophantine() satisfy the original
equation. Hope to generalize this so we can remove functions like check_ternay_quadratic,
check_solutions_normal, check_solutions()
"""
s = diophantine(eq)
factors = Mul.make_args(eq)
var = list(eq.free_symbols)
var.sort(key=default_sort_key)
while s:
solution = s.pop()
for f in factors:
if diop_simplify(f.subs(zip(var, solution))) == 0:
break
else:
return False
return True
def test_diopcoverage():
eq = (2*x + y + 1)**2
assert diop_solve(eq) == set([(t_0, -2*t_0 - 1)])
eq = 2*x**2 + 6*x*y + 12*x + 4*y**2 + 18*y + 18
assert diop_solve(eq) == set([(t_0, -t_0 - 3), (2*t_0 - 3, -t_0)])
assert diop_quadratic(x + y**2 - 3) == set([(-t**2 + 3, -t)])
assert diop_linear(x + y - 3) == (t_0, 3 - t_0)
assert base_solution_linear(0, 1, 2, t=None) == (0, 0)
ans = (3*t - 1, -2*t + 1)
assert base_solution_linear(4, 8, 12, t) == ans
assert base_solution_linear(4, 8, 12, t=None) == tuple(_.subs(t, 0) for _ in ans)
assert cornacchia(1, 1, 20) is None
assert cornacchia(1, 1, 5) == set([(2, 1)])
assert cornacchia(1, 2, 17) == set([(3, 2)])
raises(ValueError, lambda: reconstruct(4, 20, 1))
assert gaussian_reduce(4, 1, 3) == (1, 1)
eq = -w**2 - x**2 - y**2 + z**2
assert diop_general_pythagorean(eq) == \
diop_general_pythagorean(-eq) == \
(m1**2 + m2**2 - m3**2, 2*m1*m3,
2*m2*m3, m1**2 + m2**2 + m3**2)
assert check_param(S(3) + x/3, S(4) + x/2, S(2), x) == (None, None)
assert check_param(S(3)/2, S(4) + x, S(2), x) == (None, None)
assert check_param(S(4) + x, S(3)/2, S(2), x) == (None, None)
assert _nint_or_floor(16, 10) == 2
assert _odd(1) == (not _even(1)) == True
assert _odd(0) == (not _even(0)) == False
assert _remove_gcd(2, 4, 6) == (1, 2, 3)
raises(TypeError, lambda: _remove_gcd((2, 4, 6)))
assert sqf_normal(2 * 3**2 * 5, 2 * 5 * 11, 2 * 7**2 * 11) == \
(11, 1, 5)
# it's ok if these pass some day when the solvers are implemented
raises(NotImplementedError, lambda: diophantine(x**2 + y**2 + x*y + 2*y*z - 12))
raises(NotImplementedError, lambda: diophantine(x**3 + y**2))
assert diop_quadratic(x**2 + y**2 - 1**2 - 3**4) == \
set([(-9, -1), (-9, 1), (-1, -9), (-1, 9), (1, -9), (1, 9), (9, -1), (9, 1)])
def test_holzer():
# if the input is good, don't let it diverge in holzer()
# (but see test_fail_holzer below)
assert holzer(2, 7, 13, 4, 79, 23) == (2, 7, 13)
# None in uv condition met; solution is not Holzer reduced
# so this will hopefully change but is here for coverage
assert holzer(2, 6, 2, 1, 1, 10) == (2, 6, 2)
raises(ValueError, lambda: holzer(2, 7, 14, 4, 79, 23))
@XFAIL
def test_fail_holzer():
eq = lambda x, y, z: a*x**2 + b*y**2 - c*z**2
a, b, c = 4, 79, 23
x, y, z = xyz = 26, 1, 11
X, Y, Z = ans = 2, 7, 13
assert eq(*xyz) == 0
assert eq(*ans) == 0
assert max(a*x**2, b*y**2, c*z**2) <= a*b*c
assert max(a*X**2, b*Y**2, c*Z**2) <= a*b*c
h = holzer(x, y, z, a, b, c)
assert h == ans # it would be nice to get the smaller soln
def test_issue_9539():
assert diophantine(6*w + 9*y + 20*x - z) == \
set([(t_0, t_1, t_1 + t_2, 6*t_0 + 29*t_1 + 9*t_2)])
def test_issue_8943():
assert diophantine(
(3*(x**2 + y**2 + z**2) - 14*(x*y + y*z + z*x))) == \
set([(0, 0, 0)])
def test_diop_sum_of_even_powers():
eq = x**4 + y**4 + z**4 - 2673
assert diop_solve(eq) == set([(3, 6, 6), (2, 4, 7)])
assert diop_general_sum_of_even_powers(eq, 2) == set(
[(3, 6, 6), (2, 4, 7)])
raises(NotImplementedError, lambda: diop_general_sum_of_even_powers(-eq, 2))
neg = symbols('neg', negative=True)
eq = x**4 + y**4 + neg**4 - 2673
assert diop_general_sum_of_even_powers(eq) == set([(-3, 6, 6)])
assert diophantine(x**4 + y**4 + 2) == set()
assert diop_general_sum_of_even_powers(x**4 + y**4 - 2, limit=0) == set()
def test_sum_of_squares_powers():
tru = set([
(0, 0, 1, 1, 11), (0, 0, 5, 7, 7), (0, 1, 3, 7, 8), (0, 1, 4, 5, 9),
(0, 3, 4, 7, 7), (0, 3, 5, 5, 8), (1, 1, 2, 6, 9), (1, 1, 6, 6, 7),
(1, 2, 3, 3, 10), (1, 3, 4, 4, 9), (1, 5, 5, 6, 6), (2, 2, 3, 5, 9),
(2, 3, 5, 6, 7), (3, 3, 4, 5, 8)])
eq = u**2 + v**2 + x**2 + y**2 + z**2 - 123
ans = diop_general_sum_of_squares(eq, oo) # allow oo to be used
assert len(ans) == 14
raises(ValueError, lambda: list(sum_of_squares(10, -1)))
assert list(sum_of_squares(-10, 2)) == []
assert list(sum_of_squares(2, 3)) == []
assert list(sum_of_squares(0, 3, True)) == [(0, 0, 0)]
assert list(sum_of_squares(0, 3)) == []
assert list(sum_of_squares(4, 1)) == [(2,)]
assert list(sum_of_squares(5, 1)) == []
assert list(sum_of_squares(50, 2)) == [(5, 5), (1, 7)]
assert list(sum_of_squares(11, 5, True)) == [
(1, 1, 1, 2, 2), (0, 0, 1, 1, 3)]
assert list(sum_of_squares(8, 8)) == [(1, 1, 1, 1, 1, 1, 1, 1)]
assert [len(list(sum_of_squares(i, 5, True))) for i in range(30)] == [
1, 1, 1, 1, 2,
2, 1, 1, 2, 2,
2, 2, 2, 3, 2,
1, 3, 3, 3, 3,
4, 3, 3, 2, 2,
4, 4, 4, 4, 5]
assert [len(list(sum_of_squares(i, 5))) for i in range(30)] == [
0, 0, 0, 0, 0,
1, 0, 0, 1, 0,
0, 1, 0, 1, 1,
0, 1, 1, 0, 1,
2, 1, 1, 1, 1,
1, 1, 1, 1, 3]
for i in range(30):
s1 = set(sum_of_squares(i, 5, True))
assert not s1 or all(sum(j**2 for j in t) == i for t in s1)
s2 = set(sum_of_squares(i, 5))
assert all(sum(j**2 for j in t) == i for t in s2)
raises(ValueError, lambda: list(sum_of_powers(2, -1, 1)))
raises(ValueError, lambda: list(sum_of_powers(2, 1, -1)))
assert list(sum_of_powers(-2, 3, 2)) == [(-1, -1)]
assert list(sum_of_powers(-2, 4, 2)) == []
assert list(sum_of_powers(2, 1, 1)) == [(2,)]
assert list(sum_of_powers(2, 1, 3, True)) == [(0, 0, 2), (0, 1, 1)]
assert list(sum_of_powers(5, 1, 2, True)) == [(0, 5), (1, 4), (2, 3)]
assert list(sum_of_powers(6, 2, 2)) == []
assert list(sum_of_powers(3**5, 3, 1)) == []
assert list(sum_of_powers(3**6, 3, 1)) == [(9,)] and (9**3 == 3**6)
assert list(sum_of_powers(2**1000, 5, 2)) == []
def test__can_do_sum_of_squares():
assert _can_do_sum_of_squares(3, -1) is False
assert _can_do_sum_of_squares(-3, 1) is False
assert _can_do_sum_of_squares(0, 1)
assert _can_do_sum_of_squares(4, 1)
assert _can_do_sum_of_squares(1, 2)
assert _can_do_sum_of_squares(2, 2)
assert _can_do_sum_of_squares(3, 2) is False
def test_diophantine_permute_sign():
from sympy.abc import a, b, c, d, e
eq = a**4 + b**4 - (2**4 + 3**4)
base_sol = set([(2, 3)])
assert diophantine(eq) == base_sol
complete_soln = set(signed_permutations(base_sol.pop()))
assert diophantine(eq, permute=True) == complete_soln
eq = a**2 + b**2 + c**2 + d**2 + e**2 - 234
assert len(diophantine(eq)) == 35
assert len(diophantine(eq, permute=True)) == 62000
soln = set([(-1, -1), (-1, 2), (1, -2), (1, 1)])
assert diophantine(10*x**2 + 12*x*y + 12*y**2 - 34, permute=True) == soln
@XFAIL
def test_not_implemented():
eq = x**2 + y**4 - 1**2 - 3**4
assert diophantine(eq, syms=[x, y]) == set([(9, 1), (1, 3)])
def test_issue_9538():
eq = x - 3*y + 2
assert diophantine(eq, syms=[y,x]) == set([(t_0, 3*t_0 - 2)])
raises(TypeError, lambda: diophantine(eq, syms=set([y,x])))
|
7ef1c419c05c0e716808fca7062cd85f1974c0ded6d0624d7dd0d7c92614271d
|
from sympy import (acos, acosh, asinh, atan, cos, Derivative, diff, dsolve,
Dummy, Eq, Ne, erf, erfi, exp, Function, I, Integral, LambertW, log, O, pi,
Rational, rootof, S, simplify, sin, sqrt, Subs, Symbol, tan, asin, sinh,
Piecewise, symbols, Poly, sec, Ei, re, im)
from sympy.solvers.ode import (_undetermined_coefficients_match,
checkodesol, classify_ode, classify_sysode, constant_renumber,
constantsimp, homogeneous_order, infinitesimals, checkinfsol,
checksysodesol, solve_ics, dsolve, get_numbered_constants)
from sympy.solvers.deutils import ode_order
from sympy.utilities.pytest import XFAIL, skip, raises, slow, ON_TRAVIS
C0, C1, C2, C3, C4, C5, C6, C7, C8, C9, C10 = symbols('C0:11')
u, x, y, z = symbols('u,x:z', real=True)
f = Function('f')
g = Function('g')
h = Function('h')
# Note: the tests below may fail (but still be correct) if ODE solver,
# the integral engine, solve(), or even simplify() changes. Also, in
# differently formatted solutions, the arbitrary constants might not be
# equal. Using specific hints in tests can help to avoid this.
# Tests of order higher than 1 should run the solutions through
# constant_renumber because it will normalize it (constant_renumber causes
# dsolve() to return different results on different machines)
def test_linear_2eq_order1():
x, y, z = symbols('x, y, z', cls=Function)
k, l, m, n = symbols('k, l, m, n', Integer=True)
t = Symbol('t')
x0, y0 = symbols('x0, y0', cls=Function)
eq1 = (Eq(diff(x(t),t), 9*y(t)), Eq(diff(y(t),t), 12*x(t)))
sol1 = [Eq(x(t), 9*C1*exp(6*sqrt(3)*t) + 9*C2*exp(-6*sqrt(3)*t)), \
Eq(y(t), 6*sqrt(3)*C1*exp(6*sqrt(3)*t) - 6*sqrt(3)*C2*exp(-6*sqrt(3)*t))]
assert checksysodesol(eq1, sol1) == (True, [0, 0])
eq2 = (Eq(diff(x(t),t), 2*x(t) + 4*y(t)), Eq(diff(y(t),t), 12*x(t) + 41*y(t)))
sol2 = [Eq(x(t), 4*C1*exp(t*(sqrt(1713)/2 + S(43)/2)) + 4*C2*exp(t*(-sqrt(1713)/2 + S(43)/2))), \
Eq(y(t), C1*(S(39)/2 + sqrt(1713)/2)*exp(t*(sqrt(1713)/2 + S(43)/2)) + \
C2*(-sqrt(1713)/2 + S(39)/2)*exp(t*(-sqrt(1713)/2 + S(43)/2)))]
assert checksysodesol(eq2, sol2) == (True, [0, 0])
eq3 = (Eq(diff(x(t),t), x(t) + y(t)), Eq(diff(y(t),t), -2*x(t) + 2*y(t)))
sol3 = [Eq(x(t), (C1*cos(sqrt(7)*t/2) + C2*sin(sqrt(7)*t/2))*exp(3*t/2)), \
Eq(y(t), (C1*(-sqrt(7)*sin(sqrt(7)*t/2)/2 + cos(sqrt(7)*t/2)/2) + \
C2*(sin(sqrt(7)*t/2)/2 + sqrt(7)*cos(sqrt(7)*t/2)/2))*exp(3*t/2))]
assert checksysodesol(eq3, sol3) == (True, [0, 0])
eq4 = (Eq(diff(x(t),t), x(t) + y(t) + 9), Eq(diff(y(t),t), 2*x(t) + 5*y(t) + 23))
sol4 = [Eq(x(t), C1*exp(t*(sqrt(6) + 3)) + C2*exp(t*(-sqrt(6) + 3)) - S(22)/3), \
Eq(y(t), C1*(2 + sqrt(6))*exp(t*(sqrt(6) + 3)) + C2*(-sqrt(6) + 2)*exp(t*(-sqrt(6) + 3)) - S(5)/3)]
assert checksysodesol(eq4, sol4) == (True, [0, 0])
eq5 = (Eq(diff(x(t),t), x(t) + y(t) + 81), Eq(diff(y(t),t), -2*x(t) + y(t) + 23))
sol5 = [Eq(x(t), (C1*cos(sqrt(2)*t) + C2*sin(sqrt(2)*t))*exp(t) - S(58)/3), \
Eq(y(t), (-sqrt(2)*C1*sin(sqrt(2)*t) + sqrt(2)*C2*cos(sqrt(2)*t))*exp(t) - S(185)/3)]
assert checksysodesol(eq5, sol5) == (True, [0, 0])
eq6 = (Eq(diff(x(t),t), 5*t*x(t) + 2*y(t)), Eq(diff(y(t),t), 2*x(t) + 5*t*y(t)))
sol6 = [Eq(x(t), (C1*exp(2*t) + C2*exp(-2*t))*exp(S(5)/2*t**2)), \
Eq(y(t), (C1*exp(2*t) - C2*exp(-2*t))*exp(S(5)/2*t**2))]
s = dsolve(eq6)
assert checksysodesol(eq6, sol6) == (True, [0, 0])
eq7 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + 5*t*y(t)))
sol7 = [Eq(x(t), (C1*cos((t**3)/3) + C2*sin((t**3)/3))*exp(S(5)/2*t**2)), \
Eq(y(t), (-C1*sin((t**3)/3) + C2*cos((t**3)/3))*exp(S(5)/2*t**2))]
assert checksysodesol(eq7, sol7) == (True, [0, 0])
eq8 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + (5*t+9*t**2)*y(t)))
sol8 = [Eq(x(t), (C1*exp((sqrt(77)/2 + S(9)/2)*(t**3)/3) + \
C2*exp((-sqrt(77)/2 + S(9)/2)*(t**3)/3))*exp(S(5)/2*t**2)), \
Eq(y(t), (C1*(sqrt(77)/2 + S(9)/2)*exp((sqrt(77)/2 + S(9)/2)*(t**3)/3) + \
C2*(-sqrt(77)/2 + S(9)/2)*exp((-sqrt(77)/2 + S(9)/2)*(t**3)/3))*exp(S(5)/2*t**2))]
assert checksysodesol(eq8, sol8) == (True, [0, 0])
eq10 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), (1-t**2)*x(t) + (5*t+9*t**2)*y(t)))
sol10 = [Eq(x(t), C1*x0(t) + C2*x0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t)), \
Eq(y(t), C1*y0(t) + C2*(y0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t) + \
exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)))]
s = dsolve(eq10)
assert s == sol10 # too complicated to test with subs and simplify
# assert checksysodesol(eq10, sol10) == (True, [0, 0]) # this one fails
def test_linear_2eq_order1_nonhomog_linear():
e = [Eq(diff(f(x), x), f(x) + g(x) + 5*x),
Eq(diff(g(x), x), f(x) - g(x))]
raises(NotImplementedError, lambda: dsolve(e))
def test_linear_2eq_order1_nonhomog():
# Note: once implemented, add some tests esp. with resonance
e = [Eq(diff(f(x), x), f(x) + exp(x)),
Eq(diff(g(x), x), f(x) + g(x) + x*exp(x))]
raises(NotImplementedError, lambda: dsolve(e))
def test_linear_2eq_order1_type2_degen():
e = [Eq(diff(f(x), x), f(x) + 5),
Eq(diff(g(x), x), f(x) + 7)]
s1 = [Eq(f(x), C1*exp(x) - 5), Eq(g(x), C1*exp(x) - C2 + 2*x - 5)]
assert checksysodesol(e, s1) == (True, [0, 0])
def test_dsolve_linear_2eq_order1_diag_triangular():
e = [Eq(diff(f(x), x), f(x)),
Eq(diff(g(x), x), g(x))]
s1 = [Eq(f(x), C1*exp(x)), Eq(g(x), C2*exp(x))]
assert checksysodesol(e, s1) == (True, [0, 0])
e = [Eq(diff(f(x), x), 2*f(x)),
Eq(diff(g(x), x), 3*f(x) + 7*g(x))]
s1 = [Eq(f(x), -5*C2*exp(2*x)),
Eq(g(x), 5*C1*exp(7*x) + 3*C2*exp(2*x))]
assert checksysodesol(e, s1) == (True, [0, 0])
def test_sysode_linear_2eq_order1_type1_D_lt_0():
e = [Eq(diff(f(x), x), -9*I*f(x) - 4*g(x)),
Eq(diff(g(x), x), -4*I*g(x))]
s1 = [Eq(f(x), -4*C1*exp(-4*I*x) - 4*C2*exp(-9*I*x)), \
Eq(g(x), 5*I*C1*exp(-4*I*x))]
assert checksysodesol(e, s1) == (True, [0, 0])
def test_sysode_linear_2eq_order1_type1_D_lt_0_b_eq_0():
e = [Eq(diff(f(x), x), -9*I*f(x)),
Eq(diff(g(x), x), -4*I*g(x))]
s1 = [Eq(f(x), -5*I*C2*exp(-9*I*x)), Eq(g(x), 5*I*C1*exp(-4*I*x))]
assert checksysodesol(e, s1) == (True, [0, 0])
def test_sysode_linear_2eq_order1_many_zeros():
t = Symbol('t')
corner_cases = [(0, 0, 0, 0), (1, 0, 0, 0), (0, 1, 0, 0),
(0, 0, 1, 0), (0, 0, 0, 1), (1, 0, 0, I),
(I, 0, 0, -I), (0, I, 0, 0), (0, I, I, 0)]
s1 = [[Eq(f(t), C1), Eq(g(t), C2)],
[Eq(f(t), C1*exp(t)), Eq(g(t), -C2)],
[Eq(f(t), C1 + C2*t), Eq(g(t), C2)],
[Eq(f(t), C2), Eq(g(t), C1 + C2*t)],
[Eq(f(t), -C2), Eq(g(t), C1*exp(t))],
[Eq(f(t), C1*(1 - I)*exp(t)), Eq(g(t), C2*(-1 + I)*exp(I*t))],
[Eq(f(t), 2*I*C1*exp(I*t)), Eq(g(t), -2*I*C2*exp(-I*t))],
[Eq(f(t), I*C1 + I*C2*t), Eq(g(t), C2)],
[Eq(f(t), I*C1*exp(I*t) + I*C2*exp(-I*t)), \
Eq(g(t), I*C1*exp(I*t) - I*C2*exp(-I*t))]
]
for r, sol in zip(corner_cases, s1):
eq = [Eq(diff(f(t), t), r[0]*f(t) + r[1]*g(t)),
Eq(diff(g(t), t), r[2]*f(t) + r[3]*g(t))]
assert checksysodesol(eq, sol) == (True, [0, 0])
def test_dsolve_linsystem_symbol_piecewise():
u = Symbol('u') # XXX it's more complicated with real u
eq = (Eq(diff(f(x), x), 2*f(x) + g(x)),
Eq(diff(g(x), x), u*f(x)))
s1 = [Eq(f(x), Piecewise((C1*exp(x*(sqrt(4*u + 4)/2 + 1)) +
C2*exp(x*(-sqrt(4*u + 4)/2 + 1)), Ne(4*u + 4, 0)), ((C1 + C2*(x +
Piecewise((0, Eq(sqrt(4*u + 4)/2 + 1, 2)), (1/(-sqrt(4*u + 4)/2 + 1),
True))))*exp(x*(sqrt(4*u + 4)/2 + 1)), True))), Eq(g(x),
Piecewise((C1*(sqrt(4*u + 4)/2 - 1)*exp(x*(sqrt(4*u + 4)/2 + 1)) +
C2*(-sqrt(4*u + 4)/2 - 1)*exp(x*(-sqrt(4*u + 4)/2 + 1)), Ne(4*u + 4,
0)), ((C1*(sqrt(4*u + 4)/2 - 1) + C2*(x*(sqrt(4*u + 4)/2 - 1) +
Piecewise((1, Eq(sqrt(4*u + 4)/2 + 1, 2)), (0,
True))))*exp(x*(sqrt(4*u + 4)/2 + 1)), True)))]
assert dsolve(eq) == s1
# FIXME: assert checksysodesol(eq, s) == (True, [0, 0])
# Remove lines below when checksysodesol works
s = [(l.lhs, l.rhs) for l in s1]
for v in [0, 7, -42, 5*I, 3 + 4*I]:
assert eq[0].subs(s).subs(u, v).doit().simplify()
assert eq[1].subs(s).subs(u, v).doit().simplify()
# example from https://groups.google.com/d/msg/sympy/xmzoqW6tWaE/sf0bgQrlCgAJ
i, r1, c1, r2, c2, t = symbols('i, r1, c1, r2, c2, t')
x1 = Function('x1')
x2 = Function('x2')
eq1 = r1*c1*Derivative(x1(t), t) + x1(t) - x2(t) - r1*i
eq2 = r2*c1*Derivative(x1(t), t) + r2*c2*Derivative(x2(t), t) + x2(t) - r2*i
sol = dsolve((eq1, eq2))
# FIXME: assert checksysodesol(eq, sol) == (True, [0, 0])
# Remove line below when checksysodesol works
assert all(s.has(Piecewise) for s in sol)
@slow
def test_linear_2eq_order2():
x, y, z = symbols('x, y, z', cls=Function)
k, l, m, n = symbols('k, l, m, n', Integer=True)
t, l = symbols('t, l')
x0, y0 = symbols('x0, y0', cls=Function)
eq1 = (Eq(diff(x(t),t,t), 5*x(t) + 43*y(t)), Eq(diff(y(t),t,t), x(t) + 9*y(t)))
sol1 = [Eq(x(t), 43*C1*exp(t*rootof(l**4 - 14*l**2 + 2, 0)) + 43*C2*exp(t*rootof(l**4 - 14*l**2 + 2, 1)) + \
43*C3*exp(t*rootof(l**4 - 14*l**2 + 2, 2)) + 43*C4*exp(t*rootof(l**4 - 14*l**2 + 2, 3))), \
Eq(y(t), C1*(rootof(l**4 - 14*l**2 + 2, 0)**2 - 5)*exp(t*rootof(l**4 - 14*l**2 + 2, 0)) + \
C2*(rootof(l**4 - 14*l**2 + 2, 1)**2 - 5)*exp(t*rootof(l**4 - 14*l**2 + 2, 1)) + \
C3*(rootof(l**4 - 14*l**2 + 2, 2)**2 - 5)*exp(t*rootof(l**4 - 14*l**2 + 2, 2)) + \
C4*(rootof(l**4 - 14*l**2 + 2, 3)**2 - 5)*exp(t*rootof(l**4 - 14*l**2 + 2, 3)))]
assert dsolve(eq1) == sol1
# FIXME: assert checksysodesol(eq1, sol1) == (True, [0, 0]) # this one fails
eq2 = (Eq(diff(x(t),t,t), 8*x(t)+3*y(t)+31), Eq(diff(y(t),t,t), 9*x(t)+7*y(t)+12))
sol2 = [Eq(x(t), 3*C1*exp(t*rootof(l**4 - 15*l**2 + 29, 0)) + 3*C2*exp(t*rootof(l**4 - 15*l**2 + 29, 1)) + \
3*C3*exp(t*rootof(l**4 - 15*l**2 + 29, 2)) + 3*C4*exp(t*rootof(l**4 - 15*l**2 + 29, 3)) - S(181)/29), \
Eq(y(t), C1*(rootof(l**4 - 15*l**2 + 29, 0)**2 - 8)*exp(t*rootof(l**4 - 15*l**2 + 29, 0)) + \
C2*(rootof(l**4 - 15*l**2 + 29, 1)**2 - 8)*exp(t*rootof(l**4 - 15*l**2 + 29, 1)) + \
C3*(rootof(l**4 - 15*l**2 + 29, 2)**2 - 8)*exp(t*rootof(l**4 - 15*l**2 + 29, 2)) + \
C4*(rootof(l**4 - 15*l**2 + 29, 3)**2 - 8)*exp(t*rootof(l**4 - 15*l**2 + 29, 3)) + S(183)/29)]
assert dsolve(eq2) == sol2
# FIXME: assert checksysodesol(eq2, sol2) == (True, [0, 0]) # this one fails
eq3 = (Eq(diff(x(t),t,t) - 9*diff(y(t),t) + 7*x(t),0), Eq(diff(y(t),t,t) + 9*diff(x(t),t) + 7*y(t),0))
sol3 = [Eq(x(t), C1*cos(t*(S(9)/2 + sqrt(109)/2)) + C2*sin(t*(S(9)/2 + sqrt(109)/2)) + C3*cos(t*(-sqrt(109)/2 + S(9)/2)) + \
C4*sin(t*(-sqrt(109)/2 + S(9)/2))), Eq(y(t), -C1*sin(t*(S(9)/2 + sqrt(109)/2)) + C2*cos(t*(S(9)/2 + sqrt(109)/2)) - \
C3*sin(t*(-sqrt(109)/2 + S(9)/2)) + C4*cos(t*(-sqrt(109)/2 + S(9)/2)))]
assert dsolve(eq3) == sol3
assert checksysodesol(eq3, sol3) == (True, [0, 0])
eq4 = (Eq(diff(x(t),t,t), 9*t*diff(y(t),t)-9*y(t)), Eq(diff(y(t),t,t),7*t*diff(x(t),t)-7*x(t)))
sol4 = [Eq(x(t), C3*t + t*Integral((9*C1*exp(3*sqrt(7)*t**2/2) + 9*C2*exp(-3*sqrt(7)*t**2/2))/t**2, t)), \
Eq(y(t), C4*t + t*Integral((3*sqrt(7)*C1*exp(3*sqrt(7)*t**2/2) - 3*sqrt(7)*C2*exp(-3*sqrt(7)*t**2/2))/t**2, t))]
assert dsolve(eq4) == sol4
assert checksysodesol(eq4, sol4) == (True, [0, 0])
eq5 = (Eq(diff(x(t),t,t), (log(t)+t**2)*diff(x(t),t)+(log(t)+t**2)*3*diff(y(t),t)), Eq(diff(y(t),t,t), \
(log(t)+t**2)*2*diff(x(t),t)+(log(t)+t**2)*9*diff(y(t),t)))
sol5 = [Eq(x(t), -sqrt(22)*(C1*Integral(exp((-sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) + C2 - \
C3*Integral(exp((sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) - C4 - \
(sqrt(22) + 5)*(C1*Integral(exp((-sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) + C2) + \
(-sqrt(22) + 5)*(C3*Integral(exp((sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) + C4))/88), \
Eq(y(t), -sqrt(22)*(C1*Integral(exp((-sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) + \
C2 - C3*Integral(exp((sqrt(22) + 5)*Integral(t**2 + log(t), t)), t) - C4)/44)]
assert dsolve(eq5) == sol5
assert checksysodesol(eq5, sol5) == (True, [0, 0])
eq6 = (Eq(diff(x(t),t,t), log(t)*t*diff(y(t),t) - log(t)*y(t)), Eq(diff(y(t),t,t), log(t)*t*diff(x(t),t) - log(t)*x(t)))
sol6 = [Eq(x(t), C3*t + t*Integral((C1*exp(Integral(t*log(t), t)) + \
C2*exp(-Integral(t*log(t), t)))/t**2, t)), Eq(y(t), C4*t + t*Integral((C1*exp(Integral(t*log(t), t)) - \
C2*exp(-Integral(t*log(t), t)))/t**2, t))]
assert dsolve(eq6) == sol6
assert checksysodesol(eq6, sol6) == (True, [0, 0])
eq7 = (Eq(diff(x(t),t,t), log(t)*(t*diff(x(t),t) - x(t)) + exp(t)*(t*diff(y(t),t) - y(t))), \
Eq(diff(y(t),t,t), (t**2)*(t*diff(x(t),t) - x(t)) + (t)*(t*diff(y(t),t) - y(t))))
sol7 = [Eq(x(t), C3*t + t*Integral((C1*x0(t) + C2*x0(t)*Integral(t*exp(t)*exp(Integral(t**2, t))*\
exp(Integral(t*log(t), t))/x0(t)**2, t))/t**2, t)), Eq(y(t), C4*t + t*Integral((C1*y0(t) + \
C2*(y0(t)*Integral(t*exp(t)*exp(Integral(t**2, t))*exp(Integral(t*log(t), t))/x0(t)**2, t) + \
exp(Integral(t**2, t))*exp(Integral(t*log(t), t))/x0(t)))/t**2, t))]
assert dsolve(eq7) == sol7
# FIXME: assert checksysodesol(eq7, sol7) == (True, [0, 0])
eq8 = (Eq(diff(x(t),t,t), t*(4*x(t) + 9*y(t))), Eq(diff(y(t),t,t), t*(12*x(t) - 6*y(t))))
sol8 = ("[Eq(x(t), -sqrt(133)*((-sqrt(133) - 1)*(C2*(133*t**8/24 - t**3/6 + sqrt(133)*t**3/2 + 1) + "
"C1*t*(sqrt(133)*t**4/6 - t**3/12 + 1) + O(t**6)) - (-1 + sqrt(133))*(C2*(-sqrt(133)*t**3/6 - t**3/6 + 1) + "
"C1*t*(-sqrt(133)*t**3/12 - t**3/12 + 1) + O(t**6)) - 4*C2*(133*t**8/24 - t**3/6 + sqrt(133)*t**3/2 + 1) + "
"4*C2*(-sqrt(133)*t**3/6 - t**3/6 + 1) - 4*C1*t*(sqrt(133)*t**4/6 - t**3/12 + 1) + "
"4*C1*t*(-sqrt(133)*t**3/12 - t**3/12 + 1) + O(t**6))/3192), Eq(y(t), -sqrt(133)*(-C2*(133*t**8/24 - t**3/6 + "
"sqrt(133)*t**3/2 + 1) + C2*(-sqrt(133)*t**3/6 - t**3/6 + 1) - C1*t*(sqrt(133)*t**4/6 - t**3/12 + 1) + "
"C1*t*(-sqrt(133)*t**3/12 - t**3/12 + 1) + O(t**6))/266)]")
assert str(dsolve(eq8)) == sol8
# FIXME: assert checksysodesol(eq8, sol8) == (True, [0, 0])
eq9 = (Eq(diff(x(t),t,t), t*(4*diff(x(t),t) + 9*diff(y(t),t))), Eq(diff(y(t),t,t), t*(12*diff(x(t),t) - 6*diff(y(t),t))))
sol9 = [Eq(x(t), -sqrt(133)*(4*C1*Integral(exp((-sqrt(133) - 1)*Integral(t, t)), t) + 4*C2 - \
4*C3*Integral(exp((-1 + sqrt(133))*Integral(t, t)), t) - 4*C4 - (-1 + sqrt(133))*(C1*Integral(exp((-sqrt(133) - \
1)*Integral(t, t)), t) + C2) + (-sqrt(133) - 1)*(C3*Integral(exp((-1 + sqrt(133))*Integral(t, t)), t) + \
C4))/3192), Eq(y(t), -sqrt(133)*(C1*Integral(exp((-sqrt(133) - 1)*Integral(t, t)), t) + C2 - \
C3*Integral(exp((-1 + sqrt(133))*Integral(t, t)), t) - C4)/266)]
assert dsolve(eq9) == sol9
assert checksysodesol(eq9, sol9) == (True, [0, 0])
eq10 = (t**2*diff(x(t),t,t) + 3*t*diff(x(t),t) + 4*t*diff(y(t),t) + 12*x(t) + 9*y(t), \
t**2*diff(y(t),t,t) + 2*t*diff(x(t),t) - 5*t*diff(y(t),t) + 15*x(t) + 8*y(t))
sol10 = [Eq(x(t), -C1*(-2*sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 13 + 2*sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + \
346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))))*exp((-sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))/2 + 1 + sqrt(-284/sqrt(-346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)))/2)*log(t)) - \
C2*(-2*sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
13 - 2*sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))))*exp((-sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))/2 + 1 - sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2)*log(t)) - C3*t**(1 + sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))/2 + sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2)*(2*sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 13 + 2*sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))) - C4*t**(-sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2 + 1 + sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))/2)*(-2*sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))) + 2*sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 13)), Eq(y(t), C1*(-sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 14 + (-sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))/2 + 1 + sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2)**2 + sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))))*exp((-sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))/2 + 1 + sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2)*log(t)) + C2*(-sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 14 - sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))) + (-sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))/2 + 1 - sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2)**2)*exp((-sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))/2 + 1 - sqrt(-284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) - 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2)*log(t)) + C3*t**(1 + sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + \
2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))/2 + sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2)*(sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))) + 14 + (1 + sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3))/2 + sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + 346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)))/2)**2) + C4*t**(-sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + \
346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)))/2 + 1 + sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))/2)*(-sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + \
8 + 346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))) + (-sqrt(-2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3) + 8 + \
346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 284/sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)))/2 + 1 + sqrt(-346/(3*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + \
4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3))/2)**2 + sqrt(-346/(3*(S(4333)/4 + \
5*sqrt(70771857)/36)**(S(1)/3)) + 4 + 2*(S(4333)/4 + 5*sqrt(70771857)/36)**(S(1)/3)) + 14))]
assert dsolve(eq10) == sol10
# FIXME: assert checksysodesol(eq10, sol10) == (True, [0, 0]) # this hangs or at least takes a while...
def test_linear_3eq_order1():
x, y, z = symbols('x, y, z', cls=Function)
t = Symbol('t')
eq1 = (Eq(diff(x(t),t), 21*x(t)), Eq(diff(y(t),t), 17*x(t)+3*y(t)), Eq(diff(z(t),t), 5*x(t)+7*y(t)+9*z(t)))
sol1 = [Eq(x(t), C1*exp(21*t)), Eq(y(t), 17*C1*exp(21*t)/18 + C2*exp(3*t)), \
Eq(z(t), 209*C1*exp(21*t)/216 - 7*C2*exp(3*t)/6 + C3*exp(9*t))]
assert checksysodesol(eq1, sol1) == (True, [0, 0, 0])
eq2 = (Eq(diff(x(t),t),3*y(t)-11*z(t)),Eq(diff(y(t),t),7*z(t)-3*x(t)),Eq(diff(z(t),t),11*x(t)-7*y(t)))
sol2 = [Eq(x(t), 7*C0 + sqrt(179)*C1*cos(sqrt(179)*t) + (77*C1/3 + 130*C2/3)*sin(sqrt(179)*t)), \
Eq(y(t), 11*C0 + sqrt(179)*C2*cos(sqrt(179)*t) + (-58*C1/3 - 77*C2/3)*sin(sqrt(179)*t)), \
Eq(z(t), 3*C0 + sqrt(179)*(-7*C1/3 - 11*C2/3)*cos(sqrt(179)*t) + (11*C1 - 7*C2)*sin(sqrt(179)*t))]
assert checksysodesol(eq2, sol2) == (True, [0, 0, 0])
eq3 = (Eq(3*diff(x(t),t),4*5*(y(t)-z(t))),Eq(4*diff(y(t),t),3*5*(z(t)-x(t))),Eq(5*diff(z(t),t),3*4*(x(t)-y(t))))
sol3 = [Eq(x(t), C0 + 5*sqrt(2)*C1*cos(5*sqrt(2)*t) + (12*C1/5 + 164*C2/15)*sin(5*sqrt(2)*t)), \
Eq(y(t), C0 + 5*sqrt(2)*C2*cos(5*sqrt(2)*t) + (-51*C1/10 - 12*C2/5)*sin(5*sqrt(2)*t)), \
Eq(z(t), C0 + 5*sqrt(2)*(-9*C1/25 - 16*C2/25)*cos(5*sqrt(2)*t) + (12*C1/5 - 12*C2/5)*sin(5*sqrt(2)*t))]
assert checksysodesol(eq3, sol3) == (True, [0, 0, 0])
f = t**3 + log(t)
g = t**2 + sin(t)
eq4 = (Eq(diff(x(t),t),(4*f+g)*x(t)-f*y(t)-2*f*z(t)), Eq(diff(y(t),t),2*f*x(t)+(f+g)*y(t)-2*f*z(t)), Eq(diff(z(t),t),5*f*x(t)+f*y(t)+(-3*f+g)*z(t)))
sol4 = [Eq(x(t), (C1*exp(-2*Integral(t**3 + log(t), t)) + C2*(sqrt(3)*sin(sqrt(3)*Integral(t**3 + log(t), t))/6 \
+ cos(sqrt(3)*Integral(t**3 + log(t), t))/2) + C3*(sin(sqrt(3)*Integral(t**3 + log(t), t))/2 - \
sqrt(3)*cos(sqrt(3)*Integral(t**3 + log(t), t))/6))*exp(Integral(-t**2 - sin(t), t))), Eq(y(t), \
(C2*(sqrt(3)*sin(sqrt(3)*Integral(t**3 + log(t), t))/6 + cos(sqrt(3)*Integral(t**3 + log(t), t))/2) + \
C3*(sin(sqrt(3)*Integral(t**3 + log(t), t))/2 - sqrt(3)*cos(sqrt(3)*Integral(t**3 + log(t), t))/6))*\
exp(Integral(-t**2 - sin(t), t))), Eq(z(t), (C1*exp(-2*Integral(t**3 + log(t), t)) + C2*cos(sqrt(3)*\
Integral(t**3 + log(t), t)) + C3*sin(sqrt(3)*Integral(t**3 + log(t), t)))*exp(Integral(-t**2 - sin(t), t)))]
assert dsolve(eq4) == sol4
# FIXME: assert checksysodesol(eq4, sol4) == (True, [0, 0, 0]) # this one fails
eq5 = (Eq(diff(x(t),t),4*x(t) - z(t)),Eq(diff(y(t),t),2*x(t)+2*y(t)-z(t)),Eq(diff(z(t),t),3*x(t)+y(t)))
sol5 = [Eq(x(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t) + C3*exp(2*t)), \
Eq(y(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t)), \
Eq(z(t), 2*C1*exp(2*t) + 2*C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t) + C3*t*exp(2*t) + C3*exp(2*t))]
assert checksysodesol(eq5, sol5) == (True, [0, 0, 0])
eq6 = (Eq(diff(x(t),t),4*x(t) - y(t) - 2*z(t)),Eq(diff(y(t),t),2*x(t) + y(t)- 2*z(t)),Eq(diff(z(t),t),5*x(t)-3*z(t)))
sol6 = [Eq(x(t), C1*exp(2*t) + C2*(-sin(t)/5 + 3*cos(t)/5) + C3*(3*sin(t)/5 + cos(t)/5)),
Eq(y(t), C2*(-sin(t)/5 + 3*cos(t)/5) + C3*(3*sin(t)/5 + cos(t)/5)),
Eq(z(t), C1*exp(2*t) + C2*cos(t) + C3*sin(t))]
assert checksysodesol(eq5, sol5) == (True, [0, 0, 0])
def test_linear_3eq_order1_nonhomog():
e = [Eq(diff(f(x), x), -9*f(x) - 4*g(x)),
Eq(diff(g(x), x), -4*g(x)),
Eq(diff(h(x), x), h(x) + exp(x))]
raises(NotImplementedError, lambda: dsolve(e))
@XFAIL
def test_linear_3eq_order1_diagonal():
# code makes assumptions about coefficients being nonzero, breaks when assumptions are not true
e = [Eq(diff(f(x), x), f(x)),
Eq(diff(g(x), x), g(x)),
Eq(diff(h(x), x), h(x))]
s1 = [Eq(f(x), C1*exp(x)), Eq(g(x), C2*exp(x)), Eq(h(x), C3*exp(x))]
s = dsolve(e)
assert s == s1
assert checksysodesol(e, s1) == (True, [0, 0, 0])
def test_nonlinear_2eq_order1():
x, y, z = symbols('x, y, z', cls=Function)
t = Symbol('t')
eq1 = (Eq(diff(x(t),t),x(t)*y(t)**3), Eq(diff(y(t),t),y(t)**5))
sol1 = [
Eq(x(t), C1*exp((-1/(4*C2 + 4*t))**(-S(1)/4))),
Eq(y(t), -(-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), C1*exp(-1/(-1/(4*C2 + 4*t))**(S(1)/4))),
Eq(y(t), (-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), C1*exp(-I/(-1/(4*C2 + 4*t))**(S(1)/4))),
Eq(y(t), -I*(-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), C1*exp(I/(-1/(4*C2 + 4*t))**(S(1)/4))),
Eq(y(t), I*(-1/(4*C2 + 4*t))**(S(1)/4))]
assert dsolve(eq1) == sol1
assert checksysodesol(eq1, sol1) == (True, [0, 0])
eq2 = (Eq(diff(x(t),t), exp(3*x(t))*y(t)**3),Eq(diff(y(t),t), y(t)**5))
sol2 = [
Eq(x(t), -log(C1 - 3/(-1/(4*C2 + 4*t))**(S(1)/4))/3),
Eq(y(t), -(-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), -log(C1 + 3/(-1/(4*C2 + 4*t))**(S(1)/4))/3),
Eq(y(t), (-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), -log(C1 + 3*I/(-1/(4*C2 + 4*t))**(S(1)/4))/3),
Eq(y(t), -I*(-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), -log(C1 - 3*I/(-1/(4*C2 + 4*t))**(S(1)/4))/3),
Eq(y(t), I*(-1/(4*C2 + 4*t))**(S(1)/4))]
assert dsolve(eq2) == sol2
assert checksysodesol(eq2, sol2) == (True, [0, 0])
eq3 = (Eq(diff(x(t),t), y(t)*x(t)), Eq(diff(y(t),t), x(t)**3))
tt = S(2)/3
sol3 = [
Eq(x(t), 6**tt/(6*(-sinh(sqrt(C1)*(C2 + t)/2)/sqrt(C1))**tt)),
Eq(y(t), sqrt(C1 + C1/sinh(sqrt(C1)*(C2 + t)/2)**2)/3)]
assert dsolve(eq3) == sol3
# FIXME: assert checksysodesol(eq3, sol3) == (True, [0, 0])
eq4 = (Eq(diff(x(t),t),x(t)*y(t)*sin(t)**2), Eq(diff(y(t),t),y(t)**2*sin(t)**2))
sol4 = set([Eq(x(t), -2*exp(C1)/(C2*exp(C1) + t - sin(2*t)/2)), Eq(y(t), -2/(C1 + t - sin(2*t)/2))])
assert dsolve(eq4) == sol4
# FIXME: assert checksysodesol(eq4, sol4) == (True, [0, 0])
eq5 = (Eq(x(t),t*diff(x(t),t)+diff(x(t),t)*diff(y(t),t)), Eq(y(t),t*diff(y(t),t)+diff(y(t),t)**2))
sol5 = set([Eq(x(t), C1*C2 + C1*t), Eq(y(t), C2**2 + C2*t)])
assert dsolve(eq5) == sol5
assert checksysodesol(eq5, sol5) == (True, [0, 0])
eq6 = (Eq(diff(x(t),t),x(t)**2*y(t)**3), Eq(diff(y(t),t),y(t)**5))
sol6 = [
Eq(x(t), 1/(C1 - 1/(-1/(4*C2 + 4*t))**(S(1)/4))),
Eq(y(t), -(-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), 1/(C1 + (-1/(4*C2 + 4*t))**(-S(1)/4))),
Eq(y(t), (-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), 1/(C1 + I/(-1/(4*C2 + 4*t))**(S(1)/4))),
Eq(y(t), -I*(-1/(4*C2 + 4*t))**(S(1)/4)),
Eq(x(t), 1/(C1 - I/(-1/(4*C2 + 4*t))**(S(1)/4))),
Eq(y(t), I*(-1/(4*C2 + 4*t))**(S(1)/4))]
assert dsolve(eq6) == sol6
assert checksysodesol(eq6, sol6) == (True, [0, 0])
def test_checksysodesol():
x, y, z = symbols('x, y, z', cls=Function)
t = Symbol('t')
eq = (Eq(diff(x(t),t), 9*y(t)), Eq(diff(y(t),t), 12*x(t)))
sol = [Eq(x(t), 9*C1*exp(-6*sqrt(3)*t) + 9*C2*exp(6*sqrt(3)*t)), \
Eq(y(t), -6*sqrt(3)*C1*exp(-6*sqrt(3)*t) + 6*sqrt(3)*C2*exp(6*sqrt(3)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 2*x(t) + 4*y(t)), Eq(diff(y(t),t), 12*x(t) + 41*y(t)))
sol = [Eq(x(t), 4*C1*exp(t*(-sqrt(1713)/2 + S(43)/2)) + 4*C2*exp(t*(sqrt(1713)/2 + \
S(43)/2))), Eq(y(t), C1*(-sqrt(1713)/2 + S(39)/2)*exp(t*(-sqrt(1713)/2 + \
S(43)/2)) + C2*(S(39)/2 + sqrt(1713)/2)*exp(t*(sqrt(1713)/2 + S(43)/2)))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), x(t) + y(t)), Eq(diff(y(t),t), -2*x(t) + 2*y(t)))
sol = [Eq(x(t), (C1*sin(sqrt(7)*t/2) + C2*cos(sqrt(7)*t/2))*exp(3*t/2)), \
Eq(y(t), ((C1/2 - sqrt(7)*C2/2)*sin(sqrt(7)*t/2) + (sqrt(7)*C1/2 + \
C2/2)*cos(sqrt(7)*t/2))*exp(3*t/2))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), x(t) + y(t) + 9), Eq(diff(y(t),t), 2*x(t) + 5*y(t) + 23))
sol = [Eq(x(t), C1*exp(t*(-sqrt(6) + 3)) + C2*exp(t*(sqrt(6) + 3)) - \
S(22)/3), Eq(y(t), C1*(-sqrt(6) + 2)*exp(t*(-sqrt(6) + 3)) + C2*(2 + \
sqrt(6))*exp(t*(sqrt(6) + 3)) - S(5)/3)]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), x(t) + y(t) + 81), Eq(diff(y(t),t), -2*x(t) + y(t) + 23))
sol = [Eq(x(t), (C1*sin(sqrt(2)*t) + C2*cos(sqrt(2)*t))*exp(t) - S(58)/3), \
Eq(y(t), (sqrt(2)*C1*cos(sqrt(2)*t) - sqrt(2)*C2*sin(sqrt(2)*t))*exp(t) - S(185)/3)]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 5*t*x(t) + 2*y(t)), Eq(diff(y(t),t), 2*x(t) + 5*t*y(t)))
sol = [Eq(x(t), (C1*exp((Integral(2, t).doit())) + C2*exp(-(Integral(2, t)).doit()))*\
exp((Integral(5*t, t)).doit())), Eq(y(t), (C1*exp((Integral(2, t)).doit()) - \
C2*exp(-(Integral(2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + 5*t*y(t)))
sol = [Eq(x(t), (C1*cos((Integral(t**2, t)).doit()) + C2*sin((Integral(t**2, t)).doit()))*\
exp((Integral(5*t, t)).doit())), Eq(y(t), (-C1*sin((Integral(t**2, t)).doit()) + \
C2*cos((Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + (5*t+9*t**2)*y(t)))
sol = [Eq(x(t), (C1*exp((-sqrt(77)/2 + S(9)/2)*(Integral(t**2, t)).doit()) + \
C2*exp((sqrt(77)/2 + S(9)/2)*(Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit())), \
Eq(y(t), (C1*(-sqrt(77)/2 + S(9)/2)*exp((-sqrt(77)/2 + S(9)/2)*(Integral(t**2, t)).doit()) + \
C2*(sqrt(77)/2 + S(9)/2)*exp((sqrt(77)/2 + S(9)/2)*(Integral(t**2, t)).doit()))*exp((Integral(5*t, t)).doit()))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t,t), 5*x(t) + 43*y(t)), Eq(diff(y(t),t,t), x(t) + 9*y(t)))
root0 = -sqrt(-sqrt(47) + 7)
root1 = sqrt(-sqrt(47) + 7)
root2 = -sqrt(sqrt(47) + 7)
root3 = sqrt(sqrt(47) + 7)
sol = [Eq(x(t), 43*C1*exp(t*root0) + 43*C2*exp(t*root1) + 43*C3*exp(t*root2) + 43*C4*exp(t*root3)), \
Eq(y(t), C1*(root0**2 - 5)*exp(t*root0) + C2*(root1**2 - 5)*exp(t*root1) + \
C3*(root2**2 - 5)*exp(t*root2) + C4*(root3**2 - 5)*exp(t*root3))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t,t), 8*x(t)+3*y(t)+31), Eq(diff(y(t),t,t), 9*x(t)+7*y(t)+12))
root0 = -sqrt(-sqrt(109)/2 + S(15)/2)
root1 = sqrt(-sqrt(109)/2 + S(15)/2)
root2 = -sqrt(sqrt(109)/2 + S(15)/2)
root3 = sqrt(sqrt(109)/2 + S(15)/2)
sol = [Eq(x(t), 3*C1*exp(t*root0) + 3*C2*exp(t*root1) + 3*C3*exp(t*root2) + 3*C4*exp(t*root3) - S(181)/29), \
Eq(y(t), C1*(root0**2 - 8)*exp(t*root0) + C2*(root1**2 - 8)*exp(t*root1) + \
C3*(root2**2 - 8)*exp(t*root2) + C4*(root3**2 - 8)*exp(t*root3) + S(183)/29)]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t,t) - 9*diff(y(t),t) + 7*x(t),0), Eq(diff(y(t),t,t) + 9*diff(x(t),t) + 7*y(t),0))
sol = [Eq(x(t), C1*cos(t*(S(9)/2 + sqrt(109)/2)) + C2*sin(t*(S(9)/2 + sqrt(109)/2)) + \
C3*cos(t*(-sqrt(109)/2 + S(9)/2)) + C4*sin(t*(-sqrt(109)/2 + S(9)/2))), Eq(y(t), -C1*sin(t*(S(9)/2 + sqrt(109)/2)) \
+ C2*cos(t*(S(9)/2 + sqrt(109)/2)) - C3*sin(t*(-sqrt(109)/2 + S(9)/2)) + C4*cos(t*(-sqrt(109)/2 + S(9)/2)))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t,t), 9*t*diff(y(t),t)-9*y(t)), Eq(diff(y(t),t,t),7*t*diff(x(t),t)-7*x(t)))
I1 = sqrt(6)*7**(S(1)/4)*sqrt(pi)*erfi(sqrt(6)*7**(S(1)/4)*t/2)/2 - exp(3*sqrt(7)*t**2/2)/t
I2 = -sqrt(6)*7**(S(1)/4)*sqrt(pi)*erf(sqrt(6)*7**(S(1)/4)*t/2)/2 - exp(-3*sqrt(7)*t**2/2)/t
sol = [Eq(x(t), C3*t + t*(9*C1*I1 + 9*C2*I2)), Eq(y(t), C4*t + t*(3*sqrt(7)*C1*I1 - 3*sqrt(7)*C2*I2))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), 21*x(t)), Eq(diff(y(t),t), 17*x(t)+3*y(t)), Eq(diff(z(t),t), 5*x(t)+7*y(t)+9*z(t)))
sol = [Eq(x(t), C1*exp(21*t)), Eq(y(t), 17*C1*exp(21*t)/18 + C2*exp(3*t)), \
Eq(z(t), 209*C1*exp(21*t)/216 - 7*C2*exp(3*t)/6 + C3*exp(9*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t),t),3*y(t)-11*z(t)),Eq(diff(y(t),t),7*z(t)-3*x(t)),Eq(diff(z(t),t),11*x(t)-7*y(t)))
sol = [Eq(x(t), 7*C0 + sqrt(179)*C1*cos(sqrt(179)*t) + (77*C1/3 + 130*C2/3)*sin(sqrt(179)*t)), \
Eq(y(t), 11*C0 + sqrt(179)*C2*cos(sqrt(179)*t) + (-58*C1/3 - 77*C2/3)*sin(sqrt(179)*t)), \
Eq(z(t), 3*C0 + sqrt(179)*(-7*C1/3 - 11*C2/3)*cos(sqrt(179)*t) + (11*C1 - 7*C2)*sin(sqrt(179)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(3*diff(x(t),t),4*5*(y(t)-z(t))),Eq(4*diff(y(t),t),3*5*(z(t)-x(t))),Eq(5*diff(z(t),t),3*4*(x(t)-y(t))))
sol = [Eq(x(t), C0 + 5*sqrt(2)*C1*cos(5*sqrt(2)*t) + (12*C1/5 + 164*C2/15)*sin(5*sqrt(2)*t)), \
Eq(y(t), C0 + 5*sqrt(2)*C2*cos(5*sqrt(2)*t) + (-51*C1/10 - 12*C2/5)*sin(5*sqrt(2)*t)), \
Eq(z(t), C0 + 5*sqrt(2)*(-9*C1/25 - 16*C2/25)*cos(5*sqrt(2)*t) + (12*C1/5 - 12*C2/5)*sin(5*sqrt(2)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t),t),4*x(t) - z(t)),Eq(diff(y(t),t),2*x(t)+2*y(t)-z(t)),Eq(diff(z(t),t),3*x(t)+y(t)))
sol = [Eq(x(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t) + C3*exp(2*t)), \
Eq(y(t), C1*exp(2*t) + C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t)/2 + C3*t*exp(2*t)), \
Eq(z(t), 2*C1*exp(2*t) + 2*C2*t*exp(2*t) + C2*exp(2*t) + C3*t**2*exp(2*t) + C3*t*exp(2*t) + C3*exp(2*t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t),t),4*x(t) - y(t) - 2*z(t)),Eq(diff(y(t),t),2*x(t) + y(t)- 2*z(t)),Eq(diff(z(t),t),5*x(t)-3*z(t)))
sol = [Eq(x(t), C1*exp(2*t) + C2*(-sin(t) + 3*cos(t)) + C3*(3*sin(t) + cos(t))), \
Eq(y(t), C2*(-sin(t) + 3*cos(t)) + C3*(3*sin(t) + cos(t))), Eq(z(t), C1*exp(2*t) + 5*C2*cos(t) + 5*C3*sin(t))]
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
eq = (Eq(diff(x(t),t),x(t)*y(t)**3), Eq(diff(y(t),t),y(t)**5))
sol = [Eq(x(t), C1*exp((-1/(4*C2 + 4*t))**(-S(1)/4))), Eq(y(t), -(-1/(4*C2 + 4*t))**(S(1)/4)), \
Eq(x(t), C1*exp(-1/(-1/(4*C2 + 4*t))**(S(1)/4))), Eq(y(t), (-1/(4*C2 + 4*t))**(S(1)/4)), \
Eq(x(t), C1*exp(-I/(-1/(4*C2 + 4*t))**(S(1)/4))), Eq(y(t), -I*(-1/(4*C2 + 4*t))**(S(1)/4)), \
Eq(x(t), C1*exp(I/(-1/(4*C2 + 4*t))**(S(1)/4))), Eq(y(t), I*(-1/(4*C2 + 4*t))**(S(1)/4))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(diff(x(t),t), exp(3*x(t))*y(t)**3),Eq(diff(y(t),t), y(t)**5))
sol = [Eq(x(t), -log(C1 - 3/(-1/(4*C2 + 4*t))**(S(1)/4))/3), Eq(y(t), -(-1/(4*C2 + 4*t))**(S(1)/4)), \
Eq(x(t), -log(C1 + 3/(-1/(4*C2 + 4*t))**(S(1)/4))/3), Eq(y(t), (-1/(4*C2 + 4*t))**(S(1)/4)), \
Eq(x(t), -log(C1 + 3*I/(-1/(4*C2 + 4*t))**(S(1)/4))/3), Eq(y(t), -I*(-1/(4*C2 + 4*t))**(S(1)/4)), \
Eq(x(t), -log(C1 - 3*I/(-1/(4*C2 + 4*t))**(S(1)/4))/3), Eq(y(t), I*(-1/(4*C2 + 4*t))**(S(1)/4))]
assert checksysodesol(eq, sol) == (True, [0, 0])
eq = (Eq(x(t),t*diff(x(t),t)+diff(x(t),t)*diff(y(t),t)), Eq(y(t),t*diff(y(t),t)+diff(y(t),t)**2))
sol = set([Eq(x(t), C1*C2 + C1*t), Eq(y(t), C2**2 + C2*t)])
assert checksysodesol(eq, sol) == (True, [0, 0])
@slow
def test_nonlinear_3eq_order1():
x, y, z = symbols('x, y, z', cls=Function)
t, u = symbols('t u')
eq1 = (4*diff(x(t),t) + 2*y(t)*z(t), 3*diff(y(t),t) - z(t)*x(t), 5*diff(z(t),t) - x(t)*y(t))
sol1 = [Eq(4*Integral(1/(sqrt(-4*u**2 - 3*C1 + C2)*sqrt(-4*u**2 + 5*C1 - C2)), (u, x(t))),
C3 - sqrt(15)*t/15), Eq(3*Integral(1/(sqrt(-6*u**2 - C1 + 5*C2)*sqrt(3*u**2 + C1 - 4*C2)),
(u, y(t))), C3 + sqrt(5)*t/10), Eq(5*Integral(1/(sqrt(-10*u**2 - 3*C1 + C2)*
sqrt(5*u**2 + 4*C1 - C2)), (u, z(t))), C3 + sqrt(3)*t/6)]
assert [i.dummy_eq(j) for i, j in zip(dsolve(eq1), sol1)]
# FIXME: assert checksysodesol(eq1, sol1) == (True, [0, 0, 0])
eq2 = (4*diff(x(t),t) + 2*y(t)*z(t)*sin(t), 3*diff(y(t),t) - z(t)*x(t)*sin(t), 5*diff(z(t),t) - x(t)*y(t)*sin(t))
sol2 = [Eq(3*Integral(1/(sqrt(-6*u**2 - C1 + 5*C2)*sqrt(3*u**2 + C1 - 4*C2)), (u, x(t))), C3 +
sqrt(5)*cos(t)/10), Eq(4*Integral(1/(sqrt(-4*u**2 - 3*C1 + C2)*sqrt(-4*u**2 + 5*C1 - C2)),
(u, y(t))), C3 - sqrt(15)*cos(t)/15), Eq(5*Integral(1/(sqrt(-10*u**2 - 3*C1 + C2)*
sqrt(5*u**2 + 4*C1 - C2)), (u, z(t))), C3 + sqrt(3)*cos(t)/6)]
assert [i.dummy_eq(j) for i, j in zip(dsolve(eq2), sol2)]
# FIXME: assert checksysodesol(eq2, sol2) == (True, [0, 0, 0])
@slow
def test_checkodesol():
from sympy import Ei
# For the most part, checkodesol is well tested in the tests below.
# These tests only handle cases not checked below.
raises(ValueError, lambda: checkodesol(f(x, y).diff(x), Eq(f(x, y), x)))
raises(ValueError, lambda: checkodesol(f(x).diff(x), Eq(f(x, y),
x), f(x, y)))
assert checkodesol(f(x).diff(x), Eq(f(x, y), x)) == \
(False, -f(x).diff(x) + f(x, y).diff(x) - 1)
assert checkodesol(f(x).diff(x), Eq(f(x), x)) is not True
assert checkodesol(f(x).diff(x), Eq(f(x), x)) == (False, 1)
sol1 = Eq(f(x)**5 + 11*f(x) - 2*f(x) + x, 0)
assert checkodesol(diff(sol1.lhs, x), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x)*exp(f(x)), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x, 2), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x, 2)*exp(f(x)), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x, 3), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x, 3)*exp(f(x)), sol1) == (True, 0)
assert checkodesol(diff(sol1.lhs, x, 3), Eq(f(x), x*log(x))) == \
(False, 60*x**4*((log(x) + 1)**2 + log(x))*(
log(x) + 1)*log(x)**2 - 5*x**4*log(x)**4 - 9)
assert checkodesol(diff(exp(f(x)) + x, x)*x, Eq(exp(f(x)) + x)) == \
(True, 0)
assert checkodesol(diff(exp(f(x)) + x, x)*x, Eq(exp(f(x)) + x),
solve_for_func=False) == (True, 0)
assert checkodesol(f(x).diff(x, 2), [Eq(f(x), C1 + C2*x),
Eq(f(x), C2 + C1*x), Eq(f(x), C1*x + C2*x**2)]) == \
[(True, 0), (True, 0), (False, C2)]
assert checkodesol(f(x).diff(x, 2), set([Eq(f(x), C1 + C2*x),
Eq(f(x), C2 + C1*x), Eq(f(x), C1*x + C2*x**2)])) == \
set([(True, 0), (True, 0), (False, C2)])
assert checkodesol(f(x).diff(x) - 1/f(x)/2, Eq(f(x)**2, x)) == \
[(True, 0), (True, 0)]
assert checkodesol(f(x).diff(x) - f(x), Eq(C1*exp(x), f(x))) == (True, 0)
# Based on test_1st_homogeneous_coeff_ode2_eq3sol. Make sure that
# checkodesol tries back substituting f(x) when it can.
eq3 = x*exp(f(x)/x) + f(x) - x*f(x).diff(x)
sol3 = Eq(f(x), log(log(C1/x)**(-x)))
assert not checkodesol(eq3, sol3)[1].has(f(x))
# This case was failing intermittently depending on hash-seed:
eqn = Eq(Derivative(x*Derivative(f(x), x), x)/x, exp(x))
sol = Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))
assert checkodesol(eqn, sol, order=2, solve_for_func=False)[0]
@slow
def test_dsolve_options():
eq = x*f(x).diff(x) + f(x)
a = dsolve(eq, hint='all')
b = dsolve(eq, hint='all', simplify=False)
c = dsolve(eq, hint='all_Integral')
keys = ['1st_exact', '1st_exact_Integral', '1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_dep_div_indep',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral', '1st_linear',
'1st_linear_Integral', 'almost_linear', 'almost_linear_Integral',
'best', 'best_hint', 'default', 'lie_group',
'nth_linear_euler_eq_homogeneous', 'order',
'separable', 'separable_Integral']
Integral_keys = ['1st_exact_Integral',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral', '1st_linear_Integral',
'almost_linear_Integral', 'best', 'best_hint', 'default',
'nth_linear_euler_eq_homogeneous',
'order', 'separable_Integral']
assert sorted(a.keys()) == keys
assert a['order'] == ode_order(eq, f(x))
assert a['best'] == Eq(f(x), C1/x)
assert dsolve(eq, hint='best') == Eq(f(x), C1/x)
assert a['default'] == 'separable'
assert a['best_hint'] == 'separable'
assert not a['1st_exact'].has(Integral)
assert not a['separable'].has(Integral)
assert not a['1st_homogeneous_coeff_best'].has(Integral)
assert not a['1st_homogeneous_coeff_subs_dep_div_indep'].has(Integral)
assert not a['1st_homogeneous_coeff_subs_indep_div_dep'].has(Integral)
assert not a['1st_linear'].has(Integral)
assert a['1st_linear_Integral'].has(Integral)
assert a['1st_exact_Integral'].has(Integral)
assert a['1st_homogeneous_coeff_subs_dep_div_indep_Integral'].has(Integral)
assert a['1st_homogeneous_coeff_subs_indep_div_dep_Integral'].has(Integral)
assert a['separable_Integral'].has(Integral)
assert sorted(b.keys()) == keys
assert b['order'] == ode_order(eq, f(x))
assert b['best'] == Eq(f(x), C1/x)
assert dsolve(eq, hint='best', simplify=False) == Eq(f(x), C1/x)
assert b['default'] == 'separable'
assert b['best_hint'] == '1st_linear'
assert a['separable'] != b['separable']
assert a['1st_homogeneous_coeff_subs_dep_div_indep'] != \
b['1st_homogeneous_coeff_subs_dep_div_indep']
assert a['1st_homogeneous_coeff_subs_indep_div_dep'] != \
b['1st_homogeneous_coeff_subs_indep_div_dep']
assert not b['1st_exact'].has(Integral)
assert not b['separable'].has(Integral)
assert not b['1st_homogeneous_coeff_best'].has(Integral)
assert not b['1st_homogeneous_coeff_subs_dep_div_indep'].has(Integral)
assert not b['1st_homogeneous_coeff_subs_indep_div_dep'].has(Integral)
assert not b['1st_linear'].has(Integral)
assert b['1st_linear_Integral'].has(Integral)
assert b['1st_exact_Integral'].has(Integral)
assert b['1st_homogeneous_coeff_subs_dep_div_indep_Integral'].has(Integral)
assert b['1st_homogeneous_coeff_subs_indep_div_dep_Integral'].has(Integral)
assert b['separable_Integral'].has(Integral)
assert sorted(c.keys()) == Integral_keys
raises(ValueError, lambda: dsolve(eq, hint='notarealhint'))
raises(ValueError, lambda: dsolve(eq, hint='Liouville'))
assert dsolve(f(x).diff(x) - 1/f(x)**2, hint='all')['best'] == \
dsolve(f(x).diff(x) - 1/f(x)**2, hint='best')
assert dsolve(f(x) + f(x).diff(x) + sin(x).diff(x) + 1, f(x),
hint="1st_linear_Integral") == \
Eq(f(x), (C1 + Integral((-sin(x).diff(x) - 1)*
exp(Integral(1, x)), x))*exp(-Integral(1, x)))
def test_classify_ode():
assert classify_ode(f(x).diff(x, 2), f(x)) == \
(
'nth_algebraic',
'nth_linear_constant_coeff_homogeneous',
'nth_linear_euler_eq_homogeneous',
'Liouville',
'2nd_power_series_ordinary',
'nth_algebraic_Integral',
'Liouville_Integral',
)
assert classify_ode(f(x), f(x)) == ()
assert classify_ode(Eq(f(x).diff(x), 0), f(x)) == (
'nth_algebraic',
'separable',
'1st_linear', '1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep',
'1st_homogeneous_coeff_subs_dep_div_indep',
'1st_power_series', 'lie_group',
'nth_linear_constant_coeff_homogeneous',
'nth_linear_euler_eq_homogeneous',
'nth_algebraic_Integral',
'separable_Integral',
'1st_linear_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral')
assert classify_ode(f(x).diff(x)**2, f(x)) == (
'nth_algebraic',
'lie_group',
'nth_algebraic_Integral')
# issue 4749: f(x) should be cleared from highest derivative before classifying
a = classify_ode(Eq(f(x).diff(x) + f(x), x), f(x))
b = classify_ode(f(x).diff(x)*f(x) + f(x)*f(x) - x*f(x), f(x))
c = classify_ode(f(x).diff(x)/f(x) + f(x)/f(x) - x/f(x), f(x))
assert a == ('1st_linear',
'Bernoulli',
'almost_linear',
'1st_power_series', "lie_group",
'nth_linear_constant_coeff_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'1st_linear_Integral',
'Bernoulli_Integral',
'almost_linear_Integral',
'nth_linear_constant_coeff_variation_of_parameters_Integral')
assert b == c != ()
assert classify_ode(
2*x*f(x)*f(x).diff(x) + (1 + x)*f(x)**2 - exp(x), f(x)
) == ('Bernoulli', 'almost_linear', 'lie_group',
'Bernoulli_Integral', 'almost_linear_Integral')
assert 'Riccati_special_minus2' in \
classify_ode(2*f(x).diff(x) + f(x)**2 - f(x)/x + 3*x**(-2), f(x))
raises(ValueError, lambda: classify_ode(x + f(x, y).diff(x).diff(
y), f(x, y)))
# issue 5176
k = Symbol('k')
assert classify_ode(f(x).diff(x)/(k*f(x) + k*x*f(x)) + 2*f(x)/(k*f(x) +
k*x*f(x)) + x*f(x).diff(x)/(k*f(x) + k*x*f(x)) + z, f(x)) == \
('separable', '1st_exact', '1st_power_series', 'lie_group',
'separable_Integral', '1st_exact_Integral')
# preprocessing
ans = ('nth_algebraic', 'separable', '1st_exact', '1st_linear', 'Bernoulli',
'1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep',
'1st_homogeneous_coeff_subs_dep_div_indep',
'1st_power_series', 'lie_group',
'nth_linear_constant_coeff_undetermined_coefficients',
'nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters',
'nth_algebraic_Integral',
'separable_Integral', '1st_exact_Integral',
'1st_linear_Integral',
'Bernoulli_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral',
'nth_linear_constant_coeff_variation_of_parameters_Integral',
'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters_Integral')
# w/o f(x) given
assert classify_ode(diff(f(x) + x, x) + diff(f(x), x)) == ans
# w/ f(x) and prep=True
assert classify_ode(diff(f(x) + x, x) + diff(f(x), x), f(x),
prep=True) == ans
assert classify_ode(Eq(2*x**3*f(x).diff(x), 0), f(x)) == \
('nth_algebraic', 'separable', '1st_linear', '1st_power_series',
'lie_group', 'nth_linear_euler_eq_homogeneous',
'nth_algebraic_Integral', 'separable_Integral',
'1st_linear_Integral')
assert classify_ode(Eq(2*f(x)**3*f(x).diff(x), 0), f(x)) == \
('nth_algebraic', 'separable', '1st_power_series', 'lie_group',
'nth_algebraic_Integral', 'separable_Integral')
# test issue 13864
assert classify_ode(Eq(diff(f(x), x) - f(x)**x, 0), f(x)) == \
('1st_power_series', 'lie_group')
assert isinstance(classify_ode(Eq(f(x), 5), f(x), dict=True), dict)
def test_classify_ode_ics():
# Dummy
eq = f(x).diff(x, x) - f(x)
# Not f(0) or f'(0)
ics = {x: 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
############################
# f(0) type (AppliedUndef) #
############################
# Wrong function
ics = {g(0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Contains x
ics = {f(x): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Too many args
ics = {f(0, 0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# point contains f
# XXX: Should be NotImplementedError
ics = {f(0): f(1)}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Does not raise
ics = {f(0): 1}
classify_ode(eq, f(x), ics=ics)
#####################
# f'(0) type (Subs) #
#####################
# Wrong function
ics = {g(x).diff(x).subs(x, 0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Contains x
ics = {f(y).diff(y).subs(y, x): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Wrong variable
ics = {f(y).diff(y).subs(y, 0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Too many args
ics = {f(x, y).diff(x).subs(x, 0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Derivative wrt wrong vars
ics = {Derivative(f(x), x, y).subs(x, 0): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# point contains f
# XXX: Should be NotImplementedError
ics = {f(x).diff(x).subs(x, 0): f(0)}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Does not raise
ics = {f(x).diff(x).subs(x, 0): 1}
classify_ode(eq, f(x), ics=ics)
###########################
# f'(y) type (Derivative) #
###########################
# Wrong function
ics = {g(x).diff(x).subs(x, y): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Contains x
ics = {f(y).diff(y).subs(y, x): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Too many args
ics = {f(x, y).diff(x).subs(x, y): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Derivative wrt wrong vars
ics = {Derivative(f(x), x, z).subs(x, y): 1}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# point contains f
# XXX: Should be NotImplementedError
ics = {f(x).diff(x).subs(x, y): f(0)}
raises(ValueError, lambda: classify_ode(eq, f(x), ics=ics))
# Does not raise
ics = {f(x).diff(x).subs(x, y): 1}
classify_ode(eq, f(x), ics=ics)
def test_classify_sysode():
# Here x is assumed to be x(t) and y as y(t) for simplicity.
# Similarly diff(x,t) and diff(y,y) is assumed to be x1 and y1 respectively.
k, l, m, n = symbols('k, l, m, n', Integer=True)
k1, k2, k3, l1, l2, l3, m1, m2, m3 = symbols('k1, k2, k3, l1, l2, l3, m1, m2, m3', Integer=True)
P, Q, R, p, q, r = symbols('P, Q, R, p, q, r', cls=Function)
P1, P2, P3, Q1, Q2, R1, R2 = symbols('P1, P2, P3, Q1, Q2, R1, R2', cls=Function)
x, y, z = symbols('x, y, z', cls=Function)
t = symbols('t')
x1 = diff(x(t),t) ; y1 = diff(y(t),t) ; z1 = diff(z(t),t)
x2 = diff(x(t),t,t) ; y2 = diff(y(t),t,t) ; z2 = diff(z(t),t,t)
eq1 = (Eq(diff(x(t),t), 5*t*x(t) + 2*y(t)), Eq(diff(y(t),t), 2*x(t) + 5*t*y(t)))
sol1 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): -5*t, (1, x(t), 1): 0, (0, x(t), 1): 1, \
(1, y(t), 0): -5*t, (1, x(t), 0): -2, (0, y(t), 1): 0, (0, y(t), 0): -2, (1, y(t), 1): 1}, \
'type_of_equation': 'type3', 'func': [x(t), y(t)], 'is_linear': True, 'eq': [-5*t*x(t) - 2*y(t) + \
Derivative(x(t), t), -5*t*y(t) - 2*x(t) + Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq1) == sol1
eq2 = (Eq(x2, k*x(t) - l*y1), Eq(y2, l*x1 + k*y(t)))
sol2 = {'order': {y(t): 2, x(t): 2}, 'type_of_equation': 'type3', 'is_linear': True, 'eq': \
[-k*x(t) + l*Derivative(y(t), t) + Derivative(x(t), t, t), -k*y(t) - l*Derivative(x(t), t) + \
Derivative(y(t), t, t)], 'no_of_equation': 2, 'func_coeff': {(0, y(t), 0): 0, (0, x(t), 2): 1, \
(1, y(t), 1): 0, (1, y(t), 2): 1, (1, x(t), 2): 0, (0, y(t), 2): 0, (0, x(t), 0): -k, (1, x(t), 1): \
-l, (0, x(t), 1): 0, (0, y(t), 1): l, (1, x(t), 0): 0, (1, y(t), 0): -k}, 'func': [x(t), y(t)]}
assert classify_sysode(eq2) == sol2
eq3 = (Eq(x2+4*x1+3*y1+9*x(t)+7*y(t), 11*exp(I*t)), Eq(y2+5*x1+8*y1+3*x(t)+12*y(t), 2*exp(I*t)))
sol3 = {'no_of_equation': 2, 'func_coeff': {(1, x(t), 2): 0, (0, y(t), 2): 0, (0, x(t), 0): 9, \
(1, x(t), 1): 5, (0, x(t), 1): 4, (0, y(t), 1): 3, (1, x(t), 0): 3, (1, y(t), 0): 12, (0, y(t), 0): 7, \
(0, x(t), 2): 1, (1, y(t), 2): 1, (1, y(t), 1): 8}, 'type_of_equation': 'type4', 'func': [x(t), y(t)], \
'is_linear': True, 'eq': [9*x(t) + 7*y(t) - 11*exp(I*t) + 4*Derivative(x(t), t) + 3*Derivative(y(t), t) + \
Derivative(x(t), t, t), 3*x(t) + 12*y(t) - 2*exp(I*t) + 5*Derivative(x(t), t) + 8*Derivative(y(t), t) + \
Derivative(y(t), t, t)], 'order': {y(t): 2, x(t): 2}}
assert classify_sysode(eq3) == sol3
eq4 = (Eq((4*t**2 + 7*t + 1)**2*x2, 5*x(t) + 35*y(t)), Eq((4*t**2 + 7*t + 1)**2*y2, x(t) + 9*y(t)))
sol4 = {'no_of_equation': 2, 'func_coeff': {(1, x(t), 2): 0, (0, y(t), 2): 0, (0, x(t), 0): -5, \
(1, x(t), 1): 0, (0, x(t), 1): 0, (0, y(t), 1): 0, (1, x(t), 0): -1, (1, y(t), 0): -9, (0, y(t), 0): -35, \
(0, x(t), 2): 16*t**4 + 56*t**3 + 57*t**2 + 14*t + 1, (1, y(t), 2): 16*t**4 + 56*t**3 + 57*t**2 + 14*t + 1, \
(1, y(t), 1): 0}, 'type_of_equation': 'type10', 'func': [x(t), y(t)], 'is_linear': True, \
'eq': [(4*t**2 + 7*t + 1)**2*Derivative(x(t), t, t) - 5*x(t) - 35*y(t), (4*t**2 + 7*t + 1)**2*Derivative(y(t), t, t)\
- x(t) - 9*y(t)], 'order': {y(t): 2, x(t): 2}}
assert classify_sysode(eq4) == sol4
eq5 = (Eq(diff(x(t),t), x(t) + y(t) + 9), Eq(diff(y(t),t), 2*x(t) + 5*y(t) + 23))
sol5 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): -1, (1, x(t), 1): 0, (0, x(t), 1): 1, (1, y(t), 0): -5, \
(1, x(t), 0): -2, (0, y(t), 1): 0, (0, y(t), 0): -1, (1, y(t), 1): 1}, 'type_of_equation': 'type2', \
'func': [x(t), y(t)], 'is_linear': True, 'eq': [-x(t) - y(t) + Derivative(x(t), t) - 9, -2*x(t) - 5*y(t) + \
Derivative(y(t), t) - 23], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq5) == sol5
eq6 = (Eq(x1, exp(k*x(t))*P(x(t),y(t))), Eq(y1,r(y(t))*P(x(t),y(t))))
sol6 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): 0, (1, x(t), 1): 0, (0, x(t), 1): 1, (1, y(t), 0): 0, \
(1, x(t), 0): 0, (0, y(t), 1): 0, (0, y(t), 0): 0, (1, y(t), 1): 1}, 'type_of_equation': 'type2', 'func': \
[x(t), y(t)], 'is_linear': False, 'eq': [-P(x(t), y(t))*exp(k*x(t)) + Derivative(x(t), t), -P(x(t), \
y(t))*r(y(t)) + Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq6) == sol6
eq7 = (Eq(x1, x(t)**2+y(t)/x(t)), Eq(y1, x(t)/y(t)))
sol7 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): 0, (1, x(t), 1): 0, (0, x(t), 1): 1, (1, y(t), 0): 0, \
(1, x(t), 0): -1/y(t), (0, y(t), 1): 0, (0, y(t), 0): -1/x(t), (1, y(t), 1): 1}, 'type_of_equation': 'type3', \
'func': [x(t), y(t)], 'is_linear': False, 'eq': [-x(t)**2 + Derivative(x(t), t) - y(t)/x(t), -x(t)/y(t) + \
Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq7) == sol7
eq8 = (Eq(x1, P1(x(t))*Q1(y(t))*R(x(t),y(t),t)), Eq(y1, P1(x(t))*Q1(y(t))*R(x(t),y(t),t)))
sol8 = {'func': [x(t), y(t)], 'is_linear': False, 'type_of_equation': 'type4', 'eq': \
[-P1(x(t))*Q1(y(t))*R(x(t), y(t), t) + Derivative(x(t), t), -P1(x(t))*Q1(y(t))*R(x(t), y(t), t) + \
Derivative(y(t), t)], 'func_coeff': {(0, y(t), 1): 0, (1, y(t), 1): 1, (1, x(t), 1): 0, (0, y(t), 0): 0, \
(1, x(t), 0): 0, (0, x(t), 0): 0, (1, y(t), 0): 0, (0, x(t), 1): 1}, 'order': {y(t): 1, x(t): 1}, 'no_of_equation': 2}
assert classify_sysode(eq8) == sol8
eq9 = (Eq(x1,3*y(t)-11*z(t)),Eq(y1,7*z(t)-3*x(t)),Eq(z1,11*x(t)-7*y(t)))
sol9 = {'no_of_equation': 3, 'func_coeff': {(1, y(t), 0): 0, (2, y(t), 1): 0, (2, z(t), 1): 1, \
(0, x(t), 0): 0, (2, x(t), 1): 0, (1, x(t), 1): 0, (2, y(t), 0): 7, (0, x(t), 1): 1, (1, z(t), 1): 0, \
(0, y(t), 1): 0, (1, x(t), 0): 3, (0, z(t), 0): 11, (0, y(t), 0): -3, (1, z(t), 0): -7, (0, z(t), 1): 0, \
(2, x(t), 0): -11, (2, z(t), 0): 0, (1, y(t), 1): 1}, 'type_of_equation': 'type2', 'func': [x(t), y(t), z(t)], \
'is_linear': True, 'eq': [-3*y(t) + 11*z(t) + Derivative(x(t), t), 3*x(t) - 7*z(t) + Derivative(y(t), t), \
-11*x(t) + 7*y(t) + Derivative(z(t), t)], 'order': {z(t): 1, y(t): 1, x(t): 1}}
assert classify_sysode(eq9) == sol9
eq10 = (x2 + log(t)*(t*x1 - x(t)) + exp(t)*(t*y1 - y(t)), y2 + (t**2)*(t*x1 - x(t)) + (t)*(t*y1 - y(t)))
sol10 = {'no_of_equation': 2, 'func_coeff': {(1, x(t), 2): 0, (0, y(t), 2): 0, (0, x(t), 0): -log(t), \
(1, x(t), 1): t**3, (0, x(t), 1): t*log(t), (0, y(t), 1): t*exp(t), (1, x(t), 0): -t**2, (1, y(t), 0): -t, \
(0, y(t), 0): -exp(t), (0, x(t), 2): 1, (1, y(t), 2): 1, (1, y(t), 1): t**2}, 'type_of_equation': 'type11', \
'func': [x(t), y(t)], 'is_linear': True, 'eq': [(t*Derivative(x(t), t) - x(t))*log(t) + (t*Derivative(y(t), t) - \
y(t))*exp(t) + Derivative(x(t), t, t), t**2*(t*Derivative(x(t), t) - x(t)) + t*(t*Derivative(y(t), t) - y(t)) \
+ Derivative(y(t), t, t)], 'order': {y(t): 2, x(t): 2}}
assert classify_sysode(eq10) == sol10
eq11 = (Eq(x1,x(t)*y(t)**3), Eq(y1,y(t)**5))
sol11 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): -y(t)**3, (1, x(t), 1): 0, (0, x(t), 1): 1, \
(1, y(t), 0): 0, (1, x(t), 0): 0, (0, y(t), 1): 0, (0, y(t), 0): 0, (1, y(t), 1): 1}, 'type_of_equation': \
'type1', 'func': [x(t), y(t)], 'is_linear': False, 'eq': [-x(t)*y(t)**3 + Derivative(x(t), t), \
-y(t)**5 + Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq11) == sol11
eq12 = (Eq(x1, y(t)), Eq(y1, x(t)))
sol12 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): 0, (1, x(t), 1): 0, (0, x(t), 1): 1, (1, y(t), 0): 0, \
(1, x(t), 0): -1, (0, y(t), 1): 0, (0, y(t), 0): -1, (1, y(t), 1): 1}, 'type_of_equation': 'type1', 'func': \
[x(t), y(t)], 'is_linear': True, 'eq': [-y(t) + Derivative(x(t), t), -x(t) + Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq12) == sol12
eq13 = (Eq(x1,x(t)*y(t)*sin(t)**2), Eq(y1,y(t)**2*sin(t)**2))
sol13 = {'no_of_equation': 2, 'func_coeff': {(0, x(t), 0): -y(t)*sin(t)**2, (1, x(t), 1): 0, (0, x(t), 1): 1, \
(1, y(t), 0): 0, (1, x(t), 0): 0, (0, y(t), 1): 0, (0, y(t), 0): -x(t)*sin(t)**2, (1, y(t), 1): 1}, \
'type_of_equation': 'type4', 'func': [x(t), y(t)], 'is_linear': False, 'eq': [-x(t)*y(t)*sin(t)**2 + \
Derivative(x(t), t), -y(t)**2*sin(t)**2 + Derivative(y(t), t)], 'order': {y(t): 1, x(t): 1}}
assert classify_sysode(eq13) == sol13
eq14 = (Eq(x1, 21*x(t)), Eq(y1, 17*x(t)+3*y(t)), Eq(z1, 5*x(t)+7*y(t)+9*z(t)))
sol14 = {'no_of_equation': 3, 'func_coeff': {(1, y(t), 0): -3, (2, y(t), 1): 0, (2, z(t), 1): 1, \
(0, x(t), 0): -21, (2, x(t), 1): 0, (1, x(t), 1): 0, (2, y(t), 0): -7, (0, x(t), 1): 1, (1, z(t), 1): 0, \
(0, y(t), 1): 0, (1, x(t), 0): -17, (0, z(t), 0): 0, (0, y(t), 0): 0, (1, z(t), 0): 0, (0, z(t), 1): 0, \
(2, x(t), 0): -5, (2, z(t), 0): -9, (1, y(t), 1): 1}, 'type_of_equation': 'type1', 'func': [x(t), y(t), z(t)], \
'is_linear': True, 'eq': [-21*x(t) + Derivative(x(t), t), -17*x(t) - 3*y(t) + Derivative(y(t), t), -5*x(t) - \
7*y(t) - 9*z(t) + Derivative(z(t), t)], 'order': {z(t): 1, y(t): 1, x(t): 1}}
assert classify_sysode(eq14) == sol14
eq15 = (Eq(x1,4*x(t)+5*y(t)+2*z(t)),Eq(y1,x(t)+13*y(t)+9*z(t)),Eq(z1,32*x(t)+41*y(t)+11*z(t)))
sol15 = {'no_of_equation': 3, 'func_coeff': {(1, y(t), 0): -13, (2, y(t), 1): 0, (2, z(t), 1): 1, \
(0, x(t), 0): -4, (2, x(t), 1): 0, (1, x(t), 1): 0, (2, y(t), 0): -41, (0, x(t), 1): 1, (1, z(t), 1): 0, \
(0, y(t), 1): 0, (1, x(t), 0): -1, (0, z(t), 0): -2, (0, y(t), 0): -5, (1, z(t), 0): -9, (0, z(t), 1): 0, \
(2, x(t), 0): -32, (2, z(t), 0): -11, (1, y(t), 1): 1}, 'type_of_equation': 'type6', 'func': \
[x(t), y(t), z(t)], 'is_linear': True, 'eq': [-4*x(t) - 5*y(t) - 2*z(t) + Derivative(x(t), t), -x(t) - 13*y(t) - \
9*z(t) + Derivative(y(t), t), -32*x(t) - 41*y(t) - 11*z(t) + Derivative(z(t), t)], 'order': {z(t): 1, y(t): 1, x(t): 1}}
assert classify_sysode(eq15) == sol15
eq16 = (Eq(3*x1,4*5*(y(t)-z(t))),Eq(4*y1,3*5*(z(t)-x(t))),Eq(5*z1,3*4*(x(t)-y(t))))
sol16 = {'no_of_equation': 3, 'func_coeff': {(1, y(t), 0): 0, (2, y(t), 1): 0, (2, z(t), 1): 5, \
(0, x(t), 0): 0, (2, x(t), 1): 0, (1, x(t), 1): 0, (2, y(t), 0): 12, (0, x(t), 1): 3, (1, z(t), 1): 0, \
(0, y(t), 1): 0, (1, x(t), 0): 15, (0, z(t), 0): 20, (0, y(t), 0): -20, (1, z(t), 0): -15, (0, z(t), 1): 0, \
(2, x(t), 0): -12, (2, z(t), 0): 0, (1, y(t), 1): 4}, 'type_of_equation': 'type3', 'func': [x(t), y(t), z(t)], \
'is_linear': True, 'eq': [-20*y(t) + 20*z(t) + 3*Derivative(x(t), t), 15*x(t) - 15*z(t) + 4*Derivative(y(t), t), \
-12*x(t) + 12*y(t) + 5*Derivative(z(t), t)], 'order': {z(t): 1, y(t): 1, x(t): 1}}
assert classify_sysode(eq16) == sol16
# issue 8193: funcs parameter for classify_sysode has to actually work
assert classify_sysode(eq1, funcs=[x(t), y(t)]) == sol1
def test_solve_ics():
# Basic tests that things work from dsolve.
assert dsolve(f(x).diff(x) - 1/f(x), f(x), ics={f(1): 2}) == \
Eq(f(x), sqrt(2 * x + 2))
assert dsolve(f(x).diff(x) - f(x), f(x), ics={f(0): 1}) == Eq(f(x), exp(x))
assert dsolve(f(x).diff(x) - f(x), f(x), ics={f(x).diff(x).subs(x, 0): 1}) == Eq(f(x), exp(x))
assert dsolve(f(x).diff(x, x) + f(x), f(x), ics={f(0): 1,
f(x).diff(x).subs(x, 0): 1}) == Eq(f(x), sin(x) + cos(x))
assert dsolve([f(x).diff(x) - f(x) + g(x), g(x).diff(x) - g(x) - f(x)],
[f(x), g(x)], ics={f(0): 1, g(0): 0}) == [Eq(f(x), exp(x)*cos(x)),
Eq(g(x), exp(x)*sin(x))]
# Test cases where dsolve returns two solutions.
eq = (x**2*f(x)**2 - x).diff(x)
assert dsolve(eq, f(x), ics={f(1): 0}) == [Eq(f(x),
-sqrt(x - 1)/x), Eq(f(x), sqrt(x - 1)/x)]
assert dsolve(eq, f(x), ics={f(x).diff(x).subs(x, 1): 0}) == [Eq(f(x),
-sqrt(x - S(1)/2)/x), Eq(f(x), sqrt(x - S(1)/2)/x)]
eq = cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x)
assert dsolve(eq, f(x),
ics={f(0):1}, hint='1st_exact', simplify=False) == Eq(x*cos(f(x)) + f(x)**3/3, S(1)/3)
assert dsolve(eq, f(x),
ics={f(0):1}, hint='1st_exact', simplify=True) == Eq(x*cos(f(x)) + f(x)**3/3, S(1)/3)
assert solve_ics([Eq(f(x), C1*exp(x))], [f(x)], [C1], {f(0): 1}) == {C1: 1}
assert solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2],
{f(0): 1, f(pi/2): 1}) == {C1: 1, C2: 1}
assert solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2],
{f(0): 1, f(x).diff(x).subs(x, 0): 1}) == {C1: 1, C2: 1}
assert solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2], {f(0): 1}) == \
{C2: 1}
# Some more complicated tests Refer to PR #16098
assert dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0, f(x).diff(x).subs(x, 1):0}) == \
[Eq(f(x), 0), Eq(f(x), x ** 3 / 6 - x / 2)]
assert dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0}) == \
[Eq(f(x), 0), Eq(f(x), C2*x + x**3/6)]
K, r, f0 = symbols('K r f0')
sol = Eq(f(x), -K*f0*exp(r*x)/((K - f0)*(-f0*exp(r*x)/(K - f0) - 1)))
assert (dsolve(Eq(f(x).diff(x), r * f(x) * (1 - f(x) / K)), f(x), ics={f(0): f0})) == sol
#Order dependent issues Refer to PR #16098
assert dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(x).diff(x).subs(x,0):0, f(0):0}) == \
[Eq(f(x), 0), Eq(f(x), x ** 3 / 6)]
assert dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0, f(x).diff(x).subs(x,0):0}) == \
[Eq(f(x), 0), Eq(f(x), x ** 3 / 6)]
# XXX: Ought to be ValueError
raises(ValueError, lambda: solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2], {f(0): 1, f(pi): 1}))
# Degenerate case. f'(0) is identically 0.
raises(ValueError, lambda: solve_ics([Eq(f(x), sqrt(C1 - x**2))], [f(x)], [C1], {f(x).diff(x).subs(x, 0): 0}))
EI, q, L = symbols('EI q L')
# eq = Eq(EI*diff(f(x), x, 4), q)
sols = [Eq(f(x), C1 + C2*x + C3*x**2 + C4*x**3 + q*x**4/(24*EI))]
funcs = [f(x)]
constants = [C1, C2, C3, C4]
# Test both cases, Derivative (the default from f(x).diff(x).subs(x, L)),
# and Subs
ics1 = {f(0): 0,
f(x).diff(x).subs(x, 0): 0,
f(L).diff(L, 2): 0,
f(L).diff(L, 3): 0}
ics2 = {f(0): 0,
f(x).diff(x).subs(x, 0): 0,
Subs(f(x).diff(x, 2), x, L): 0,
Subs(f(x).diff(x, 3), x, L): 0}
solved_constants1 = solve_ics(sols, funcs, constants, ics1)
solved_constants2 = solve_ics(sols, funcs, constants, ics2)
assert solved_constants1 == solved_constants2 == {
C1: 0,
C2: 0,
C3: L**2*q/(4*EI),
C4: -L*q/(6*EI)}
def test_ode_order():
f = Function('f')
g = Function('g')
x = Symbol('x')
assert ode_order(3*x*exp(f(x)), f(x)) == 0
assert ode_order(x*diff(f(x), x) + 3*x*f(x) - sin(x)/x, f(x)) == 1
assert ode_order(x**2*f(x).diff(x, x) + x*diff(f(x), x) - f(x), f(x)) == 2
assert ode_order(diff(x*exp(f(x)), x, x), f(x)) == 2
assert ode_order(diff(x*diff(x*exp(f(x)), x, x), x), f(x)) == 3
assert ode_order(diff(f(x), x, x), g(x)) == 0
assert ode_order(diff(f(x), x, x)*diff(g(x), x), f(x)) == 2
assert ode_order(diff(f(x), x, x)*diff(g(x), x), g(x)) == 1
assert ode_order(diff(x*diff(x*exp(f(x)), x, x), x), g(x)) == 0
# issue 5835: ode_order has to also work for unevaluated derivatives
# (ie, without using doit()).
assert ode_order(Derivative(x*f(x), x), f(x)) == 1
assert ode_order(x*sin(Derivative(x*f(x)**2, x, x)), f(x)) == 2
assert ode_order(Derivative(x*Derivative(x*exp(f(x)), x, x), x), g(x)) == 0
assert ode_order(Derivative(f(x), x, x), g(x)) == 0
assert ode_order(Derivative(x*exp(f(x)), x, x), f(x)) == 2
assert ode_order(Derivative(f(x), x, x)*Derivative(g(x), x), g(x)) == 1
assert ode_order(Derivative(x*Derivative(f(x), x, x), x), f(x)) == 3
assert ode_order(
x*sin(Derivative(x*Derivative(f(x), x)**2, x, x)), f(x)) == 3
# In all tests below, checkodesol has the order option set to prevent
# superfluous calls to ode_order(), and the solve_for_func flag set to False
# because dsolve() already tries to solve for the function, unless the
# simplify=False option is set.
def test_old_ode_tests():
# These are simple tests from the old ode module
eq1 = Eq(f(x).diff(x), 0)
eq2 = Eq(3*f(x).diff(x) - 5, 0)
eq3 = Eq(3*f(x).diff(x), 5)
eq4 = Eq(9*f(x).diff(x, x) + f(x), 0)
eq5 = Eq(9*f(x).diff(x, x), f(x))
# Type: a(x)f'(x)+b(x)*f(x)+c(x)=0
eq6 = Eq(x**2*f(x).diff(x) + 3*x*f(x) - sin(x)/x, 0)
eq7 = Eq(f(x).diff(x, x) - 3*diff(f(x), x) + 2*f(x), 0)
# Type: 2nd order, constant coefficients (two real different roots)
eq8 = Eq(f(x).diff(x, x) - 4*diff(f(x), x) + 4*f(x), 0)
# Type: 2nd order, constant coefficients (two real equal roots)
eq9 = Eq(f(x).diff(x, x) + 2*diff(f(x), x) + 3*f(x), 0)
# Type: 2nd order, constant coefficients (two complex roots)
eq10 = Eq(3*f(x).diff(x) - 1, 0)
eq11 = Eq(x*f(x).diff(x) - 1, 0)
sol1 = Eq(f(x), C1)
sol2 = Eq(f(x), C1 + 5*x/3)
sol3 = Eq(f(x), C1 + 5*x/3)
sol4 = Eq(f(x), C1*sin(x/3) + C2*cos(x/3))
sol5 = Eq(f(x), C1*exp(-x/3) + C2*exp(x/3))
sol6 = Eq(f(x), (C1 - cos(x))/x**3)
sol7 = Eq(f(x), (C1 + C2*exp(x))*exp(x))
sol8 = Eq(f(x), (C1 + C2*x)*exp(2*x))
sol9 = Eq(f(x), (C1*sin(x*sqrt(2)) + C2*cos(x*sqrt(2)))*exp(-x))
sol10 = Eq(f(x), C1 + x/3)
sol11 = Eq(f(x), C1 + log(x))
assert dsolve(eq1) == sol1
assert dsolve(eq1.lhs) == sol1
assert dsolve(eq2) == sol2
assert dsolve(eq3) == sol3
assert dsolve(eq4) == sol4
assert dsolve(eq5) == sol5
assert dsolve(eq6) == sol6
assert dsolve(eq7) == sol7
assert dsolve(eq8) == sol8
assert dsolve(eq9) == sol9
assert dsolve(eq10) == sol10
assert dsolve(eq11) == sol11
assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=2, solve_for_func=False)[0]
assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
assert checkodesol(eq6, sol6, order=1, solve_for_func=False)[0]
assert checkodesol(eq7, sol7, order=2, solve_for_func=False)[0]
assert checkodesol(eq8, sol8, order=2, solve_for_func=False)[0]
assert checkodesol(eq9, sol9, order=2, solve_for_func=False)[0]
assert checkodesol(eq10, sol10, order=1, solve_for_func=False)[0]
assert checkodesol(eq11, sol11, order=1, solve_for_func=False)[0]
def test_1st_linear():
# Type: first order linear form f'(x)+p(x)f(x)=q(x)
eq = Eq(f(x).diff(x) + x*f(x), x**2)
sol = Eq(f(x), (C1 + x*exp(x**2/2)
- sqrt(2)*sqrt(pi)*erfi(sqrt(2)*x/2)/2)*exp(-x**2/2))
assert dsolve(eq, hint='1st_linear') == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_Bernoulli():
# Type: Bernoulli, f'(x) + p(x)*f(x) == q(x)*f(x)**n
eq = Eq(x*f(x).diff(x) + f(x) - f(x)**2, 0)
sol = dsolve(eq, f(x), hint='Bernoulli')
assert sol == Eq(f(x), 1/(x*(C1 + 1/x)))
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_Riccati_special_minus2():
# Type: Riccati special alpha = -2, a*dy/dx + b*y**2 + c*y/x +d/x**2
eq = 2*f(x).diff(x) + f(x)**2 - f(x)/x + 3*x**(-2)
sol = dsolve(eq, f(x), hint='Riccati_special_minus2')
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
@slow
def test_1st_exact1():
# Type: Exact differential equation, p(x,f) + q(x,f)*f' == 0,
# where dp/df == dq/dx
eq1 = sin(x)*cos(f(x)) + cos(x)*sin(f(x))*f(x).diff(x)
eq2 = (2*x*f(x) + 1)/f(x) + (f(x) - x)/f(x)**2*f(x).diff(x)
eq3 = 2*x + f(x)*cos(x) + (2*f(x) + sin(x) - sin(f(x)))*f(x).diff(x)
eq4 = cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x)
eq5 = 2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x)
sol1 = [Eq(f(x), -acos(C1/cos(x)) + 2*pi), Eq(f(x), acos(C1/cos(x)))]
sol2 = Eq(f(x), exp(C1 - x**2 + LambertW(-x*exp(-C1 + x**2))))
sol2b = Eq(log(f(x)) + x/f(x) + x**2, C1)
sol3 = Eq(f(x)*sin(x) + cos(f(x)) + x**2 + f(x)**2, C1)
sol4 = Eq(x*cos(f(x)) + f(x)**3/3, C1)
sol5 = Eq(x**2*f(x) + f(x)**3/3, C1)
assert dsolve(eq1, f(x), hint='1st_exact') == sol1
assert dsolve(eq2, f(x), hint='1st_exact') == sol2
assert dsolve(eq3, f(x), hint='1st_exact') == sol3
assert dsolve(eq4, hint='1st_exact') == sol4
assert dsolve(eq5, hint='1st_exact', simplify=False) == sol5
assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
# issue 5080 blocks the testing of this solution
# FIXME: assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
assert checkodesol(eq2, sol2b, order=1, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=1, solve_for_func=False)[0]
assert checkodesol(eq5, sol5, order=1, solve_for_func=False)[0]
@slow
@XFAIL
def test_1st_exact2():
"""
This is an exact equation that fails under the exact engine. It is caught
by first order homogeneous albeit with a much contorted solution. The
exact engine fails because of a poorly simplified integral of q(0,y)dy,
where q is the function multiplying f'. The solutions should be
Eq(sqrt(x**2+f(x)**2)**3+y**3, C1). The equation below is
equivalent, but it is so complex that checkodesol fails, and takes a long
time to do so.
"""
if ON_TRAVIS:
skip("Too slow for travis.")
eq = (x*sqrt(x**2 + f(x)**2) - (x**2*f(x)/(f(x) -
sqrt(x**2 + f(x)**2)))*f(x).diff(x))
sol = dsolve(eq)
assert sol == Eq(log(x),
C1 - 9*sqrt(1 + f(x)**2/x**2)*asinh(f(x)/x)/(-27*f(x)/x +
27*sqrt(1 + f(x)**2/x**2)) - 9*sqrt(1 + f(x)**2/x**2)*
log(1 - sqrt(1 + f(x)**2/x**2)*f(x)/x + 2*f(x)**2/x**2)/
(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2)) +
9*asinh(f(x)/x)*f(x)/(x*(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2))) +
9*f(x)*log(1 - sqrt(1 + f(x)**2/x**2)*f(x)/x + 2*f(x)**2/x**2)/
(x*(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2))))
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_separable1():
# test_separable1-5 are from Ordinary Differential Equations, Tenenbaum and
# Pollard, pg. 55
eq1 = f(x).diff(x) - f(x)
eq2 = x*f(x).diff(x) - f(x)
eq3 = f(x).diff(x) + sin(x)
eq4 = f(x)**2 + 1 - (x**2 + 1)*f(x).diff(x)
eq5 = f(x).diff(x)/tan(x) - f(x) - 2
eq6 = f(x).diff(x) * (1 - sin(f(x))) - 1
sol1 = Eq(f(x), C1*exp(x))
sol2 = Eq(f(x), C1*x)
sol3 = Eq(f(x), C1 + cos(x))
sol4 = Eq(f(x), tan(C1 + atan(x)))
sol5 = Eq(f(x), C1/cos(x) - 2)
sol6 = Eq(-x + f(x) + cos(f(x)), C1)
assert dsolve(eq1, hint='separable') == sol1
assert dsolve(eq2, hint='separable') == sol2
assert dsolve(eq3, hint='separable') == sol3
assert dsolve(eq4, hint='separable') == sol4
assert dsolve(eq5, hint='separable') == sol5
assert dsolve(eq6, hint='separable') == sol6
assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=1, solve_for_func=False)[0]
assert checkodesol(eq5, sol5, order=1, solve_for_func=False)[0]
assert checkodesol(eq6, sol6, order=1, solve_for_func=False)[0]
@slow
def test_separable2():
a = Symbol('a')
eq6 = f(x)*x**2*f(x).diff(x) - f(x)**3 - 2*x**2*f(x).diff(x)
eq7 = f(x)**2 - 1 - (2*f(x) + x*f(x))*f(x).diff(x)
eq8 = x*log(x)*f(x).diff(x) + sqrt(1 + f(x)**2)
eq9 = exp(x + 1)*tan(f(x)) + cos(f(x))*f(x).diff(x)
eq10 = (x*cos(f(x)) + x**2*sin(f(x))*f(x).diff(x) -
a**2*sin(f(x))*f(x).diff(x))
sol6 = Eq(Integral((u - 2)/u**3, (u, f(x))),
C1 + Integral(x**(-2), x))
sol7 = Eq(-log(-1 + f(x)**2)/2, C1 - log(2 + x))
sol8 = Eq(asinh(f(x)), C1 - log(log(x)))
# integrate cannot handle the integral on the lhs (cos/tan)
sol9 = Eq(Integral(cos(u)/tan(u), (u, f(x))),
C1 + Integral(-exp(1)*exp(x), x))
sol10 = Eq(-log(cos(f(x))), C1 - log(- a**2 + x**2)/2)
assert dsolve(eq6, hint='separable_Integral').dummy_eq(sol6)
assert dsolve(eq7, hint='separable', simplify=False) == sol7
assert dsolve(eq8, hint='separable', simplify=False) == sol8
assert dsolve(eq9, hint='separable_Integral').dummy_eq(sol9)
assert dsolve(eq10, hint='separable', simplify=False) == sol10
assert checkodesol(eq6, sol6, order=1, solve_for_func=False)[0]
assert checkodesol(eq7, sol7, order=1, solve_for_func=False)[0]
assert checkodesol(eq8, sol8, order=1, solve_for_func=False)[0]
assert checkodesol(eq9, sol9, order=1, solve_for_func=False)[0]
assert checkodesol(eq10, sol10, order=1, solve_for_func=False)[0]
def test_separable3():
eq11 = f(x).diff(x) - f(x)*tan(x)
eq12 = (x - 1)*cos(f(x))*f(x).diff(x) - 2*x*sin(f(x))
eq13 = f(x).diff(x) - f(x)*log(f(x))/tan(x)
sol11 = Eq(f(x), C1/cos(x))
sol12 = Eq(log(sin(f(x))), C1 + 2*x + 2*log(x - 1))
sol13 = Eq(log(log(f(x))), C1 + log(sin(x)))
assert dsolve(eq11, hint='separable') == sol11
assert dsolve(eq12, hint='separable', simplify=False) == sol12
assert dsolve(eq13, hint='separable', simplify=False) == sol13
assert checkodesol(eq11, sol11, order=1, solve_for_func=False)[0]
assert checkodesol(eq12, sol12, order=1, solve_for_func=False)[0]
assert checkodesol(eq13, sol13, order=1, solve_for_func=False)[0]
def test_separable4():
# This has a slow integral (1/((1 + y**2)*atan(y))), so we isolate it.
eq14 = x*f(x).diff(x) + (1 + f(x)**2)*atan(f(x))
sol14 = Eq(log(atan(f(x))), C1 - log(x))
assert dsolve(eq14, hint='separable', simplify=False) == sol14
assert checkodesol(eq14, sol14, order=1, solve_for_func=False)[0]
def test_separable5():
eq15 = f(x).diff(x) + x*(f(x) + 1)
eq16 = exp(f(x)**2)*(x**2 + 2*x + 1) + (x*f(x) + f(x))*f(x).diff(x)
eq17 = f(x).diff(x) + f(x)
eq18 = sin(x)*cos(2*f(x)) + cos(x)*sin(2*f(x))*f(x).diff(x)
eq19 = (1 - x)*f(x).diff(x) - x*(f(x) + 1)
eq20 = f(x)*diff(f(x), x) + x - 3*x*f(x)**2
eq21 = f(x).diff(x) - exp(x + f(x))
sol15 = Eq(f(x), -1 + C1*exp(-x**2/2))
sol16 = Eq(-exp(-f(x)**2)/2, C1 - x - x**2/2)
sol17 = Eq(f(x), C1*exp(-x))
sol18 = Eq(-log(cos(2*f(x)))/2, C1 + log(cos(x)))
sol19 = Eq(f(x), (C1*exp(-x) - x + 1)/(x - 1))
sol20 = Eq(log(-1 + 3*f(x)**2)/6, C1 + x**2/2)
sol21 = Eq(-exp(-f(x)), C1 + exp(x))
assert dsolve(eq15, hint='separable') == sol15
assert dsolve(eq16, hint='separable', simplify=False) == sol16
assert dsolve(eq17, hint='separable') == sol17
assert dsolve(eq18, hint='separable', simplify=False) == sol18
assert dsolve(eq19, hint='separable') == sol19
assert dsolve(eq20, hint='separable', simplify=False) == sol20
assert dsolve(eq21, hint='separable', simplify=False) == sol21
assert checkodesol(eq15, sol15, order=1, solve_for_func=False)[0]
assert checkodesol(eq16, sol16, order=1, solve_for_func=False)[0]
assert checkodesol(eq17, sol17, order=1, solve_for_func=False)[0]
assert checkodesol(eq18, sol18, order=1, solve_for_func=False)[0]
assert checkodesol(eq19, sol19, order=1, solve_for_func=False)[0]
assert checkodesol(eq20, sol20, order=1, solve_for_func=False)[0]
assert checkodesol(eq21, sol21, order=1, solve_for_func=False)[0]
def test_separable_1_5_checkodesol():
eq12 = (x - 1)*cos(f(x))*f(x).diff(x) - 2*x*sin(f(x))
sol12 = Eq(-log(1 - cos(f(x))**2)/2, C1 - 2*x - 2*log(1 - x))
assert checkodesol(eq12, sol12, order=1, solve_for_func=False)[0]
def test_homogeneous_order():
assert homogeneous_order(exp(y/x) + tan(y/x), x, y) == 0
assert homogeneous_order(x**2 + sin(x)*cos(y), x, y) is None
assert homogeneous_order(x - y - x*sin(y/x), x, y) == 1
assert homogeneous_order((x*y + sqrt(x**4 + y**4) + x**2*(log(x) - log(y)))/
(pi*x**Rational(2, 3)*sqrt(y)**3), x, y) == Rational(-1, 6)
assert homogeneous_order(y/x*cos(y/x) - x/y*sin(y/x) + cos(y/x), x, y) == 0
assert homogeneous_order(f(x), x, f(x)) == 1
assert homogeneous_order(f(x)**2, x, f(x)) == 2
assert homogeneous_order(x*y*z, x, y) == 2
assert homogeneous_order(x*y*z, x, y, z) == 3
assert homogeneous_order(x**2*f(x)/sqrt(x**2 + f(x)**2), f(x)) is None
assert homogeneous_order(f(x, y)**2, x, f(x, y), y) == 2
assert homogeneous_order(f(x, y)**2, x, f(x), y) is None
assert homogeneous_order(f(x, y)**2, x, f(x, y)) is None
assert homogeneous_order(f(y, x)**2, x, y, f(x, y)) is None
assert homogeneous_order(f(y), f(x), x) is None
assert homogeneous_order(-f(x)/x + 1/sin(f(x)/ x), f(x), x) == 0
assert homogeneous_order(log(1/y) + log(x**2), x, y) is None
assert homogeneous_order(log(1/y) + log(x), x, y) == 0
assert homogeneous_order(log(x/y), x, y) == 0
assert homogeneous_order(2*log(1/y) + 2*log(x), x, y) == 0
a = Symbol('a')
assert homogeneous_order(a*log(1/y) + a*log(x), x, y) == 0
assert homogeneous_order(f(x).diff(x), x, y) is None
assert homogeneous_order(-f(x).diff(x) + x, x, y) is None
assert homogeneous_order(O(x), x, y) is None
assert homogeneous_order(x + O(x**2), x, y) is None
assert homogeneous_order(x**pi, x) == pi
assert homogeneous_order(x**x, x) is None
raises(ValueError, lambda: homogeneous_order(x*y))
@slow
def test_1st_homogeneous_coeff_ode():
# Type: First order homogeneous, y'=f(y/x)
eq1 = f(x)/x*cos(f(x)/x) - (x/f(x)*sin(f(x)/x) + cos(f(x)/x))*f(x).diff(x)
eq2 = x*f(x).diff(x) - f(x) - x*sin(f(x)/x)
eq3 = f(x) + (x*log(f(x)/x) - 2*x)*diff(f(x), x)
eq4 = 2*f(x)*exp(x/f(x)) + f(x)*f(x).diff(x) - 2*x*exp(x/f(x))*f(x).diff(x)
eq5 = 2*x**2*f(x) + f(x)**3 + (x*f(x)**2 - 2*x**3)*f(x).diff(x)
eq6 = x*exp(f(x)/x) - f(x)*sin(f(x)/x) + x*sin(f(x)/x)*f(x).diff(x)
eq7 = (x + sqrt(f(x)**2 - x*f(x)))*f(x).diff(x) - f(x)
eq8 = x + f(x) - (x - f(x))*f(x).diff(x)
sol1 = Eq(log(x), C1 - log(f(x)*sin(f(x)/x)/x))
sol2 = Eq(log(x), log(C1) + log(cos(f(x)/x) - 1)/2 - log(cos(f(x)/x) + 1)/2)
sol3 = Eq(f(x), -exp(C1)*LambertW(-x*exp(-C1 + 1)))
sol4 = Eq(log(f(x)), C1 - 2*exp(x/f(x)))
sol5 = Eq(f(x), exp(2*C1 + LambertW(-2*x**4*exp(-4*C1))/2)/x)
sol6 = Eq(log(x), C1 + exp(-f(x)/x)*sin(f(x)/x)/2 + exp(-f(x)/x)*cos(f(x)/x)/2)
sol7 = Eq(log(f(x)), C1 - 2*sqrt(-x/f(x) + 1))
sol8 = Eq(log(x), C1 - log(sqrt(1 + f(x)**2/x**2)) + atan(f(x)/x))
# indep_div_dep actually has a simpler solution for eq2,
# but it runs too slow
assert dsolve(eq1, hint='1st_homogeneous_coeff_subs_dep_div_indep') == sol1
assert dsolve(eq2, hint='1st_homogeneous_coeff_subs_dep_div_indep', simplify=False) == sol2
assert dsolve(eq3, hint='1st_homogeneous_coeff_best') == sol3
assert dsolve(eq4, hint='1st_homogeneous_coeff_best') == sol4
assert dsolve(eq5, hint='1st_homogeneous_coeff_best') == sol5
assert dsolve(eq6, hint='1st_homogeneous_coeff_subs_dep_div_indep') == sol6
assert dsolve(eq7, hint='1st_homogeneous_coeff_best') == sol7
assert dsolve(eq8, hint='1st_homogeneous_coeff_best') == sol8
# FIXME: sol3 and sol5 don't work with checkodesol (because of LambertW?)
# previous code was testing with these other solutions:
sol3b = Eq(-f(x)/(1 + log(x/f(x))), C1)
sol5b = Eq(log(C1*x*sqrt(1/x)*sqrt(f(x))) + x**2/(2*f(x)**2), 0)
assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
assert checkodesol(eq3, sol3b, order=1, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=1, solve_for_func=False)[0]
assert checkodesol(eq5, sol5b, order=1, solve_for_func=False)[0]
assert checkodesol(eq6, sol6, order=1, solve_for_func=False)[0]
assert checkodesol(eq8, sol8, order=1, solve_for_func=False)[0]
def test_1st_homogeneous_coeff_ode_check2():
eq2 = x*f(x).diff(x) - f(x) - x*sin(f(x)/x)
sol2 = Eq(x/tan(f(x)/(2*x)), C1)
assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
@XFAIL
def test_1st_homogeneous_coeff_ode_check3():
skip('This is a known issue.')
# checker cannot determine that the following expression is zero:
# (False,
# x*(log(exp(-LambertW(C1*x))) +
# LambertW(C1*x))*exp(-LambertW(C1*x) + 1))
# This is blocked by issue 5080.
eq3 = f(x) + (x*log(f(x)/x) - 2*x)*diff(f(x), x)
sol3a = Eq(f(x), x*exp(1 - LambertW(C1*x)))
assert checkodesol(eq3, sol3a, solve_for_func=True)[0]
# Checker can't verify this form either
# (False,
# C1*(log(C1*LambertW(C2*x)/x) + LambertW(C2*x) - 1)*LambertW(C2*x))
# It is because a = W(a)*exp(W(a)), so log(a) == log(W(a)) + W(a) and C2 =
# -E/C1 (which can be verified by solving with simplify=False).
sol3b = Eq(f(x), C1*LambertW(C2*x))
assert checkodesol(eq3, sol3b, solve_for_func=True)[0]
def test_1st_homogeneous_coeff_ode_check7():
eq7 = (x + sqrt(f(x)**2 - x*f(x)))*f(x).diff(x) - f(x)
sol7 = Eq(log(C1*f(x)) + 2*sqrt(1 - x/f(x)), 0)
assert checkodesol(eq7, sol7, order=1, solve_for_func=False)[0]
def test_1st_homogeneous_coeff_ode2():
eq1 = f(x).diff(x) - f(x)/x + 1/sin(f(x)/x)
eq2 = x**2 + f(x)**2 - 2*x*f(x)*f(x).diff(x)
eq3 = x*exp(f(x)/x) + f(x) - x*f(x).diff(x)
sol1 = [Eq(f(x), x*(-acos(C1 + log(x)) + 2*pi)), Eq(f(x), x*acos(C1 + log(x)))]
sol2 = Eq(log(f(x)), log(C1) + log(x/f(x)) - log(x**2/f(x)**2 - 1))
sol3 = Eq(f(x), log((1/(C1 - log(x)))**x))
# specific hints are applied for speed reasons
assert dsolve(eq1, hint='1st_homogeneous_coeff_subs_dep_div_indep') == sol1
assert dsolve(eq2, hint='1st_homogeneous_coeff_best', simplify=False) == sol2
assert dsolve(eq3, hint='1st_homogeneous_coeff_subs_dep_div_indep') == sol3
# FIXME: sol3 doesn't work with checkodesol (because of **x?)
# previous code was testing with this other solution:
sol3b = Eq(f(x), log(log(C1/x)**(-x)))
assert checkodesol(eq1, sol1, order=1, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=1, solve_for_func=False)[0]
assert checkodesol(eq3, sol3b, order=1, solve_for_func=False)[0]
def test_1st_homogeneous_coeff_ode_check9():
_u2 = Dummy('u2')
__a = Dummy('a')
eq9 = f(x)**2 + (x*sqrt(f(x)**2 - x**2) - x*f(x))*f(x).diff(x)
sol9 = Eq(-Integral(-1/(-(1 - sqrt(1 - _u2**2))*_u2 + _u2), (_u2, __a,
x/f(x))) + log(C1*f(x)), 0)
assert checkodesol(eq9, sol9, order=1, solve_for_func=False)[0]
def test_1st_homogeneous_coeff_ode3():
# The standard integration engine cannot handle one of the integrals
# involved (see issue 4551). meijerg code comes up with an answer, but in
# unconventional form.
# checkodesol fails for this equation, so its test is in
# test_1st_homogeneous_coeff_ode_check9 above. It has to compare string
# expressions because u2 is a dummy variable.
eq = f(x)**2 + (x*sqrt(f(x)**2 - x**2) - x*f(x))*f(x).diff(x)
sol = Eq(log(f(x)), C1 + Piecewise(
(acosh(f(x)/x), abs(f(x)**2)/x**2 > 1),
(-I*asin(f(x)/x), True)))
assert dsolve(eq, hint='1st_homogeneous_coeff_subs_indep_div_dep') == sol
def test_1st_homogeneous_coeff_corner_case():
eq1 = f(x).diff(x) - f(x)/x
c1 = classify_ode(eq1, f(x))
eq2 = x*f(x).diff(x) - f(x)
c2 = classify_ode(eq2, f(x))
sdi = "1st_homogeneous_coeff_subs_dep_div_indep"
sid = "1st_homogeneous_coeff_subs_indep_div_dep"
assert sid not in c1 and sdi not in c1
assert sid not in c2 and sdi not in c2
@slow
def test_nth_linear_constant_coeff_homogeneous():
# From Exercise 20, in Ordinary Differential Equations,
# Tenenbaum and Pollard, pg. 220
a = Symbol('a', positive=True)
k = Symbol('k', real=True)
eq1 = f(x).diff(x, 2) + 2*f(x).diff(x)
eq2 = f(x).diff(x, 2) - 3*f(x).diff(x) + 2*f(x)
eq3 = f(x).diff(x, 2) - f(x)
eq4 = f(x).diff(x, 3) + f(x).diff(x, 2) - 6*f(x).diff(x)
eq5 = 6*f(x).diff(x, 2) - 11*f(x).diff(x) + 4*f(x)
eq6 = Eq(f(x).diff(x, 2) + 2*f(x).diff(x) - f(x), 0)
eq7 = diff(f(x), x, 3) + diff(f(x), x, 2) - 10*diff(f(x), x) - 6*f(x)
eq8 = f(x).diff(x, 4) - f(x).diff(x, 3) - 4*f(x).diff(x, 2) + \
4*f(x).diff(x)
eq9 = f(x).diff(x, 4) + 4*f(x).diff(x, 3) + f(x).diff(x, 2) - \
4*f(x).diff(x) - 2*f(x)
eq10 = f(x).diff(x, 4) - a**2*f(x)
eq11 = f(x).diff(x, 2) - 2*k*f(x).diff(x) - 2*f(x)
eq12 = f(x).diff(x, 2) + 4*k*f(x).diff(x) - 12*k**2*f(x)
eq13 = f(x).diff(x, 4)
eq14 = f(x).diff(x, 2) + 4*f(x).diff(x) + 4*f(x)
eq15 = 3*f(x).diff(x, 3) + 5*f(x).diff(x, 2) + f(x).diff(x) - f(x)
eq16 = f(x).diff(x, 3) - 6*f(x).diff(x, 2) + 12*f(x).diff(x) - 8*f(x)
eq17 = f(x).diff(x, 2) - 2*a*f(x).diff(x) + a**2*f(x)
eq18 = f(x).diff(x, 4) + 3*f(x).diff(x, 3)
eq19 = f(x).diff(x, 4) - 2*f(x).diff(x, 2)
eq20 = f(x).diff(x, 4) + 2*f(x).diff(x, 3) - 11*f(x).diff(x, 2) - \
12*f(x).diff(x) + 36*f(x)
eq21 = 36*f(x).diff(x, 4) - 37*f(x).diff(x, 2) + 4*f(x).diff(x) + 5*f(x)
eq22 = f(x).diff(x, 4) - 8*f(x).diff(x, 2) + 16*f(x)
eq23 = f(x).diff(x, 2) - 2*f(x).diff(x) + 5*f(x)
eq24 = f(x).diff(x, 2) - f(x).diff(x) + f(x)
eq25 = f(x).diff(x, 4) + 5*f(x).diff(x, 2) + 6*f(x)
eq26 = f(x).diff(x, 2) - 4*f(x).diff(x) + 20*f(x)
eq27 = f(x).diff(x, 4) + 4*f(x).diff(x, 2) + 4*f(x)
eq28 = f(x).diff(x, 3) + 8*f(x)
eq29 = f(x).diff(x, 4) + 4*f(x).diff(x, 2)
eq30 = f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x)
eq31 = f(x).diff(x, 4) + f(x).diff(x, 2) + f(x)
eq32 = f(x).diff(x, 4) + 4*f(x).diff(x, 2) + f(x)
sol1 = Eq(f(x), C1 + C2*exp(-2*x))
sol2 = Eq(f(x), (C1 + C2*exp(x))*exp(x))
sol3 = Eq(f(x), C1*exp(x) + C2*exp(-x))
sol4 = Eq(f(x), C1 + C2*exp(-3*x) + C3*exp(2*x))
sol5 = Eq(f(x), C1*exp(x/2) + C2*exp(4*x/3))
sol6 = Eq(f(x), C1*exp(x*(-1 + sqrt(2))) + C2*exp(x*(-sqrt(2) - 1)))
sol7 = Eq(f(x),
C1*exp(3*x) + C2*exp(x*(-2 - sqrt(2))) + C3*exp(x*(-2 + sqrt(2))))
sol8 = Eq(f(x), C1 + C2*exp(x) + C3*exp(-2*x) + C4*exp(2*x))
sol9 = Eq(f(x),
C1*exp(x) + C2*exp(-x) + C3*exp(x*(-2 + sqrt(2))) +
C4*exp(x*(-2 - sqrt(2))))
sol10 = Eq(f(x),
C1*sin(x*sqrt(a)) + C2*cos(x*sqrt(a)) + C3*exp(x*sqrt(a)) +
C4*exp(-x*sqrt(a)))
sol11 = Eq(f(x),
C1*exp(x*(k - sqrt(k**2 + 2))) + C2*exp(x*(k + sqrt(k**2 + 2))))
sol12 = Eq(f(x), C1*exp(-6*k*x) + C2*exp(2*k*x))
sol13 = Eq(f(x), C1 + C2*x + C3*x**2 + C4*x**3)
sol14 = Eq(f(x), (C1 + C2*x)*exp(-2*x))
sol15 = Eq(f(x), (C1 + C2*x)*exp(-x) + C3*exp(x/3))
sol16 = Eq(f(x), (C1 + C2*x + C3*x**2)*exp(2*x))
sol17 = Eq(f(x), (C1 + C2*x)*exp(a*x))
sol18 = Eq(f(x), C1 + C2*x + C3*x**2 + C4*exp(-3*x))
sol19 = Eq(f(x), C1 + C2*x + C3*exp(x*sqrt(2)) + C4*exp(-x*sqrt(2)))
sol20 = Eq(f(x), (C1 + C2*x)*exp(-3*x) + (C3 + C4*x)*exp(2*x))
sol21 = Eq(f(x), C1*exp(x/2) + C2*exp(-x) + C3*exp(-x/3) + C4*exp(5*x/6))
sol22 = Eq(f(x), (C1 + C2*x)*exp(-2*x) + (C3 + C4*x)*exp(2*x))
sol23 = Eq(f(x), (C1*sin(2*x) + C2*cos(2*x))*exp(x))
sol24 = Eq(f(x), (C1*sin(x*sqrt(3)/2) + C2*cos(x*sqrt(3)/2))*exp(x/2))
sol25 = Eq(f(x),
C1*cos(x*sqrt(3)) + C2*sin(x*sqrt(3)) + C3*sin(x*sqrt(2)) +
C4*cos(x*sqrt(2)))
sol26 = Eq(f(x), (C1*sin(4*x) + C2*cos(4*x))*exp(2*x))
sol27 = Eq(f(x), (C1 + C2*x)*sin(x*sqrt(2)) + (C3 + C4*x)*cos(x*sqrt(2)))
sol28 = Eq(f(x),
(C1*sin(x*sqrt(3)) + C2*cos(x*sqrt(3)))*exp(x) + C3*exp(-2*x))
sol29 = Eq(f(x), C1 + C2*sin(2*x) + C3*cos(2*x) + C4*x)
sol30 = Eq(f(x), C1 + (C2 + C3*x)*sin(x) + (C4 + C5*x)*cos(x))
sol31 = Eq(f(x), (C1*sin(sqrt(3)*x/2) + C2*cos(sqrt(3)*x/2))/sqrt(exp(x))
+ (C3*sin(sqrt(3)*x/2) + C4*cos(sqrt(3)*x/2))*sqrt(exp(x)))
sol32 = Eq(f(x), C1*sin(x*sqrt(-sqrt(3) + 2)) + C2*sin(x*sqrt(sqrt(3) + 2))
+ C3*cos(x*sqrt(-sqrt(3) + 2)) + C4*cos(x*sqrt(sqrt(3) + 2)))
sol1s = constant_renumber(sol1)
sol2s = constant_renumber(sol2)
sol3s = constant_renumber(sol3)
sol4s = constant_renumber(sol4)
sol5s = constant_renumber(sol5)
sol6s = constant_renumber(sol6)
sol7s = constant_renumber(sol7)
sol8s = constant_renumber(sol8)
sol9s = constant_renumber(sol9)
sol10s = constant_renumber(sol10)
sol11s = constant_renumber(sol11)
sol12s = constant_renumber(sol12)
sol13s = constant_renumber(sol13)
sol14s = constant_renumber(sol14)
sol15s = constant_renumber(sol15)
sol16s = constant_renumber(sol16)
sol17s = constant_renumber(sol17)
sol18s = constant_renumber(sol18)
sol19s = constant_renumber(sol19)
sol20s = constant_renumber(sol20)
sol21s = constant_renumber(sol21)
sol22s = constant_renumber(sol22)
sol23s = constant_renumber(sol23)
sol24s = constant_renumber(sol24)
sol25s = constant_renumber(sol25)
sol26s = constant_renumber(sol26)
sol27s = constant_renumber(sol27)
sol28s = constant_renumber(sol28)
sol29s = constant_renumber(sol29)
sol30s = constant_renumber(sol30)
assert dsolve(eq1) in (sol1, sol1s)
assert dsolve(eq2) in (sol2, sol2s)
assert dsolve(eq3) in (sol3, sol3s)
assert dsolve(eq4) in (sol4, sol4s)
assert dsolve(eq5) in (sol5, sol5s)
assert dsolve(eq6) in (sol6, sol6s)
assert dsolve(eq7) in (sol7, sol7s)
assert dsolve(eq8) in (sol8, sol8s)
assert dsolve(eq9) in (sol9, sol9s)
assert dsolve(eq10) in (sol10, sol10s)
assert dsolve(eq11) in (sol11, sol11s)
assert dsolve(eq12) in (sol12, sol12s)
assert dsolve(eq13) in (sol13, sol13s)
assert dsolve(eq14) in (sol14, sol14s)
assert dsolve(eq15) in (sol15, sol15s)
assert dsolve(eq16) in (sol16, sol16s)
assert dsolve(eq17) in (sol17, sol17s)
assert dsolve(eq18) in (sol18, sol18s)
assert dsolve(eq19) in (sol19, sol19s)
assert dsolve(eq20) in (sol20, sol20s)
assert dsolve(eq21) in (sol21, sol21s)
assert dsolve(eq22) in (sol22, sol22s)
assert dsolve(eq23) in (sol23, sol23s)
assert dsolve(eq24) in (sol24, sol24s)
assert dsolve(eq25) in (sol25, sol25s)
assert dsolve(eq26) in (sol26, sol26s)
assert dsolve(eq27) in (sol27, sol27s)
assert dsolve(eq28) in (sol28, sol28s)
assert dsolve(eq29) in (sol29, sol29s)
assert dsolve(eq30) in (sol30, sol30s)
assert dsolve(eq31) in (sol31,)
assert dsolve(eq32) in (sol32,)
assert checkodesol(eq1, sol1, order=2, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=2, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=2, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=3, solve_for_func=False)[0]
assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
assert checkodesol(eq6, sol6, order=2, solve_for_func=False)[0]
assert checkodesol(eq7, sol7, order=3, solve_for_func=False)[0]
assert checkodesol(eq8, sol8, order=4, solve_for_func=False)[0]
assert checkodesol(eq9, sol9, order=4, solve_for_func=False)[0]
assert checkodesol(eq10, sol10, order=4, solve_for_func=False)[0]
assert checkodesol(eq11, sol11, order=2, solve_for_func=False)[0]
assert checkodesol(eq12, sol12, order=2, solve_for_func=False)[0]
assert checkodesol(eq13, sol13, order=4, solve_for_func=False)[0]
assert checkodesol(eq14, sol14, order=2, solve_for_func=False)[0]
assert checkodesol(eq15, sol15, order=3, solve_for_func=False)[0]
assert checkodesol(eq16, sol16, order=3, solve_for_func=False)[0]
assert checkodesol(eq17, sol17, order=2, solve_for_func=False)[0]
assert checkodesol(eq18, sol18, order=4, solve_for_func=False)[0]
assert checkodesol(eq19, sol19, order=4, solve_for_func=False)[0]
assert checkodesol(eq20, sol20, order=4, solve_for_func=False)[0]
assert checkodesol(eq21, sol21, order=4, solve_for_func=False)[0]
assert checkodesol(eq22, sol22, order=4, solve_for_func=False)[0]
assert checkodesol(eq23, sol23, order=2, solve_for_func=False)[0]
assert checkodesol(eq24, sol24, order=2, solve_for_func=False)[0]
assert checkodesol(eq25, sol25, order=4, solve_for_func=False)[0]
assert checkodesol(eq26, sol26, order=2, solve_for_func=False)[0]
assert checkodesol(eq27, sol27, order=4, solve_for_func=False)[0]
assert checkodesol(eq28, sol28, order=3, solve_for_func=False)[0]
assert checkodesol(eq29, sol29, order=4, solve_for_func=False)[0]
assert checkodesol(eq30, sol30, order=5, solve_for_func=False)[0]
assert checkodesol(eq31, sol31, order=4, solve_for_func=False)[0]
assert checkodesol(eq32, sol32, order=4, solve_for_func=False)[0]
# Issue #15237
eqn = Derivative(x*f(x), x, x, x)
hint = 'nth_linear_constant_coeff_homogeneous'
raises(ValueError, lambda: dsolve(eqn, f(x), hint, prep=True))
raises(ValueError, lambda: dsolve(eqn, f(x), hint, prep=False))
def test_nth_linear_constant_coeff_homogeneous_rootof():
# One real root, two complex conjugate pairs
eq = f(x).diff(x, 5) + 11*f(x).diff(x) - 2*f(x)
r1, r2, r3, r4, r5 = [rootof(x**5 + 11*x - 2, n) for n in range(5)]
sol = Eq(f(x),
C5*exp(r1*x)
+ exp(re(r2)*x) * (C1*sin(im(r2)*x) + C2*cos(im(r2)*x))
+ exp(re(r4)*x) * (C3*sin(im(r4)*x) + C4*cos(im(r4)*x))
)
assert dsolve(eq) == sol
# FIXME: assert checkodesol(eq, sol) == (True, [0]) # Hangs...
# Three real roots, one complex conjugate pair
eq = f(x).diff(x,5) - 3*f(x).diff(x) + f(x)
r1, r2, r3, r4, r5 = [rootof(x**5 - 3*x + 1, n) for n in range(5)]
sol = Eq(f(x),
C3*exp(r1*x) + C4*exp(r2*x) + C5*exp(r3*x)
+ exp(re(r4)*x) * (C1*sin(im(r4)*x) + C2*cos(im(r4)*x))
)
assert dsolve(eq) == sol
# FIXME: assert checkodesol(eq, sol) == (True, [0]) # Hangs...
# Five distinct real roots
eq = f(x).diff(x,5) - 100*f(x).diff(x,3) + 1000*f(x).diff(x) + f(x)
r1, r2, r3, r4, r5 = [rootof(x**5 - 100*x**3 + 1000*x + 1, n) for n in range(5)]
sol = Eq(f(x), C1*exp(r1*x) + C2*exp(r2*x) + C3*exp(r3*x) + C4*exp(r4*x) + C5*exp(r5*x))
assert dsolve(eq) == sol
# FIXME: assert checkodesol(eq, sol) == (True, [0]) # Hangs...
# Rational root and unsolvable quintic
eq = f(x).diff(x, 6) - 6*f(x).diff(x, 5) + 5*f(x).diff(x, 4) + 10*f(x).diff(x) - 50 * f(x)
r2, r3, r4, r5, r6 = [rootof(x**5 - x**4 + 10, n) for n in range(5)]
sol = Eq(f(x),
C5*exp(5*x)
+ C6*exp(x*r2)
+ exp(re(r3)*x) * (C1*sin(im(r3)*x) + C2*cos(im(r3)*x))
+ exp(re(r5)*x) * (C3*sin(im(r5)*x) + C4*cos(im(r5)*x))
)
assert dsolve(eq) == sol
# FIXME: assert checkodesol(eq, sol) == (True, [0]) # Hangs...
# Five double roots (this is (x**5 - x + 1)**2)
eq = f(x).diff(x, 10) - 2*f(x).diff(x, 6) + 2*f(x).diff(x, 5) + f(x).diff(x, 2) - 2*f(x).diff(x, 1) + f(x)
r1, r2, r3, r4, r5 = [rootof(x**5 - x + 1, n) for n in range(5)]
sol = Eq(f(x),
(C1 + C2 *x)*exp(r1*x)
+ exp(re(r2)*x) * ((C3 + C4*x)*sin(im(r2)*x) + (C5 + C6 *x)*cos(im(r2)*x))
+ exp(re(r4)*x) * ((C7 + C8*x)*sin(im(r4)*x) + (C9 + C10*x)*cos(im(r4)*x))
)
assert dsolve(eq) == sol
# FIXME: assert checkodesol(eq, sol) == (True, [0]) # Hangs...
def test_nth_linear_constant_coeff_homogeneous_irrational():
our_hint='nth_linear_constant_coeff_homogeneous'
eq = Eq(sqrt(2) * f(x).diff(x,x,x) + f(x).diff(x), 0)
sol = Eq(f(x), C1 + C2*sin(2**(S(3)/4)*x/2) + C3*cos(2**(S(3)/4)*x/2))
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint) == sol
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=3, solve_for_func=False)[0]
E = exp(1)
eq = Eq(E * f(x).diff(x,x,x) + f(x).diff(x), 0)
sol = Eq(f(x), C1 + C2*sin(x/sqrt(E)) + C3*cos(x/sqrt(E)))
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint) == sol
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=3, solve_for_func=False)[0]
eq = Eq(pi * f(x).diff(x,x,x) + f(x).diff(x), 0)
sol = Eq(f(x), C1 + C2*sin(x/sqrt(pi)) + C3*cos(x/sqrt(pi)))
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint) == sol
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=3, solve_for_func=False)[0]
eq = Eq(I * f(x).diff(x,x,x) + f(x).diff(x), 0)
sol = Eq(f(x), C1 + C2*exp(-sqrt(I)*x) + C3*exp(sqrt(I)*x))
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint) == sol
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=3, solve_for_func=False)[0]
@XFAIL
@slow
def test_nth_linear_constant_coeff_homogeneous_rootof_sol():
if ON_TRAVIS:
skip("Too slow for travis.")
eq = f(x).diff(x, 5) + 11*f(x).diff(x) - 2*f(x)
sol = Eq(f(x),
C1*exp(x*rootof(x**5 + 11*x - 2, 0)) +
C2*exp(x*rootof(x**5 + 11*x - 2, 1)) +
C3*exp(x*rootof(x**5 + 11*x - 2, 2)) +
C4*exp(x*rootof(x**5 + 11*x - 2, 3)) +
C5*exp(x*rootof(x**5 + 11*x - 2, 4)))
assert checkodesol(eq, sol, order=5, solve_for_func=False)[0]
@XFAIL
def test_noncircularized_real_imaginary_parts():
# If this passes, lines numbered 3878-3882 (at the time of this commit)
# of sympy/solvers/ode.py for nth_linear_constant_coeff_homogeneous
# should be removed.
y = sqrt(1+x)
i, r = im(y), re(y)
assert not (i.has(atan2) and r.has(atan2))
@XFAIL
def test_collect_respecting_exponentials():
# If this test passes, lines 1306-1311 (at the time of this commit)
# of sympy/solvers/ode.py should be removed.
sol = 1 + exp(x/2)
assert sol == collect( sol, exp(x/3))
def test_undetermined_coefficients_match():
assert _undetermined_coefficients_match(g(x), x) == {'test': False}
assert _undetermined_coefficients_match(sin(2*x + sqrt(5)), x) == \
{'test': True, 'trialset':
set([cos(2*x + sqrt(5)), sin(2*x + sqrt(5))])}
assert _undetermined_coefficients_match(sin(x)*cos(x), x) == \
{'test': False}
s = set([cos(x), x*cos(x), x**2*cos(x), x**2*sin(x), x*sin(x), sin(x)])
assert _undetermined_coefficients_match(sin(x)*(x**2 + x + 1), x) == \
{'test': True, 'trialset': s}
assert _undetermined_coefficients_match(
sin(x)*x**2 + sin(x)*x + sin(x), x) == {'test': True, 'trialset': s}
assert _undetermined_coefficients_match(
exp(2*x)*sin(x)*(x**2 + x + 1), x
) == {
'test': True, 'trialset': set([exp(2*x)*sin(x), x**2*exp(2*x)*sin(x),
cos(x)*exp(2*x), x**2*cos(x)*exp(2*x), x*cos(x)*exp(2*x),
x*exp(2*x)*sin(x)])}
assert _undetermined_coefficients_match(1/sin(x), x) == {'test': False}
assert _undetermined_coefficients_match(log(x), x) == {'test': False}
assert _undetermined_coefficients_match(2**(x)*(x**2 + x + 1), x) == \
{'test': True, 'trialset': set([2**x, x*2**x, x**2*2**x])}
assert _undetermined_coefficients_match(x**y, x) == {'test': False}
assert _undetermined_coefficients_match(exp(x)*exp(2*x + 1), x) == \
{'test': True, 'trialset': set([exp(1 + 3*x)])}
assert _undetermined_coefficients_match(sin(x)*(x**2 + x + 1), x) == \
{'test': True, 'trialset': set([x*cos(x), x*sin(x), x**2*cos(x),
x**2*sin(x), cos(x), sin(x)])}
assert _undetermined_coefficients_match(sin(x)*(x + sin(x)), x) == \
{'test': False}
assert _undetermined_coefficients_match(sin(x)*(x + sin(2*x)), x) == \
{'test': False}
assert _undetermined_coefficients_match(sin(x)*tan(x), x) == \
{'test': False}
assert _undetermined_coefficients_match(
x**2*sin(x)*exp(x) + x*sin(x) + x, x
) == {
'test': True, 'trialset': set([x**2*cos(x)*exp(x), x, cos(x), S(1),
exp(x)*sin(x), sin(x), x*exp(x)*sin(x), x*cos(x), x*cos(x)*exp(x),
x*sin(x), cos(x)*exp(x), x**2*exp(x)*sin(x)])}
assert _undetermined_coefficients_match(4*x*sin(x - 2), x) == {
'trialset': set([x*cos(x - 2), x*sin(x - 2), cos(x - 2), sin(x - 2)]),
'test': True,
}
assert _undetermined_coefficients_match(2**x*x, x) == \
{'test': True, 'trialset': set([2**x, x*2**x])}
assert _undetermined_coefficients_match(2**x*exp(2*x), x) == \
{'test': True, 'trialset': set([2**x*exp(2*x)])}
assert _undetermined_coefficients_match(exp(-x)/x, x) == \
{'test': False}
# Below are from Ordinary Differential Equations,
# Tenenbaum and Pollard, pg. 231
assert _undetermined_coefficients_match(S(4), x) == \
{'test': True, 'trialset': set([S(1)])}
assert _undetermined_coefficients_match(12*exp(x), x) == \
{'test': True, 'trialset': set([exp(x)])}
assert _undetermined_coefficients_match(exp(I*x), x) == \
{'test': True, 'trialset': set([exp(I*x)])}
assert _undetermined_coefficients_match(sin(x), x) == \
{'test': True, 'trialset': set([cos(x), sin(x)])}
assert _undetermined_coefficients_match(cos(x), x) == \
{'test': True, 'trialset': set([cos(x), sin(x)])}
assert _undetermined_coefficients_match(8 + 6*exp(x) + 2*sin(x), x) == \
{'test': True, 'trialset': set([S(1), cos(x), sin(x), exp(x)])}
assert _undetermined_coefficients_match(x**2, x) == \
{'test': True, 'trialset': set([S(1), x, x**2])}
assert _undetermined_coefficients_match(9*x*exp(x) + exp(-x), x) == \
{'test': True, 'trialset': set([x*exp(x), exp(x), exp(-x)])}
assert _undetermined_coefficients_match(2*exp(2*x)*sin(x), x) == \
{'test': True, 'trialset': set([exp(2*x)*sin(x), cos(x)*exp(2*x)])}
assert _undetermined_coefficients_match(x - sin(x), x) == \
{'test': True, 'trialset': set([S(1), x, cos(x), sin(x)])}
assert _undetermined_coefficients_match(x**2 + 2*x, x) == \
{'test': True, 'trialset': set([S(1), x, x**2])}
assert _undetermined_coefficients_match(4*x*sin(x), x) == \
{'test': True, 'trialset': set([x*cos(x), x*sin(x), cos(x), sin(x)])}
assert _undetermined_coefficients_match(x*sin(2*x), x) == \
{'test': True, 'trialset':
set([x*cos(2*x), x*sin(2*x), cos(2*x), sin(2*x)])}
assert _undetermined_coefficients_match(x**2*exp(-x), x) == \
{'test': True, 'trialset': set([x*exp(-x), x**2*exp(-x), exp(-x)])}
assert _undetermined_coefficients_match(2*exp(-x) - x**2*exp(-x), x) == \
{'test': True, 'trialset': set([x*exp(-x), x**2*exp(-x), exp(-x)])}
assert _undetermined_coefficients_match(exp(-2*x) + x**2, x) == \
{'test': True, 'trialset': set([S(1), x, x**2, exp(-2*x)])}
assert _undetermined_coefficients_match(x*exp(-x), x) == \
{'test': True, 'trialset': set([x*exp(-x), exp(-x)])}
assert _undetermined_coefficients_match(x + exp(2*x), x) == \
{'test': True, 'trialset': set([S(1), x, exp(2*x)])}
assert _undetermined_coefficients_match(sin(x) + exp(-x), x) == \
{'test': True, 'trialset': set([cos(x), sin(x), exp(-x)])}
assert _undetermined_coefficients_match(exp(x), x) == \
{'test': True, 'trialset': set([exp(x)])}
# converted from sin(x)**2
assert _undetermined_coefficients_match(S(1)/2 - cos(2*x)/2, x) == \
{'test': True, 'trialset': set([S(1), cos(2*x), sin(2*x)])}
# converted from exp(2*x)*sin(x)**2
assert _undetermined_coefficients_match(
exp(2*x)*(S(1)/2 + cos(2*x)/2), x
) == {
'test': True, 'trialset': set([exp(2*x)*sin(2*x), cos(2*x)*exp(2*x),
exp(2*x)])}
assert _undetermined_coefficients_match(2*x + sin(x) + cos(x), x) == \
{'test': True, 'trialset': set([S(1), x, cos(x), sin(x)])}
# converted from sin(2*x)*sin(x)
assert _undetermined_coefficients_match(cos(x)/2 - cos(3*x)/2, x) == \
{'test': True, 'trialset': set([cos(x), cos(3*x), sin(x), sin(3*x)])}
assert _undetermined_coefficients_match(cos(x**2), x) == {'test': False}
assert _undetermined_coefficients_match(2**(x**2), x) == {'test': False}
@slow
def test_nth_linear_constant_coeff_undetermined_coefficients():
hint = 'nth_linear_constant_coeff_undetermined_coefficients'
g = exp(-x)
f2 = f(x).diff(x, 2)
c = 3*f(x).diff(x, 3) + 5*f2 + f(x).diff(x) - f(x) - x
eq1 = c - x*g
eq2 = c - g
# 3-27 below are from Ordinary Differential Equations,
# Tenenbaum and Pollard, pg. 231
eq3 = f2 + 3*f(x).diff(x) + 2*f(x) - 4
eq4 = f2 + 3*f(x).diff(x) + 2*f(x) - 12*exp(x)
eq5 = f2 + 3*f(x).diff(x) + 2*f(x) - exp(I*x)
eq6 = f2 + 3*f(x).diff(x) + 2*f(x) - sin(x)
eq7 = f2 + 3*f(x).diff(x) + 2*f(x) - cos(x)
eq8 = f2 + 3*f(x).diff(x) + 2*f(x) - (8 + 6*exp(x) + 2*sin(x))
eq9 = f2 + f(x).diff(x) + f(x) - x**2
eq10 = f2 - 2*f(x).diff(x) - 8*f(x) - 9*x*exp(x) - 10*exp(-x)
eq11 = f2 - 3*f(x).diff(x) - 2*exp(2*x)*sin(x)
eq12 = f(x).diff(x, 4) - 2*f2 + f(x) - x + sin(x)
eq13 = f2 + f(x).diff(x) - x**2 - 2*x
eq14 = f2 + f(x).diff(x) - x - sin(2*x)
eq15 = f2 + f(x) - 4*x*sin(x)
eq16 = f2 + 4*f(x) - x*sin(2*x)
eq17 = f2 + 2*f(x).diff(x) + f(x) - x**2*exp(-x)
eq18 = f(x).diff(x, 3) + 3*f2 + 3*f(x).diff(x) + f(x) - 2*exp(-x) + \
x**2*exp(-x)
eq19 = f2 + 3*f(x).diff(x) + 2*f(x) - exp(-2*x) - x**2
eq20 = f2 - 3*f(x).diff(x) + 2*f(x) - x*exp(-x)
eq21 = f2 + f(x).diff(x) - 6*f(x) - x - exp(2*x)
eq22 = f2 + f(x) - sin(x) - exp(-x)
eq23 = f(x).diff(x, 3) - 3*f2 + 3*f(x).diff(x) - f(x) - exp(x)
# sin(x)**2
eq24 = f2 + f(x) - S(1)/2 - cos(2*x)/2
# exp(2*x)*sin(x)**2
eq25 = f(x).diff(x, 3) - f(x).diff(x) - exp(2*x)*(S(1)/2 - cos(2*x)/2)
eq26 = (f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x -
sin(x) - cos(x))
# sin(2*x)*sin(x), skip 3127 for now, match bug
eq27 = f2 + f(x) - cos(x)/2 + cos(3*x)/2
eq28 = f(x).diff(x) - 1
sol1 = Eq(f(x),
-1 - x + (C1 + C2*x - 3*x**2/32 - x**3/24)*exp(-x) + C3*exp(x/3))
sol2 = Eq(f(x), -1 - x + (C1 + C2*x - x**2/8)*exp(-x) + C3*exp(x/3))
sol3 = Eq(f(x), 2 + C1*exp(-x) + C2*exp(-2*x))
sol4 = Eq(f(x), 2*exp(x) + C1*exp(-x) + C2*exp(-2*x))
sol5 = Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + exp(I*x)/10 - 3*I*exp(I*x)/10)
sol6 = Eq(f(x), -3*cos(x)/10 + sin(x)/10 + C1*exp(-x) + C2*exp(-2*x))
sol7 = Eq(f(x), cos(x)/10 + 3*sin(x)/10 + C1*exp(-x) + C2*exp(-2*x))
sol8 = Eq(f(x),
4 - 3*cos(x)/5 + sin(x)/5 + exp(x) + C1*exp(-x) + C2*exp(-2*x))
sol9 = Eq(f(x),
-2*x + x**2 + (C1*sin(x*sqrt(3)/2) + C2*cos(x*sqrt(3)/2))*exp(-x/2))
sol10 = Eq(f(x), -x*exp(x) - 2*exp(-x) + C1*exp(-2*x) + C2*exp(4*x))
sol11 = Eq(f(x), C1 + C2*exp(3*x) + (-3*sin(x) - cos(x))*exp(2*x)/5)
sol12 = Eq(f(x), x - sin(x)/4 + (C1 + C2*x)*exp(-x) + (C3 + C4*x)*exp(x))
sol13 = Eq(f(x), C1 + x**3/3 + C2*exp(-x))
sol14 = Eq(f(x), C1 - x - sin(2*x)/5 - cos(2*x)/10 + x**2/2 + C2*exp(-x))
sol15 = Eq(f(x), (C1 + x)*sin(x) + (C2 - x**2)*cos(x))
sol16 = Eq(f(x), (C1 + x/16)*sin(2*x) + (C2 - x**2/8)*cos(2*x))
sol17 = Eq(f(x), (C1 + C2*x + x**4/12)*exp(-x))
sol18 = Eq(f(x), (C1 + C2*x + C3*x**2 - x**5/60 + x**3/3)*exp(-x))
sol19 = Eq(f(x), S(7)/4 - 3*x/2 + x**2/2 + C1*exp(-x) + (C2 - x)*exp(-2*x))
sol20 = Eq(f(x), C1*exp(x) + C2*exp(2*x) + (6*x + 5)*exp(-x)/36)
sol21 = Eq(f(x), -S(1)/36 - x/6 + C1*exp(-3*x) + (C2 + x/5)*exp(2*x))
sol22 = Eq(f(x), C1*sin(x) + (C2 - x/2)*cos(x) + exp(-x)/2)
sol23 = Eq(f(x), (C1 + C2*x + C3*x**2 + x**3/6)*exp(x))
sol24 = Eq(f(x), S(1)/2 - cos(2*x)/6 + C1*sin(x) + C2*cos(x))
sol25 = Eq(f(x), C1 + C2*exp(-x) + C3*exp(x) +
(-21*sin(2*x) + 27*cos(2*x) + 130)*exp(2*x)/1560)
sol26 = Eq(f(x),
C1 + (C2 + C3*x - x**2/8)*sin(x) + (C4 + C5*x + x**2/8)*cos(x) + x**2)
sol27 = Eq(f(x), cos(3*x)/16 + C1*cos(x) + (C2 + x/4)*sin(x))
sol28 = Eq(f(x), C1 + x)
sol1s = constant_renumber(sol1)
sol2s = constant_renumber(sol2)
sol3s = constant_renumber(sol3)
sol4s = constant_renumber(sol4)
sol5s = constant_renumber(sol5)
sol6s = constant_renumber(sol6)
sol7s = constant_renumber(sol7)
sol8s = constant_renumber(sol8)
sol9s = constant_renumber(sol9)
sol10s = constant_renumber(sol10)
sol11s = constant_renumber(sol11)
sol12s = constant_renumber(sol12)
sol13s = constant_renumber(sol13)
sol14s = constant_renumber(sol14)
sol15s = constant_renumber(sol15)
sol16s = constant_renumber(sol16)
sol17s = constant_renumber(sol17)
sol18s = constant_renumber(sol18)
sol19s = constant_renumber(sol19)
sol20s = constant_renumber(sol20)
sol21s = constant_renumber(sol21)
sol22s = constant_renumber(sol22)
sol23s = constant_renumber(sol23)
sol24s = constant_renumber(sol24)
sol25s = constant_renumber(sol25)
sol26s = constant_renumber(sol26)
sol27s = constant_renumber(sol27)
assert dsolve(eq1, hint=hint) in (sol1, sol1s)
assert dsolve(eq2, hint=hint) in (sol2, sol2s)
assert dsolve(eq3, hint=hint) in (sol3, sol3s)
assert dsolve(eq4, hint=hint) in (sol4, sol4s)
assert dsolve(eq5, hint=hint) in (sol5, sol5s)
assert dsolve(eq6, hint=hint) in (sol6, sol6s)
assert dsolve(eq7, hint=hint) in (sol7, sol7s)
assert dsolve(eq8, hint=hint) in (sol8, sol8s)
assert dsolve(eq9, hint=hint) in (sol9, sol9s)
assert dsolve(eq10, hint=hint) in (sol10, sol10s)
assert dsolve(eq11, hint=hint) in (sol11, sol11s)
assert dsolve(eq12, hint=hint) in (sol12, sol12s)
assert dsolve(eq13, hint=hint) in (sol13, sol13s)
assert dsolve(eq14, hint=hint) in (sol14, sol14s)
assert dsolve(eq15, hint=hint) in (sol15, sol15s)
assert dsolve(eq16, hint=hint) in (sol16, sol16s)
assert dsolve(eq17, hint=hint) in (sol17, sol17s)
assert dsolve(eq18, hint=hint) in (sol18, sol18s)
assert dsolve(eq19, hint=hint) in (sol19, sol19s)
assert dsolve(eq20, hint=hint) in (sol20, sol20s)
assert dsolve(eq21, hint=hint) in (sol21, sol21s)
assert dsolve(eq22, hint=hint) in (sol22, sol22s)
assert dsolve(eq23, hint=hint) in (sol23, sol23s)
assert dsolve(eq24, hint=hint) in (sol24, sol24s)
assert dsolve(eq25, hint=hint) in (sol25, sol25s)
assert dsolve(eq26, hint=hint) in (sol26, sol26s)
assert dsolve(eq27, hint=hint) in (sol27, sol27s)
assert dsolve(eq28, hint=hint) == sol28
assert checkodesol(eq1, sol1, order=3, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=3, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=2, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=2, solve_for_func=False)[0]
assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
assert checkodesol(eq6, sol6, order=2, solve_for_func=False)[0]
assert checkodesol(eq7, sol7, order=2, solve_for_func=False)[0]
assert checkodesol(eq8, sol8, order=2, solve_for_func=False)[0]
assert checkodesol(eq9, sol9, order=2, solve_for_func=False)[0]
assert checkodesol(eq10, sol10, order=2, solve_for_func=False)[0]
assert checkodesol(eq11, sol11, order=2, solve_for_func=False)[0]
assert checkodesol(eq12, sol12, order=4, solve_for_func=False)[0]
assert checkodesol(eq13, sol13, order=2, solve_for_func=False)[0]
assert checkodesol(eq14, sol14, order=2, solve_for_func=False)[0]
assert checkodesol(eq15, sol15, order=2, solve_for_func=False)[0]
assert checkodesol(eq16, sol16, order=2, solve_for_func=False)[0]
assert checkodesol(eq17, sol17, order=2, solve_for_func=False)[0]
assert checkodesol(eq18, sol18, order=3, solve_for_func=False)[0]
assert checkodesol(eq19, sol19, order=2, solve_for_func=False)[0]
assert checkodesol(eq20, sol20, order=2, solve_for_func=False)[0]
assert checkodesol(eq21, sol21, order=2, solve_for_func=False)[0]
assert checkodesol(eq22, sol22, order=2, solve_for_func=False)[0]
assert checkodesol(eq23, sol23, order=3, solve_for_func=False)[0]
assert checkodesol(eq24, sol24, order=2, solve_for_func=False)[0]
assert checkodesol(eq25, sol25, order=3, solve_for_func=False)[0]
assert checkodesol(eq26, sol26, order=5, solve_for_func=False)[0]
assert checkodesol(eq27, sol27, order=2, solve_for_func=False)[0]
assert checkodesol(eq28, sol28, order=1, solve_for_func=False)[0]
def test_issue_5787():
# This test case is to show the classification of imaginary constants under
# nth_linear_constant_coeff_undetermined_coefficients
eq = Eq(diff(f(x), x), I*f(x) + S(1)/2 - I)
our_hint = 'nth_linear_constant_coeff_undetermined_coefficients'
assert our_hint in classify_ode(eq)
@XFAIL
def test_nth_linear_constant_coeff_undetermined_coefficients_imaginary_exp():
# Equivalent to eq26 in
# test_nth_linear_constant_coeff_undetermined_coefficients above.
# This fails because the algorithm for undetermined coefficients
# doesn't know to multiply exp(I*x) by sufficient x because it is linearly
# dependent on sin(x) and cos(x).
hint = 'nth_linear_constant_coeff_undetermined_coefficients'
eq26a = f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x - exp(I*x)
sol26 = Eq(f(x),
C1 + (C2 + C3*x - x**2/8)*sin(x) + (C4 + C5*x + x**2/8)*cos(x) + x**2)
assert dsolve(eq26a, hint=hint) == sol26
assert checkodesol(eq26a, sol26, order=5, solve_for_func=False)[0]
@slow
def test_nth_linear_constant_coeff_variation_of_parameters():
hint = 'nth_linear_constant_coeff_variation_of_parameters'
g = exp(-x)
f2 = f(x).diff(x, 2)
c = 3*f(x).diff(x, 3) + 5*f2 + f(x).diff(x) - f(x) - x
eq1 = c - x*g
eq2 = c - g
eq3 = f(x).diff(x) - 1
eq4 = f2 + 3*f(x).diff(x) + 2*f(x) - 4
eq5 = f2 + 3*f(x).diff(x) + 2*f(x) - 12*exp(x)
eq6 = f2 - 2*f(x).diff(x) - 8*f(x) - 9*x*exp(x) - 10*exp(-x)
eq7 = f2 + 2*f(x).diff(x) + f(x) - x**2*exp(-x)
eq8 = f2 - 3*f(x).diff(x) + 2*f(x) - x*exp(-x)
eq9 = f(x).diff(x, 3) - 3*f2 + 3*f(x).diff(x) - f(x) - exp(x)
eq10 = f2 + 2*f(x).diff(x) + f(x) - exp(-x)/x
eq11 = f2 + f(x) - 1/sin(x)*1/cos(x)
eq12 = f(x).diff(x, 4) - 1/x
sol1 = Eq(f(x),
-1 - x + (C1 + C2*x - 3*x**2/32 - x**3/24)*exp(-x) + C3*exp(x/3))
sol2 = Eq(f(x), -1 - x + (C1 + C2*x - x**2/8)*exp(-x) + C3*exp(x/3))
sol3 = Eq(f(x), C1 + x)
sol4 = Eq(f(x), 2 + C1*exp(-x) + C2*exp(-2*x))
sol5 = Eq(f(x), 2*exp(x) + C1*exp(-x) + C2*exp(-2*x))
sol6 = Eq(f(x), -x*exp(x) - 2*exp(-x) + C1*exp(-2*x) + C2*exp(4*x))
sol7 = Eq(f(x), (C1 + C2*x + x**4/12)*exp(-x))
sol8 = Eq(f(x), C1*exp(x) + C2*exp(2*x) + (6*x + 5)*exp(-x)/36)
sol9 = Eq(f(x), (C1 + C2*x + C3*x**2 + x**3/6)*exp(x))
sol10 = Eq(f(x), (C1 + x*(C2 + log(x)))*exp(-x))
sol11 = Eq(f(x), cos(x)*(C2 - Integral(1/cos(x), x)) + sin(x)*(C1 +
Integral(1/sin(x), x)))
sol12 = Eq(f(x), C1 + C2*x + x**3*(C3 + log(x)/6) + C4*x**2)
sol1s = constant_renumber(sol1)
sol2s = constant_renumber(sol2)
sol3s = constant_renumber(sol3)
sol4s = constant_renumber(sol4)
sol5s = constant_renumber(sol5)
sol6s = constant_renumber(sol6)
sol7s = constant_renumber(sol7)
sol8s = constant_renumber(sol8)
sol9s = constant_renumber(sol9)
sol10s = constant_renumber(sol10)
sol11s = constant_renumber(sol11)
sol12s = constant_renumber(sol12)
assert dsolve(eq1, hint=hint) in (sol1, sol1s)
assert dsolve(eq2, hint=hint) in (sol2, sol2s)
assert dsolve(eq3, hint=hint) in (sol3, sol3s)
assert dsolve(eq4, hint=hint) in (sol4, sol4s)
assert dsolve(eq5, hint=hint) in (sol5, sol5s)
assert dsolve(eq6, hint=hint) in (sol6, sol6s)
assert dsolve(eq7, hint=hint) in (sol7, sol7s)
assert dsolve(eq8, hint=hint) in (sol8, sol8s)
assert dsolve(eq9, hint=hint) in (sol9, sol9s)
assert dsolve(eq10, hint=hint) in (sol10, sol10s)
assert dsolve(eq11, hint=hint + '_Integral') in (sol11, sol11s)
assert dsolve(eq12, hint=hint) in (sol12, sol12s)
assert checkodesol(eq1, sol1, order=3, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=3, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=1, solve_for_func=False)[0]
assert checkodesol(eq4, sol4, order=2, solve_for_func=False)[0]
assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
assert checkodesol(eq6, sol6, order=2, solve_for_func=False)[0]
assert checkodesol(eq7, sol7, order=2, solve_for_func=False)[0]
assert checkodesol(eq8, sol8, order=2, solve_for_func=False)[0]
assert checkodesol(eq9, sol9, order=3, solve_for_func=False)[0]
assert checkodesol(eq10, sol10, order=2, solve_for_func=False)[0]
assert checkodesol(eq12, sol12, order=4, solve_for_func=False)[0]
@slow
def test_nth_linear_constant_coeff_variation_of_parameters_simplify_False():
# solve_variation_of_parameters shouldn't attempt to simplify the
# Wronskian if simplify=False. If wronskian() ever gets good enough
# to simplify the result itself, this test might fail.
our_hint = 'nth_linear_constant_coeff_variation_of_parameters_Integral'
eq = f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x - exp(I*x)
sol_simp = dsolve(eq, f(x), hint=our_hint, simplify=True)
sol_nsimp = dsolve(eq, f(x), hint=our_hint, simplify=False)
assert sol_simp != sol_nsimp
assert checkodesol(eq, sol_simp, order=5, solve_for_func=False)[0]
assert checkodesol(eq, sol_nsimp, order=5, solve_for_func=False)[0]
def test_Liouville_ODE():
hint = 'Liouville'
# The first part here used to be test_ODE_1() from test_solvers.py
eq1 = diff(f(x), x)/x + diff(f(x), x, x)/2 - diff(f(x), x)**2/2
eq1a = diff(x*exp(-f(x)), x, x)
# compare to test_unexpanded_Liouville_ODE() below
eq2 = (eq1*exp(-f(x))/exp(f(x))).expand()
eq3 = diff(f(x), x, x) + 1/f(x)*(diff(f(x), x))**2 + 1/x*diff(f(x), x)
eq4 = x*diff(f(x), x, x) + x/f(x)*diff(f(x), x)**2 + x*diff(f(x), x)
eq5 = Eq((x*exp(f(x))).diff(x, x), 0)
sol1 = Eq(f(x), log(x/(C1 + C2*x)))
sol1a = Eq(C1 + C2/x - exp(-f(x)), 0)
sol2 = sol1
sol3 = set(
[Eq(f(x), -sqrt(C1 + C2*log(x))),
Eq(f(x), sqrt(C1 + C2*log(x)))])
sol4 = set([Eq(f(x), sqrt(C1 + C2*exp(x))*exp(-x/2)),
Eq(f(x), -sqrt(C1 + C2*exp(x))*exp(-x/2))])
sol5 = Eq(f(x), log(C1 + C2/x))
sol1s = constant_renumber(sol1)
sol2s = constant_renumber(sol2)
sol3s = constant_renumber(sol3)
sol4s = constant_renumber(sol4)
sol5s = constant_renumber(sol5)
assert dsolve(eq1, hint=hint) in (sol1, sol1s)
assert dsolve(eq1a, hint=hint) in (sol1, sol1s)
assert dsolve(eq2, hint=hint) in (sol2, sol2s)
assert set(dsolve(eq3, hint=hint)) in (sol3, sol3s)
assert set(dsolve(eq4, hint=hint)) in (sol4, sol4s)
assert dsolve(eq5, hint=hint) in (sol5, sol5s)
assert checkodesol(eq1, sol1, order=2, solve_for_func=False)[0]
assert checkodesol(eq1a, sol1a, order=2, solve_for_func=False)[0]
assert checkodesol(eq2, sol2, order=2, solve_for_func=False)[0]
assert checkodesol(eq3, sol3, order=2, solve_for_func=False) == {(True, 0)}
assert checkodesol(eq4, sol4, order=2, solve_for_func=False) == {(True, 0)}
assert checkodesol(eq5, sol5, order=2, solve_for_func=False)[0]
not_Liouville1 = classify_ode(diff(f(x), x)/x + f(x)*diff(f(x), x, x)/2 -
diff(f(x), x)**2/2, f(x))
not_Liouville2 = classify_ode(diff(f(x), x)/x + diff(f(x), x, x)/2 -
x*diff(f(x), x)**2/2, f(x))
assert hint not in not_Liouville1
assert hint not in not_Liouville2
assert hint + '_Integral' not in not_Liouville1
assert hint + '_Integral' not in not_Liouville2
def test_unexpanded_Liouville_ODE():
# This is the same as eq1 from test_Liouville_ODE() above.
eq1 = diff(f(x), x)/x + diff(f(x), x, x)/2 - diff(f(x), x)**2/2
eq2 = eq1*exp(-f(x))/exp(f(x))
sol2 = Eq(f(x), log(x/(C1 + C2*x)))
sol2s = constant_renumber(sol2)
assert dsolve(eq2) in (sol2, sol2s)
assert checkodesol(eq2, sol2, order=2, solve_for_func=False)[0]
def test_issue_4785():
from sympy.abc import A
eq = x + A*(x + diff(f(x), x) + f(x)) + diff(f(x), x) + f(x) + 2
assert classify_ode(eq, f(x)) == ('1st_linear', 'almost_linear',
'1st_power_series', 'lie_group',
'nth_linear_constant_coeff_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'1st_linear_Integral', 'almost_linear_Integral',
'nth_linear_constant_coeff_variation_of_parameters_Integral')
# issue 4864
eq = (x**2 + f(x)**2)*f(x).diff(x) - 2*x*f(x)
assert classify_ode(eq, f(x)) == ('1st_exact',
'1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep',
'1st_homogeneous_coeff_subs_dep_div_indep',
'1st_power_series',
'lie_group', '1st_exact_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral')
def test_issue_4825():
raises(ValueError, lambda: dsolve(f(x, y).diff(x) - y*f(x, y), f(x)))
assert classify_ode(f(x, y).diff(x) - y*f(x, y), f(x), dict=True) == \
{'default': None, 'order': 0}
# See also issue 3793, test Z13.
raises(ValueError, lambda: dsolve(f(x).diff(x), f(y)))
assert classify_ode(f(x).diff(x), f(y), dict=True) == \
{'default': None, 'order': 0}
def test_constant_renumber_order_issue_5308():
from sympy.utilities.iterables import variations
eq = f(x).diff(x) - y - x
assert constant_renumber(C1*x + C2*y) == \
constant_renumber(C1*y + C2*x) == \
C1*x + C2*y
e = C1*(C2 + x)*(C3 + y)
for a, b, c in variations([C1, C2, C3], 3):
assert constant_renumber(a*(b + x)*(c + y)) == e
def test_issue_5770():
k = Symbol("k", real=True)
t = Symbol('t')
w = Function('w')
sol = dsolve(w(t).diff(t, 6) - k**6*w(t), w(t))
assert len([s for s in sol.free_symbols if s.name.startswith('C')]) == 6
assert constantsimp((C1*cos(x) + C2*cos(x))*exp(x), set([C1, C2])) == \
C1*cos(x)*exp(x)
assert constantsimp(C1*cos(x) + C2*cos(x) + C3*sin(x), set([C1, C2, C3])) == \
C1*cos(x) + C3*sin(x)
assert constantsimp(exp(C1 + x), set([C1])) == C1*exp(x)
assert constantsimp(x + C1 + y, set([C1, y])) == C1 + x
assert constantsimp(x + C1 + Integral(x, (x, 1, 2)), set([C1])) == C1 + x
def test_issue_5112_5430():
assert homogeneous_order(-log(x) + acosh(x), x) is None
assert homogeneous_order(y - log(x), x, y) is None
def test_nth_order_linear_euler_eq_homogeneous():
x, t, a, b, c = symbols('x t a b c')
y = Function('y')
our_hint = "nth_linear_euler_eq_homogeneous"
eq = diff(f(t), t, 4)*t**4 - 13*diff(f(t), t, 2)*t**2 + 36*f(t)
assert our_hint in classify_ode(eq)
eq = a*y(t) + b*t*diff(y(t), t) + c*t**2*diff(y(t), t, 2)
assert our_hint in classify_ode(eq)
eq = Eq(-3*diff(f(x), x)*x + 2*x**2*diff(f(x), x, x), 0)
sol = C1 + C2*x**Rational(5, 2)
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(3*f(x) - 5*diff(f(x), x)*x + 2*x**2*diff(f(x), x, x), 0)
sol = C1*sqrt(x) + C2*x**3
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(4*f(x) + 5*diff(f(x), x)*x + x**2*diff(f(x), x, x), 0)
sol = (C1 + C2*log(x))/x**2
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(6*f(x) - 6*diff(f(x), x)*x + 1*x**2*diff(f(x), x, x) + x**3*diff(f(x), x, x, x), 0)
sol = dsolve(eq, f(x), hint=our_hint)
sol = C1/x**2 + C2*x + C3*x**3
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(-125*f(x) + 61*diff(f(x), x)*x - 12*x**2*diff(f(x), x, x) + x**3*diff(f(x), x, x, x), 0)
sol = x**5*(C1 + C2*log(x) + C3*log(x)**2)
sols = [sol, constant_renumber(sol)]
sols += [sols[-1].expand()]
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs in sols
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = t**2*diff(y(t), t, 2) + t*diff(y(t), t) - 9*y(t)
sol = C1*t**3 + C2*t**-3
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, y(t), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = sin(x)*x**2*f(x).diff(x, 2) + sin(x)*x*f(x).diff(x) + sin(x)*f(x)
sol = C1*sin(log(x)) + C2*cos(log(x))
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
def test_nth_order_linear_euler_eq_nonhomogeneous_undetermined_coefficients():
x, t = symbols('x t')
a, b, c, d = symbols('a b c d', integer=True)
our_hint = "nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients"
eq = x**4*diff(f(x), x, 4) - 13*x**2*diff(f(x), x, 2) + 36*f(x) + x
assert our_hint in classify_ode(eq, f(x))
eq = a*x**2*diff(f(x), x, 2) + b*x*diff(f(x), x) + c*f(x) + d*log(x)
assert our_hint in classify_ode(eq, f(x))
eq = Eq(x**2*diff(f(x), x, x) + x*diff(f(x), x), 1)
sol = C1 + C2*log(x) + log(x)**2/2
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq, f(x))
assert dsolve(eq, f(x), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(x**2*diff(f(x), x, x) - 2*x*diff(f(x), x) + 2*f(x), x**3)
sol = x*(C1 + C2*x + Rational(1, 2)*x**2)
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq, f(x))
assert dsolve(eq, f(x), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(x**2*diff(f(x), x, x) - x*diff(f(x), x) - 3*f(x), log(x)/x)
sol = C1/x + C2*x**3 - Rational(1, 16)*log(x)/x - Rational(1, 8)*log(x)**2/x
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq, f(x))
assert dsolve(eq, f(x), hint=our_hint).rhs.expand() in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(x**2*diff(f(x), x, x) + 3*x*diff(f(x), x) - 8*f(x), log(x)**3 - log(x))
sol = C1/x**4 + C2*x**2 - Rational(1,8)*log(x)**3 - Rational(3,32)*log(x)**2 - Rational(1,64)*log(x) - Rational(7, 256)
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs.expand() in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(x**3*diff(f(x), x, x, x) - 3*x**2*diff(f(x), x, x) + 6*x*diff(f(x), x) - 6*f(x), log(x))
sol = C1*x + C2*x**2 + C3*x**3 - Rational(1, 6)*log(x) - Rational(11, 36)
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs.expand() in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
def test_nth_order_linear_euler_eq_nonhomogeneous_variation_of_parameters():
x, t = symbols('x, t')
a, b, c, d = symbols('a, b, c, d', integer=True)
our_hint = "nth_linear_euler_eq_nonhomogeneous_variation_of_parameters"
eq = Eq(x**2*diff(f(x),x,2) - 8*x*diff(f(x),x) + 12*f(x), x**2)
assert our_hint in classify_ode(eq, f(x))
eq = Eq(a*x**3*diff(f(x),x,3) + b*x**2*diff(f(x),x,2) + c*x*diff(f(x),x) + d*f(x), x*log(x))
assert our_hint in classify_ode(eq, f(x))
eq = Eq(x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x), x**4)
sol = C1*x + C2*x**2 + x**4/6
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs.expand() in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(3*x**2*diff(f(x), x, x) + 6*x*diff(f(x), x) - 6*f(x), x**3*exp(x))
sol = C1/x**2 + C2*x + x*exp(x)/3 - 4*exp(x)/3 + 8*exp(x)/(3*x) - 8*exp(x)/(3*x**2)
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs.expand() in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = Eq(x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x), x**4*exp(x))
sol = C1*x + C2*x**2 + x**2*exp(x) - 2*x*exp(x)
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs.expand() in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x) - log(x)
sol = C1*x + C2*x**2 + log(x)/2 + S(3)/4
sols = constant_renumber(sol)
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint).rhs in (sol, sols)
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
eq = -exp(x) + (x*Derivative(f(x), (x, 2)) + Derivative(f(x), x))/x
sol = Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))
assert our_hint in classify_ode(eq)
assert dsolve(eq, f(x), hint=our_hint) == sol
assert checkodesol(eq, sol, order=2, solve_for_func=False)[0]
def test_issue_5095():
f = Function('f')
raises(ValueError, lambda: dsolve(f(x).diff(x)**2, f(x), 'separable'))
raises(ValueError, lambda: dsolve(f(x).diff(x)**2, f(x), 'fdsjf'))
def test_almost_linear():
from sympy import Ei
A = Symbol('A', positive=True)
our_hint = 'almost_linear'
f = Function('f')
d = f(x).diff(x)
eq = x**2*f(x)**2*d + f(x)**3 + 1
sol = dsolve(eq, f(x), hint = 'almost_linear')
assert sol[0].rhs == (C1*exp(3/x) - 1)**(S(1)/3)
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
eq = x*f(x)*d + 2*x*f(x)**2 + 1
sol = dsolve(eq, f(x), hint = 'almost_linear')
assert sol[0].rhs == -sqrt(C1 - 2*Ei(4*x))*exp(-2*x)
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
eq = x*d + x*f(x) + 1
sol = dsolve(eq, f(x), hint = 'almost_linear')
assert sol.rhs == (C1 - Ei(x))*exp(-x)
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
assert our_hint in classify_ode(eq, f(x))
eq = x*exp(f(x))*d + exp(f(x)) + 3*x
sol = dsolve(eq, f(x), hint = 'almost_linear')
assert sol.rhs == log(C1/x - 3*x/2)
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
eq = x + A*(x + diff(f(x), x) + f(x)) + diff(f(x), x) + f(x) + 2
sol = dsolve(eq, f(x), hint = 'almost_linear')
assert sol.rhs == (C1 + Piecewise(
(x, Eq(A + 1, 0)), ((-A*x + A - x - 1)*exp(x)/(A + 1), True)))*exp(-x)
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_exact_enhancement():
f = Function('f')(x)
df = Derivative(f, x)
eq = f/x**2 + ((f*x - 1)/x)*df
sol = [Eq(f, (i*sqrt(C1*x**2 + 1) + 1)/x) for i in (-1, 1)]
assert set(dsolve(eq, f)) == set(sol)
assert checkodesol(eq, sol, order=1, solve_for_func=False) == [(True, 0), (True, 0)]
eq = (x*f - 1) + df*(x**2 - x*f)
sol = [Eq(f, x - sqrt(C1 + x**2 - 2*log(x))),
Eq(f, x + sqrt(C1 + x**2 - 2*log(x)))]
assert set(dsolve(eq, f)) == set(sol)
assert checkodesol(eq, sol, order=1, solve_for_func=False) == [(True, 0), (True, 0)]
eq = (x + 2)*sin(f) + df*x*cos(f)
sol = [Eq(f, -asin(C1*exp(-x)/x**2) + pi),
Eq(f, asin(C1*exp(-x)/x**2))]
assert set(dsolve(eq, f)) == set(sol)
assert checkodesol(eq, sol, order=1, solve_for_func=False) == [(True, 0), (True, 0)]
@slow
def test_separable_reduced():
f = Function('f')
x = Symbol('x')
df = f(x).diff(x)
eq = (x / f(x))*df + tan(x**2*f(x) / (x**2*f(x) - 1))
assert classify_ode(eq) == ('separable_reduced', 'lie_group',
'separable_reduced_Integral')
eq = x* df + f(x)* (1 / (x**2*f(x) - 1))
assert classify_ode(eq) == ('separable_reduced', 'lie_group',
'separable_reduced_Integral')
sol = dsolve(eq, hint = 'separable_reduced', simplify=False)
assert sol.lhs == log(x**2*f(x))/3 + log(x**2*f(x) - S(3)/2)/6
assert sol.rhs == C1 + log(x)
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
eq = f(x).diff(x) + (f(x) / (x**4*f(x) - x))
assert classify_ode(eq) == ('separable_reduced', 'lie_group',
'separable_reduced_Integral')
sol = dsolve(eq, hint = 'separable_reduced')
# FIXME: This one hangs
#assert checkodesol(eq, sol, order=1, solve_for_func=False) == [(True, 0)] * 4
assert len(sol) == 4
eq = x*df + f(x)*(x**2*f(x))
sol = dsolve(eq, hint = 'separable_reduced', simplify=False)
assert sol == Eq(log(x**2*f(x))/2 - log(x**2*f(x) - 2)/2, C1 + log(x))
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_homogeneous_function():
f = Function('f')
eq1 = tan(x + f(x))
eq2 = sin((3*x)/(4*f(x)))
eq3 = cos(3*x/4*f(x))
eq4 = log((3*x + 4*f(x))/(5*f(x) + 7*x))
eq5 = exp((2*x**2)/(3*f(x)**2))
eq6 = log((3*x + 4*f(x))/(5*f(x) + 7*x) + exp((2*x**2)/(3*f(x)**2)))
eq7 = sin((3*x)/(5*f(x) + x**2))
assert homogeneous_order(eq1, x, f(x)) == None
assert homogeneous_order(eq2, x, f(x)) == 0
assert homogeneous_order(eq3, x, f(x)) == None
assert homogeneous_order(eq4, x, f(x)) == 0
assert homogeneous_order(eq5, x, f(x)) == 0
assert homogeneous_order(eq6, x, f(x)) == 0
assert homogeneous_order(eq7, x, f(x)) == None
def test_linear_coeff_match():
from sympy.solvers.ode import _linear_coeff_match
n, d = z*(2*x + 3*f(x) + 5), z*(7*x + 9*f(x) + 11)
rat = n/d
eq1 = sin(rat) + cos(rat.expand())
eq2 = rat
eq3 = log(sin(rat))
ans = (4, -S(13)/3)
assert _linear_coeff_match(eq1, f(x)) == ans
assert _linear_coeff_match(eq2, f(x)) == ans
assert _linear_coeff_match(eq3, f(x)) == ans
# no c
eq4 = (3*x)/f(x)
# not x and f(x)
eq5 = (3*x + 2)/x
# denom will be zero
eq6 = (3*x + 2*f(x) + 1)/(3*x + 2*f(x) + 5)
# not rational coefficient
eq7 = (3*x + 2*f(x) + sqrt(2))/(3*x + 2*f(x) + 5)
assert _linear_coeff_match(eq4, f(x)) is None
assert _linear_coeff_match(eq5, f(x)) is None
assert _linear_coeff_match(eq6, f(x)) is None
assert _linear_coeff_match(eq7, f(x)) is None
def test_linear_coefficients():
f = Function('f')
sol = Eq(f(x), C1/(x**2 + 6*x + 9) - S(3)/2)
eq = f(x).diff(x) + (3 + 2*f(x))/(x + 3)
assert dsolve(eq, hint='linear_coefficients') == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_constantsimp_take_problem():
c = exp(C1) + 2
assert len(Poly(constantsimp(exp(C1) + c + c*x, [C1])).gens) == 2
def test_issue_6879():
f = Function('f')
eq = Eq(Derivative(f(x), x, 2) - 2*Derivative(f(x), x) + f(x), sin(x))
sol = (C1 + C2*x)*exp(x) + cos(x)/2
assert dsolve(eq).rhs == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_issue_6989():
f = Function('f')
k = Symbol('k')
eq = f(x).diff(x) - x*exp(-k*x)
sol = Eq(f(x), C1 + Piecewise(
((-k*x - 1)*exp(-k*x)/k**2, Ne(k**2, 0)),
(x**2/2, True)
))
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
eq = -f(x).diff(x) + x*exp(-k*x)
sol = Eq(f(x), C1 + Piecewise(
((-k*x - 1)*exp(-k*x)/k**2, Ne(k**2, 0)),
(+x**2/2, True)
))
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False)[0]
def test_heuristic1():
y, a, b, c, a4, a3, a2, a1, a0 = symbols("y a b c a4 a3 a2 a1 a0")
y = Symbol('y')
f = Function('f')
xi = Function('xi')
eta = Function('eta')
df = f(x).diff(x)
eq = Eq(df, x**2*f(x))
eq1 = f(x).diff(x) + a*f(x) - c*exp(b*x)
eq2 = f(x).diff(x) + 2*x*f(x) - x*exp(-x**2)
eq3 = (1 + 2*x)*df + 2 - 4*exp(-f(x))
eq4 = f(x).diff(x) - (a4*x**4 + a3*x**3 + a2*x**2 + a1*x + a0)**(S(-1)/2)
eq5 = x**2*df - f(x) + x**2*exp(x - (1/x))
eqlist = [eq, eq1, eq2, eq3, eq4, eq5]
i = infinitesimals(eq, hint='abaco1_simple')
assert i == [{eta(x, f(x)): exp(x**3/3), xi(x, f(x)): 0},
{eta(x, f(x)): f(x), xi(x, f(x)): 0},
{eta(x, f(x)): 0, xi(x, f(x)): x**(-2)}]
i1 = infinitesimals(eq1, hint='abaco1_simple')
assert i1 == [{eta(x, f(x)): exp(-a*x), xi(x, f(x)): 0}]
i2 = infinitesimals(eq2, hint='abaco1_simple')
assert i2 == [{eta(x, f(x)): exp(-x**2), xi(x, f(x)): 0}]
i3 = infinitesimals(eq3, hint='abaco1_simple')
assert i3 == [{eta(x, f(x)): 0, xi(x, f(x)): 2*x + 1},
{eta(x, f(x)): 0, xi(x, f(x)): 1/(exp(f(x)) - 2)}]
i4 = infinitesimals(eq4, hint='abaco1_simple')
assert i4 == [{eta(x, f(x)): 1, xi(x, f(x)): 0},
{eta(x, f(x)): 0,
xi(x, f(x)): sqrt(a0 + a1*x + a2*x**2 + a3*x**3 + a4*x**4)}]
i5 = infinitesimals(eq5, hint='abaco1_simple')
assert i5 == [{xi(x, f(x)): 0, eta(x, f(x)): exp(-1/x)}]
ilist = [i, i1, i2, i3, i4, i5]
for eq, i in (zip(eqlist, ilist)):
check = checkinfsol(eq, i)
assert check[0]
def test_issue_6247():
eq = x**2*f(x)**2 + x*Derivative(f(x), x)
sol = Eq(f(x), 2*C1/(C1*x**2 - 1))
assert dsolve(eq, hint = 'separable_reduced') == sol
assert checkodesol(eq, sol, order=1)[0]
eq = f(x).diff(x, x) + 4*f(x)
sol = Eq(f(x), C1*sin(2*x) + C2*cos(2*x))
assert dsolve(eq) == sol
assert checkodesol(eq, sol, order=1)[0]
def test_heuristic2():
y = Symbol('y')
xi = Function('xi')
eta = Function('eta')
df = f(x).diff(x)
# This ODE can be solved by the Lie Group method, when there are
# better assumptions
eq = df - (f(x)/x)*(x*log(x**2/f(x)) + 2)
i = infinitesimals(eq, hint='abaco1_product')
assert i == [{eta(x, f(x)): f(x)*exp(-x), xi(x, f(x)): 0}]
assert checkinfsol(eq, i)[0]
@slow
def test_heuristic3():
y = Symbol('y')
xi = Function('xi')
eta = Function('eta')
a, b = symbols("a b")
df = f(x).diff(x)
eq = x**2*df + x*f(x) + f(x)**2 + x**2
i = infinitesimals(eq, hint='bivariate')
assert i == [{eta(x, f(x)): f(x), xi(x, f(x)): x}]
assert checkinfsol(eq, i)[0]
eq = x**2*(-f(x)**2 + df)- a*x**2*f(x) + 2 - a*x
i = infinitesimals(eq, hint='bivariate')
assert checkinfsol(eq, i)[0]
def test_heuristic_4():
y, a = symbols("y a")
xi = Function('xi')
eta = Function('eta')
eq = x*(f(x).diff(x)) + 1 - f(x)**2
i = infinitesimals(eq, hint='chi')
assert checkinfsol(eq, i)[0]
def test_heuristic_function_sum():
xi = Function('xi')
eta = Function('eta')
eq = f(x).diff(x) - (3*(1 + x**2/f(x)**2)*atan(f(x)/x) + (1 - 2*f(x))/x +
(1 - 3*f(x))*(x/f(x)**2))
i = infinitesimals(eq, hint='function_sum')
assert i == [{eta(x, f(x)): f(x)**(-2) + x**(-2), xi(x, f(x)): 0}]
assert checkinfsol(eq, i)[0]
def test_heuristic_abaco2_similar():
xi = Function('xi')
eta = Function('eta')
F = Function('F')
a, b = symbols("a b")
eq = f(x).diff(x) - F(a*x + b*f(x))
i = infinitesimals(eq, hint='abaco2_similar')
assert i == [{eta(x, f(x)): -a/b, xi(x, f(x)): 1}]
assert checkinfsol(eq, i)[0]
eq = f(x).diff(x) - (f(x)**2 / (sin(f(x) - x) - x**2 + 2*x*f(x)))
i = infinitesimals(eq, hint='abaco2_similar')
assert i == [{eta(x, f(x)): f(x)**2, xi(x, f(x)): f(x)**2}]
assert checkinfsol(eq, i)[0]
def test_heuristic_abaco2_unique_unknown():
xi = Function('xi')
eta = Function('eta')
F = Function('F')
a, b = symbols("a b")
x = Symbol("x", positive=True)
eq = f(x).diff(x) - x**(a - 1)*(f(x)**(1 - b))*F(x**a/a + f(x)**b/b)
i = infinitesimals(eq, hint='abaco2_unique_unknown')
assert i == [{eta(x, f(x)): -f(x)*f(x)**(-b), xi(x, f(x)): x*x**(-a)}]
assert checkinfsol(eq, i)[0]
eq = f(x).diff(x) + tan(F(x**2 + f(x)**2) + atan(x/f(x)))
i = infinitesimals(eq, hint='abaco2_unique_unknown')
assert i == [{eta(x, f(x)): x, xi(x, f(x)): -f(x)}]
assert checkinfsol(eq, i)[0]
eq = (x*f(x).diff(x) + f(x) + 2*x)**2 -4*x*f(x) -4*x**2 -4*a
i = infinitesimals(eq, hint='abaco2_unique_unknown')
assert checkinfsol(eq, i)[0]
def test_heuristic_linear():
xi = Function('xi')
eta = Function('eta')
F = Function('F')
a, b, m, n = symbols("a b m n")
eq = x**(n*(m + 1) - m)*(f(x).diff(x)) - a*f(x)**n -b*x**(n*(m + 1))
i = infinitesimals(eq, hint='linear')
assert checkinfsol(eq, i)[0]
@XFAIL
def test_kamke():
a, b, alpha, c = symbols("a b alpha c")
eq = x**2*(a*f(x)**2+(f(x).diff(x))) + b*x**alpha + c
i = infinitesimals(eq, hint='sum_function')
assert checkinfsol(eq, i)[0]
def test_series():
# FIXME: Maybe there should be a way to check series solutions
# checkodesol doesn't work with them.
C1 = Symbol("C1")
eq = f(x).diff(x) - f(x)
assert dsolve(eq, hint='1st_power_series') == Eq(f(x),
C1 + C1*x + C1*x**2/2 + C1*x**3/6 + C1*x**4/24 +
C1*x**5/120 + O(x**6))
eq = f(x).diff(x) - x*f(x)
assert dsolve(eq, hint='1st_power_series') == Eq(f(x),
C1*x**4/8 + C1*x**2/2 + C1 + O(x**6))
eq = f(x).diff(x) - sin(x*f(x))
sol = Eq(f(x), (x - 2)**2*(1+ sin(4))*cos(4) + (x - 2)*sin(4) + 2 + O(x**3))
assert dsolve(eq, hint='1st_power_series', ics={f(2): 2}, n=3) == sol
@slow
def test_lie_group():
C1 = Symbol("C1")
x = Symbol("x") # assuming x is real generates an error!
a, b, c = symbols("a b c")
eq = f(x).diff(x)**2
sol = dsolve(eq, f(x), hint='lie_group')
assert checkodesol(eq, sol)[0]
eq = Eq(f(x).diff(x), x**2*f(x))
sol = dsolve(eq, f(x), hint='lie_group')
assert sol == Eq(f(x), C1*exp(x**3)**(S(1)/3))
assert checkodesol(eq, sol)[0]
eq = f(x).diff(x) + a*f(x) - c*exp(b*x)
sol = dsolve(eq, f(x), hint='lie_group')
assert checkodesol(eq, sol)[0]
eq = f(x).diff(x) + 2*x*f(x) - x*exp(-x**2)
sol = dsolve(eq, f(x), hint='lie_group')
actual_sol = Eq(f(x), (C1 + x**2/2)*exp(-x**2))
errstr = str(eq)+' : '+str(sol)+' == '+str(actual_sol)
assert sol == actual_sol, errstr
assert checkodesol(eq, sol)[0]
eq = (1 + 2*x)*(f(x).diff(x)) + 2 - 4*exp(-f(x))
sol = dsolve(eq, f(x), hint='lie_group')
assert sol == Eq(f(x), log(C1/(2*x + 1) + 2))
assert checkodesol(eq, sol)[0]
eq = x**2*(f(x).diff(x)) - f(x) + x**2*exp(x - (1/x))
sol = dsolve(eq, f(x), hint='lie_group')
assert checkodesol(eq, sol)[0]
eq = x**2*f(x)**2 + x*Derivative(f(x), x)
sol = dsolve(eq, f(x), hint='lie_group')
assert sol == Eq(f(x), 2/(C1 + x**2))
assert checkodesol(eq, sol)[0]
@XFAIL
def test_lie_group_issue15219():
eqn = exp(f(x).diff(x)-f(x))
assert 'lie_group' not in classify_ode(eqn, f(x))
def test_user_infinitesimals():
x = Symbol("x") # assuming x is real generates an error
eq = x*(f(x).diff(x)) + 1 - f(x)**2
sol = Eq(f(x), (C1 + x**2)/(C1 - x**2))
infinitesimals = {'xi':sqrt(f(x) - 1)/sqrt(f(x) + 1), 'eta':0}
assert dsolve(eq, hint='lie_group', **infinitesimals) == sol
assert checkodesol(eq, sol) == (True, 0)
raises(ValueError, lambda: dsolve(eq, hint='lie_group', xi=0, eta=f(x)))
def test_issue_7081():
eq = x*(f(x).diff(x)) + 1 - f(x)**2
s = Eq(f(x), -1/(-C1 + x**2)*(C1 + x**2))
assert dsolve(eq) == s
assert checkodesol(eq, s) == (True, 0)
@slow
def test_2nd_power_series_ordinary():
# FIXME: Maybe there should be a way to check series solutions
# checkodesol doesn't work with them.
C1, C2 = symbols("C1 C2")
eq = f(x).diff(x, 2) - x*f(x)
assert classify_ode(eq) == ('2nd_power_series_ordinary',)
assert dsolve(eq) == Eq(f(x),
C2*(x**3/6 + 1) + C1*x*(x**3/12 + 1) + O(x**6))
assert dsolve(eq, x0=-2) == Eq(f(x),
C2*((x + 2)**4/6 + (x + 2)**3/6 - (x + 2)**2 + 1)
+ C1*(x + (x + 2)**4/12 - (x + 2)**3/3 + S(2))
+ O(x**6))
assert dsolve(eq, n=2) == Eq(f(x), C2*x + C1 + O(x**2))
eq = (1 + x**2)*(f(x).diff(x, 2)) + 2*x*(f(x).diff(x)) -2*f(x)
assert classify_ode(eq) == ('2nd_power_series_ordinary',)
assert dsolve(eq) == Eq(f(x), C2*(-x**4/3 + x**2 + 1) + C1*x
+ O(x**6))
eq = f(x).diff(x, 2) + x*(f(x).diff(x)) + f(x)
assert classify_ode(eq) == ('2nd_power_series_ordinary',)
assert dsolve(eq) == Eq(f(x), C2*(
x**4/8 - x**2/2 + 1) + C1*x*(-x**2/3 + 1) + O(x**6))
eq = f(x).diff(x, 2) + f(x).diff(x) - x*f(x)
assert classify_ode(eq) == ('2nd_power_series_ordinary',)
assert dsolve(eq) == Eq(f(x), C2*(
-x**4/24 + x**3/6 + 1) + C1*x*(x**3/24 + x**2/6 - x/2
+ 1) + O(x**6))
eq = f(x).diff(x, 2) + x*f(x)
assert classify_ode(eq) == ('2nd_power_series_ordinary',)
assert dsolve(eq, n=7) == Eq(f(x), C2*(
x**6/180 - x**3/6 + 1) + C1*x*(-x**3/12 + 1) + O(x**7))
def test_2nd_power_series_regular():
# FIXME: Maybe there should be a way to check series solutions
# checkodesol doesn't work with them.
C1, C2 = symbols("C1 C2")
eq = x**2*(f(x).diff(x, 2)) - 3*x*(f(x).diff(x)) + (4*x + 4)*f(x)
assert dsolve(eq) == Eq(f(x), C1*x**2*(-16*x**3/9 +
4*x**2 - 4*x + 1) + O(x**6))
eq = 4*x**2*(f(x).diff(x, 2)) -8*x**2*(f(x).diff(x)) + (4*x**2 +
1)*f(x)
assert dsolve(eq) == Eq(f(x), C1*sqrt(x)*(
x**4/24 + x**3/6 + x**2/2 + x + 1) + O(x**6))
eq = x**2*(f(x).diff(x, 2)) - x**2*(f(x).diff(x)) + (
x**2 - 2)*f(x)
assert dsolve(eq) == Eq(f(x), C1*(-x**6/720 - 3*x**5/80 - x**4/8 +
x**2/2 + x/2 + 1)/x + C2*x**2*(-x**3/60 + x**2/20 + x/2 + 1)
+ O(x**6))
eq = x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**2 - S(1)/4)*f(x)
assert dsolve(eq) == Eq(f(x), C1*(x**4/24 - x**2/2 + 1)/sqrt(x) +
C2*sqrt(x)*(x**4/120 - x**2/6 + 1) + O(x**6))
eq = x*(f(x).diff(x, 2)) - f(x).diff(x) + 4*x**3*f(x)
assert dsolve(eq) == Eq(f(x), C2*(-x**4/2 + 1) + C1*x**2 + O(x**6))
def test_issue_7093():
x = Symbol("x") # assuming x is real leads to an error
sol = [Eq(f(x), C1 - 2*x*sqrt(x**3)/5),
Eq(f(x), C1 + 2*x*sqrt(x**3)/5)]
eq = Derivative(f(x), x)**2 - x**3
assert set(dsolve(eq)) == set(sol)
assert checkodesol(eq, sol) == [(True, 0)] * 2
def test_dsolve_linsystem_symbol():
eps = Symbol('epsilon', positive=True)
eq1 = (Eq(diff(f(x), x), -eps*g(x)), Eq(diff(g(x), x), eps*f(x)))
sol1 = [Eq(f(x), -C1*eps*cos(eps*x) - C2*eps*sin(eps*x)),
Eq(g(x), -C1*eps*sin(eps*x) + C2*eps*cos(eps*x))]
assert checksysodesol(eq1, sol1) == (True, [0, 0])
def test_C1_function_9239():
t = Symbol('t')
C1 = Function('C1')
C2 = Function('C2')
C3 = Symbol('C3')
C4 = Symbol('C4')
eq = (Eq(diff(C1(t), t), 9*C2(t)), Eq(diff(C2(t), t), 12*C1(t)))
sol = [Eq(C1(t), 9*C3*exp(6*sqrt(3)*t) + 9*C4*exp(-6*sqrt(3)*t)),
Eq(C2(t), 6*sqrt(3)*C3*exp(6*sqrt(3)*t) - 6*sqrt(3)*C4*exp(-6*sqrt(3)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0])
def test_issue_15056():
t = Symbol('t')
C3 = Symbol('C3')
assert get_numbered_constants(Symbol('C1') * Function('C2')(t)) == C3
def test_issue_10379():
t,y = symbols('t,y')
eq = f(t).diff(t)-(1-51.05*y*f(t))
sol = Eq(f(t), (0.019588638589618*exp(y*(C1 - 51.05*t)) + 0.019588638589618)/y)
dsolve_sol = dsolve(eq, rational=False)
assert str(dsolve_sol) == str(sol)
assert checkodesol(eq, dsolve_sol)[0]
def test_issue_10867():
x = Symbol('x')
eq = Eq(g(x).diff(x).diff(x), (x-2)**2 + (x-3)**3)
sol = Eq(g(x), C1 + C2*x + x**5/20 - 2*x**4/3 + 23*x**3/6 - 23*x**2/2)
assert dsolve(eq, g(x)) == sol
assert checkodesol(eq, sol, order=2, solve_for_func=False) == (True, 0)
def test_issue_11290():
eq = cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x)
sol_1 = dsolve(eq, f(x), simplify=False, hint='1st_exact_Integral')
sol_0 = dsolve(eq, f(x), simplify=False, hint='1st_exact')
assert sol_1.dummy_eq(Eq(Subs(
Integral(u**2 - x*sin(u) - Integral(-sin(u), x), u) +
Integral(cos(u), x), u, f(x)), C1))
assert sol_1.doit() == sol_0
assert checkodesol(eq, sol_0, order=1, solve_for_func=False)
assert checkodesol(eq, sol_1, order=1, solve_for_func=False)
def test_issue_4838():
# Issue #15999
eq = f(x).diff(x) - C1*f(x)
sol = Eq(f(x), C2*exp(C1*x))
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False) == (True, 0)
# Issue #13691
eq = f(x).diff(x) - C1*g(x).diff(x)
sol = Eq(f(x), C2 + C1*g(x))
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, f(x), order=1, solve_for_func=False) == (True, 0)
# Issue #4838
eq = f(x).diff(x) - 3*C1 - 3*x**2
sol = Eq(f(x), C2 + 3*C1*x + x**3)
assert dsolve(eq, f(x)) == sol
assert checkodesol(eq, sol, order=1, solve_for_func=False) == (True, 0)
@slow
def test_issue_14395():
eq = Derivative(f(x), x, x) + 9*f(x) - sec(x)
sol = Eq(f(x), (C1 - x/3 + sin(2*x)/3)*sin(3*x) + (C2 + log(cos(x))
- 2*log(cos(x)**2)/3 + 2*cos(x)**2/3)*cos(3*x))
assert dsolve(eq, f(x)) == sol
# FIXME: assert checkodesol(eq, sol, order=2, solve_for_func=False) == (True, 0)
def test_sysode_linear_neq_order1():
from sympy.abc import t
Z0 = Function('Z0')
Z1 = Function('Z1')
Z2 = Function('Z2')
Z3 = Function('Z3')
k01, k10, k20, k21, k23, k30 = symbols('k01 k10 k20 k21 k23 k30')
eq = (Eq(Derivative(Z0(t), t), -k01*Z0(t) + k10*Z1(t) + k20*Z2(t) + k30*Z3(t)), Eq(Derivative(Z1(t), t),
k01*Z0(t) - k10*Z1(t) + k21*Z2(t)), Eq(Derivative(Z2(t), t), -(k20 + k21 + k23)*Z2(t)), Eq(Derivative(Z3(t),
t), k23*Z2(t) - k30*Z3(t)))
sols_eq = [Eq(Z0(t), C1*k10/k01 + C2*(-k10 + k30)*exp(-k30*t)/(k01 + k10 - k30) - C3*exp(t*(-
k01 - k10)) + C4*(k10*k20 + k10*k21 - k10*k30 - k20**2 - k20*k21 - k20*k23 + k20*k30 +
k23*k30)*exp(t*(-k20 - k21 - k23))/(k23*(k01 + k10 - k20 - k21 - k23))),
Eq(Z1(t), C1 - C2*k01*exp(-k30*t)/(k01 + k10 - k30) + C3*exp(t*(-k01 - k10)) + C4*(k01*k20 + k01*k21
- k01*k30 - k20*k21 - k21**2 - k21*k23 + k21*k30)*exp(t*(-k20 - k21 - k23))/(k23*(k01 + k10 - k20 -
k21 - k23))),
Eq(Z2(t), C4*(-k20 - k21 - k23 + k30)*exp(t*(-k20 - k21 - k23))/k23),
Eq(Z3(t), C2*exp(-k30*t) + C4*exp(t*(-k20 - k21 - k23)))]
assert dsolve(eq, simplify=False) == sols_eq
assert checksysodesol(eq, sols_eq) == (True, [0, 0, 0, 0])
def test_order_reducible():
from sympy.solvers.ode import _order_reducible_match
eqn = Eq(x*Derivative(f(x), x)**2 + Derivative(f(x), x, 2))
sol = Eq(f(x),
C1 - sqrt(-1/C2)*log(-C2*sqrt(-1/C2) + x) + sqrt(-1/C2)*log(C2*sqrt(-1/C2) + x))
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert sol == dsolve(eqn, f(x), hint='order_reducible')
assert sol == dsolve(eqn, f(x))
F = lambda eq: _order_reducible_match(eq, f(x))
D = Derivative
assert F(D(y*f(x), x, y) + D(f(x), x)) is None
assert F(D(y*f(y), y, y) + D(f(y), y)) is None
assert F(f(x)*D(f(x), x) + D(f(x), x, 2)) is None
assert F(D(x*f(y), y, 2) + D(u*y*f(x), x, 3)) is None # no simplification by design
assert F(D(f(y), y, 2) + D(f(y), y, 3) + D(f(x), x, 4)) is None
assert F(D(f(x), x, 2) + D(f(x), x, 3)) == dict(n=2)
eqn = -exp(x) + (x*Derivative(f(x), (x, 2)) + Derivative(f(x), x))/x
sol = Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert sol == dsolve(eqn, f(x))
assert sol == dsolve(eqn, f(x), hint='order_reducible')
eqn = Eq(sqrt(2) * f(x).diff(x,x,x) + f(x).diff(x), 0)
sol = Eq(f(x), C1 + C2*sin(2**(S(3)/4)*x/2) + C3*cos(2**(S(3)/4)*x/2))
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert sol == dsolve(eqn, f(x))
assert sol == dsolve(eqn, f(x), hint='order_reducible')
eqn = f(x).diff(x, 2) + 2*f(x).diff(x)
sol = Eq(f(x), C1 + C2*exp(-2*x))
sols = constant_renumber(sol)
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert dsolve(eqn, f(x)) in (sol, sols)
assert dsolve(eqn, f(x), hint='order_reducible') in (sol, sols)
eqn = f(x).diff(x, 3) + f(x).diff(x, 2) - 6*f(x).diff(x)
sol = Eq(f(x), C1 + C2*exp(-3*x) + C3*exp(2*x))
sols = constant_renumber(sol)
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert dsolve(eqn, f(x)) in (sol, sols)
assert dsolve(eqn, f(x), hint='order_reducible') in (sol, sols)
eqn = f(x).diff(x, 4) - f(x).diff(x, 3) - 4*f(x).diff(x, 2) + \
4*f(x).diff(x)
sol = Eq(f(x), C1 + C2*exp(x) + C3*exp(-2*x) + C4*exp(2*x))
sols = constant_renumber(sol)
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert dsolve(eqn, f(x)) in (sol, sols)
assert dsolve(eqn, f(x), hint='order_reducible') in (sol, sols)
eqn = f(x).diff(x, 4) + 3*f(x).diff(x, 3)
sol = Eq(f(x), C1 + C2*x + C3*x**2 + C4*exp(-3*x))
sols = constant_renumber(sol)
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert dsolve(eqn, f(x)) in (sol, sols)
assert dsolve(eqn, f(x), hint='order_reducible') in (sol, sols)
eqn = f(x).diff(x, 4) - 2*f(x).diff(x, 2)
sol = Eq(f(x), C1 + C2*x + C3*exp(x*sqrt(2)) + C4*exp(-x*sqrt(2)))
sols = constant_renumber(sol)
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert dsolve(eqn, f(x)) in (sol, sols)
assert dsolve(eqn, f(x), hint='order_reducible') in (sol, sols)
eqn = f(x).diff(x, 4) + 4*f(x).diff(x, 2)
sol = Eq(f(x), C1 + C2*sin(2*x) + C3*cos(2*x) + C4*x)
sols = constant_renumber(sol)
assert checkodesol(eqn, sol, order=2, solve_for_func=False) == (True, 0)
assert dsolve(eqn, f(x)) in (sol, sols)
assert dsolve(eqn, f(x), hint='order_reducible') in (sol, sols)
eqn = f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x)
# These are equivalent:
sol1 = Eq(f(x), C1 + (C2 + C3*x)*sin(x) + (C4 + C5*x)*cos(x))
sol2 = Eq(f(x), C1 + C2*(x*sin(x) + cos(x)) + C3*(-x*cos(x) + sin(x)) + C4*sin(x) + C5*cos(x))
sol1s = constant_renumber(sol1)
sol2s = constant_renumber(sol2)
assert checkodesol(eqn, sol1, order=2, solve_for_func=False) == (True, 0)
assert checkodesol(eqn, sol2, order=2, solve_for_func=False) == (True, 0)
assert dsolve(eqn, f(x)) in (sol1, sol1s)
assert dsolve(eqn, f(x), hint='order_reducible') in (sol2, sol2s)
def test_nth_algebraic():
eqn = Eq(Derivative(f(x), x), Derivative(g(x), x))
sol = Eq(f(x), C1 + g(x))
assert checkodesol(eqn, sol, order=1, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
eqn = (diff(f(x)) - x)*(diff(f(x)) + x)
sol = [Eq(f(x), C1 - x**2/2), Eq(f(x), C1 + x**2/2)]
assert checkodesol(eqn, sol, order=1, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
eqn = (1 - sin(f(x))) * f(x).diff(x)
sol = Eq(f(x), C1)
assert checkodesol(eqn, sol, order=1, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
M, m, r, t = symbols('M m r t')
phi = Function('phi')
eqn = Eq(-M * phi(t).diff(t),
Rational(3, 2) * m * r**2 * phi(t).diff(t) * phi(t).diff(t,t))
solns = [Eq(phi(t), C1), Eq(phi(t), C1 + C2*t - M*t**2/(3*m*r**2))]
assert checkodesol(eqn, solns[0], order=2, solve_for_func=False)[0]
assert checkodesol(eqn, solns[1], order=2, solve_for_func=False)[0]
assert set(solns) == set(dsolve(eqn, phi(t), hint='nth_algebraic'))
assert set(solns) == set(dsolve(eqn, phi(t)))
eqn = f(x) * f(x).diff(x) * f(x).diff(x, x)
sol = Eq(f(x), C1 + C2*x)
assert checkodesol(eqn, sol, order=1, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
eqn = f(x) * f(x).diff(x) * f(x).diff(x, x) * (f(x) - 1)
sol = Eq(f(x), C1 + C2*x)
assert checkodesol(eqn, sol, order=1, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
eqn = f(x) * f(x).diff(x) * f(x).diff(x, x) * (f(x) - 1) * (f(x).diff(x) - x)
solns = [Eq(f(x), C1 + x**2/2), Eq(f(x), C1 + C2*x)]
assert checkodesol(eqn, solns[0], order=2, solve_for_func=False)[0]
assert checkodesol(eqn, solns[1], order=2, solve_for_func=False)[0]
assert set(solns) == set(dsolve(eqn, f(x), hint='nth_algebraic'))
assert set(solns) == set(dsolve(eqn, f(x)))
def test_nth_algebraic_issue15999():
eqn = f(x).diff(x) - C1
sol = Eq(f(x), C1*x + C2) # Correct solution
assert checkodesol(eqn, sol, order=1, solve_for_func=False) == (True, 0)
assert dsolve(eqn, f(x), hint='nth_algebraic') == sol
assert dsolve(eqn, f(x)) == sol
def test_nth_algebraic_redundant_solutions():
# This one has a redundant solution that should be removed
eqn = f(x)*f(x).diff(x)
soln = Eq(f(x), C1)
assert checkodesol(eqn, soln, order=1, solve_for_func=False)[0]
assert soln == dsolve(eqn, f(x), hint='nth_algebraic')
assert soln == dsolve(eqn, f(x))
# This has two integral solutions and no algebraic solutions
eqn = (diff(f(x)) - x)*(diff(f(x)) + x)
sol = [Eq(f(x), C1 - x**2/2), Eq(f(x), C1 + x**2/2)]
assert all(c[0] for c in checkodesol(eqn, sol, order=1, solve_for_func=False))
assert set(sol) == set(dsolve(eqn, f(x), hint='nth_algebraic'))
assert set(sol) == set(dsolve(eqn, f(x)))
# This one doesn't work with dsolve at the time of writing but the
# redundancy checking code should not remove the algebraic solution.
from sympy.solvers.ode import _nth_algebraic_remove_redundant_solutions
eqn = f(x) + f(x)*f(x).diff(x)
solns = [Eq(f(x), 0),
Eq(f(x), C1 - x)]
solns_final = _nth_algebraic_remove_redundant_solutions(eqn, solns, 1, x)
assert all(c[0] for c in checkodesol(eqn, solns, order=1, solve_for_func=False))
assert set(solns) == set(solns_final)
solns = [Eq(f(x), exp(x)),
Eq(f(x), C1*exp(C2*x))]
solns_final = _nth_algebraic_remove_redundant_solutions(eqn, solns, 2, x)
assert solns_final == [Eq(f(x), C1*exp(C2*x))]
# This one needs a substitution f' = g.
eqn = -exp(x) + (x*Derivative(f(x), (x, 2)) + Derivative(f(x), x))/x
sol = Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))
assert checkodesol(eqn, sol, order=2, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x))
#
# These tests can be combined with the above test if they get fixed
# so that dsolve actually works in all these cases.
#
# Fails due to division by f(x) eliminating the solution before nth_algebraic
# is called.
@XFAIL
def test_nth_algebraic_find_multiple1():
eqn = f(x) + f(x)*f(x).diff(x)
solns = [Eq(f(x), 0),
Eq(f(x), C1 - x)]
assert all(c[0] for c in checkodesol(eqn, solns, order=1, solve_for_func=False))
assert set(solns) == set(dsolve(eqn, f(x)))
# prep = True breaks this
def test_nth_algebraic_noprep1():
eqn = Derivative(x*f(x), x, x, x)
sol = Eq(f(x), (C1 + C2*x + C3*x**2) / x)
assert checkodesol(eqn, sol, order=3, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), prep=False, hint='nth_algebraic')
@XFAIL
def test_nth_algebraic_prep1():
eqn = Derivative(x*f(x), x, x, x)
sol = Eq(f(x), (C1 + C2*x + C3*x**2) / x)
assert checkodesol(eqn, sol, order=3, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), prep=True, hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
# prep = True breaks this
def test_nth_algebraic_noprep2():
eqn = Eq(Derivative(x*Derivative(f(x), x), x)/x, exp(x))
sol = Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))
assert checkodesol(eqn, sol, order=2, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), prep=False, hint='nth_algebraic')
@XFAIL
def test_nth_algebraic_prep2():
eqn = Eq(Derivative(x*Derivative(f(x), x), x)/x, exp(x))
sol = Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))
assert checkodesol(eqn, sol, order=2, solve_for_func=False)[0]
assert sol == dsolve(eqn, f(x), prep=True, hint='nth_algebraic')
assert sol == dsolve(eqn, f(x))
# This needs a combination of solutions from nth_algebraic and some other
# method from dsolve
@XFAIL
def test_nth_algebraic_find_multiple2():
eqn = f(x)**2 + f(x)*f(x).diff(x)
solns = [Eq(f(x), 0),
Eq(f(x), C1*exp(-x))]
assert all(c[0] for c in checkodesol(eqn, solns, order=1, solve_for_func=False))
assert set(solns) == dsolve(eqn, f(x))
# Needs to be a way to know how to combine derivatives in the expression
@XFAIL
def test_factoring_ode():
eqn = Derivative(x*f(x), x, x, x) + Derivative(f(x), x, x, x)
soln = Eq(f(x), (C1*x**2/2 + C2*x + C3 - x)/(1 + x))
assert checkodesol(eqn, soln, order=2, solve_for_func=False)[0]
assert soln == dsolve(eqn, f(x))
def test_issue_15913():
eq = -C1/x - 2*x*f(x) - f(x) + Derivative(f(x), x)
sol = C2*exp(x**2 + x) + exp(x**2 + x)*Integral(C1*exp(-x**2 - x)/x, x)
assert checkodesol(eq, sol) == (True, 0)
sol = C1 + C2*exp(-x*y)
eq = Derivative(y*f(x), x) + f(x).diff(x, 2)
assert checkodesol(eq, sol, f(x)) == (True, 0)
def test_issue_16146():
raises(ValueError, lambda: dsolve([f(x).diff(x), g(x).diff(x)], [f(x), g(x), h(x)]))
raises(ValueError, lambda: dsolve([f(x).diff(x), g(x).diff(x)], [f(x)]))
|
b3aa9eac6ce49547bba7bac52619adecf33617e78ea053fa57f9cbcd932f948b
|
from sympy import Eq, factorial, Function, Lambda, rf, S, sqrt, symbols, I, expand_func, binomial, gamma
from sympy.solvers.recurr import rsolve, rsolve_hyper, rsolve_poly, rsolve_ratio
from sympy.utilities.pytest import raises, slow
from sympy.core.compatibility import range
from sympy.abc import a, b
y = Function('y')
n, k = symbols('n,k', integer=True)
C0, C1, C2 = symbols('C0,C1,C2')
def test_rsolve_poly():
assert rsolve_poly([-1, -1, 1], 0, n) == 0
assert rsolve_poly([-1, -1, 1], 1, n) == -1
assert rsolve_poly([-1, n + 1], n, n) == 1
assert rsolve_poly([-1, 1], n, n) == C0 + (n**2 - n)/2
assert rsolve_poly([-n - 1, n], 1, n) == C1*n - 1
assert rsolve_poly([-4*n - 2, 1], 4*n + 1, n) == -1
assert rsolve_poly([-1, 1], n**5 + n**3, n) == \
C0 - n**3 / 2 - n**5 / 2 + n**2 / 6 + n**6 / 6 + 2*n**4 / 3
def test_rsolve_ratio():
solution = rsolve_ratio([-2*n**3 + n**2 + 2*n - 1, 2*n**3 + n**2 - 6*n,
-2*n**3 - 11*n**2 - 18*n - 9, 2*n**3 + 13*n**2 + 22*n + 8], 0, n)
assert solution in [
C1*((-2*n + 3)/(n**2 - 1))/3,
(S(1)/2)*(C1*(-3 + 2*n)/(-1 + n**2)),
(S(1)/2)*(C1*( 3 - 2*n)/( 1 - n**2)),
(S(1)/2)*(C2*(-3 + 2*n)/(-1 + n**2)),
(S(1)/2)*(C2*( 3 - 2*n)/( 1 - n**2)),
]
def test_rsolve_hyper():
assert rsolve_hyper([-1, -1, 1], 0, n) in [
C0*(S.Half - S.Half*sqrt(5))**n + C1*(S.Half + S.Half*sqrt(5))**n,
C1*(S.Half - S.Half*sqrt(5))**n + C0*(S.Half + S.Half*sqrt(5))**n,
]
assert rsolve_hyper([n**2 - 2, -2*n - 1, 1], 0, n) in [
C0*rf(sqrt(2), n) + C1*rf(-sqrt(2), n),
C1*rf(sqrt(2), n) + C0*rf(-sqrt(2), n),
]
assert rsolve_hyper([n**2 - k, -2*n - 1, 1], 0, n) in [
C0*rf(sqrt(k), n) + C1*rf(-sqrt(k), n),
C1*rf(sqrt(k), n) + C0*rf(-sqrt(k), n),
]
assert rsolve_hyper(
[2*n*(n + 1), -n**2 - 3*n + 2, n - 1], 0, n) == C1*factorial(n) + C0*2**n
assert rsolve_hyper(
[n + 2, -(2*n + 3)*(17*n**2 + 51*n + 39), n + 1], 0, n) == None
assert rsolve_hyper([-n - 1, -1, 1], 0, n) == None
assert rsolve_hyper([-1, 1], n, n).expand() == C0 + n**2/2 - n/2
assert rsolve_hyper([-1, 1], 1 + n, n).expand() == C0 + n**2/2 + n/2
assert rsolve_hyper([-1, 1], 3*(n + n**2), n).expand() == C0 + n**3 - n
assert rsolve_hyper([-a, 1],0,n).expand() == C0*a**n
assert rsolve_hyper([-a, 0, 1], 0, n).expand() == (-1)**n*C1*a**(n/2) + C0*a**(n/2)
assert rsolve_hyper([1, 1, 1], 0, n).expand() == \
C0*(-S(1)/2 - sqrt(3)*I/2)**n + C1*(-S(1)/2 + sqrt(3)*I/2)**n
assert rsolve_hyper([1, -2*n/a - 2/a, 1], 0, n) is None
def recurrence_term(c, f):
"""Compute RHS of recurrence in f(n) with coefficients in c."""
return sum(c[i]*f.subs(n, n + i) for i in range(len(c)))
def test_rsolve_bulk():
"""Some bulk-generated tests."""
funcs = [ n, n + 1, n**2, n**3, n**4, n + n**2, 27*n + 52*n**2 - 3*
n**3 + 12*n**4 - 52*n**5 ]
coeffs = [ [-2, 1], [-2, -1, 1], [-1, 1, 1, -1, 1], [-n, 1], [n**2 -
n + 12, 1] ]
for p in funcs:
# compute difference
for c in coeffs:
q = recurrence_term(c, p)
if p.is_polynomial(n):
assert rsolve_poly(c, q, n) == p
# See issue 3956:
#if p.is_hypergeometric(n):
# assert rsolve_hyper(c, q, n) == p
def test_rsolve():
f = y(n + 2) - y(n + 1) - y(n)
h = sqrt(5)*(S.Half + S.Half*sqrt(5))**n \
- sqrt(5)*(S.Half - S.Half*sqrt(5))**n
assert rsolve(f, y(n)) in [
C0*(S.Half - S.Half*sqrt(5))**n + C1*(S.Half + S.Half*sqrt(5))**n,
C1*(S.Half - S.Half*sqrt(5))**n + C0*(S.Half + S.Half*sqrt(5))**n,
]
assert rsolve(f, y(n), [0, 5]) == h
assert rsolve(f, y(n), {0: 0, 1: 5}) == h
assert rsolve(f, y(n), {y(0): 0, y(1): 5}) == h
assert rsolve(y(n) - y(n - 1) - y(n - 2), y(n), [0, 5]) == h
assert rsolve(Eq(y(n), y(n - 1) + y(n - 2)), y(n), [0, 5]) == h
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = (n - 1)*y(n + 2) - (n**2 + 3*n - 2)*y(n + 1) + 2*n*(n + 1)*y(n)
g = C1*factorial(n) + C0*2**n
h = -3*factorial(n) + 3*2**n
assert rsolve(f, y(n)) == g
assert rsolve(f, y(n), []) == g
assert rsolve(f, y(n), {}) == g
assert rsolve(f, y(n), [0, 3]) == h
assert rsolve(f, y(n), {0: 0, 1: 3}) == h
assert rsolve(f, y(n), {y(0): 0, y(1): 3}) == h
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = y(n) - y(n - 1) - 2
assert rsolve(f, y(n), {y(0): 0}) == 2*n
assert rsolve(f, y(n), {y(0): 1}) == 2*n + 1
assert rsolve(f, y(n), {y(0): 0, y(1): 1}) is None
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = 3*y(n - 1) - y(n) - 1
assert rsolve(f, y(n), {y(0): 0}) == -3**n/2 + S.Half
assert rsolve(f, y(n), {y(0): 1}) == 3**n/2 + S.Half
assert rsolve(f, y(n), {y(0): 2}) == 3*3**n/2 + S.Half
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = y(n) - 1/n*y(n - 1)
assert rsolve(f, y(n)) == C0/factorial(n)
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = y(n) - 1/n*y(n - 1) - 1
assert rsolve(f, y(n)) is None
f = 2*y(n - 1) + (1 - n)*y(n)/n
assert rsolve(f, y(n), {y(1): 1}) == 2**(n - 1)*n
assert rsolve(f, y(n), {y(1): 2}) == 2**(n - 1)*n*2
assert rsolve(f, y(n), {y(1): 3}) == 2**(n - 1)*n*3
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = (n - 1)*(n - 2)*y(n + 2) - (n + 1)*(n + 2)*y(n)
assert rsolve(f, y(n), {y(3): 6, y(4): 24}) == n*(n - 1)*(n - 2)
assert rsolve(
f, y(n), {y(3): 6, y(4): -24}) == -n*(n - 1)*(n - 2)*(-1)**(n)
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
assert rsolve(Eq(y(n + 1), a*y(n)), y(n), {y(1): a}).simplify() == a**n
assert rsolve(y(n) - a*y(n-2),y(n), \
{y(1): sqrt(a)*(a + b), y(2): a*(a - b)}).simplify() == \
a**(n/2)*(-(-1)**n*b + a)
f = (-16*n**2 + 32*n - 12)*y(n - 1) + (4*n**2 - 12*n + 9)*y(n)
assert expand_func(rsolve(f, y(n), \
{y(1): binomial(2*n + 1, 3)}).rewrite(gamma)).simplify() == \
2**(2*n)*n*(2*n - 1)*(4*n**2 - 1)/12
assert (rsolve(y(n) + a*(y(n + 1) + y(n - 1))/2, y(n)) -
(C0*((sqrt(-a**2 + 1) - 1)/a)**n +
C1*((-sqrt(-a**2 + 1) - 1)/a)**n)).simplify() == 0
assert rsolve((k + 1)*y(k), y(k)) is None
assert (rsolve((k + 1)*y(k) + (k + 3)*y(k + 1) + (k + 5)*y(k + 2), y(k))
is None)
def test_rsolve_raises():
x = Function('x')
raises(ValueError, lambda: rsolve(y(n) - y(k + 1), y(n)))
raises(ValueError, lambda: rsolve(y(n) - y(n + 1), x(n)))
raises(ValueError, lambda: rsolve(y(n) - x(n + 1), y(n)))
raises(ValueError, lambda: rsolve(y(n) - sqrt(n)*y(n + 1), y(n)))
raises(ValueError, lambda: rsolve(y(n) - y(n + 1), y(n), {x(0): 0}))
def test_issue_6844():
f = y(n + 2) - y(n + 1) + y(n)/4
assert rsolve(f, y(n)) == 2**(-n)*(C0 + C1*n)
assert rsolve(f, y(n), {y(0): 0, y(1): 1}) == 2*2**(-n)*n
@slow
def test_issue_15751():
f = y(n) + 21*y(n + 1) - 273*y(n + 2) - 1092*y(n + 3) + 1820*y(n + 4) + 1092*y(n + 5) - 273*y(n + 6) - 21*y(n + 7) + y(n + 8)
assert rsolve(f, y(n)) is not None
|
a224c700e2b08ddc7e3c62a36a6122b7530150afc0072ac8b38fe4b5f12d5ca6
|
"""
If the arbitrary constant class from issue 4435 is ever implemented, this
should serve as a set of test cases.
"""
from sympy import (acos, cos, cosh, Eq, exp, Function, I, Integral, log, Pow,
S, sin, sinh, sqrt, Symbol, Add)
from sympy.solvers.ode import constantsimp, constant_renumber
from sympy.utilities.pytest import XFAIL
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
u2 = Symbol('u2')
_a = Symbol('_a')
C1 = Symbol('C1')
C2 = Symbol('C2')
C3 = Symbol('C3')
f = Function('f')
def test_constant_mul():
# We want C1 (Constant) below to absorb the y's, but not the x's
assert constant_renumber(constantsimp(y*C1, [C1])) == C1*y
assert constant_renumber(constantsimp(C1*y, [C1])) == C1*y
assert constant_renumber(constantsimp(x*C1, [C1])) == x*C1
assert constant_renumber(constantsimp(C1*x, [C1])) == x*C1
assert constant_renumber(constantsimp(2*C1, [C1])) == C1
assert constant_renumber(constantsimp(C1*2, [C1])) == C1
assert constant_renumber(constantsimp(y*C1*x, [C1, y])) == C1*x
assert constant_renumber(constantsimp(x*y*C1, [C1, y])) == x*C1
assert constant_renumber(constantsimp(y*x*C1, [C1, y])) == x*C1
assert constant_renumber(constantsimp(C1*x*y, [C1, y])) == C1*x
assert constant_renumber(constantsimp(x*C1*y, [C1, y])) == x*C1
assert constant_renumber(constantsimp(C1*y*(y + 1), [C1])) == C1*y*(y+1)
assert constant_renumber(constantsimp(y*C1*(y + 1), [C1])) == C1*y*(y+1)
assert constant_renumber(constantsimp(x*(y*C1), [C1])) == x*y*C1
assert constant_renumber(constantsimp(x*(C1*y), [C1])) == x*y*C1
assert constant_renumber(constantsimp(C1*(x*y), [C1, y])) == C1*x
assert constant_renumber(constantsimp((x*y)*C1, [C1, y])) == x*C1
assert constant_renumber(constantsimp((y*x)*C1, [C1, y])) == x*C1
assert constant_renumber(constantsimp(y*(y + 1)*C1, [C1, y])) == C1
assert constant_renumber(constantsimp((C1*x)*y, [C1, y])) == C1*x
assert constant_renumber(constantsimp(y*(x*C1), [C1, y])) == x*C1
assert constant_renumber(constantsimp((x*C1)*y, [C1, y])) == x*C1
assert constant_renumber(constantsimp(C1*x*y*x*y*2, [C1, y])) == C1*x**2
assert constant_renumber(constantsimp(C1*x*y*z, [C1, y, z])) == C1*x
assert constant_renumber(constantsimp(C1*x*y**2*sin(z), [C1, y, z])) == C1*x
assert constant_renumber(constantsimp(C1*C1, [C1])) == C1
assert constant_renumber(constantsimp(C1*C2, [C1, C2])) == C1
assert constant_renumber(constantsimp(C2*C2, [C1, C2])) == C1
assert constant_renumber(constantsimp(C1*C1*C2, [C1, C2])) == C1
assert constant_renumber(constantsimp(C1*x*2**x, [C1])) == C1*x*2**x
def test_constant_add():
assert constant_renumber(constantsimp(C1 + C1, [C1])) == C1
assert constant_renumber(constantsimp(C1 + 2, [C1])) == C1
assert constant_renumber(constantsimp(2 + C1, [C1])) == C1
assert constant_renumber(constantsimp(C1 + y, [C1, y])) == C1
assert constant_renumber(constantsimp(C1 + x, [C1])) == C1 + x
assert constant_renumber(constantsimp(C1 + C1, [C1])) == C1
assert constant_renumber(constantsimp(C1 + C2, [C1, C2])) == C1
assert constant_renumber(constantsimp(C2 + C1, [C1, C2])) == C1
assert constant_renumber(constantsimp(C1 + C2 + C1, [C1, C2])) == C1
def test_constant_power_as_base():
assert constant_renumber(constantsimp(C1**C1, [C1])) == C1
assert constant_renumber(constantsimp(Pow(C1, C1), [C1])) == C1
assert constant_renumber(constantsimp(C1**C1, [C1])) == C1
assert constant_renumber(constantsimp(C1**C2, [C1, C2])) == C1
assert constant_renumber(constantsimp(C2**C1, [C1, C2])) == C1
assert constant_renumber(constantsimp(C2**C2, [C1, C2])) == C1
assert constant_renumber(constantsimp(C1**y, [C1, y])) == C1
assert constant_renumber(constantsimp(C1**x, [C1])) == C1**x
assert constant_renumber(constantsimp(C1**2, [C1])) == C1
assert constant_renumber(
constantsimp(C1**(x*y), [C1])) == C1**(x*y)
def test_constant_power_as_exp():
assert constant_renumber(constantsimp(x**C1, [C1])) == x**C1
assert constant_renumber(constantsimp(y**C1, [C1, y])) == C1
assert constant_renumber(constantsimp(x**y**C1, [C1, y])) == x**C1
assert constant_renumber(
constantsimp((x**y)**C1, [C1])) == (x**y)**C1
assert constant_renumber(
constantsimp(x**(y**C1), [C1, y])) == x**C1
assert constant_renumber(constantsimp(x**C1**y, [C1, y])) == x**C1
assert constant_renumber(
constantsimp(x**(C1**y), [C1, y])) == x**C1
assert constant_renumber(
constantsimp((x**C1)**y, [C1])) == (x**C1)**y
assert constant_renumber(constantsimp(2**C1, [C1])) == C1
assert constant_renumber(constantsimp(S(2)**C1, [C1])) == C1
assert constant_renumber(constantsimp(exp(C1), [C1])) == C1
assert constant_renumber(
constantsimp(exp(C1 + x), [C1])) == C1*exp(x)
assert constant_renumber(constantsimp(Pow(2, C1), [C1])) == C1
def test_constant_function():
assert constant_renumber(constantsimp(sin(C1), [C1])) == C1
assert constant_renumber(constantsimp(f(C1), [C1])) == C1
assert constant_renumber(constantsimp(f(C1, C1), [C1])) == C1
assert constant_renumber(constantsimp(f(C1, C2), [C1, C2])) == C1
assert constant_renumber(constantsimp(f(C2, C1), [C1, C2])) == C1
assert constant_renumber(constantsimp(f(C2, C2), [C1, C2])) == C1
assert constant_renumber(
constantsimp(f(C1, x), [C1])) == f(C1, x)
assert constant_renumber(constantsimp(f(C1, y), [C1, y])) == C1
assert constant_renumber(constantsimp(f(y, C1), [C1, y])) == C1
assert constant_renumber(constantsimp(f(C1, y, C2), [C1, C2, y])) == C1
def test_constant_function_multiple():
# The rules to not renumber in this case would be too complicated, and
# dsolve is not likely to ever encounter anything remotely like this.
assert constant_renumber(
constantsimp(f(C1, C1, x), [C1])) == f(C1, C1, x)
def test_constant_multiple():
assert constant_renumber(constantsimp(C1*2 + 2, [C1])) == C1
assert constant_renumber(constantsimp(x*2/C1, [C1])) == C1*x
assert constant_renumber(constantsimp(C1**2*2 + 2, [C1])) == C1
assert constant_renumber(
constantsimp(sin(2*C1) + x + sqrt(2), [C1])) == C1 + x
assert constant_renumber(constantsimp(2*C1 + C2, [C1, C2])) == C1
def test_constant_repeated():
assert C1 + C1*x == constant_renumber( C1 + C1*x)
def test_ode_solutions():
# only a few examples here, the rest will be tested in the actual dsolve tests
assert constant_renumber(constantsimp(C1*exp(2*x) + exp(x)*(C2 + C3), [C1, C2, C3])) == \
constant_renumber((C1*exp(x) + C2*exp(2*x)))
assert constant_renumber(
constantsimp(Eq(f(x), I*C1*sinh(x/3) + C2*cosh(x/3)), [C1, C2])
) == constant_renumber(Eq(f(x), C1*sinh(x/3) + C2*cosh(x/3)))
assert constant_renumber(constantsimp(Eq(f(x), acos((-C1)/cos(x))), [C1])) == \
Eq(f(x), acos(C1/cos(x)))
assert constant_renumber(
constantsimp(Eq(log(f(x)/C1) + 2*exp(x/f(x)), 0), [C1])
) == Eq(log(C1*f(x)) + 2*exp(x/f(x)), 0)
assert constant_renumber(constantsimp(Eq(log(x*sqrt(2)*sqrt(1/x)*sqrt(f(x))
/C1) + x**2/(2*f(x)**2), 0), [C1])) == \
Eq(log(C1*sqrt(x)*sqrt(f(x))) + x**2/(2*f(x)**2), 0)
assert constant_renumber(constantsimp(Eq(-exp(-f(x)/x)*sin(f(x)/x)/2 + log(x/C1) -
cos(f(x)/x)*exp(-f(x)/x)/2, 0), [C1])) == \
Eq(-exp(-f(x)/x)*sin(f(x)/x)/2 + log(C1*x) - cos(f(x)/x)*
exp(-f(x)/x)/2, 0)
assert constant_renumber(constantsimp(Eq(-Integral(-1/(sqrt(1 - u2**2)*u2),
(u2, _a, x/f(x))) + log(f(x)/C1), 0), [C1])) == \
Eq(-Integral(-1/(u2*sqrt(1 - u2**2)), (u2, _a, x/f(x))) +
log(C1*f(x)), 0)
assert [constantsimp(i, [C1]) for i in [Eq(f(x), sqrt(-C1*x + x**2)), Eq(f(x), -sqrt(-C1*x + x**2))]] == \
[Eq(f(x), sqrt(x*(C1 + x))), Eq(f(x), -sqrt(x*(C1 + x)))]
@XFAIL
def test_nonlocal_simplification():
assert constantsimp(C1 + C2+x*C2, [C1, C2]) == C1 + C2*x
def test_constant_Eq():
# C1 on the rhs is well-tested, but the lhs is only tested here
assert constantsimp(Eq(C1, 3 + f(x)*x), [C1]) == Eq(x*f(x), C1)
assert constantsimp(Eq(C1, 3 * f(x)*x), [C1]) == Eq(f(x)*x, C1)
|
60494b36364502128c2d50e3f93e2c6705027e95095f8561274bf706b7efe1c1
|
from sympy import sqrt
from sympy.core import S, Symbol, symbols, I
from sympy.core.compatibility import range
from sympy.discrete import (fft, ifft, ntt, intt, fwht, ifwht,
mobius_transform, inverse_mobius_transform)
from sympy.utilities.pytest import raises
def test_fft_ifft():
assert all(tf(ls) == ls for tf in (fft, ifft)
for ls in ([], [S(5)/3]))
ls = list(range(6))
fls = [15, -7*sqrt(2)/2 - 4 - sqrt(2)*I/2 + 2*I, 2 + 3*I,
-4 + 7*sqrt(2)/2 - 2*I - sqrt(2)*I/2, -3,
-4 + 7*sqrt(2)/2 + sqrt(2)*I/2 + 2*I,
2 - 3*I, -7*sqrt(2)/2 - 4 - 2*I + sqrt(2)*I/2]
assert fft(ls) == fls
assert ifft(fls) == ls + [S.Zero]*2
ls = [1 + 2*I, 3 + 4*I, 5 + 6*I]
ifls = [S(9)/4 + 3*I, -7*I/4, S(3)/4 + I, -2 - I/4]
assert ifft(ls) == ifls
assert fft(ifls) == ls + [S.Zero]
x = Symbol('x', real=True)
raises(TypeError, lambda: fft(x))
raises(ValueError, lambda: ifft([x, 2*x, 3*x**2, 4*x**3]))
def test_ntt_intt():
# prime moduli of the form (m*2**k + 1), sequence length
# should be a divisor of 2**k
p = 7*17*2**23 + 1
q = 2*500000003 + 1 # only for sequences of length 1 or 2
r = 2*3*5*7 # composite modulus
assert all(tf(ls, p) == ls for tf in (ntt, intt)
for ls in ([], [5]))
ls = list(range(6))
nls = [15, 801133602, 738493201, 334102277, 998244350, 849020224,
259751156, 12232587]
assert ntt(ls, p) == nls
assert intt(nls, p) == ls + [0]*2
ls = [1 + 2*I, 3 + 4*I, 5 + 6*I]
x = Symbol('x', integer=True)
raises(TypeError, lambda: ntt(x, p))
raises(ValueError, lambda: intt([x, 2*x, 3*x**2, 4*x**3], p))
raises(ValueError, lambda: intt(ls, p))
raises(ValueError, lambda: ntt([1.2, 2.1, 3.5], p))
raises(ValueError, lambda: ntt([3, 5, 6], q))
raises(ValueError, lambda: ntt([4, 5, 7], r))
raises(ValueError, lambda: ntt([1.0, 2.0, 3.0], p))
def test_fwht_ifwht():
assert all(tf(ls) == ls for tf in (fwht, ifwht) \
for ls in ([], [S(7)/4]))
ls = [213, 321, 43235, 5325, 312, 53]
fls = [49459, 38061, -47661, -37759, 48729, 37543, -48391, -38277]
assert fwht(ls) == fls
assert ifwht(fls) == ls + [S.Zero]*2
ls = [S(1)/2 + 2*I, S(3)/7 + 4*I, S(5)/6 + 6*I, S(7)/3, S(9)/4]
ifls = [S(533)/672 + 3*I/2, S(23)/224 + I/2, S(1)/672, S(107)/224 - I,
S(155)/672 + 3*I/2, -S(103)/224 + I/2, -S(377)/672, -S(19)/224 - I]
assert ifwht(ls) == ifls
assert fwht(ifls) == ls + [S.Zero]*3
x, y = symbols('x y')
raises(TypeError, lambda: fwht(x))
ls = [x, 2*x, 3*x**2, 4*x**3]
ifls = [x**3 + 3*x**2/4 + 3*x/4,
-x**3 + 3*x**2/4 - x/4,
-x**3 - 3*x**2/4 + 3*x/4,
x**3 - 3*x**2/4 - x/4]
assert ifwht(ls) == ifls
assert fwht(ifls) == ls
ls = [x, y, x**2, y**2, x*y]
fls = [x**2 + x*y + x + y**2 + y,
x**2 + x*y + x - y**2 - y,
-x**2 + x*y + x - y**2 + y,
-x**2 + x*y + x + y**2 - y,
x**2 - x*y + x + y**2 + y,
x**2 - x*y + x - y**2 - y,
-x**2 - x*y + x - y**2 + y,
-x**2 - x*y + x + y**2 - y]
assert fwht(ls) == fls
assert ifwht(fls) == ls + [S.Zero]*3
ls = list(range(6))
assert fwht(ls) == [x*8 for x in ifwht(ls)]
def test_mobius_transform():
assert all(tf(ls, subset=subset) == ls
for ls in ([], [S(7)/4]) for subset in (True, False)
for tf in (mobius_transform, inverse_mobius_transform))
w, x, y, z = symbols('w x y z')
assert mobius_transform([x, y]) == [x, x + y]
assert inverse_mobius_transform([x, x + y]) == [x, y]
assert mobius_transform([x, y], subset=False) == [x + y, y]
assert inverse_mobius_transform([x + y, y], subset=False) == [x, y]
assert mobius_transform([w, x, y, z]) == [w, w + x, w + y, w + x + y + z]
assert inverse_mobius_transform([w, w + x, w + y, w + x + y + z]) == \
[w, x, y, z]
assert mobius_transform([w, x, y, z], subset=False) == \
[w + x + y + z, x + z, y + z, z]
assert inverse_mobius_transform([w + x + y + z, x + z, y + z, z], subset=False) == \
[w, x, y, z]
ls = [S(2)/3, S(6)/7, S(5)/8, 9, S(5)/3 + 7*I]
mls = [S(2)/3, S(32)/21, S(31)/24, S(1873)/168,
S(7)/3 + 7*I, S(67)/21 + 7*I, S(71)/24 + 7*I,
S(2153)/168 + 7*I]
assert mobius_transform(ls) == mls
assert inverse_mobius_transform(mls) == ls + [S.Zero]*3
mls = [S(2153)/168 + 7*I, S(69)/7, S(77)/8, 9, S(5)/3 + 7*I, 0, 0, 0]
assert mobius_transform(ls, subset=False) == mls
assert inverse_mobius_transform(mls, subset=False) == ls + [S.Zero]*3
ls = ls[:-1]
mls = [S(2)/3, S(32)/21, S(31)/24, S(1873)/168]
assert mobius_transform(ls) == mls
assert inverse_mobius_transform(mls) == ls
mls = [S(1873)/168, S(69)/7, S(77)/8, 9]
assert mobius_transform(ls, subset=False) == mls
assert inverse_mobius_transform(mls, subset=False) == ls
raises(TypeError, lambda: mobius_transform(x, subset=True))
raises(TypeError, lambda: inverse_mobius_transform(y, subset=False))
|
a71921b9a3dc376ee00ce4b07e334de40be2ab809383fcf970822b42353104a4
|
from sympy import (Symbol, S, exp, log, sqrt, oo, E, zoo, pi, tan, sin, cos,
cot, sec, csc, Abs, symbols, I, re)
from sympy.calculus.util import (function_range, continuous_domain, not_empty_in,
periodicity, lcim, AccumBounds, is_convex)
from sympy.core import Add, Mul, Pow
from sympy.sets.sets import Interval, FiniteSet, Complement, Union
from sympy.utilities.pytest import raises
from sympy.abc import x
a = Symbol('a', real=True)
def test_function_range():
x, y, a, b = symbols('x y a b')
assert function_range(sin(x), x, Interval(-pi/2, pi/2)
) == Interval(-1, 1)
assert function_range(sin(x), x, Interval(0, pi)
) == Interval(0, 1)
assert function_range(tan(x), x, Interval(0, pi)
) == Interval(-oo, oo)
assert function_range(tan(x), x, Interval(pi/2, pi)
) == Interval(-oo, 0)
assert function_range((x + 3)/(x - 2), x, Interval(-5, 5)
) == Union(Interval(-oo, S(2)/7), Interval(S(8)/3, oo))
assert function_range(1/(x**2), x, Interval(-1, 1)
) == Interval(1, oo)
assert function_range(exp(x), x, Interval(-1, 1)
) == Interval(exp(-1), exp(1))
assert function_range(log(x) - x, x, S.Reals
) == Interval(-oo, -1)
assert function_range(sqrt(3*x - 1), x, Interval(0, 2)
) == Interval(0, sqrt(5))
assert function_range(x*(x - 1) - (x**2 - x), x, S.Reals
) == FiniteSet(0)
assert function_range(x*(x - 1) - (x**2 - x) + y, x, S.Reals
) == FiniteSet(y)
assert function_range(sin(x), x, Union(Interval(-5, -3), FiniteSet(4))
) == Union(Interval(-sin(3), 1), FiniteSet(sin(4)))
assert function_range(cos(x), x, Interval(-oo, -4)
) == Interval(-1, 1)
raises(NotImplementedError, lambda : function_range(
exp(x)*(sin(x) - cos(x))/2 - x, x, S.Reals))
raises(NotImplementedError, lambda : function_range(
log(x), x, S.Integers))
raises(NotImplementedError, lambda : function_range(
sin(x)/2, x, S.Naturals))
def test_continuous_domain():
x = Symbol('x')
assert continuous_domain(sin(x), x, Interval(0, 2*pi)) == Interval(0, 2*pi)
assert continuous_domain(tan(x), x, Interval(0, 2*pi)) == \
Union(Interval(0, pi/2, False, True), Interval(pi/2, 3*pi/2, True, True),
Interval(3*pi/2, 2*pi, True, False))
assert continuous_domain((x - 1)/((x - 1)**2), x, S.Reals) == \
Union(Interval(-oo, 1, True, True), Interval(1, oo, True, True))
assert continuous_domain(log(x) + log(4*x - 1), x, S.Reals) == \
Interval(S(1)/4, oo, True, True)
assert continuous_domain(1/sqrt(x - 3), x, S.Reals) == Interval(3, oo, True, True)
assert continuous_domain(1/x - 2, x, S.Reals) == \
Union(Interval.open(-oo, 0), Interval.open(0, oo))
assert continuous_domain(1/(x**2 - 4) + 2, x, S.Reals) == \
Union(Interval.open(-oo, -2), Interval.open(-2, 2), Interval.open(2, oo))
def test_not_empty_in():
assert not_empty_in(FiniteSet(x, 2*x).intersect(Interval(1, 2, True, False)), x) == \
Interval(S(1)/2, 2, True, False)
assert not_empty_in(FiniteSet(x, x**2).intersect(Interval(1, 2)), x) == \
Union(Interval(-sqrt(2), -1), Interval(1, 2))
assert not_empty_in(FiniteSet(x**2 + x, x).intersect(Interval(2, 4)), x) == \
Union(Interval(-sqrt(17)/2 - S(1)/2, -2),
Interval(1, -S(1)/2 + sqrt(17)/2), Interval(2, 4))
assert not_empty_in(FiniteSet(x/(x - 1)).intersect(S.Reals), x) == \
Complement(S.Reals, FiniteSet(1))
assert not_empty_in(FiniteSet(a/(a - 1)).intersect(S.Reals), a) == \
Complement(S.Reals, FiniteSet(1))
assert not_empty_in(FiniteSet((x**2 - 3*x + 2)/(x - 1)).intersect(S.Reals), x) == \
Complement(S.Reals, FiniteSet(1))
assert not_empty_in(FiniteSet(3, 4, x/(x - 1)).intersect(Interval(2, 3)), x) == \
Union(Interval(S(3)/2, 2), FiniteSet(3))
assert not_empty_in(FiniteSet(x/(x**2 - 1)).intersect(S.Reals), x) == \
Complement(S.Reals, FiniteSet(-1, 1))
assert not_empty_in(FiniteSet(x, x**2).intersect(Union(Interval(1, 3, True, True),
Interval(4, 5))), x) == \
Union(Interval(-sqrt(5), -2), Interval(-sqrt(3), -1, True, True),
Interval(1, 3, True, True), Interval(4, 5))
assert not_empty_in(FiniteSet(1).intersect(Interval(3, 4)), x) == S.EmptySet
assert not_empty_in(FiniteSet(x**2/(x + 2)).intersect(Interval(1, oo)), x) == \
Union(Interval(-2, -1, True, False), Interval(2, oo))
def test_periodicity():
x = Symbol('x')
y = Symbol('y')
z = Symbol('z', real=True)
assert periodicity(sin(2*x), x) == pi
assert periodicity((-2)*tan(4*x), x) == pi/4
assert periodicity(sin(x)**2, x) == 2*pi
assert periodicity(3**tan(3*x), x) == pi/3
assert periodicity(tan(x)*cos(x), x) == 2*pi
assert periodicity(sin(x)**(tan(x)), x) == 2*pi
assert periodicity(tan(x)*sec(x), x) == 2*pi
assert periodicity(sin(2*x)*cos(2*x) - y, x) == pi/2
assert periodicity(tan(x) + cot(x), x) == pi
assert periodicity(sin(x) - cos(2*x), x) == 2*pi
assert periodicity(sin(x) - 1, x) == 2*pi
assert periodicity(sin(4*x) + sin(x)*cos(x), x) == pi
assert periodicity(exp(sin(x)), x) == 2*pi
assert periodicity(log(cot(2*x)) - sin(cos(2*x)), x) == pi
assert periodicity(sin(2*x)*exp(tan(x) - csc(2*x)), x) == pi
assert periodicity(cos(sec(x) - csc(2*x)), x) == 2*pi
assert periodicity(tan(sin(2*x)), x) == pi
assert periodicity(2*tan(x)**2, x) == pi
assert periodicity(sin(x%4), x) == 4
assert periodicity(sin(x)%4, x) == 2*pi
assert periodicity(tan((3*x-2)%4), x) == S(4)/3
assert periodicity((sqrt(2)*(x+1)+x) % 3, x) == 3 / (sqrt(2)+1)
assert periodicity((x**2+1) % x, x) == None
assert periodicity(sin(re(x)), x) == 2*pi
assert periodicity(sin(x)**2 + cos(x)**2, x) == S.Zero
assert periodicity(tan(x), y) == S.Zero
assert periodicity(sin(x) + I*cos(x), x) == 2*pi
assert periodicity(x - sin(2*y), y) == pi
assert periodicity(exp(x), x) is None
assert periodicity(exp(I*x), x) == 2*pi
assert periodicity(exp(I*z), z) == 2*pi
assert periodicity(exp(z), z) is None
assert periodicity(exp(log(sin(z) + I*cos(2*z)), evaluate=False), z) == 2*pi
assert periodicity(exp(log(sin(2*z) + I*cos(z)), evaluate=False), z) == 2*pi
assert periodicity(exp(sin(z)), z) == 2*pi
assert periodicity(exp(2*I*z), z) == pi
assert periodicity(exp(z + I*sin(z)), z) is None
assert periodicity(exp(cos(z/2) + sin(z)), z) == 4*pi
assert periodicity(log(x), x) is None
assert periodicity(exp(x)**sin(x), x) is None
assert periodicity(sin(x)**y, y) is None
assert periodicity(Abs(sin(Abs(sin(x)))), x) == pi
assert all(periodicity(Abs(f(x)), x) == pi for f in (
cos, sin, sec, csc, tan, cot))
assert periodicity(Abs(sin(tan(x))), x) == pi
assert periodicity(Abs(sin(sin(x) + tan(x))), x) == 2*pi
assert periodicity(sin(x) > S.Half, x) is 2*pi
assert periodicity(x > 2, x) is None
assert periodicity(x**3 - x**2 + 1, x) is None
assert periodicity(Abs(x), x) is None
assert periodicity(Abs(x**2 - 1), x) is None
assert periodicity((x**2 + 4)%2, x) is None
assert periodicity((E**x)%3, x) is None
def test_periodicity_check():
x = Symbol('x')
y = Symbol('y')
assert periodicity(tan(x), x, check=True) == pi
assert periodicity(sin(x) + cos(x), x, check=True) == 2*pi
assert periodicity(sec(x), x) == 2*pi
assert periodicity(sin(x*y), x) == 2*pi/abs(y)
assert periodicity(Abs(sec(sec(x))), x) == pi
def test_lcim():
from sympy import pi
assert lcim([S(1)/2, S(2), S(3)]) == 6
assert lcim([pi/2, pi/4, pi]) == pi
assert lcim([2*pi, pi/2]) == 2*pi
assert lcim([S(1), 2*pi]) is None
assert lcim([S(2) + 2*E, E/3 + S(1)/3, S(1) + E]) == S(2) + 2*E
def test_is_convex():
assert is_convex(1/x, x, domain=Interval(0, oo)) == True
assert is_convex(1/x, x, domain=Interval(-oo, 0)) == False
assert is_convex(x**2, x, domain=Interval(0, oo)) == True
assert is_convex(log(x), x) == False
def test_AccumBounds():
assert AccumBounds(1, 2).args == (1, 2)
assert AccumBounds(1, 2).delta == S(1)
assert AccumBounds(1, 2).mid == S(3)/2
assert AccumBounds(1, 3).is_real == True
assert AccumBounds(1, 1) == S(1)
assert AccumBounds(1, 2) + 1 == AccumBounds(2, 3)
assert 1 + AccumBounds(1, 2) == AccumBounds(2, 3)
assert AccumBounds(1, 2) + AccumBounds(2, 3) == AccumBounds(3, 5)
assert -AccumBounds(1, 2) == AccumBounds(-2, -1)
assert AccumBounds(1, 2) - 1 == AccumBounds(0, 1)
assert 1 - AccumBounds(1, 2) == AccumBounds(-1, 0)
assert AccumBounds(2, 3) - AccumBounds(1, 2) == AccumBounds(0, 2)
assert x + AccumBounds(1, 2) == Add(AccumBounds(1, 2), x)
assert a + AccumBounds(1, 2) == AccumBounds(1 + a, 2 + a)
assert AccumBounds(1, 2) - x == Add(AccumBounds(1, 2), -x)
assert AccumBounds(-oo, 1) + oo == AccumBounds(-oo, oo)
assert AccumBounds(1, oo) + oo == oo
assert AccumBounds(1, oo) - oo == AccumBounds(-oo, oo)
assert (-oo - AccumBounds(-1, oo)) == -oo
assert AccumBounds(-oo, 1) - oo == -oo
assert AccumBounds(1, oo) - oo == AccumBounds(-oo, oo)
assert AccumBounds(-oo, 1) - (-oo) == AccumBounds(-oo, oo)
assert (oo - AccumBounds(1, oo)) == AccumBounds(-oo, oo)
assert (-oo - AccumBounds(1, oo)) == -oo
assert AccumBounds(1, 2)/2 == AccumBounds(S(1)/2, 1)
assert 2/AccumBounds(2, 3) == AccumBounds(S(2)/3, 1)
assert 1/AccumBounds(-1, 1) == AccumBounds(-oo, oo)
assert abs(AccumBounds(1, 2)) == AccumBounds(1, 2)
assert abs(AccumBounds(-2, -1)) == AccumBounds(1, 2)
assert abs(AccumBounds(-2, 1)) == AccumBounds(0, 2)
assert abs(AccumBounds(-1, 2)) == AccumBounds(0, 2)
def test_AccumBounds_mul():
assert AccumBounds(1, 2)*2 == AccumBounds(2, 4)
assert 2*AccumBounds(1, 2) == AccumBounds(2, 4)
assert AccumBounds(1, 2)*AccumBounds(2, 3) == AccumBounds(2, 6)
assert AccumBounds(1, 2)*0 == 0
assert AccumBounds(1, oo)*0 == AccumBounds(0, oo)
assert AccumBounds(-oo, 1)*0 == AccumBounds(-oo, 0)
assert AccumBounds(-oo, oo)*0 == AccumBounds(-oo, oo)
assert AccumBounds(1, 2)*x == Mul(AccumBounds(1, 2), x, evaluate=False)
assert AccumBounds(0, 2)*oo == AccumBounds(0, oo)
assert AccumBounds(-2, 0)*oo == AccumBounds(-oo, 0)
assert AccumBounds(0, 2)*(-oo) == AccumBounds(-oo, 0)
assert AccumBounds(-2, 0)*(-oo) == AccumBounds(0, oo)
assert AccumBounds(-1, 1)*oo == AccumBounds(-oo, oo)
assert AccumBounds(-1, 1)*(-oo) == AccumBounds(-oo, oo)
assert AccumBounds(-oo, oo)*oo == AccumBounds(-oo, oo)
def test_AccumBounds_div():
assert AccumBounds(-1, 3)/AccumBounds(3, 4) == AccumBounds(-S(1)/3, 1)
assert AccumBounds(-2, 4)/AccumBounds(-3, 4) == AccumBounds(-oo, oo)
assert AccumBounds(-3, -2)/AccumBounds(-4, 0) == AccumBounds(S(1)/2, oo)
# these two tests can have a better answer
# after Union of AccumBounds is improved
assert AccumBounds(-3, -2)/AccumBounds(-2, 1) == AccumBounds(-oo, oo)
assert AccumBounds(2, 3)/AccumBounds(-2, 2) == AccumBounds(-oo, oo)
assert AccumBounds(-3, -2)/AccumBounds(0, 4) == AccumBounds(-oo, -S(1)/2)
assert AccumBounds(2, 4)/AccumBounds(-3, 0) == AccumBounds(-oo, -S(2)/3)
assert AccumBounds(2, 4)/AccumBounds(0, 3) == AccumBounds(S(2)/3, oo)
assert AccumBounds(0, 1)/AccumBounds(0, 1) == AccumBounds(0, oo)
assert AccumBounds(-1, 0)/AccumBounds(0, 1) == AccumBounds(-oo, 0)
assert AccumBounds(-1, 2)/AccumBounds(-2, 2) == AccumBounds(-oo, oo)
assert 1/AccumBounds(-1, 2) == AccumBounds(-oo, oo)
assert 1/AccumBounds(0, 2) == AccumBounds(S(1)/2, oo)
assert (-1)/AccumBounds(0, 2) == AccumBounds(-oo, -S(1)/2)
assert 1/AccumBounds(-oo, 0) == AccumBounds(-oo, 0)
assert 1/AccumBounds(-1, 0) == AccumBounds(-oo, -1)
assert (-2)/AccumBounds(-oo, 0) == AccumBounds(0, oo)
assert 1/AccumBounds(-oo, -1) == AccumBounds(-1, 0)
assert AccumBounds(1, 2)/a == Mul(AccumBounds(1, 2), 1/a, evaluate=False)
assert AccumBounds(1, 2)/0 == AccumBounds(1, 2)*zoo
assert AccumBounds(1, oo)/oo == AccumBounds(0, oo)
assert AccumBounds(1, oo)/(-oo) == AccumBounds(-oo, 0)
assert AccumBounds(-oo, -1)/oo == AccumBounds(-oo, 0)
assert AccumBounds(-oo, -1)/(-oo) == AccumBounds(0, oo)
assert AccumBounds(-oo, oo)/oo == AccumBounds(-oo, oo)
assert AccumBounds(-oo, oo)/(-oo) == AccumBounds(-oo, oo)
assert AccumBounds(-1, oo)/oo == AccumBounds(0, oo)
assert AccumBounds(-1, oo)/(-oo) == AccumBounds(-oo, 0)
assert AccumBounds(-oo, 1)/oo == AccumBounds(-oo, 0)
assert AccumBounds(-oo, 1)/(-oo) == AccumBounds(0, oo)
def test_AccumBounds_func():
assert (x**2 + 2*x + 1).subs(x, AccumBounds(-1, 1)) == AccumBounds(-1, 4)
assert exp(AccumBounds(0, 1)) == AccumBounds(1, E)
assert exp(AccumBounds(-oo, oo)) == AccumBounds(0, oo)
assert log(AccumBounds(3, 6)) == AccumBounds(log(3), log(6))
def test_AccumBounds_pow():
assert AccumBounds(0, 2)**2 == AccumBounds(0, 4)
assert AccumBounds(-1, 1)**2 == AccumBounds(0, 1)
assert AccumBounds(1, 2)**2 == AccumBounds(1, 4)
assert AccumBounds(-1, 2)**3 == AccumBounds(-1, 8)
assert AccumBounds(-1, 1)**0 == 1
assert AccumBounds(1, 2)**(S(5)/2) == AccumBounds(1, 4*sqrt(2))
assert AccumBounds(-1, 2)**(S(1)/3) == AccumBounds(-1, 2**(S(1)/3))
assert AccumBounds(0, 2)**(S(1)/2) == AccumBounds(0, sqrt(2))
assert AccumBounds(-4, 2)**(S(2)/3) == AccumBounds(0, 2*2**(S(1)/3))
assert AccumBounds(-1, 5)**(S(1)/2) == AccumBounds(0, sqrt(5))
assert AccumBounds(-oo, 2)**(S(1)/2) == AccumBounds(0, sqrt(2))
assert AccumBounds(-2, 3)**(S(-1)/4) == AccumBounds(0, oo)
assert AccumBounds(1, 5)**(-2) == AccumBounds(S(1)/25, 1)
assert AccumBounds(-1, 3)**(-2) == AccumBounds(0, oo)
assert AccumBounds(0, 2)**(-2) == AccumBounds(S(1)/4, oo)
assert AccumBounds(-1, 2)**(-3) == AccumBounds(-oo, oo)
assert AccumBounds(-3, -2)**(-3) == AccumBounds(S(-1)/8, -S(1)/27)
assert AccumBounds(-3, -2)**(-2) == AccumBounds(S(1)/9, S(1)/4)
assert AccumBounds(0, oo)**(S(1)/2) == AccumBounds(0, oo)
assert AccumBounds(-oo, -1)**(S(1)/3) == AccumBounds(-oo, -1)
assert AccumBounds(-2, 3)**(-S(1)/3) == AccumBounds(-oo, oo)
assert AccumBounds(-oo, 0)**(-2) == AccumBounds(0, oo)
assert AccumBounds(-2, 0)**(-2) == AccumBounds(S(1)/4, oo)
assert AccumBounds(S(1)/3, S(1)/2)**oo == S(0)
assert AccumBounds(0, S(1)/2)**oo == S(0)
assert AccumBounds(S(1)/2, 1)**oo == AccumBounds(0, oo)
assert AccumBounds(0, 1)**oo == AccumBounds(0, oo)
assert AccumBounds(2, 3)**oo == oo
assert AccumBounds(1, 2)**oo == AccumBounds(0, oo)
assert AccumBounds(S(1)/2, 3)**oo == AccumBounds(0, oo)
assert AccumBounds(-S(1)/3, -S(1)/4)**oo == S(0)
assert AccumBounds(-1, -S(1)/2)**oo == AccumBounds(-oo, oo)
assert AccumBounds(-3, -2)**oo == FiniteSet(-oo, oo)
assert AccumBounds(-2, -1)**oo == AccumBounds(-oo, oo)
assert AccumBounds(-2, -S(1)/2)**oo == AccumBounds(-oo, oo)
assert AccumBounds(-S(1)/2, S(1)/2)**oo == S(0)
assert AccumBounds(-S(1)/2, 1)**oo == AccumBounds(0, oo)
assert AccumBounds(-S(2)/3, 2)**oo == AccumBounds(0, oo)
assert AccumBounds(-1, 1)**oo == AccumBounds(-oo, oo)
assert AccumBounds(-1, S(1)/2)**oo == AccumBounds(-oo, oo)
assert AccumBounds(-1, 2)**oo == AccumBounds(-oo, oo)
assert AccumBounds(-2, S(1)/2)**oo == AccumBounds(-oo, oo)
assert AccumBounds(1, 2)**x == Pow(AccumBounds(1, 2), x, evaluate=False)
assert AccumBounds(2, 3)**(-oo) == S(0)
assert AccumBounds(0, 2)**(-oo) == AccumBounds(0, oo)
assert AccumBounds(-1, 2)**(-oo) == AccumBounds(-oo, oo)
assert (tan(x)**sin(2*x)).subs(x, AccumBounds(0, pi/2)) == \
Pow(AccumBounds(-oo, oo), AccumBounds(0, 1), evaluate=False)
def test_comparison_AccumBounds():
assert (AccumBounds(1, 3) < 4) == S.true
assert (AccumBounds(1, 3) < -1) == S.false
assert (AccumBounds(1, 3) < 2).rel_op == '<'
assert (AccumBounds(1, 3) <= 2).rel_op == '<='
assert (AccumBounds(1, 3) > 4) == S.false
assert (AccumBounds(1, 3) > -1) == S.true
assert (AccumBounds(1, 3) > 2).rel_op == '>'
assert (AccumBounds(1, 3) >= 2).rel_op == '>='
assert (AccumBounds(1, 3) < AccumBounds(4, 6)) == S.true
assert (AccumBounds(1, 3) < AccumBounds(2, 4)).rel_op == '<'
assert (AccumBounds(1, 3) < AccumBounds(-2, 0)) == S.false
# issue 13499
assert (cos(x) > 0).subs(x, oo) == (AccumBounds(-1, 1) > 0)
def test_contains_AccumBounds():
assert (1 in AccumBounds(1, 2)) == S.true
raises(TypeError, lambda: a in AccumBounds(1, 2))
assert 0 in AccumBounds(-1, 0)
raises(TypeError, lambda:
(cos(1)**2 + sin(1)**2 - 1) in AccumBounds(-1, 0))
assert (-oo in AccumBounds(1, oo)) == S.true
assert (oo in AccumBounds(-oo, 0)) == S.true
# issue 13159
assert Mul(0, AccumBounds(-1, 1)) == Mul(AccumBounds(-1, 1), 0) == 0
import itertools
for perm in itertools.permutations([0, AccumBounds(-1, 1), x]):
assert Mul(*perm) == 0
|
3e904f1adf5d4431603b0744bfc2291544cb7e452f46c585dbf4bba938cff82f
|
from __future__ import (absolute_import, division, print_function)
import os
import re
import shutil
import subprocess
import sys
import tempfile
import warnings
from distutils.sysconfig import get_config_var, get_config_vars
from .util import (
get_abspath, make_dirs, copy, Glob, ArbitraryDepthGlob,
glob_at_depth, CompileError,
import_module_from_file, pyx_is_cplus,
sha256_of_string, sha256_of_file
)
from .runners import (
CCompilerRunner,
CppCompilerRunner,
FortranCompilerRunner
)
sharedext = get_config_var('EXT_SUFFIX' if sys.version_info >= (3, 3) else 'SO')
if os.name == 'posix':
objext = '.o'
elif os.name == 'nt':
objext = '.obj'
else:
warning.warng("Unknown os.name: {}".format(os.name))
objext = '.o'
def compile_sources(files, Runner=None, destdir=None, cwd=None, keep_dir_struct=False,
per_file_kwargs=None, **kwargs):
""" Compile source code files to object files.
Parameters
==========
files : iterable of str
Paths to source files, if ``cwd`` is given, the paths are taken as relative.
Runner: CompilerRunner subclass (optional)
Could be e.g. ``FortranCompilerRunner``. Will be inferred from filename
extensions if missing.
destdir: str
Output directory, if cwd is given, the path is taken as relative.
cwd: str
Working directory. Specify to have compiler run in other directory.
also used as root of relative paths.
keep_dir_struct: bool
Reproduce directory structure in `destdir`. default: ``False``
per_file_kwargs: dict
Dict mapping instances in ``files`` to keyword arguments.
\\*\\*kwargs: dict
Default keyword arguments to pass to ``Runner``.
"""
_per_file_kwargs = {}
if per_file_kwargs is not None:
for k, v in per_file_kwargs.items():
if isinstance(k, Glob):
for path in glob.glob(k.pathname):
_per_file_kwargs[path] = v
elif isinstance(k, ArbitraryDepthGlob):
for path in glob_at_depth(k.filename, cwd):
_per_file_kwargs[path] = v
else:
_per_file_kwargs[k] = v
# Set up destination directory
destdir = destdir or '.'
if not os.path.isdir(destdir):
if os.path.exists(destdir):
raise IOError("{} is not a directory".format(destdir))
else:
make_dirs(destdir)
if cwd is None:
cwd = '.'
for f in files:
copy(f, destdir, only_update=True, dest_is_dir=True)
# Compile files and return list of paths to the objects
dstpaths = []
for f in files:
if keep_dir_struct:
name, ext = os.path.splitext(f)
else:
name, ext = os.path.splitext(os.path.basename(f))
file_kwargs = kwargs.copy()
file_kwargs.update(_per_file_kwargs.get(f, {}))
dstpaths.append(src2obj(f, Runner, cwd=cwd, **file_kwargs))
return dstpaths
def get_mixed_fort_c_linker(vendor=None, cplus=False, cwd=None):
vendor = vendor or os.environ.get('SYMPY_COMPILER_VENDOR', 'gnu')
if vendor.lower() == 'intel':
if cplus:
return (FortranCompilerRunner,
{'flags': ['-nofor_main', '-cxxlib']}, vendor)
else:
return (FortranCompilerRunner,
{'flags': ['-nofor_main']}, vendor)
elif vendor.lower() == 'gnu' or 'llvm':
if cplus:
return (CppCompilerRunner,
{'lib_options': ['fortran']}, vendor)
else:
return (FortranCompilerRunner,
{}, vendor)
else:
raise ValueError("No vendor found.")
def link(obj_files, out_file=None, shared=False, Runner=None,
cwd=None, cplus=False, fort=False, **kwargs):
""" Link object files.
Parameters
==========
obj_files: iterable of str
Paths to object files.
out_file: str (optional)
Path to executable/shared library, if ``None`` it will be
deduced from the last item in obj_files.
shared: bool
Generate a shared library?
Runner: CompilerRunner subclass (optional)
If not given the ``cplus`` and ``fort`` flags will be inspected
(fallback is the C compiler).
cwd: str
Path to the root of relative paths and working directory for compiler.
cplus: bool
C++ objects? default: ``False``.
fort: bool
Fortran objects? default: ``False``.
\\*\\*kwargs: dict
Keyword arguments passed to ``Runner``.
Returns
=======
The absolute path to the generated shared object / executable.
"""
if out_file is None:
out_file, ext = os.path.splitext(os.path.basename(obj_files[-1]))
if shared:
out_file += sharedext
if not Runner:
if fort:
Runner, extra_kwargs, vendor = \
get_mixed_fort_c_linker(
vendor=kwargs.get('vendor', None),
cplus=cplus,
cwd=cwd,
)
for k, v in extra_kwargs.items():
if k in kwargs:
kwargs[k].expand(v)
else:
kwargs[k] = v
else:
if cplus:
Runner = CppCompilerRunner
else:
Runner = CCompilerRunner
flags = kwargs.pop('flags', [])
if shared:
if '-shared' not in flags:
flags.append('-shared')
run_linker = kwargs.pop('run_linker', True)
if not run_linker:
raise ValueError("run_linker was set to False (nonsensical).")
out_file = get_abspath(out_file, cwd=cwd)
runner = Runner(obj_files, out_file, flags, cwd=cwd, **kwargs)
runner.run()
return out_file
def link_py_so(obj_files, so_file=None, cwd=None, libraries=None,
cplus=False, fort=False, **kwargs):
""" Link python extension module (shared object) for importing
Parameters
==========
obj_files: iterable of str
Paths to object files to be linked.
so_file: str
Name (path) of shared object file to create. If not specified it will
have the basname of the last object file in `obj_files` but with the
extension '.so' (Unix).
cwd: path string
Root of relative paths and working directory of linker.
libraries: iterable of strings
Libraries to link against, e.g. ['m'].
cplus: bool
Any C++ objects? default: ``False``.
fort: bool
Any Fortran objects? default: ``False``.
kwargs**: dict
Keyword arguments passed to ``link(...)``.
Returns
=======
Absolute path to the generate shared object.
"""
libraries = libraries or []
include_dirs = kwargs.pop('include_dirs', [])
library_dirs = kwargs.pop('library_dirs', [])
# from distutils/command/build_ext.py:
if sys.platform == "win32":
warnings.warn("Windows not yet supported.")
elif sys.platform == 'darwin':
# Don't use the default code below
pass
elif sys.platform[:3] == 'aix':
# Don't use the default code below
pass
else:
from distutils import sysconfig
if sysconfig.get_config_var('Py_ENABLE_SHARED'):
ABIFLAGS = sysconfig.get_config_var('ABIFLAGS')
pythonlib = 'python{}.{}{}'.format(
sys.hexversion >> 24, (sys.hexversion >> 16) & 0xff,
ABIFLAGS or '')
libraries += [pythonlib]
else:
pass
flags = kwargs.pop('flags', [])
needed_flags = ('-pthread',)
for flag in needed_flags:
if flag not in flags:
flags.append(flag)
return link(obj_files, shared=True, flags=flags, cwd=cwd,
cplus=cplus, fort=fort, include_dirs=include_dirs,
libraries=libraries, library_dirs=library_dirs, **kwargs)
def simple_cythonize(src, destdir=None, cwd=None, **cy_kwargs):
""" Generates a C file from a Cython source file.
Parameters
==========
src: str
Path to Cython source.
destdir: str (optional)
Path to output directory (default: '.').
cwd: path string (optional)
Root of relative paths (default: '.').
**cy_kwargs:
Second argument passed to cy_compile. Generates a .cpp file if ``cplus=True`` in ``cy_kwargs``,
else a .c file.
"""
from Cython.Compiler.Main import (
default_options, CompilationOptions
)
from Cython.Compiler.Main import compile as cy_compile
assert src.lower().endswith('.pyx') or src.lower().endswith('.py')
cwd = cwd or '.'
destdir = destdir or '.'
ext = '.cpp' if cy_kwargs.get('cplus', False) else '.c'
c_name = os.path.splitext(os.path.basename(src))[0] + ext
dstfile = os.path.join(destdir, c_name)
if cwd:
ori_dir = os.getcwd()
else:
ori_dir = '.'
os.chdir(cwd)
try:
cy_options = CompilationOptions(default_options)
cy_options.__dict__.update(cy_kwargs)
cy_result = cy_compile([src], cy_options)
if cy_result.num_errors > 0:
raise ValueError("Cython compilation failed.")
if os.path.abspath(os.path.dirname(src)) != os.path.abspath(destdir):
if os.path.exists(dstfile):
os.unlink(dstfile)
shutil.move(os.path.join(os.path.dirname(src), c_name), destdir)
finally:
os.chdir(ori_dir)
return dstfile
extension_mapping = {
'.c': (CCompilerRunner, None),
'.cpp': (CppCompilerRunner, None),
'.cxx': (CppCompilerRunner, None),
'.f': (FortranCompilerRunner, None),
'.for': (FortranCompilerRunner, None),
'.ftn': (FortranCompilerRunner, None),
'.f90': (FortranCompilerRunner, None), # ifort only knows about .f90
'.f95': (FortranCompilerRunner, 'f95'),
'.f03': (FortranCompilerRunner, 'f2003'),
'.f08': (FortranCompilerRunner, 'f2008'),
}
def src2obj(srcpath, Runner=None, objpath=None, cwd=None, inc_py=False, **kwargs):
""" Compiles a source code file to an object file.
Files ending with '.pyx' assumed to be cython files and
are dispatched to pyx2obj.
Parameters
==========
srcpath: str
Path to source file.
Runner: CompilerRunner subclass (optional)
If ``None``: deduced from extension of srcpath.
objpath : str (optional)
Path to generated object. If ``None``: deduced from ``srcpath``.
cwd: str (optional)
Working directory and root of relative paths. If ``None``: current dir.
inc_py: bool
Add Python include path to kwarg "include_dirs". Default: False
\\*\\*kwargs: dict
keyword arguments passed to Runner or pyx2obj
"""
name, ext = os.path.splitext(os.path.basename(srcpath))
if objpath is None:
if os.path.isabs(srcpath):
objpath = '.'
else:
objpath = os.path.dirname(srcpath)
objpath = objpath or '.' # avoid objpath == ''
if os.path.isdir(objpath):
objpath = os.path.join(objpath, name+objext)
include_dirs = kwargs.pop('include_dirs', [])
if inc_py:
from distutils.sysconfig import get_python_inc
py_inc_dir = get_python_inc()
if py_inc_dir not in include_dirs:
include_dirs.append(py_inc_dir)
if ext.lower() == '.pyx':
return pyx2obj(srcpath, objpath=objpath, include_dirs=include_dirs, cwd=cwd,
**kwargs)
if Runner is None:
Runner, std = extension_mapping[ext.lower()]
if 'std' not in kwargs:
kwargs['std'] = std
flags = kwargs.pop('flags', [])
needed_flags = ('-fPIC',)
for flag in needed_flags:
if flag not in flags:
flags.append(flag)
# src2obj implies not running the linker...
run_linker = kwargs.pop('run_linker', False)
if run_linker:
raise CompileError("src2obj called with run_linker=True")
runner = Runner([srcpath], objpath, include_dirs=include_dirs,
run_linker=run_linker, cwd=cwd, flags=flags, **kwargs)
runner.run()
return objpath
def pyx2obj(pyxpath, objpath=None, destdir=None, cwd=None,
include_dirs=None, cy_kwargs=None, cplus=None, **kwargs):
"""
Convenience function
If cwd is specified, pyxpath and dst are taken to be relative
If only_update is set to `True` the modification time is checked
and compilation is only run if the source is newer than the
destination
Parameters
==========
pyxpath: str
Path to Cython source file.
objpath: str (optional)
Path to object file to generate.
destdir: str (optional)
Directory to put generated C file. When ``None``: directory of ``objpath``.
cwd: str (optional)
Working directory and root of relative paths.
include_dirs: iterable of path strings (optional)
Passed onto src2obj and via cy_kwargs['include_path']
to simple_cythonize.
cy_kwargs: dict (optional)
Keyword arguments passed onto `simple_cythonize`
cplus: bool (optional)
Indicate whether C++ is used. default: auto-detect using ``.util.pyx_is_cplus``.
compile_kwargs: dict
keyword arguments passed onto src2obj
Returns
=======
Absolute path of generated object file.
"""
assert pyxpath.endswith('.pyx')
cwd = cwd or '.'
objpath = objpath or '.'
destdir = destdir or os.path.dirname(objpath)
abs_objpath = get_abspath(objpath, cwd=cwd)
if os.path.isdir(abs_objpath):
pyx_fname = os.path.basename(pyxpath)
name, ext = os.path.splitext(pyx_fname)
objpath = os.path.join(objpath, name+objext)
cy_kwargs = cy_kwargs or {}
cy_kwargs['output_dir'] = cwd
if cplus is None:
cplus = pyx_is_cplus(pyxpath)
cy_kwargs['cplus'] = cplus
interm_c_file = simple_cythonize(pyxpath, destdir=destdir, cwd=cwd, **cy_kwargs)
include_dirs = include_dirs or []
flags = kwargs.pop('flags', [])
needed_flags = ('-fwrapv', '-pthread', '-fPIC')
for flag in needed_flags:
if flag not in flags:
flags.append(flag)
options = kwargs.pop('options', [])
if kwargs.pop('strict_aliasing', False):
raise CompileError("Cython requires strict aliasing to be disabled.")
# Let's be explicit about standard
if cplus:
std = kwargs.pop('std', 'c++98')
else:
std = kwargs.pop('std', 'c99')
return src2obj(interm_c_file, objpath=objpath, cwd=cwd,
include_dirs=include_dirs, flags=flags, std=std,
options=options, inc_py=True, strict_aliasing=False,
**kwargs)
def _any_X(srcs, cls):
for src in srcs:
name, ext = os.path.splitext(src)
key = ext.lower()
if key in extension_mapping:
if extension_mapping[key][0] == cls:
return True
return False
def any_fortran_src(srcs):
return _any_X(srcs, FortranCompilerRunner)
def any_cplus_src(srcs):
return _any_X(srcs, CppCompilerRunner)
def compile_link_import_py_ext(sources, extname=None, build_dir='.', compile_kwargs=None,
link_kwargs=None):
""" Compiles sources to a shared object (python extension) and imports it
Sources in ``sources`` which is imported. If shared object is newer than the sources, they
are not recompiled but instead it is imported.
Parameters
==========
sources : string
List of paths to sources.
extname : string
Name of extension (default: ``None``).
If ``None``: taken from the last file in ``sources`` without extension.
build_dir: str
Path to directory in which objects files etc. are generated.
compile_kwargs: dict
keyword arguments passed to ``compile_sources``
link_kwargs: dict
keyword arguments passed to ``link_py_so``
Returns
=======
The imported module from of the python extension.
Examples
========
>>> mod = compile_link_import_py_ext(['fft.f90', 'conv.cpp', '_fft.pyx']) # doctest: +SKIP
>>> Aprim = mod.fft(A) # doctest: +SKIP
"""
if extname is None:
extname = os.path.splitext(os.path.basename(sources[-1]))[0]
compile_kwargs = compile_kwargs or {}
link_kwargs = link_kwargs or {}
try:
mod = import_module_from_file(os.path.join(build_dir, extname), sources)
except ImportError:
objs = compile_sources(list(map(get_abspath, sources)), destdir=build_dir,
cwd=build_dir, **compile_kwargs)
so = link_py_so(objs, cwd=build_dir, fort=any_fortran_src(sources),
cplus=any_cplus_src(sources), **link_kwargs)
mod = import_module_from_file(so)
return mod
def _write_sources_to_build_dir(sources, build_dir):
build_dir = build_dir or tempfile.mkdtemp()
if not os.path.isdir(build_dir):
raise OSError("Non-existent directory: ", build_dir)
source_files = []
for name, src in sources:
dest = os.path.join(build_dir, name)
differs = True
sha256_in_mem = sha256_of_string(src.encode('utf-8')).hexdigest()
if os.path.exists(dest):
if os.path.exists(dest+'.sha256'):
sha256_on_disk = open(dest+'.sha256', 'rt').read()
else:
sha256_on_disk = sha256_of_file(dest).hexdigest()
differs = sha256_on_disk != sha256_in_mem
if differs:
with open(dest, 'wt') as fh:
fh.write(src)
open(dest+'.sha256', 'wt').write(sha256_in_mem)
source_files.append(dest)
return source_files, build_dir
def compile_link_import_strings(sources, build_dir=None, **kwargs):
""" Compiles, links and imports extension module from source.
Parameters
==========
sources : iterable of name/source pair tuples
build_dir : string (default: None)
Path. ``None`` implies use a temporary directory.
**kwargs:
Keyword arguments passed onto `compile_link_import_py_ext`.
Returns
=======
mod : module
The compiled and imported extension module.
info : dict
Containing ``build_dir`` as 'build_dir'.
"""
source_files, build_dir = _write_sources_to_build_dir(sources, build_dir)
mod = compile_link_import_py_ext(source_files, build_dir=build_dir, **kwargs)
info = dict(build_dir=build_dir)
return mod, info
def compile_run_strings(sources, build_dir=None, clean=False, compile_kwargs=None, link_kwargs=None):
""" Compiles, links and runs a program built from sources.
Parameters
==========
sources : iterable of name/source pair tuples
build_dir : string (default: None)
Path. ``None`` implies use a temporary directory.
clean : bool
Whether to remove build_dir after use. This will only have an
effect if ``build_dir`` is ``None`` (which creates a temporary directory).
Passing ``clean == True`` and ``build_dir != None`` raises a ``ValueError``.
This will also set ``build_dir`` in returned info dictionary to ``None``.
compile_kwargs: dict
Keyword arguments passed onto ``compile_sources``
link_kwargs: dict
Keyword arguments passed onto ``link``
Returns
=======
(stdout, stderr): pair of strings
info: dict
Containing exit status as 'exit_status' and ``build_dir`` as 'build_dir'
"""
if clean and build_dir is not None:
raise ValueError("Automatic removal of build_dir is only available for temporary directory.")
try:
source_files, build_dir = _write_sources_to_build_dir(sources, build_dir)
objs = compile_sources(list(map(get_abspath, source_files)), destdir=build_dir,
cwd=build_dir, **(compile_kwargs or {}))
prog = link(objs, cwd=build_dir,
fort=any_fortran_src(source_files),
cplus=any_cplus_src(source_files), **(link_kwargs or {}))
p = subprocess.Popen([prog], stdout=subprocess.PIPE, stderr=subprocess.PIPE)
exit_status = p.wait()
stdout, stderr = [txt.decode('utf-8') for txt in p.communicate()]
finally:
if clean and os.path.isdir(build_dir):
shutil.rmtree(build_dir)
build_dir = None
info = dict(exit_status=exit_status, build_dir=build_dir)
return (stdout, stderr), info
|
b051c067405c229704fdbeb7cf2d0bc76a9a0d6bd9f197bbbce2259d48badc23
|
from __future__ import print_function, division, absolute_import
from collections import OrderedDict
import os
import re
import subprocess
import sys
from .util import (
get_abspath, FileNotFoundError,
find_binary_of_command, unique_list,
CompileError
)
from sympy.core.compatibility import string_types
class CompilerRunner(object):
""" CompilerRunner base class.
Parameters
==========
sources : list of str
Paths to sources.
out : str
flags : iterable of str
Compiler flags.
run_linker : bool
compiler_name_exe : (str, str) tuple
Tuple of compiler name & command to call.
cwd : str
Path of root of relative paths.
include_dirs : list of str
Include directories.
libraries : list of str
Libraries to link against.
library_dirs : list of str
Paths to search for shared libraries.
std : str
Standard string, e.g. ``'c++11'``, ``'c99'``, ``'f2003'``.
define: iterable of strings
macros to define
undef : iterable of strings
macros to undefine
preferred_vendor : string
name of preferred vendor e.g. 'gnu' or 'intel'
Methods
=======
run():
Invoke compilation as a subprocess.
"""
compiler_dict = None # Subclass to vendor/binary dict
# Standards should be a tuple of supported standards
# (first one will be the default)
standards = None
std_formater = None # Subclass to dict of binary/formater-callback
# subclass to be e.g. {'gcc': 'gnu', ...}
compiler_name_vendor_mapping = None
def __init__(self, sources, out, flags=None, run_linker=True, compiler=None, cwd='.',
include_dirs=None, libraries=None, library_dirs=None, std=None, define=None,
undef=None, strict_aliasing=None, preferred_vendor=None, **kwargs):
if isinstance(sources, string_types):
raise ValueError("Expected argument sources to be a list of strings.")
self.sources = list(sources)
self.out = out
self.flags = flags or []
self.cwd = cwd
if compiler:
self.compiler_name, self.compiler_binary = compiler
else:
# Find a compiler
if preferred_vendor is None:
preferred_vendor = os.environ.get('SYMPY_COMPILER_VENDOR', None)
self.compiler_name, self.compiler_binary, self.compiler_vendor = self.find_compiler(preferred_vendor)
if self.compiler_binary is None:
raise ValueError("No compiler found (searched: {0})".format(', '.join(self.compiler_dict.values())))
self.define = define or []
self.undef = undef or []
self.include_dirs = include_dirs or []
self.libraries = libraries or []
self.library_dirs = library_dirs or []
self.std = std or self.standards[0]
self.run_linker = run_linker
if self.run_linker:
# both gnu and intel compilers use '-c' for disabling linker
self.flags = list(filter(lambda x: x != '-c', self.flags))
else:
if '-c' not in self.flags:
self.flags.append('-c')
if self.std:
self.flags.append(self.std_formater[
self.compiler_name](self.std))
self.linkline = []
if strict_aliasing is not None:
nsa_re = re.compile("no-strict-aliasing$")
sa_re = re.compile("strict-aliasing$")
if strict_aliasing is True:
if any(map(nsa_re.match, flags)):
raise CompileError("Strict aliasing cannot be both enforced and disabled")
elif any(map(sa_re.match, flags)):
pass # already enforced
else:
flags.append('-fstrict-aliasing')
elif strict_aliasing is False:
if any(map(nsa_re.match, flags)):
pass # already disabled
else:
if any(map(sa_re.match, flags)):
raise CompileError("Strict aliasing cannot be both enforced and disabled")
else:
flags.append('-fno-strict-aliasing')
else:
msg = "Expected argument strict_aliasing to be True/False, got {}"
raise ValueError(msg.format(strict_aliasing))
@classmethod
def find_compiler(cls, preferred_vendor=None):
""" Identify a suitable C/fortran/other compiler. """
candidates = list(cls.compiler_dict.keys())
if preferred_vendor:
if preferred_vendor in candidates:
candidates = [preferred_vendor]+candidates
else:
raise ValueError("Unknown vendor {}".format(preferred_vendor))
name, path = find_binary_of_command([cls.compiler_dict[x] for x in candidates])
return name, path, cls.compiler_name_vendor_mapping[name]
def cmd(self):
""" List of arguments (str) to be passed to e.g. ``subprocess.Popen``. """
cmd = (
[self.compiler_binary] +
self.flags +
['-U'+x for x in self.undef] +
['-D'+x for x in self.define] +
['-I'+x for x in self.include_dirs] +
self.sources
)
if self.run_linker:
cmd += (['-L'+x for x in self.library_dirs] +
['-l'+x for x in self.libraries] +
self.linkline)
counted = []
for envvar in re.findall(r'\$\{(\w+)\}', ' '.join(cmd)):
if os.getenv(envvar) is None:
if envvar not in counted:
counted.append(envvar)
msg = "Environment variable '{}' undefined.".format(envvar)
raise CompileError(msg)
return cmd
def run(self):
self.flags = unique_list(self.flags)
# Append output flag and name to tail of flags
self.flags.extend(['-o', self.out])
env = os.environ.copy()
env['PWD'] = self.cwd
# NOTE: intel compilers seems to need shell=True
p = subprocess.Popen(' '.join(self.cmd()),
shell=True,
cwd=self.cwd,
stdin=subprocess.PIPE,
stdout=subprocess.PIPE,
stderr=subprocess.STDOUT,
env=env)
comm = p.communicate()
if sys.version_info[0] == 2:
self.cmd_outerr = comm[0]
else:
try:
self.cmd_outerr = comm[0].decode('utf-8')
except UnicodeDecodeError:
self.cmd_outerr = comm[0].decode('iso-8859-1') # win32
self.cmd_returncode = p.returncode
# Error handling
if self.cmd_returncode != 0:
msg = "Error executing '{0}' in {1} (exited status {2}):\n {3}\n".format(
' '.join(self.cmd()), self.cwd, str(self.cmd_returncode), self.cmd_outerr
)
raise CompileError(msg)
return self.cmd_outerr, self.cmd_returncode
class CCompilerRunner(CompilerRunner):
compiler_dict = OrderedDict([
('gnu', 'gcc'),
('intel', 'icc'),
('llvm', 'clang'),
])
standards = ('c89', 'c90', 'c99', 'c11') # First is default
std_formater = {
'gcc': '-std={}'.format,
'icc': '-std={}'.format,
'clang': '-std={}'.format,
}
compiler_name_vendor_mapping = {
'gcc': 'gnu',
'icc': 'intel',
'clang': 'llvm'
}
def _mk_flag_filter(cmplr_name): # helper for class initialization
not_welcome = {'g++': ("Wimplicit-interface",)} # "Wstrict-prototypes",)}
if cmplr_name in not_welcome:
def fltr(x):
for nw in not_welcome[cmplr_name]:
if nw in x:
return False
return True
else:
def fltr(x):
return True
return fltr
class CppCompilerRunner(CompilerRunner):
compiler_dict = OrderedDict([
('gnu', 'g++'),
('intel', 'icpc'),
('llvm', 'clang++'),
])
# First is the default, c++0x == c++11
standards = ('c++98', 'c++0x')
std_formater = {
'g++': '-std={}'.format,
'icpc': '-std={}'.format,
'clang++': '-std={}'.format,
}
compiler_name_vendor_mapping = {
'g++': 'gnu',
'icpc': 'intel',
'clang++': 'llvm'
}
class FortranCompilerRunner(CompilerRunner):
standards = (None, 'f77', 'f95', 'f2003', 'f2008')
std_formater = {
'gfortran': lambda x: '-std=gnu' if x is None else '-std=legacy' if x == 'f77' else '-std={}'.format(x),
'ifort': lambda x: '-stand f08' if x is None else '-stand f{}'.format(x[-2:]), # f2008 => f08
}
compiler_dict = OrderedDict([
('gnu', 'gfortran'),
('intel', 'ifort'),
])
compiler_name_vendor_mapping = {
'gfortran': 'gnu',
'ifort': 'intel',
}
|
2375aa2c9285262617f9225e5839411449d6743bbbbe4489ac2f833bc02ba98a
|
from __future__ import (absolute_import, division, print_function)
""" This sub-module is private, i.e. external code should not depend on it.
These functions are used by tests run as part of continuous integration.
Once the implementation is mature (it should support the major
platforms: Windows, OS X & Linux) it may become official API which
may be relied upon by downstream libraries. Until then API may break
without prior notice.
TODO:
- (optionally) clean up after tempfile.mkdtemp()
- cross-platform testing
- caching of compiler choice and intermediate files
"""
from .compilation import compile_link_import_strings, compile_run_strings
from .availability import has_fortran, has_c, has_cxx
|
b7c2cd31f8ee85c48b5960aba3957f1fb44fc2a0c71475a1f2af672671c26782
|
from __future__ import (absolute_import, division, print_function)
from collections import namedtuple
from contextlib import contextmanager
from distutils.errors import CompileError
from hashlib import sha256
import glob
import io
import os
import shutil
import sys
import tempfile
from sympy.utilities.pytest import XFAIL
def may_xfail(func):
if sys.platform.lower() == 'darwin' or os.name == 'nt':
# sympy.utilities._compilation needs more testing on Windows and macOS
# once those two platforms are reliably supported this xfail decorator
# may be removed.
return XFAIL(func)
else:
return func
if sys.version_info[0] == 2:
class FileNotFoundError(IOError):
pass
class TemporaryDirectory(object):
def __init__(self):
self.path = tempfile.mkdtemp()
def __enter__(self):
return self.path
def __exit__(self, exc, value, tb):
shutil.rmtree(self.path)
else:
FileNotFoundError = FileNotFoundError
TemporaryDirectory = tempfile.TemporaryDirectory
class CompilerNotFoundError(FileNotFoundError):
pass
def get_abspath(path, cwd='.'):
""" Returns the aboslute path.
Parameters
==========
path : str
(relative) path.
cwd : str
Path to root of relative path.
"""
if os.path.isabs(path):
return path
else:
if not os.path.isabs(cwd):
cwd = os.path.abspath(cwd)
return os.path.abspath(
os.path.join(cwd, path)
)
def make_dirs(path):
""" Create directories (equivalent of ``mkdir -p``). """
if path[-1] == '/':
parent = os.path.dirname(path[:-1])
else:
parent = os.path.dirname(path)
if len(parent) > 0:
if not os.path.exists(parent):
make_dirs(parent)
if not os.path.exists(path):
os.mkdir(path, 0o777)
else:
assert os.path.isdir(path)
def copy(src, dst, only_update=False, copystat=True, cwd=None,
dest_is_dir=False, create_dest_dirs=False):
""" Variation of ``shutil.copy`` with extra options.
Parameters
==========
src : str
Path to source file.
dst : str
Path to destination.
only_update : bool
Only copy if source is newer than destination
(returns None if it was newer), default: ``False``.
copystat : bool
See ``shutil.copystat``. default: ``True``.
cwd : str
Path to working directory (root of relative paths).
dest_is_dir : bool
Ensures that dst is treated as a directory. default: ``False``
create_dest_dirs : bool
Creates directories if needed.
Returns
=======
Path to the copied file.
"""
if cwd: # Handle working directory
if not os.path.isabs(src):
src = os.path.join(cwd, src)
if not os.path.isabs(dst):
dst = os.path.join(cwd, dst)
if not os.path.exists(src): # Make sure source file extists
raise FileNotFoundError("Source: `{}` does not exist".format(src))
# We accept both (re)naming destination file _or_
# passing a (possible non-existant) destination directory
if dest_is_dir:
if not dst[-1] == '/':
dst = dst+'/'
else:
if os.path.exists(dst) and os.path.isdir(dst):
dest_is_dir = True
if dest_is_dir:
dest_dir = dst
dest_fname = os.path.basename(src)
dst = os.path.join(dest_dir, dest_fname)
else:
dest_dir = os.path.dirname(dst)
dest_fname = os.path.basename(dst)
if not os.path.exists(dest_dir):
if create_dest_dirs:
make_dirs(dest_dir)
else:
raise FileNotFoundError("You must create directory first.")
if only_update:
if not missing_or_other_newer(dst, src):
return
if os.path.islink(dst):
_cwd = os.path.dirname(dst)
dst = os.path.abspath(os.path.realpath(dst), cwd=cwd)
shutil.copy(src, dst)
if copystat:
shutil.copystat(src, dst)
return dst
Glob = namedtuple('Glob', 'pathname')
ArbitraryDepthGlob = namedtuple('ArbitraryDepthGlob', 'filename')
def glob_at_depth(filename_glob, cwd=None):
if cwd is not None:
cwd = '.'
globbed = []
for root, dirs, filenames in os.walk(cwd):
for fn in filenames:
if fnmatch.fnmatch(fn, filename_glob):
globbed.append(os.path.join(root, fn))
return globbed
def sha256_of_file(path, nblocks=128):
""" Computes the SHA256 hash of a file.
Parameters
==========
path : string
Path to file to compute hash of.
nblocks : int
Number of blocks to read per iteration.
Returns
=======
hashlib sha256 hash object. Use ``.digest()`` or ``.hexdigest()``
on returned object to get binary or hex encoded string.
"""
sh = sha256()
with open(path, 'rb') as f:
for chunk in iter(lambda: f.read(nblocks*sh.block_size), b''):
sh.update(chunk)
return sh
def sha256_of_string(string):
""" Computes the SHA256 hash of a string. """
sh = sha256()
sh.update(string)
return sh
def pyx_is_cplus(path):
"""
Inspect a Cython source file (.pyx) and look for comment line like:
# distutils: language = c++
Returns True if such a file is present in the file, else False.
"""
for line in open(path, 'rt'):
if line.startswith('#') and '=' in line:
splitted = line.split('=')
if len(splitted) != 2:
continue
lhs, rhs = splitted
if lhs.strip().split()[-1].lower() == 'language' and \
rhs.strip().split()[0].lower() == 'c++':
return True
return False
def import_module_from_file(filename, only_if_newer_than=None):
""" Imports python extension (from shared object file)
Provide a list of paths in `only_if_newer_than` to check
timestamps of dependencies. import_ raises an ImportError
if any is newer.
Word of warning: The OS may cache shared objects which makes
reimporting same path of an shared object file very problematic.
It will not detect the new time stamp, nor new checksum, but will
instead silently use old module. Use unique names for this reason.
Parameters
==========
filename : str
Path to shared object.
only_if_newer_than : iterable of strings
Paths to dependencies of the shared object.
Raises
======
``ImportError`` if any of the files specified in ``only_if_newer_than`` are newer
than the file given by filename.
"""
path, name = os.path.split(filename)
name, ext = os.path.splitext(name)
name = name.split('.')[0]
if sys.version_info[0] == 2:
from imp import find_module, load_module
fobj, filename, data = find_module(name, [path])
if only_if_newer_than:
for dep in only_if_newer_than:
if os.path.getmtime(filename) < os.path.getmtime(dep):
raise ImportError("{} is newer than {}".format(dep, filename))
mod = load_module(name, fobj, filename, data)
else:
import importlib.util
spec = importlib.util.spec_from_file_location(name, filename)
if spec is None:
raise ImportError("Failed to import: '%s'" % filename)
mod = importlib.util.module_from_spec(spec)
spec.loader.exec_module(mod)
return mod
def find_binary_of_command(candidates):
""" Finds binary first matching name among candidates.
Calls `find_executable` from distuils for provided candidates and returns
first hit.
Parameters
==========
candidates : iterable of str
Names of candidate commands
Raises
======
CompilerNotFoundError if no candidates match.
"""
from distutils.spawn import find_executable
for c in candidates:
binary_path = find_executable(c)
if c and binary_path:
return c, binary_path
raise CompilerNotFoundError('No binary located for candidates: {}'.format(candidates))
def unique_list(l):
""" Uniquify a list (skip duplicate items). """
result = []
for x in l:
if x not in result:
result.append(x)
return result
|
3e14b002cc3af3631aa304c53589d0d092da89143a32ac91c928d0ba603fbabc
|
from __future__ import (absolute_import, division, print_function)
import os
from .compilation import compile_run_strings
from .util import CompilerNotFoundError
def has_fortran():
if not hasattr(has_fortran, 'result'):
try:
(stdout, stderr), info = compile_run_strings(
[('main.f90', (
'program foo\n'
'print *, "hello world"\n'
'end program'
))], clean=True
)
except CompilerNotFoundError:
has_fortran.result = False
if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1':
raise
else:
if info['exit_status'] != os.EX_OK or 'hello world' not in stdout:
if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1':
raise ValueError("Failed to compile test program:\n%s\n%s\n" % (stdout, stderr))
has_fortran.result = False
else:
has_fortran.result = True
return has_fortran.result
def has_c():
if not hasattr(has_c, 'result'):
try:
(stdout, stderr), info = compile_run_strings(
[('main.c', (
'#include <stdio.h>\n'
'int main(){\n'
'printf("hello world\\n");\n'
'return 0;\n'
'}'
))], clean=True
)
except CompilerNotFoundError:
has_c.result = False
if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1':
raise
else:
if info['exit_status'] != os.EX_OK or 'hello world' not in stdout:
if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1':
raise ValueError("Failed to compile test program:\n%s\n%s\n" % (stdout, stderr))
has_c.result = False
else:
has_c.result = True
return has_c.result
def has_cxx():
if not hasattr(has_cxx, 'result'):
try:
(stdout, stderr), info = compile_run_strings(
[('main.cxx', (
'#include <iostream>\n'
'int main(){\n'
'std::cout << "hello world" << std::endl;\n'
'}'
))], clean=True
)
except CompilerNotFoundError:
has_cxx.result = False
if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1':
raise
else:
if info['exit_status'] != os.EX_OK or 'hello world' not in stdout:
if os.environ.get('SYMPY_STRICT_COMPILER_CHECKS', '0') == '1':
raise ValueError("Failed to compile test program:\n%s\n%s\n" % (stdout, stderr))
has_cxx.result = False
else:
has_cxx.result = True
return has_cxx.result
|
02b1db6ef5c70cba5a91b062bb6cde00d386ed29020fed5d412ebcb2cc50421c
|
# Tests that require installed backends go into
# sympy/test_external/test_autowrap
import os
import tempfile
import shutil
from sympy.core import symbols, Eq
from sympy.core.compatibility import StringIO
from sympy.utilities.autowrap import (autowrap, binary_function,
CythonCodeWrapper, UfuncifyCodeWrapper, CodeWrapper)
from sympy.utilities.codegen import (
CCodeGen, C99CodeGen, CodeGenArgumentListError, make_routine
)
from sympy.utilities.pytest import raises
from sympy.utilities.tmpfiles import TmpFileManager
def get_string(dump_fn, routines, prefix="file", **kwargs):
"""Wrapper for dump_fn. dump_fn writes its results to a stream object and
this wrapper returns the contents of that stream as a string. This
auxiliary function is used by many tests below.
The header and the empty lines are not generator to facilitate the
testing of the output.
"""
output = StringIO()
dump_fn(routines, output, prefix, **kwargs)
source = output.getvalue()
output.close()
return source
def test_cython_wrapper_scalar_function():
x, y, z = symbols('x,y,z')
expr = (x + y)*z
routine = make_routine("test", expr)
code_gen = CythonCodeWrapper(CCodeGen())
source = get_string(code_gen.dump_pyx, [routine])
expected = (
"cdef extern from 'file.h':\n"
" double test(double x, double y, double z)\n"
"\n"
"def test_c(double x, double y, double z):\n"
"\n"
" return test(x, y, z)")
assert source == expected
def test_cython_wrapper_outarg():
from sympy import Equality
x, y, z = symbols('x,y,z')
code_gen = CythonCodeWrapper(C99CodeGen())
routine = make_routine("test", Equality(z, x + y))
source = get_string(code_gen.dump_pyx, [routine])
expected = (
"cdef extern from 'file.h':\n"
" void test(double x, double y, double *z)\n"
"\n"
"def test_c(double x, double y):\n"
"\n"
" cdef double z = 0\n"
" test(x, y, &z)\n"
" return z")
assert source == expected
def test_cython_wrapper_inoutarg():
from sympy import Equality
x, y, z = symbols('x,y,z')
code_gen = CythonCodeWrapper(C99CodeGen())
routine = make_routine("test", Equality(z, x + y + z))
source = get_string(code_gen.dump_pyx, [routine])
expected = (
"cdef extern from 'file.h':\n"
" void test(double x, double y, double *z)\n"
"\n"
"def test_c(double x, double y, double z):\n"
"\n"
" test(x, y, &z)\n"
" return z")
assert source == expected
def test_cython_wrapper_compile_flags():
from sympy import Equality
x, y, z = symbols('x,y,z')
routine = make_routine("test", Equality(z, x + y))
code_gen = CythonCodeWrapper(CCodeGen())
expected = """\
try:
from setuptools import setup
from setuptools import Extension
except ImportError:
from distutils.core import setup
from distutils.extension import Extension
from Cython.Build import cythonize
cy_opts = {}
ext_mods = [Extension(
'wrapper_module_%(num)s', ['wrapper_module_%(num)s.pyx', 'wrapped_code_%(num)s.c'],
include_dirs=[],
library_dirs=[],
libraries=[],
extra_compile_args=['-std=c99'],
extra_link_args=[]
)]
setup(ext_modules=cythonize(ext_mods, **cy_opts))
""" % {'num': CodeWrapper._module_counter}
temp_dir = tempfile.mkdtemp()
TmpFileManager.tmp_folder(temp_dir)
setup_file_path = os.path.join(temp_dir, 'setup.py')
code_gen._prepare_files(routine, build_dir=temp_dir)
with open(setup_file_path) as f:
setup_text = f.read()
assert setup_text == expected
code_gen = CythonCodeWrapper(CCodeGen(),
include_dirs=['/usr/local/include', '/opt/booger/include'],
library_dirs=['/user/local/lib'],
libraries=['thelib', 'nilib'],
extra_compile_args=['-slow-math'],
extra_link_args=['-lswamp', '-ltrident'],
cythonize_options={'compiler_directives': {'boundscheck': False}}
)
expected = """\
try:
from setuptools import setup
from setuptools import Extension
except ImportError:
from distutils.core import setup
from distutils.extension import Extension
from Cython.Build import cythonize
cy_opts = {'compiler_directives': {'boundscheck': False}}
ext_mods = [Extension(
'wrapper_module_%(num)s', ['wrapper_module_%(num)s.pyx', 'wrapped_code_%(num)s.c'],
include_dirs=['/usr/local/include', '/opt/booger/include'],
library_dirs=['/user/local/lib'],
libraries=['thelib', 'nilib'],
extra_compile_args=['-slow-math', '-std=c99'],
extra_link_args=['-lswamp', '-ltrident']
)]
setup(ext_modules=cythonize(ext_mods, **cy_opts))
""" % {'num': CodeWrapper._module_counter}
code_gen._prepare_files(routine, build_dir=temp_dir)
with open(setup_file_path) as f:
setup_text = f.read()
assert setup_text == expected
expected = """\
try:
from setuptools import setup
from setuptools import Extension
except ImportError:
from distutils.core import setup
from distutils.extension import Extension
from Cython.Build import cythonize
cy_opts = {'compiler_directives': {'boundscheck': False}}
import numpy as np
ext_mods = [Extension(
'wrapper_module_%(num)s', ['wrapper_module_%(num)s.pyx', 'wrapped_code_%(num)s.c'],
include_dirs=['/usr/local/include', '/opt/booger/include', np.get_include()],
library_dirs=['/user/local/lib'],
libraries=['thelib', 'nilib'],
extra_compile_args=['-slow-math', '-std=c99'],
extra_link_args=['-lswamp', '-ltrident']
)]
setup(ext_modules=cythonize(ext_mods, **cy_opts))
""" % {'num': CodeWrapper._module_counter}
code_gen._need_numpy = True
code_gen._prepare_files(routine, build_dir=temp_dir)
with open(setup_file_path) as f:
setup_text = f.read()
assert setup_text == expected
TmpFileManager.cleanup()
def test_cython_wrapper_unique_dummyvars():
from sympy import Dummy, Equality
x, y, z = Dummy('x'), Dummy('y'), Dummy('z')
x_id, y_id, z_id = [str(d.dummy_index) for d in [x, y, z]]
expr = Equality(z, x + y)
routine = make_routine("test", expr)
code_gen = CythonCodeWrapper(CCodeGen())
source = get_string(code_gen.dump_pyx, [routine])
expected_template = (
"cdef extern from 'file.h':\n"
" void test(double x_{x_id}, double y_{y_id}, double *z_{z_id})\n"
"\n"
"def test_c(double x_{x_id}, double y_{y_id}):\n"
"\n"
" cdef double z_{z_id} = 0\n"
" test(x_{x_id}, y_{y_id}, &z_{z_id})\n"
" return z_{z_id}")
expected = expected_template.format(x_id=x_id, y_id=y_id, z_id=z_id)
assert source == expected
def test_autowrap_dummy():
x, y, z = symbols('x y z')
# Uses DummyWrapper to test that codegen works as expected
f = autowrap(x + y, backend='dummy')
assert f() == str(x + y)
assert f.args == "x, y"
assert f.returns == "nameless"
f = autowrap(Eq(z, x + y), backend='dummy')
assert f() == str(x + y)
assert f.args == "x, y"
assert f.returns == "z"
f = autowrap(Eq(z, x + y + z), backend='dummy')
assert f() == str(x + y + z)
assert f.args == "x, y, z"
assert f.returns == "z"
def test_autowrap_args():
x, y, z = symbols('x y z')
raises(CodeGenArgumentListError, lambda: autowrap(Eq(z, x + y),
backend='dummy', args=[x]))
f = autowrap(Eq(z, x + y), backend='dummy', args=[y, x])
assert f() == str(x + y)
assert f.args == "y, x"
assert f.returns == "z"
raises(CodeGenArgumentListError, lambda: autowrap(Eq(z, x + y + z),
backend='dummy', args=[x, y]))
f = autowrap(Eq(z, x + y + z), backend='dummy', args=[y, x, z])
assert f() == str(x + y + z)
assert f.args == "y, x, z"
assert f.returns == "z"
f = autowrap(Eq(z, x + y + z), backend='dummy', args=(y, x, z))
assert f() == str(x + y + z)
assert f.args == "y, x, z"
assert f.returns == "z"
def test_autowrap_store_files():
x, y = symbols('x y')
tmp = tempfile.mkdtemp()
TmpFileManager.tmp_folder(tmp)
f = autowrap(x + y, backend='dummy', tempdir=tmp)
assert f() == str(x + y)
assert os.access(tmp, os.F_OK)
TmpFileManager.cleanup()
def test_autowrap_store_files_issue_gh12939():
x, y = symbols('x y')
tmp = './tmp'
try:
f = autowrap(x + y, backend='dummy', tempdir=tmp)
assert f() == str(x + y)
assert os.access(tmp, os.F_OK)
finally:
shutil.rmtree(tmp)
def test_binary_function():
x, y = symbols('x y')
f = binary_function('f', x + y, backend='dummy')
assert f._imp_() == str(x + y)
def test_ufuncify_source():
x, y, z = symbols('x,y,z')
code_wrapper = UfuncifyCodeWrapper(C99CodeGen("ufuncify"))
routine = make_routine("test", x + y + z)
source = get_string(code_wrapper.dump_c, [routine])
expected = """\
#include "Python.h"
#include "math.h"
#include "numpy/ndarraytypes.h"
#include "numpy/ufuncobject.h"
#include "numpy/halffloat.h"
#include "file.h"
static PyMethodDef wrapper_module_%(num)sMethods[] = {
{NULL, NULL, 0, NULL}
};
static void test_ufunc(char **args, npy_intp *dimensions, npy_intp* steps, void* data)
{
npy_intp i;
npy_intp n = dimensions[0];
char *in0 = args[0];
char *in1 = args[1];
char *in2 = args[2];
char *out0 = args[3];
npy_intp in0_step = steps[0];
npy_intp in1_step = steps[1];
npy_intp in2_step = steps[2];
npy_intp out0_step = steps[3];
for (i = 0; i < n; i++) {
*((double *)out0) = test(*(double *)in0, *(double *)in1, *(double *)in2);
in0 += in0_step;
in1 += in1_step;
in2 += in2_step;
out0 += out0_step;
}
}
PyUFuncGenericFunction test_funcs[1] = {&test_ufunc};
static char test_types[4] = {NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE};
static void *test_data[1] = {NULL};
#if PY_VERSION_HEX >= 0x03000000
static struct PyModuleDef moduledef = {
PyModuleDef_HEAD_INIT,
"wrapper_module_%(num)s",
NULL,
-1,
wrapper_module_%(num)sMethods,
NULL,
NULL,
NULL,
NULL
};
PyMODINIT_FUNC PyInit_wrapper_module_%(num)s(void)
{
PyObject *m, *d;
PyObject *ufunc0;
m = PyModule_Create(&moduledef);
if (!m) {
return NULL;
}
import_array();
import_umath();
d = PyModule_GetDict(m);
ufunc0 = PyUFunc_FromFuncAndData(test_funcs, test_data, test_types, 1, 3, 1,
PyUFunc_None, "wrapper_module_%(num)s", "Created in SymPy with Ufuncify", 0);
PyDict_SetItemString(d, "test", ufunc0);
Py_DECREF(ufunc0);
return m;
}
#else
PyMODINIT_FUNC initwrapper_module_%(num)s(void)
{
PyObject *m, *d;
PyObject *ufunc0;
m = Py_InitModule("wrapper_module_%(num)s", wrapper_module_%(num)sMethods);
if (m == NULL) {
return;
}
import_array();
import_umath();
d = PyModule_GetDict(m);
ufunc0 = PyUFunc_FromFuncAndData(test_funcs, test_data, test_types, 1, 3, 1,
PyUFunc_None, "wrapper_module_%(num)s", "Created in SymPy with Ufuncify", 0);
PyDict_SetItemString(d, "test", ufunc0);
Py_DECREF(ufunc0);
}
#endif""" % {'num': CodeWrapper._module_counter}
assert source == expected
def test_ufuncify_source_multioutput():
x, y, z = symbols('x,y,z')
var_symbols = (x, y, z)
expr = x + y**3 + 10*z**2
code_wrapper = UfuncifyCodeWrapper(C99CodeGen("ufuncify"))
routines = [make_routine("func{}".format(i), expr.diff(var_symbols[i]), var_symbols) for i in range(len(var_symbols))]
source = get_string(code_wrapper.dump_c, routines, funcname='multitest')
expected = """\
#include "Python.h"
#include "math.h"
#include "numpy/ndarraytypes.h"
#include "numpy/ufuncobject.h"
#include "numpy/halffloat.h"
#include "file.h"
static PyMethodDef wrapper_module_%(num)sMethods[] = {
{NULL, NULL, 0, NULL}
};
static void multitest_ufunc(char **args, npy_intp *dimensions, npy_intp* steps, void* data)
{
npy_intp i;
npy_intp n = dimensions[0];
char *in0 = args[0];
char *in1 = args[1];
char *in2 = args[2];
char *out0 = args[3];
char *out1 = args[4];
char *out2 = args[5];
npy_intp in0_step = steps[0];
npy_intp in1_step = steps[1];
npy_intp in2_step = steps[2];
npy_intp out0_step = steps[3];
npy_intp out1_step = steps[4];
npy_intp out2_step = steps[5];
for (i = 0; i < n; i++) {
*((double *)out0) = func0(*(double *)in0, *(double *)in1, *(double *)in2);
*((double *)out1) = func1(*(double *)in0, *(double *)in1, *(double *)in2);
*((double *)out2) = func2(*(double *)in0, *(double *)in1, *(double *)in2);
in0 += in0_step;
in1 += in1_step;
in2 += in2_step;
out0 += out0_step;
out1 += out1_step;
out2 += out2_step;
}
}
PyUFuncGenericFunction multitest_funcs[1] = {&multitest_ufunc};
static char multitest_types[6] = {NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE};
static void *multitest_data[1] = {NULL};
#if PY_VERSION_HEX >= 0x03000000
static struct PyModuleDef moduledef = {
PyModuleDef_HEAD_INIT,
"wrapper_module_%(num)s",
NULL,
-1,
wrapper_module_%(num)sMethods,
NULL,
NULL,
NULL,
NULL
};
PyMODINIT_FUNC PyInit_wrapper_module_%(num)s(void)
{
PyObject *m, *d;
PyObject *ufunc0;
m = PyModule_Create(&moduledef);
if (!m) {
return NULL;
}
import_array();
import_umath();
d = PyModule_GetDict(m);
ufunc0 = PyUFunc_FromFuncAndData(multitest_funcs, multitest_data, multitest_types, 1, 3, 3,
PyUFunc_None, "wrapper_module_%(num)s", "Created in SymPy with Ufuncify", 0);
PyDict_SetItemString(d, "multitest", ufunc0);
Py_DECREF(ufunc0);
return m;
}
#else
PyMODINIT_FUNC initwrapper_module_%(num)s(void)
{
PyObject *m, *d;
PyObject *ufunc0;
m = Py_InitModule("wrapper_module_%(num)s", wrapper_module_%(num)sMethods);
if (m == NULL) {
return;
}
import_array();
import_umath();
d = PyModule_GetDict(m);
ufunc0 = PyUFunc_FromFuncAndData(multitest_funcs, multitest_data, multitest_types, 1, 3, 3,
PyUFunc_None, "wrapper_module_%(num)s", "Created in SymPy with Ufuncify", 0);
PyDict_SetItemString(d, "multitest", ufunc0);
Py_DECREF(ufunc0);
}
#endif""" % {'num': CodeWrapper._module_counter}
assert source == expected
|
a199e3c60ae1eaf6a1a7808bc6345c66a1e2ccd8be5443b7db6d4e7627a0a3f9
|
from textwrap import dedent
from sympy.core.compatibility import range, unichr
from sympy.utilities.misc import translate, replace, ordinal, rawlines, strlines
def test_translate():
abc = 'abc'
translate(abc, None, 'a') == 'bc'
translate(abc, None, '') == 'abc'
translate(abc, {'a': 'x'}, 'c') == 'xb'
assert translate(abc, {'a': 'bc'}, 'c') == 'bcb'
assert translate(abc, {'ab': 'x'}, 'c') == 'x'
assert translate(abc, {'ab': ''}, 'c') == ''
assert translate(abc, {'bc': 'x'}, 'c') == 'ab'
assert translate(abc, {'abc': 'x', 'a': 'y'}) == 'x'
u = unichr(4096)
assert translate(abc, 'a', 'x', u) == 'xbc'
assert (u in translate(abc, 'a', u, u)) is True
def test_replace():
assert replace('abc', ('a', 'b')) == 'bbc'
assert replace('abc', {'a': 'Aa'}) == 'Aabc'
assert replace('abc', ('a', 'b'), ('c', 'C')) == 'bbC'
def test_ordinal():
assert ordinal(-1) == '-1st'
assert ordinal(0) == '0th'
assert ordinal(1) == '1st'
assert ordinal(2) == '2nd'
assert ordinal(3) == '3rd'
assert all(ordinal(i).endswith('th') for i in range(4, 21))
assert ordinal(100) == '100th'
assert ordinal(101) == '101st'
assert ordinal(102) == '102nd'
assert ordinal(103) == '103rd'
assert ordinal(104) == '104th'
assert ordinal(200) == '200th'
assert all(ordinal(i) == str(i) + 'th' for i in range(-220, -203))
def test_rawlines():
assert rawlines('a a\na') == "dedent('''\\\n a a\n a''')"
assert rawlines('a a') == "'a a'"
assert rawlines(strlines('\\le"ft')) == (
'(\n'
" '(\\n'\n"
' \'r\\\'\\\\le"ft\\\'\\n\'\n'
" ')'\n"
')')
def test_strlines():
q = 'this quote (") is in the middle'
# the following assert rhs was prepared with
# print(rawlines(strlines(q, 10)))
assert strlines(q, 10) == dedent('''\
(
'this quo'
'te (") i'
's in the'
' middle'
)''')
assert q == (
'this quo'
'te (") i'
's in the'
' middle'
)
q = "this quote (') is in the middle"
assert strlines(q, 20) == dedent('''\
(
"this quote (') is "
"in the middle"
)''')
assert strlines('\\left') == (
'(\n'
"r'\\left'\n"
')')
assert strlines('\\left', short=True) == r"r'\left'"
assert strlines('\\le"ft') == (
'(\n'
'r\'\\le"ft\'\n'
')')
q = 'this\nother line'
assert strlines(q) == rawlines(q)
|
474512edf6ad4040e31f2fd5de456cfc1100d601c2b1fb68127362e26b968014
|
from sympy.core import (S, symbols, Eq, pi, Catalan, EulerGamma, Lambda,
Dummy, Function)
from sympy.core.compatibility import StringIO
from sympy import erf, Integral, Piecewise
from sympy import Equality
from sympy.matrices import Matrix, MatrixSymbol
from sympy.utilities.codegen import OctaveCodeGen, codegen, make_routine
from sympy.utilities.pytest import raises
from sympy.utilities.lambdify import implemented_function
from sympy.utilities.pytest import XFAIL
import sympy
x, y, z = symbols('x,y,z')
def test_empty_m_code():
code_gen = OctaveCodeGen()
output = StringIO()
code_gen.dump_m([], output, "file", header=False, empty=False)
source = output.getvalue()
assert source == ""
def test_m_simple_code():
name_expr = ("test", (x + y)*z)
result, = codegen(name_expr, "Octave", header=False, empty=False)
assert result[0] == "test.m"
source = result[1]
expected = (
"function out1 = test(x, y, z)\n"
" out1 = z.*(x + y);\n"
"end\n"
)
assert source == expected
def test_m_simple_code_with_header():
name_expr = ("test", (x + y)*z)
result, = codegen(name_expr, "Octave", header=True, empty=False)
assert result[0] == "test.m"
source = result[1]
expected = (
"function out1 = test(x, y, z)\n"
" %TEST Autogenerated by sympy\n"
" % Code generated with sympy " + sympy.__version__ + "\n"
" %\n"
" % See http://www.sympy.org/ for more information.\n"
" %\n"
" % This file is part of 'project'\n"
" out1 = z.*(x + y);\n"
"end\n"
)
assert source == expected
def test_m_simple_code_nameout():
expr = Equality(z, (x + y))
name_expr = ("test", expr)
result, = codegen(name_expr, "Octave", header=False, empty=False)
source = result[1]
expected = (
"function z = test(x, y)\n"
" z = x + y;\n"
"end\n"
)
assert source == expected
def test_m_numbersymbol():
name_expr = ("test", pi**Catalan)
result, = codegen(name_expr, "Octave", header=False, empty=False)
source = result[1]
expected = (
"function out1 = test()\n"
" out1 = pi^%s;\n"
"end\n"
) % Catalan.evalf(17)
assert source == expected
@XFAIL
def test_m_numbersymbol_no_inline():
# FIXME: how to pass inline=False to the OctaveCodePrinter?
name_expr = ("test", [pi**Catalan, EulerGamma])
result, = codegen(name_expr, "Octave", header=False,
empty=False, inline=False)
source = result[1]
expected = (
"function [out1, out2] = test()\n"
" Catalan = 0.915965594177219; % constant\n"
" EulerGamma = 0.5772156649015329; % constant\n"
" out1 = pi^Catalan;\n"
" out2 = EulerGamma;\n"
"end\n"
)
assert source == expected
def test_m_code_argument_order():
expr = x + y
routine = make_routine("test", expr, argument_sequence=[z, x, y], language="octave")
code_gen = OctaveCodeGen()
output = StringIO()
code_gen.dump_m([routine], output, "test", header=False, empty=False)
source = output.getvalue()
expected = (
"function out1 = test(z, x, y)\n"
" out1 = x + y;\n"
"end\n"
)
assert source == expected
def test_multiple_results_m():
# Here the output order is the input order
expr1 = (x + y)*z
expr2 = (x - y)*z
name_expr = ("test", [expr1, expr2])
result, = codegen(name_expr, "Octave", header=False, empty=False)
source = result[1]
expected = (
"function [out1, out2] = test(x, y, z)\n"
" out1 = z.*(x + y);\n"
" out2 = z.*(x - y);\n"
"end\n"
)
assert source == expected
def test_results_named_unordered():
# Here output order is based on name_expr
A, B, C = symbols('A,B,C')
expr1 = Equality(C, (x + y)*z)
expr2 = Equality(A, (x - y)*z)
expr3 = Equality(B, 2*x)
name_expr = ("test", [expr1, expr2, expr3])
result, = codegen(name_expr, "Octave", header=False, empty=False)
source = result[1]
expected = (
"function [C, A, B] = test(x, y, z)\n"
" C = z.*(x + y);\n"
" A = z.*(x - y);\n"
" B = 2*x;\n"
"end\n"
)
assert source == expected
def test_results_named_ordered():
A, B, C = symbols('A,B,C')
expr1 = Equality(C, (x + y)*z)
expr2 = Equality(A, (x - y)*z)
expr3 = Equality(B, 2*x)
name_expr = ("test", [expr1, expr2, expr3])
result = codegen(name_expr, "Octave", header=False, empty=False,
argument_sequence=(x, z, y))
assert result[0][0] == "test.m"
source = result[0][1]
expected = (
"function [C, A, B] = test(x, z, y)\n"
" C = z.*(x + y);\n"
" A = z.*(x - y);\n"
" B = 2*x;\n"
"end\n"
)
assert source == expected
def test_complicated_m_codegen():
from sympy import sin, cos, tan
name_expr = ("testlong",
[ ((sin(x) + cos(y) + tan(z))**3).expand(),
cos(cos(cos(cos(cos(cos(cos(cos(x + y + z))))))))
])
result = codegen(name_expr, "Octave", header=False, empty=False)
assert result[0][0] == "testlong.m"
source = result[0][1]
expected = (
"function [out1, out2] = testlong(x, y, z)\n"
" out1 = sin(x).^3 + 3*sin(x).^2.*cos(y) + 3*sin(x).^2.*tan(z)"
" + 3*sin(x).*cos(y).^2 + 6*sin(x).*cos(y).*tan(z) + 3*sin(x).*tan(z).^2"
" + cos(y).^3 + 3*cos(y).^2.*tan(z) + 3*cos(y).*tan(z).^2 + tan(z).^3;\n"
" out2 = cos(cos(cos(cos(cos(cos(cos(cos(x + y + z))))))));\n"
"end\n"
)
assert source == expected
def test_m_output_arg_mixed_unordered():
# named outputs are alphabetical, unnamed output appear in the given order
from sympy import sin, cos, tan
a = symbols("a")
name_expr = ("foo", [cos(2*x), Equality(y, sin(x)), cos(x), Equality(a, sin(2*x))])
result, = codegen(name_expr, "Octave", header=False, empty=False)
assert result[0] == "foo.m"
source = result[1];
expected = (
'function [out1, y, out3, a] = foo(x)\n'
' out1 = cos(2*x);\n'
' y = sin(x);\n'
' out3 = cos(x);\n'
' a = sin(2*x);\n'
'end\n'
)
assert source == expected
def test_m_piecewise_():
pw = Piecewise((0, x < -1), (x**2, x <= 1), (-x+2, x > 1), (1, True), evaluate=False)
name_expr = ("pwtest", pw)
result, = codegen(name_expr, "Octave", header=False, empty=False)
source = result[1]
expected = (
"function out1 = pwtest(x)\n"
" out1 = ((x < -1).*(0) + (~(x < -1)).*( ...\n"
" (x <= 1).*(x.^2) + (~(x <= 1)).*( ...\n"
" (x > 1).*(2 - x) + (~(x > 1)).*(1))));\n"
"end\n"
)
assert source == expected
@XFAIL
def test_m_piecewise_no_inline():
# FIXME: how to pass inline=False to the OctaveCodePrinter?
pw = Piecewise((0, x < -1), (x**2, x <= 1), (-x+2, x > 1), (1, True))
name_expr = ("pwtest", pw)
result, = codegen(name_expr, "Octave", header=False, empty=False,
inline=False)
source = result[1]
expected = (
"function out1 = pwtest(x)\n"
" if (x < -1)\n"
" out1 = 0;\n"
" elseif (x <= 1)\n"
" out1 = x.^2;\n"
" elseif (x > 1)\n"
" out1 = -x + 2;\n"
" else\n"
" out1 = 1;\n"
" end\n"
"end\n"
)
assert source == expected
def test_m_multifcns_per_file():
name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ]
result = codegen(name_expr, "Octave", header=False, empty=False)
assert result[0][0] == "foo.m"
source = result[0][1];
expected = (
"function [out1, out2] = foo(x, y)\n"
" out1 = 2*x;\n"
" out2 = 3*y;\n"
"end\n"
"function [out1, out2] = bar(y)\n"
" out1 = y.^2;\n"
" out2 = 4*y;\n"
"end\n"
)
assert source == expected
def test_m_multifcns_per_file_w_header():
name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ]
result = codegen(name_expr, "Octave", header=True, empty=False)
assert result[0][0] == "foo.m"
source = result[0][1];
expected = (
"function [out1, out2] = foo(x, y)\n"
" %FOO Autogenerated by sympy\n"
" % Code generated with sympy " + sympy.__version__ + "\n"
" %\n"
" % See http://www.sympy.org/ for more information.\n"
" %\n"
" % This file is part of 'project'\n"
" out1 = 2*x;\n"
" out2 = 3*y;\n"
"end\n"
"function [out1, out2] = bar(y)\n"
" out1 = y.^2;\n"
" out2 = 4*y;\n"
"end\n"
)
assert source == expected
def test_m_filename_match_first_fcn():
name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ]
raises(ValueError, lambda: codegen(name_expr,
"Octave", prefix="bar", header=False, empty=False))
def test_m_matrix_named():
e2 = Matrix([[x, 2*y, pi*z]])
name_expr = ("test", Equality(MatrixSymbol('myout1', 1, 3), e2))
result = codegen(name_expr, "Octave", header=False, empty=False)
assert result[0][0] == "test.m"
source = result[0][1]
expected = (
"function myout1 = test(x, y, z)\n"
" myout1 = [x 2*y pi*z];\n"
"end\n"
)
assert source == expected
def test_m_matrix_named_matsym():
myout1 = MatrixSymbol('myout1', 1, 3)
e2 = Matrix([[x, 2*y, pi*z]])
name_expr = ("test", Equality(myout1, e2, evaluate=False))
result, = codegen(name_expr, "Octave", header=False, empty=False)
source = result[1]
expected = (
"function myout1 = test(x, y, z)\n"
" myout1 = [x 2*y pi*z];\n"
"end\n"
)
assert source == expected
def test_m_matrix_output_autoname():
expr = Matrix([[x, x+y, 3]])
name_expr = ("test", expr)
result, = codegen(name_expr, "Octave", header=False, empty=False)
source = result[1]
expected = (
"function out1 = test(x, y)\n"
" out1 = [x x + y 3];\n"
"end\n"
)
assert source == expected
def test_m_matrix_output_autoname_2():
e1 = (x + y)
e2 = Matrix([[2*x, 2*y, 2*z]])
e3 = Matrix([[x], [y], [z]])
e4 = Matrix([[x, y], [z, 16]])
name_expr = ("test", (e1, e2, e3, e4))
result, = codegen(name_expr, "Octave", header=False, empty=False)
source = result[1]
expected = (
"function [out1, out2, out3, out4] = test(x, y, z)\n"
" out1 = x + y;\n"
" out2 = [2*x 2*y 2*z];\n"
" out3 = [x; y; z];\n"
" out4 = [x y; z 16];\n"
"end\n"
)
assert source == expected
def test_m_results_matrix_named_ordered():
B, C = symbols('B,C')
A = MatrixSymbol('A', 1, 3)
expr1 = Equality(C, (x + y)*z)
expr2 = Equality(A, Matrix([[1, 2, x]]))
expr3 = Equality(B, 2*x)
name_expr = ("test", [expr1, expr2, expr3])
result, = codegen(name_expr, "Octave", header=False, empty=False,
argument_sequence=(x, z, y))
source = result[1]
expected = (
"function [C, A, B] = test(x, z, y)\n"
" C = z.*(x + y);\n"
" A = [1 2 x];\n"
" B = 2*x;\n"
"end\n"
)
assert source == expected
def test_m_matrixsymbol_slice():
A = MatrixSymbol('A', 2, 3)
B = MatrixSymbol('B', 1, 3)
C = MatrixSymbol('C', 1, 3)
D = MatrixSymbol('D', 2, 1)
name_expr = ("test", [Equality(B, A[0, :]),
Equality(C, A[1, :]),
Equality(D, A[:, 2])])
result, = codegen(name_expr, "Octave", header=False, empty=False)
source = result[1]
expected = (
"function [B, C, D] = test(A)\n"
" B = A(1, :);\n"
" C = A(2, :);\n"
" D = A(:, 3);\n"
"end\n"
)
assert source == expected
def test_m_matrixsymbol_slice2():
A = MatrixSymbol('A', 3, 4)
B = MatrixSymbol('B', 2, 2)
C = MatrixSymbol('C', 2, 2)
name_expr = ("test", [Equality(B, A[0:2, 0:2]),
Equality(C, A[0:2, 1:3])])
result, = codegen(name_expr, "Octave", header=False, empty=False)
source = result[1]
expected = (
"function [B, C] = test(A)\n"
" B = A(1:2, 1:2);\n"
" C = A(1:2, 2:3);\n"
"end\n"
)
assert source == expected
def test_m_matrixsymbol_slice3():
A = MatrixSymbol('A', 8, 7)
B = MatrixSymbol('B', 2, 2)
C = MatrixSymbol('C', 4, 2)
name_expr = ("test", [Equality(B, A[6:, 1::3]),
Equality(C, A[::2, ::3])])
result, = codegen(name_expr, "Octave", header=False, empty=False)
source = result[1]
expected = (
"function [B, C] = test(A)\n"
" B = A(7:end, 2:3:end);\n"
" C = A(1:2:end, 1:3:end);\n"
"end\n"
)
assert source == expected
def test_m_matrixsymbol_slice_autoname():
A = MatrixSymbol('A', 2, 3)
B = MatrixSymbol('B', 1, 3)
name_expr = ("test", [Equality(B, A[0,:]), A[1,:], A[:,0], A[:,1]])
result, = codegen(name_expr, "Octave", header=False, empty=False)
source = result[1]
expected = (
"function [B, out2, out3, out4] = test(A)\n"
" B = A(1, :);\n"
" out2 = A(2, :);\n"
" out3 = A(:, 1);\n"
" out4 = A(:, 2);\n"
"end\n"
)
assert source == expected
def test_m_loops():
# Note: an Octave programmer would probably vectorize this across one or
# more dimensions. Also, size(A) would be used rather than passing in m
# and n. Perhaps users would expect us to vectorize automatically here?
# Or is it possible to represent such things using IndexedBase?
from sympy.tensor import IndexedBase, Idx
from sympy import symbols
n, m = symbols('n m', integer=True)
A = IndexedBase('A')
x = IndexedBase('x')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
result, = codegen(('mat_vec_mult', Eq(y[i], A[i, j]*x[j])), "Octave",
header=False, empty=False)
source = result[1]
expected = (
'function y = mat_vec_mult(A, m, n, x)\n'
' for i = 1:m\n'
' y(i) = 0;\n'
' end\n'
' for i = 1:m\n'
' for j = 1:n\n'
' y(i) = %(rhs)s + y(i);\n'
' end\n'
' end\n'
'end\n'
)
assert (source == expected % {'rhs': 'A(%s, %s).*x(j)' % (i, j)} or
source == expected % {'rhs': 'x(j).*A(%s, %s)' % (i, j)})
def test_m_tensor_loops_multiple_contractions():
# see comments in previous test about vectorizing
from sympy.tensor import IndexedBase, Idx
from sympy import symbols
n, m, o, p = symbols('n m o p', integer=True)
A = IndexedBase('A')
B = IndexedBase('B')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
l = Idx('l', p)
result, = codegen(('tensorthing', Eq(y[i], B[j, k, l]*A[i, j, k, l])),
"Octave", header=False, empty=False)
source = result[1]
expected = (
'function y = tensorthing(A, B, m, n, o, p)\n'
' for i = 1:m\n'
' y(i) = 0;\n'
' end\n'
' for i = 1:m\n'
' for j = 1:n\n'
' for k = 1:o\n'
' for l = 1:p\n'
' y(i) = A(i, j, k, l).*B(j, k, l) + y(i);\n'
' end\n'
' end\n'
' end\n'
' end\n'
'end\n'
)
assert source == expected
def test_m_InOutArgument():
expr = Equality(x, x**2)
name_expr = ("mysqr", expr)
result, = codegen(name_expr, "Octave", header=False, empty=False)
source = result[1]
expected = (
"function x = mysqr(x)\n"
" x = x.^2;\n"
"end\n"
)
assert source == expected
def test_m_InOutArgument_order():
# can specify the order as (x, y)
expr = Equality(x, x**2 + y)
name_expr = ("test", expr)
result, = codegen(name_expr, "Octave", header=False,
empty=False, argument_sequence=(x,y))
source = result[1]
expected = (
"function x = test(x, y)\n"
" x = x.^2 + y;\n"
"end\n"
)
assert source == expected
# make sure it gives (x, y) not (y, x)
expr = Equality(x, x**2 + y)
name_expr = ("test", expr)
result, = codegen(name_expr, "Octave", header=False, empty=False)
source = result[1]
expected = (
"function x = test(x, y)\n"
" x = x.^2 + y;\n"
"end\n"
)
assert source == expected
def test_m_not_supported():
f = Function('f')
name_expr = ("test", [f(x).diff(x), S.ComplexInfinity])
result, = codegen(name_expr, "Octave", header=False, empty=False)
source = result[1]
expected = (
"function [out1, out2] = test(x)\n"
" % unsupported: Derivative(f(x), x)\n"
" % unsupported: zoo\n"
" out1 = Derivative(f(x), x);\n"
" out2 = zoo;\n"
"end\n"
)
assert source == expected
def test_global_vars_octave():
x, y, z, t = symbols("x y z t")
result = codegen(('f', x*y), "Octave", header=False, empty=False,
global_vars=(y,))
source = result[0][1]
expected = (
"function out1 = f(x)\n"
" global y\n"
" out1 = x.*y;\n"
"end\n"
)
assert source == expected
result = codegen(('f', x*y+z), "Octave", header=False, empty=False,
argument_sequence=(x, y), global_vars=(z, t))
source = result[0][1]
expected = (
"function out1 = f(x, y)\n"
" global t z\n"
" out1 = x.*y + z;\n"
"end\n"
)
assert source == expected
|
225fa86374f042140f15a4f8ef6e6b6092df01cd070a99730b046f6d9e653a4d
|
import sys
import inspect
import copy
import pickle
from sympy.physics.units import meter
from sympy.utilities.pytest import XFAIL
from sympy.core.basic import Atom, Basic
from sympy.core.core import BasicMeta
from sympy.core.singleton import SingletonRegistry
from sympy.core.symbol import Dummy, Symbol, Wild
from sympy.core.numbers import (E, I, pi, oo, zoo, nan, Integer,
Rational, Float)
from sympy.core.relational import (Equality, GreaterThan, LessThan, Relational,
StrictGreaterThan, StrictLessThan, Unequality)
from sympy.core.add import Add
from sympy.core.mul import Mul
from sympy.core.power import Pow
from sympy.core.function import Derivative, Function, FunctionClass, Lambda, \
WildFunction
from sympy.sets.sets import Interval
from sympy.core.multidimensional import vectorize
from sympy.core.compatibility import HAS_GMPY
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.utilities.pytest import ignore_warnings
from sympy import symbols, S
from sympy.external import import_module
cloudpickle = import_module('cloudpickle')
excluded_attrs = set(
['_assumptions', '_mhash', 'message',
# XXX Remove these after deprecation is done for issue #15887
'_cache_eigenvects', '_cache_is_diagonalizable']
)
def check(a, exclude=[], check_attr=True):
""" Check that pickling and copying round-trips.
"""
protocols = [0, 1, 2, copy.copy, copy.deepcopy]
# Python 2.x doesn't support the third pickling protocol
if sys.version_info >= (3,):
protocols.extend([3, 4])
if cloudpickle:
protocols.extend([cloudpickle])
for protocol in protocols:
if protocol in exclude:
continue
if callable(protocol):
if isinstance(a, BasicMeta):
# Classes can't be copied, but that's okay.
continue
b = protocol(a)
elif inspect.ismodule(protocol):
b = protocol.loads(protocol.dumps(a))
else:
b = pickle.loads(pickle.dumps(a, protocol))
d1 = dir(a)
d2 = dir(b)
assert set(d1) == set(d2)
if not check_attr:
continue
def c(a, b, d):
for i in d:
if i in excluded_attrs:
continue
if not hasattr(a, i):
continue
attr = getattr(a, i)
if not hasattr(attr, "__call__"):
assert hasattr(b, i), i
assert getattr(b, i) == attr, "%s != %s, protocol: %s" % (getattr(b, i), attr, protocol)
c(a, b, d1)
c(b, a, d2)
#================== core =========================
def test_core_basic():
for c in (Atom, Atom(),
Basic, Basic(),
# XXX: dynamically created types are not picklable
# BasicMeta, BasicMeta("test", (), {}),
SingletonRegistry, S):
check(c)
def test_core_symbol():
# make the Symbol a unique name that doesn't class with any other
# testing variable in this file since after this test the symbol
# having the same name will be cached as noncommutative
for c in (Dummy, Dummy("x", commutative=False), Symbol,
Symbol("_issue_3130", commutative=False), Wild, Wild("x")):
check(c)
def test_core_numbers():
for c in (Integer(2), Rational(2, 3), Float("1.2")):
check(c)
def test_core_float_copy():
# See gh-7457
y = Symbol("x") + 1.0
check(y) # does not raise TypeError ("argument is not an mpz")
def test_core_relational():
x = Symbol("x")
y = Symbol("y")
for c in (Equality, Equality(x, y), GreaterThan, GreaterThan(x, y),
LessThan, LessThan(x, y), Relational, Relational(x, y),
StrictGreaterThan, StrictGreaterThan(x, y), StrictLessThan,
StrictLessThan(x, y), Unequality, Unequality(x, y)):
check(c)
def test_core_add():
x = Symbol("x")
for c in (Add, Add(x, 4)):
check(c)
def test_core_mul():
x = Symbol("x")
for c in (Mul, Mul(x, 4)):
check(c)
def test_core_power():
x = Symbol("x")
for c in (Pow, Pow(x, 4)):
check(c)
def test_core_function():
x = Symbol("x")
for f in (Derivative, Derivative(x), Function, FunctionClass, Lambda,
WildFunction):
check(f)
def test_core_undefinedfunctions():
f = Function("f")
# Full XFAILed test below
exclude = list(range(5))
# https://github.com/cloudpipe/cloudpickle/issues/65
# https://github.com/cloudpipe/cloudpickle/issues/190
exclude.append(cloudpickle)
check(f, exclude=exclude)
@XFAIL
def test_core_undefinedfunctions_fail():
# This fails because f is assumed to be a class at sympy.basic.function.f
f = Function("f")
check(f)
def test_core_interval():
for c in (Interval, Interval(0, 2)):
check(c)
def test_core_multidimensional():
for c in (vectorize, vectorize(0)):
check(c)
def test_Singletons():
protocols = [0, 1, 2]
if sys.version_info >= (3,):
protocols.extend([3, 4])
copiers = [copy.copy, copy.deepcopy]
copiers += [lambda x: pickle.loads(pickle.dumps(x, proto))
for proto in protocols]
if cloudpickle:
copiers += [lambda x: cloudpickle.loads(cloudpickle.dumps(x))]
for obj in (Integer(-1), Integer(0), Integer(1), Rational(1, 2), pi, E, I,
oo, -oo, zoo, nan, S.GoldenRatio, S.TribonacciConstant,
S.EulerGamma, S.Catalan, S.EmptySet, S.IdentityFunction):
for func in copiers:
assert func(obj) is obj
#================== functions ===================
from sympy.functions import (Piecewise, lowergamma, acosh,
chebyshevu, chebyshevt, ln, chebyshevt_root, binomial, legendre,
Heaviside, factorial, bernoulli, coth, tanh, assoc_legendre, sign,
arg, asin, DiracDelta, re, rf, Abs, uppergamma, binomial, sinh, Ynm,
cos, cot, acos, acot, gamma, bell, hermite, harmonic,
LambertW, zeta, log, factorial, asinh, acoth, Znm,
cosh, dirichlet_eta, Eijk, loggamma, erf, ceiling, im, fibonacci,
tribonacci, conjugate, tan, chebyshevu_root, floor, atanh, sqrt,
RisingFactorial, sin, atan, ff, FallingFactorial, lucas, atan2,
polygamma, exp)
def test_functions():
one_var = (acosh, ln, Heaviside, factorial, bernoulli, coth, tanh,
sign, arg, asin, DiracDelta, re, Abs, sinh, cos, cot, acos, acot,
gamma, bell, harmonic, LambertW, zeta, log, factorial, asinh,
acoth, cosh, dirichlet_eta, loggamma, erf, ceiling, im, fibonacci,
tribonacci, conjugate, tan, floor, atanh, sin, atan, lucas, exp)
two_var = (rf, ff, lowergamma, chebyshevu, chebyshevt, binomial,
atan2, polygamma, hermite, legendre, uppergamma)
x, y, z = symbols("x,y,z")
others = (chebyshevt_root, chebyshevu_root, Eijk(x, y, z),
Piecewise( (0, x < -1), (x**2, x <= 1), (x**3, True)),
assoc_legendre)
for cls in one_var:
check(cls)
c = cls(x)
check(c)
for cls in two_var:
check(cls)
c = cls(x, y)
check(c)
for cls in others:
check(cls)
#================== geometry ====================
from sympy.geometry.entity import GeometryEntity
from sympy.geometry.point import Point
from sympy.geometry.ellipse import Circle, Ellipse
from sympy.geometry.line import Line, LinearEntity, Ray, Segment
from sympy.geometry.polygon import Polygon, RegularPolygon, Triangle
def test_geometry():
p1 = Point(1, 2)
p2 = Point(2, 3)
p3 = Point(0, 0)
p4 = Point(0, 1)
for c in (
GeometryEntity, GeometryEntity(), Point, p1, Circle, Circle(p1, 2),
Ellipse, Ellipse(p1, 3, 4), Line, Line(p1, p2), LinearEntity,
LinearEntity(p1, p2), Ray, Ray(p1, p2), Segment, Segment(p1, p2),
Polygon, Polygon(p1, p2, p3, p4), RegularPolygon,
RegularPolygon(p1, 4, 5), Triangle, Triangle(p1, p2, p3)):
check(c, check_attr=False)
#================== integrals ====================
from sympy.integrals.integrals import Integral
def test_integrals():
x = Symbol("x")
for c in (Integral, Integral(x)):
check(c)
#==================== logic =====================
from sympy.core.logic import Logic
def test_logic():
for c in (Logic, Logic(1)):
check(c)
#================== matrices ====================
from sympy.matrices import Matrix, SparseMatrix
def test_matrices():
for c in (Matrix, Matrix([1, 2, 3]), SparseMatrix, SparseMatrix([[1, 2], [3, 4]])):
check(c)
#================== ntheory =====================
from sympy.ntheory.generate import Sieve
def test_ntheory():
for c in (Sieve, Sieve()):
check(c)
#================== physics =====================
from sympy.physics.paulialgebra import Pauli
from sympy.physics.units import Unit
def test_physics():
for c in (Unit, meter, Pauli, Pauli(1)):
check(c)
#================== plotting ====================
# XXX: These tests are not complete, so XFAIL them
@XFAIL
def test_plotting():
from sympy.plotting.color_scheme import ColorGradient, ColorScheme
from sympy.plotting.managed_window import ManagedWindow
from sympy.plotting.plot import Plot, ScreenShot
from sympy.plotting.plot_axes import PlotAxes, PlotAxesBase, PlotAxesFrame, PlotAxesOrdinate
from sympy.plotting.plot_camera import PlotCamera
from sympy.plotting.plot_controller import PlotController
from sympy.plotting.plot_curve import PlotCurve
from sympy.plotting.plot_interval import PlotInterval
from sympy.plotting.plot_mode import PlotMode
from sympy.plotting.plot_modes import Cartesian2D, Cartesian3D, Cylindrical, \
ParametricCurve2D, ParametricCurve3D, ParametricSurface, Polar, Spherical
from sympy.plotting.plot_object import PlotObject
from sympy.plotting.plot_surface import PlotSurface
from sympy.plotting.plot_window import PlotWindow
for c in (
ColorGradient, ColorGradient(0.2, 0.4), ColorScheme, ManagedWindow,
ManagedWindow, Plot, ScreenShot, PlotAxes, PlotAxesBase,
PlotAxesFrame, PlotAxesOrdinate, PlotCamera, PlotController,
PlotCurve, PlotInterval, PlotMode, Cartesian2D, Cartesian3D,
Cylindrical, ParametricCurve2D, ParametricCurve3D,
ParametricSurface, Polar, Spherical, PlotObject, PlotSurface,
PlotWindow):
check(c)
@XFAIL
def test_plotting2():
from sympy.plotting.color_scheme import ColorGradient, ColorScheme
from sympy.plotting.managed_window import ManagedWindow
from sympy.plotting.plot import Plot, ScreenShot
from sympy.plotting.plot_axes import PlotAxes, PlotAxesBase, PlotAxesFrame, PlotAxesOrdinate
from sympy.plotting.plot_camera import PlotCamera
from sympy.plotting.plot_controller import PlotController
from sympy.plotting.plot_curve import PlotCurve
from sympy.plotting.plot_interval import PlotInterval
from sympy.plotting.plot_mode import PlotMode
from sympy.plotting.plot_modes import Cartesian2D, Cartesian3D, Cylindrical, \
ParametricCurve2D, ParametricCurve3D, ParametricSurface, Polar, Spherical
from sympy.plotting.plot_object import PlotObject
from sympy.plotting.plot_surface import PlotSurface
from sympy.plotting.plot_window import PlotWindow
check(ColorScheme("rainbow"))
check(Plot(1, visible=False))
check(PlotAxes())
#================== polys =======================
from sympy import Poly, ZZ, QQ, lex
def test_pickling_polys_polytools():
from sympy.polys.polytools import Poly, PurePoly, GroebnerBasis
x = Symbol('x')
for c in (Poly, Poly(x, x)):
check(c)
for c in (PurePoly, PurePoly(x)):
check(c)
# TODO: fix pickling of Options class (see GroebnerBasis._options)
# for c in (GroebnerBasis, GroebnerBasis([x**2 - 1], x, order=lex)):
# check(c)
def test_pickling_polys_polyclasses():
from sympy.polys.polyclasses import DMP, DMF, ANP
for c in (DMP, DMP([[ZZ(1)], [ZZ(2)], [ZZ(3)]], ZZ)):
check(c)
for c in (DMF, DMF(([ZZ(1), ZZ(2)], [ZZ(1), ZZ(3)]), ZZ)):
check(c)
for c in (ANP, ANP([QQ(1), QQ(2)], [QQ(1), QQ(2), QQ(3)], QQ)):
check(c)
@XFAIL
def test_pickling_polys_rings():
# NOTE: can't use protocols < 2 because we have to execute __new__ to
# make sure caching of rings works properly.
from sympy.polys.rings import PolyRing
ring = PolyRing("x,y,z", ZZ, lex)
for c in (PolyRing, ring):
check(c, exclude=[0, 1])
for c in (ring.dtype, ring.one):
check(c, exclude=[0, 1], check_attr=False) # TODO: Py3k
def test_pickling_polys_fields():
# NOTE: can't use protocols < 2 because we have to execute __new__ to
# make sure caching of fields works properly.
from sympy.polys.fields import FracField
field = FracField("x,y,z", ZZ, lex)
# TODO: AssertionError: assert id(obj) not in self.memo
# for c in (FracField, field):
# check(c, exclude=[0, 1])
# TODO: AssertionError: assert id(obj) not in self.memo
# for c in (field.dtype, field.one):
# check(c, exclude=[0, 1])
def test_pickling_polys_elements():
from sympy.polys.domains.pythonrational import PythonRational
from sympy.polys.domains.pythonfinitefield import PythonFiniteField
from sympy.polys.domains.mpelements import MPContext
for c in (PythonRational, PythonRational(1, 7)):
check(c)
gf = PythonFiniteField(17)
# TODO: fix pickling of ModularInteger
# for c in (gf.dtype, gf(5)):
# check(c)
mp = MPContext()
# TODO: fix pickling of RealElement
# for c in (mp.mpf, mp.mpf(1.0)):
# check(c)
# TODO: fix pickling of ComplexElement
# for c in (mp.mpc, mp.mpc(1.0, -1.5)):
# check(c)
def test_pickling_polys_domains():
from sympy.polys.domains.pythonfinitefield import PythonFiniteField
from sympy.polys.domains.pythonintegerring import PythonIntegerRing
from sympy.polys.domains.pythonrationalfield import PythonRationalField
# TODO: fix pickling of ModularInteger
# for c in (PythonFiniteField, PythonFiniteField(17)):
# check(c)
for c in (PythonIntegerRing, PythonIntegerRing()):
check(c, check_attr=False)
for c in (PythonRationalField, PythonRationalField()):
check(c, check_attr=False)
if HAS_GMPY:
from sympy.polys.domains.gmpyfinitefield import GMPYFiniteField
from sympy.polys.domains.gmpyintegerring import GMPYIntegerRing
from sympy.polys.domains.gmpyrationalfield import GMPYRationalField
# TODO: fix pickling of ModularInteger
# for c in (GMPYFiniteField, GMPYFiniteField(17)):
# check(c)
for c in (GMPYIntegerRing, GMPYIntegerRing()):
check(c, check_attr=False)
for c in (GMPYRationalField, GMPYRationalField()):
check(c, check_attr=False)
from sympy.polys.domains.realfield import RealField
from sympy.polys.domains.complexfield import ComplexField
from sympy.polys.domains.algebraicfield import AlgebraicField
from sympy.polys.domains.polynomialring import PolynomialRing
from sympy.polys.domains.fractionfield import FractionField
from sympy.polys.domains.expressiondomain import ExpressionDomain
# TODO: fix pickling of RealElement
# for c in (RealField, RealField(100)):
# check(c)
# TODO: fix pickling of ComplexElement
# for c in (ComplexField, ComplexField(100)):
# check(c)
for c in (AlgebraicField, AlgebraicField(QQ, sqrt(3))):
check(c, check_attr=False)
# TODO: AssertionError
# for c in (PolynomialRing, PolynomialRing(ZZ, "x,y,z")):
# check(c)
# TODO: AttributeError: 'PolyElement' object has no attribute 'ring'
# for c in (FractionField, FractionField(ZZ, "x,y,z")):
# check(c)
for c in (ExpressionDomain, ExpressionDomain()):
check(c, check_attr=False)
def test_pickling_polys_numberfields():
from sympy.polys.numberfields import AlgebraicNumber
for c in (AlgebraicNumber, AlgebraicNumber(sqrt(3))):
check(c, check_attr=False)
def test_pickling_polys_orderings():
from sympy.polys.orderings import (LexOrder, GradedLexOrder,
ReversedGradedLexOrder, ProductOrder, InverseOrder)
for c in (LexOrder, LexOrder()):
check(c)
for c in (GradedLexOrder, GradedLexOrder()):
check(c)
for c in (ReversedGradedLexOrder, ReversedGradedLexOrder()):
check(c)
# TODO: Argh, Python is so naive. No lambdas nor inner function support in
# pickling module. Maybe someone could figure out what to do with this.
#
# for c in (ProductOrder, ProductOrder((LexOrder(), lambda m: m[:2]),
# (GradedLexOrder(), lambda m: m[2:]))):
# check(c)
for c in (InverseOrder, InverseOrder(LexOrder())):
check(c)
def test_pickling_polys_monomials():
from sympy.polys.monomials import MonomialOps, Monomial
x, y, z = symbols("x,y,z")
for c in (MonomialOps, MonomialOps(3)):
check(c)
for c in (Monomial, Monomial((1, 2, 3), (x, y, z))):
check(c)
def test_pickling_polys_errors():
from sympy.polys.polyerrors import (ExactQuotientFailed, OperationNotSupported,
HeuristicGCDFailed, HomomorphismFailed, IsomorphismFailed, ExtraneousFactors,
EvaluationFailed, RefinementFailed, CoercionFailed, NotInvertible, NotReversible,
NotAlgebraic, DomainError, PolynomialError, UnificationFailed, GeneratorsError,
GeneratorsNeeded, ComputationFailed, UnivariatePolynomialError,
MultivariatePolynomialError, PolificationFailed, OptionError, FlagError)
x = Symbol('x')
# TODO: TypeError: __init__() takes at least 3 arguments (1 given)
# for c in (ExactQuotientFailed, ExactQuotientFailed(x, 3*x, ZZ)):
# check(c)
# TODO: TypeError: can't pickle instancemethod objects
# for c in (OperationNotSupported, OperationNotSupported(Poly(x), Poly.gcd)):
# check(c)
for c in (HeuristicGCDFailed, HeuristicGCDFailed()):
check(c)
for c in (HomomorphismFailed, HomomorphismFailed()):
check(c)
for c in (IsomorphismFailed, IsomorphismFailed()):
check(c)
for c in (ExtraneousFactors, ExtraneousFactors()):
check(c)
for c in (EvaluationFailed, EvaluationFailed()):
check(c)
for c in (RefinementFailed, RefinementFailed()):
check(c)
for c in (CoercionFailed, CoercionFailed()):
check(c)
for c in (NotInvertible, NotInvertible()):
check(c)
for c in (NotReversible, NotReversible()):
check(c)
for c in (NotAlgebraic, NotAlgebraic()):
check(c)
for c in (DomainError, DomainError()):
check(c)
for c in (PolynomialError, PolynomialError()):
check(c)
for c in (UnificationFailed, UnificationFailed()):
check(c)
for c in (GeneratorsError, GeneratorsError()):
check(c)
for c in (GeneratorsNeeded, GeneratorsNeeded()):
check(c)
# TODO: PicklingError: Can't pickle <function <lambda> at 0x38578c0>: it's not found as __main__.<lambda>
# for c in (ComputationFailed, ComputationFailed(lambda t: t, 3, None)):
# check(c)
for c in (UnivariatePolynomialError, UnivariatePolynomialError()):
check(c)
for c in (MultivariatePolynomialError, MultivariatePolynomialError()):
check(c)
# TODO: TypeError: __init__() takes at least 3 arguments (1 given)
# for c in (PolificationFailed, PolificationFailed({}, x, x, False)):
# check(c)
for c in (OptionError, OptionError()):
check(c)
for c in (FlagError, FlagError()):
check(c)
def test_pickling_polys_options():
from sympy.polys.polyoptions import Options
# TODO: fix pickling of `symbols' flag
# for c in (Options, Options((), dict(domain='ZZ', polys=False))):
# check(c)
# TODO: def test_pickling_polys_rootisolation():
# RealInterval
# ComplexInterval
def test_pickling_polys_rootoftools():
from sympy.polys.rootoftools import CRootOf, RootSum
x = Symbol('x')
f = x**3 + x + 3
for c in (CRootOf, CRootOf(f, 0)):
check(c)
for c in (RootSum, RootSum(f, exp)):
check(c)
#================== printing ====================
from sympy.printing.latex import LatexPrinter
from sympy.printing.mathml import MathMLContentPrinter, MathMLPresentationPrinter
from sympy.printing.pretty.pretty import PrettyPrinter
from sympy.printing.pretty.stringpict import prettyForm, stringPict
from sympy.printing.printer import Printer
from sympy.printing.python import PythonPrinter
def test_printing():
for c in (LatexPrinter, LatexPrinter(), MathMLContentPrinter,
MathMLPresentationPrinter, PrettyPrinter, prettyForm, stringPict,
stringPict("a"), Printer, Printer(), PythonPrinter,
PythonPrinter()):
check(c)
@XFAIL
def test_printing1():
check(MathMLContentPrinter())
@XFAIL
def test_printing2():
check(MathMLPresentationPrinter())
@XFAIL
def test_printing3():
check(PrettyPrinter())
#================== series ======================
from sympy.series.limits import Limit
from sympy.series.order import Order
def test_series():
e = Symbol("e")
x = Symbol("x")
for c in (Limit, Limit(e, x, 1), Order, Order(e)):
check(c)
#================== concrete ==================
from sympy.concrete.products import Product
from sympy.concrete.summations import Sum
def test_concrete():
x = Symbol("x")
for c in (Product, Product(x, (x, 2, 4)), Sum, Sum(x, (x, 2, 4))):
check(c)
def test_deprecation_warning():
w = SymPyDeprecationWarning('value', 'feature', issue=12345, deprecated_since_version='1.0')
check(w)
|
9673383dbebc5deda71aae711dc89d9d3d3c0d58ff0af925bc07e87695522340
|
from sympy.core import (S, symbols, Eq, pi, Catalan, EulerGamma, Lambda,
Dummy, Function)
from sympy.core.compatibility import StringIO
from sympy import erf, Integral, Piecewise
from sympy import Equality
from sympy.matrices import Matrix, MatrixSymbol
from sympy.printing.codeprinter import Assignment
from sympy.utilities.codegen import RustCodeGen, codegen, make_routine
from sympy.utilities.pytest import raises
from sympy.utilities.lambdify import implemented_function
from sympy.utilities.pytest import XFAIL
import sympy
x, y, z = symbols('x,y,z')
def test_empty_rust_code():
code_gen = RustCodeGen()
output = StringIO()
code_gen.dump_rs([], output, "file", header=False, empty=False)
source = output.getvalue()
assert source == ""
def test_simple_rust_code():
name_expr = ("test", (x + y)*z)
result, = codegen(name_expr, "Rust", header=False, empty=False)
assert result[0] == "test.rs"
source = result[1]
expected = (
"fn test(x: f64, y: f64, z: f64) -> f64 {\n"
" let out1 = z*(x + y);\n"
" out1\n"
"}\n"
)
assert source == expected
def test_simple_code_with_header():
name_expr = ("test", (x + y)*z)
result, = codegen(name_expr, "Rust", header=True, empty=False)
assert result[0] == "test.rs"
source = result[1]
version_str = "Code generated with sympy %s" % sympy.__version__
version_line = version_str.center(76).rstrip()
expected = (
"/*\n"
" *%(version_line)s\n"
" *\n"
" * See http://www.sympy.org/ for more information.\n"
" *\n"
" * This file is part of 'project'\n"
" */\n"
"fn test(x: f64, y: f64, z: f64) -> f64 {\n"
" let out1 = z*(x + y);\n"
" out1\n"
"}\n"
) % {'version_line': version_line}
assert source == expected
def test_simple_code_nameout():
expr = Equality(z, (x + y))
name_expr = ("test", expr)
result, = codegen(name_expr, "Rust", header=False, empty=False)
source = result[1]
expected = (
"fn test(x: f64, y: f64) -> f64 {\n"
" let z = x + y;\n"
" z\n"
"}\n"
)
assert source == expected
def test_numbersymbol():
name_expr = ("test", pi**Catalan)
result, = codegen(name_expr, "Rust", header=False, empty=False)
source = result[1]
expected = (
"fn test() -> f64 {\n"
" const Catalan: f64 = %s;\n"
" let out1 = PI.powf(Catalan);\n"
" out1\n"
"}\n"
) % Catalan.evalf(17)
assert source == expected
@XFAIL
def test_numbersymbol_inline():
# FIXME: how to pass inline to the RustCodePrinter?
name_expr = ("test", [pi**Catalan, EulerGamma])
result, = codegen(name_expr, "Rust", header=False,
empty=False, inline=True)
source = result[1]
expected = (
"fn test() -> (f64, f64) {\n"
" const Catalan: f64 = %s;\n"
" const EulerGamma: f64 = %s;\n"
" let out1 = PI.powf(Catalan);\n"
" let out2 = EulerGamma);\n"
" (out1, out2)\n"
"}\n"
) % (Catalan.evalf(17), EulerGamma.evalf(17))
assert source == expected
def test_argument_order():
expr = x + y
routine = make_routine("test", expr, argument_sequence=[z, x, y], language="rust")
code_gen = RustCodeGen()
output = StringIO()
code_gen.dump_rs([routine], output, "test", header=False, empty=False)
source = output.getvalue()
expected = (
"fn test(z: f64, x: f64, y: f64) -> f64 {\n"
" let out1 = x + y;\n"
" out1\n"
"}\n"
)
assert source == expected
def test_multiple_results_rust():
# Here the output order is the input order
expr1 = (x + y)*z
expr2 = (x - y)*z
name_expr = ("test", [expr1, expr2])
result, = codegen(name_expr, "Rust", header=False, empty=False)
source = result[1]
expected = (
"fn test(x: f64, y: f64, z: f64) -> (f64, f64) {\n"
" let out1 = z*(x + y);\n"
" let out2 = z*(x - y);\n"
" (out1, out2)\n"
"}\n"
)
assert source == expected
def test_results_named_unordered():
# Here output order is based on name_expr
A, B, C = symbols('A,B,C')
expr1 = Equality(C, (x + y)*z)
expr2 = Equality(A, (x - y)*z)
expr3 = Equality(B, 2*x)
name_expr = ("test", [expr1, expr2, expr3])
result, = codegen(name_expr, "Rust", header=False, empty=False)
source = result[1]
expected = (
"fn test(x: f64, y: f64, z: f64) -> (f64, f64, f64) {\n"
" let C = z*(x + y);\n"
" let A = z*(x - y);\n"
" let B = 2*x;\n"
" (C, A, B)\n"
"}\n"
)
assert source == expected
def test_results_named_ordered():
A, B, C = symbols('A,B,C')
expr1 = Equality(C, (x + y)*z)
expr2 = Equality(A, (x - y)*z)
expr3 = Equality(B, 2*x)
name_expr = ("test", [expr1, expr2, expr3])
result = codegen(name_expr, "Rust", header=False, empty=False,
argument_sequence=(x, z, y))
assert result[0][0] == "test.rs"
source = result[0][1]
expected = (
"fn test(x: f64, z: f64, y: f64) -> (f64, f64, f64) {\n"
" let C = z*(x + y);\n"
" let A = z*(x - y);\n"
" let B = 2*x;\n"
" (C, A, B)\n"
"}\n"
)
assert source == expected
def test_complicated_rs_codegen():
from sympy import sin, cos, tan
name_expr = ("testlong",
[ ((sin(x) + cos(y) + tan(z))**3).expand(),
cos(cos(cos(cos(cos(cos(cos(cos(x + y + z))))))))
])
result = codegen(name_expr, "Rust", header=False, empty=False)
assert result[0][0] == "testlong.rs"
source = result[0][1]
expected = (
"fn testlong(x: f64, y: f64, z: f64) -> (f64, f64) {\n"
" let out1 = x.sin().powi(3) + 3*x.sin().powi(2)*y.cos()"
" + 3*x.sin().powi(2)*z.tan() + 3*x.sin()*y.cos().powi(2)"
" + 6*x.sin()*y.cos()*z.tan() + 3*x.sin()*z.tan().powi(2)"
" + y.cos().powi(3) + 3*y.cos().powi(2)*z.tan()"
" + 3*y.cos()*z.tan().powi(2) + z.tan().powi(3);\n"
" let out2 = (x + y + z).cos().cos().cos().cos()"
".cos().cos().cos().cos();\n"
" (out1, out2)\n"
"}\n"
)
assert source == expected
def test_output_arg_mixed_unordered():
# named outputs are alphabetical, unnamed output appear in the given order
from sympy import sin, cos, tan
a = symbols("a")
name_expr = ("foo", [cos(2*x), Equality(y, sin(x)), cos(x), Equality(a, sin(2*x))])
result, = codegen(name_expr, "Rust", header=False, empty=False)
assert result[0] == "foo.rs"
source = result[1];
expected = (
"fn foo(x: f64) -> (f64, f64, f64, f64) {\n"
" let out1 = (2*x).cos();\n"
" let y = x.sin();\n"
" let out3 = x.cos();\n"
" let a = (2*x).sin();\n"
" (out1, y, out3, a)\n"
"}\n"
)
assert source == expected
def test_piecewise_():
pw = Piecewise((0, x < -1), (x**2, x <= 1), (-x+2, x > 1), (1, True), evaluate=False)
name_expr = ("pwtest", pw)
result, = codegen(name_expr, "Rust", header=False, empty=False)
source = result[1]
expected = (
"fn pwtest(x: f64) -> f64 {\n"
" let out1 = if (x < -1) {\n"
" 0\n"
" } else if (x <= 1) {\n"
" x.powi(2)\n"
" } else if (x > 1) {\n"
" 2 - x\n"
" } else {\n"
" 1\n"
" };\n"
" out1\n"
"}\n"
)
assert source == expected
@XFAIL
def test_piecewise_inline():
# FIXME: how to pass inline to the RustCodePrinter?
pw = Piecewise((0, x < -1), (x**2, x <= 1), (-x+2, x > 1), (1, True))
name_expr = ("pwtest", pw)
result, = codegen(name_expr, "Rust", header=False, empty=False,
inline=True)
source = result[1]
expected = (
"fn pwtest(x: f64) -> f64 {\n"
" let out1 = if (x < -1) { 0 } else if (x <= 1) { x.powi(2) }"
" else if (x > 1) { -x + 2 } else { 1 };\n"
" out1\n"
"}\n"
)
assert source == expected
def test_multifcns_per_file():
name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ]
result = codegen(name_expr, "Rust", header=False, empty=False)
assert result[0][0] == "foo.rs"
source = result[0][1];
expected = (
"fn foo(x: f64, y: f64) -> (f64, f64) {\n"
" let out1 = 2*x;\n"
" let out2 = 3*y;\n"
" (out1, out2)\n"
"}\n"
"fn bar(y: f64) -> (f64, f64) {\n"
" let out1 = y.powi(2);\n"
" let out2 = 4*y;\n"
" (out1, out2)\n"
"}\n"
)
assert source == expected
def test_multifcns_per_file_w_header():
name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ]
result = codegen(name_expr, "Rust", header=True, empty=False)
assert result[0][0] == "foo.rs"
source = result[0][1];
version_str = "Code generated with sympy %s" % sympy.__version__
version_line = version_str.center(76).rstrip()
expected = (
"/*\n"
" *%(version_line)s\n"
" *\n"
" * See http://www.sympy.org/ for more information.\n"
" *\n"
" * This file is part of 'project'\n"
" */\n"
"fn foo(x: f64, y: f64) -> (f64, f64) {\n"
" let out1 = 2*x;\n"
" let out2 = 3*y;\n"
" (out1, out2)\n"
"}\n"
"fn bar(y: f64) -> (f64, f64) {\n"
" let out1 = y.powi(2);\n"
" let out2 = 4*y;\n"
" (out1, out2)\n"
"}\n"
) % {'version_line': version_line}
assert source == expected
def test_filename_match_prefix():
name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ]
result, = codegen(name_expr, "Rust", prefix="baz", header=False,
empty=False)
assert result[0] == "baz.rs"
def test_InOutArgument():
expr = Equality(x, x**2)
name_expr = ("mysqr", expr)
result, = codegen(name_expr, "Rust", header=False, empty=False)
source = result[1]
expected = (
"fn mysqr(x: f64) -> f64 {\n"
" let x = x.powi(2);\n"
" x\n"
"}\n"
)
assert source == expected
def test_InOutArgument_order():
# can specify the order as (x, y)
expr = Equality(x, x**2 + y)
name_expr = ("test", expr)
result, = codegen(name_expr, "Rust", header=False,
empty=False, argument_sequence=(x,y))
source = result[1]
expected = (
"fn test(x: f64, y: f64) -> f64 {\n"
" let x = x.powi(2) + y;\n"
" x\n"
"}\n"
)
assert source == expected
# make sure it gives (x, y) not (y, x)
expr = Equality(x, x**2 + y)
name_expr = ("test", expr)
result, = codegen(name_expr, "Rust", header=False, empty=False)
source = result[1]
expected = (
"fn test(x: f64, y: f64) -> f64 {\n"
" let x = x.powi(2) + y;\n"
" x\n"
"}\n"
)
assert source == expected
def test_not_supported():
f = Function('f')
name_expr = ("test", [f(x).diff(x), S.ComplexInfinity])
result, = codegen(name_expr, "Rust", header=False, empty=False)
source = result[1]
expected = (
"fn test(x: f64) -> (f64, f64) {\n"
" // unsupported: Derivative(f(x), x)\n"
" // unsupported: zoo\n"
" let out1 = Derivative(f(x), x);\n"
" let out2 = zoo;\n"
" (out1, out2)\n"
"}\n"
)
assert source == expected
def test_global_vars_rust():
x, y, z, t = symbols("x y z t")
result = codegen(('f', x*y), "Rust", header=False, empty=False,
global_vars=(y,))
source = result[0][1]
expected = (
"fn f(x: f64) -> f64 {\n"
" let out1 = x*y;\n"
" out1\n"
"}\n"
)
assert source == expected
result = codegen(('f', x*y+z), "Rust", header=False, empty=False,
argument_sequence=(x, y), global_vars=(z, t))
source = result[0][1]
expected = (
"fn f(x: f64, y: f64) -> f64 {\n"
" let out1 = x*y + z;\n"
" out1\n"
"}\n"
)
assert source == expected
|
a2ecb4eeb6d5de01213a616b680226910536444c4f57152ae8647b9332a57d5c
|
from sympy.core.compatibility import range, zip_longest
from sympy.utilities.enumerative import (
list_visitor,
MultisetPartitionTraverser,
multiset_partitions_taocp
)
from sympy.utilities.iterables import _set_partitions
from sympy.utilities.pytest import slow
# first some functions only useful as test scaffolding - these provide
# straightforward, but slow reference implementations against which to
# compare the real versions, and also a comparison to verify that
# different versions are giving identical results.
def part_range_filter(partition_iterator, lb, ub):
"""
Filters (on the number of parts) a multiset partition enumeration
Arguments
=========
lb, and ub are a range (in the python slice sense) on the lpart
variable returned from a multiset partition enumeration. Recall
that lpart is 0-based (it points to the topmost part on the part
stack), so if you want to return parts of sizes 2,3,4,5 you would
use lb=1 and ub=5.
"""
for state in partition_iterator:
f, lpart, pstack = state
if lpart >= lb and lpart < ub:
yield state
def multiset_partitions_baseline(multiplicities, components):
"""Enumerates partitions of a multiset
Parameters
==========
multiplicities
list of integer multiplicities of the components of the multiset.
components
the components (elements) themselves
Returns
=======
Set of partitions. Each partition is tuple of parts, and each
part is a tuple of components (with repeats to indicate
multiplicity)
Notes
=====
Multiset partitions can be created as equivalence classes of set
partitions, and this function does just that. This approach is
slow and memory intensive compared to the more advanced algorithms
available, but the code is simple and easy to understand. Hence
this routine is strictly for testing -- to provide a
straightforward baseline against which to regress the production
versions. (This code is a simplified version of an earlier
production implementation.)
"""
canon = [] # list of components with repeats
for ct, elem in zip(multiplicities, components):
canon.extend([elem]*ct)
# accumulate the multiset partitions in a set to eliminate dups
cache = set()
n = len(canon)
for nc, q in _set_partitions(n):
rv = [[] for i in range(nc)]
for i in range(n):
rv[q[i]].append(canon[i])
canonical = tuple(
sorted([tuple(p) for p in rv]))
cache.add(canonical)
return cache
def compare_multiset_w_baseline(multiplicities):
"""
Enumerates the partitions of multiset with AOCP algorithm and
baseline implementation, and compare the results.
"""
letters = "abcdefghijklmnopqrstuvwxyz"
bl_partitions = multiset_partitions_baseline(multiplicities, letters)
# The partitions returned by the different algorithms may have
# their parts in different orders. Also, they generate partitions
# in different orders. Hence the sorting, and set comparison.
aocp_partitions = set()
for state in multiset_partitions_taocp(multiplicities):
p1 = tuple(sorted(
[tuple(p) for p in list_visitor(state, letters)]))
aocp_partitions.add(p1)
assert bl_partitions == aocp_partitions
def compare_multiset_states(s1, s2):
"""compare for equality two instances of multiset partition states
This is useful for comparing different versions of the algorithm
to verify correctness."""
# Comparison is physical, the only use of semantics is to ignore
# trash off the top of the stack.
f1, lpart1, pstack1 = s1
f2, lpart2, pstack2 = s2
if (lpart1 == lpart2) and (f1[0:lpart1+1] == f2[0:lpart2+1]):
if pstack1[0:f1[lpart1+1]] == pstack2[0:f2[lpart2+1]]:
return True
return False
def test_multiset_partitions_taocp():
"""Compares the output of multiset_partitions_taocp with a baseline
(set partition based) implementation."""
# Test cases should not be too large, since the baseline
# implementation is fairly slow.
multiplicities = [2,2]
compare_multiset_w_baseline(multiplicities)
multiplicities = [4,3,1]
compare_multiset_w_baseline(multiplicities)
def test_multiset_partitions_versions():
"""Compares Knuth-based versions of multiset_partitions"""
multiplicities = [5,2,2,1]
m = MultisetPartitionTraverser()
for s1, s2 in zip_longest(m.enum_all(multiplicities),
multiset_partitions_taocp(multiplicities)):
assert compare_multiset_states(s1, s2)
def subrange_exercise(mult, lb, ub):
"""Compare filter-based and more optimized subrange implementations
Helper for tests, called with both small and larger multisets.
"""
m = MultisetPartitionTraverser()
assert m.count_partitions(mult) == \
m.count_partitions_slow(mult)
# Note - multiple traversals from the same
# MultisetPartitionTraverser object cannot execute at the same
# time, hence make several instances here.
ma = MultisetPartitionTraverser()
mc = MultisetPartitionTraverser()
md = MultisetPartitionTraverser()
# Several paths to compute just the size two partitions
a_it = ma.enum_range(mult, lb, ub)
b_it = part_range_filter(multiset_partitions_taocp(mult), lb, ub)
c_it = part_range_filter(mc.enum_small(mult, ub), lb, sum(mult))
d_it = part_range_filter(md.enum_large(mult, lb), 0, ub)
for sa, sb, sc, sd in zip_longest(a_it, b_it, c_it, d_it):
assert compare_multiset_states(sa, sb)
assert compare_multiset_states(sa, sc)
assert compare_multiset_states(sa, sd)
def test_subrange():
# Quick, but doesn't hit some of the corner cases
mult = [4,4,2,1] # mississippi
lb = 1
ub = 2
subrange_exercise(mult, lb, ub)
def test_subrange_large():
# takes a second or so, depending on cpu, Python version, etc.
mult = [6,3,2,1]
lb = 4
ub = 7
subrange_exercise(mult, lb, ub)
|
5965923d12c0cf42ee6c73a6ccbe9a5efcbd35e3c8c26d2ee1f03fda12786eec
|
from distutils.version import LooseVersion as V
from itertools import product
import math
import inspect
import mpmath
from sympy.utilities.pytest import XFAIL, raises
from sympy import (
symbols, lambdify, sqrt, sin, cos, tan, pi, acos, acosh, Rational,
Float, Matrix, Lambda, Piecewise, exp, Integral, oo, I, Abs, Function,
true, false, And, Or, Not, ITE, Min, Max, floor, diff, IndexedBase, Sum,
DotProduct, Eq, Dummy, sinc, erf, erfc, factorial, gamma, loggamma,
digamma, RisingFactorial, besselj, bessely, besseli, besselk, S,
MatrixSymbol, chebyshevt, chebyshevu, legendre, hermite, laguerre,
gegenbauer, assoc_legendre, assoc_laguerre, jacobi)
from sympy.printing.lambdarepr import LambdaPrinter
from sympy.printing.pycode import NumPyPrinter
from sympy.utilities.lambdify import implemented_function, lambdastr
from sympy.utilities.pytest import skip
from sympy.utilities.decorator import conserve_mpmath_dps
from sympy.external import import_module
from sympy.functions.special.gamma_functions import uppergamma,lowergamma
import sympy
MutableDenseMatrix = Matrix
numpy = import_module('numpy')
scipy = import_module('scipy')
scipy_special = import_module('scipy.special')
numexpr = import_module('numexpr')
tensorflow = import_module('tensorflow')
if tensorflow:
# Hide Tensorflow warnings
import os
os.environ['TF_CPP_MIN_LOG_LEVEL'] = '2'
w, x, y, z = symbols('w,x,y,z')
#================== Test different arguments =======================
def test_no_args():
f = lambdify([], 1)
raises(TypeError, lambda: f(-1))
assert f() == 1
def test_single_arg():
f = lambdify(x, 2*x)
assert f(1) == 2
def test_list_args():
f = lambdify([x, y], x + y)
assert f(1, 2) == 3
def test_nested_args():
f1 = lambdify([[w, x]], [w, x])
assert f1([91, 2]) == [91, 2]
raises(TypeError, lambda: f1(1, 2))
f2 = lambdify([(w, x), (y, z)], [w, x, y, z])
assert f2((18, 12), (73, 4)) == [18, 12, 73, 4]
raises(TypeError, lambda: f2(3, 4))
f3 = lambdify([w, [[[x]], y], z], [w, x, y, z])
assert f3(10, [[[52]], 31], 44) == [10, 52, 31, 44]
def test_str_args():
f = lambdify('x,y,z', 'z,y,x')
assert f(3, 2, 1) == (1, 2, 3)
assert f(1.0, 2.0, 3.0) == (3.0, 2.0, 1.0)
# make sure correct number of args required
raises(TypeError, lambda: f(0))
def test_own_namespace_1():
myfunc = lambda x: 1
f = lambdify(x, sin(x), {"sin": myfunc})
assert f(0.1) == 1
assert f(100) == 1
def test_own_namespace_2():
def myfunc(x):
return 1
f = lambdify(x, sin(x), {'sin': myfunc})
assert f(0.1) == 1
assert f(100) == 1
def test_own_module():
f = lambdify(x, sin(x), math)
assert f(0) == 0.0
def test_bad_args():
# no vargs given
raises(TypeError, lambda: lambdify(1))
# same with vector exprs
raises(TypeError, lambda: lambdify([1, 2]))
def test_atoms():
# Non-Symbol atoms should not be pulled out from the expression namespace
f = lambdify(x, pi + x, {"pi": 3.14})
assert f(0) == 3.14
f = lambdify(x, I + x, {"I": 1j})
assert f(1) == 1 + 1j
#================== Test different modules =========================
# high precision output of sin(0.2*pi) is used to detect if precision is lost unwanted
@conserve_mpmath_dps
def test_sympy_lambda():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "sympy")
assert f(x) == sin(x)
prec = 1e-15
assert -prec < f(Rational(1, 5)).evalf() - Float(str(sin02)) < prec
# arctan is in numpy module and should not be available
raises(NameError, lambda: lambdify(x, arctan(x), "sympy"))
@conserve_mpmath_dps
def test_math_lambda():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "math")
prec = 1e-15
assert -prec < f(0.2) - sin02 < prec
raises(TypeError, lambda: f(x))
# if this succeeds, it can't be a python math function
@conserve_mpmath_dps
def test_mpmath_lambda():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "mpmath")
prec = 1e-49 # mpmath precision is around 50 decimal places
assert -prec < f(mpmath.mpf("0.2")) - sin02 < prec
raises(TypeError, lambda: f(x))
# if this succeeds, it can't be a mpmath function
@conserve_mpmath_dps
def test_number_precision():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin02, "mpmath")
prec = 1e-49 # mpmath precision is around 50 decimal places
assert -prec < f(0) - sin02 < prec
@conserve_mpmath_dps
def test_mpmath_precision():
mpmath.mp.dps = 100
assert str(lambdify((), pi.evalf(100), 'mpmath')()) == str(pi.evalf(100))
#================== Test Translations ==============================
# We can only check if all translated functions are valid. It has to be checked
# by hand if they are complete.
def test_math_transl():
from sympy.utilities.lambdify import MATH_TRANSLATIONS
for sym, mat in MATH_TRANSLATIONS.items():
assert sym in sympy.__dict__
assert mat in math.__dict__
def test_mpmath_transl():
from sympy.utilities.lambdify import MPMATH_TRANSLATIONS
for sym, mat in MPMATH_TRANSLATIONS.items():
assert sym in sympy.__dict__ or sym == 'Matrix'
assert mat in mpmath.__dict__
def test_numpy_transl():
if not numpy:
skip("numpy not installed.")
from sympy.utilities.lambdify import NUMPY_TRANSLATIONS
for sym, nump in NUMPY_TRANSLATIONS.items():
assert sym in sympy.__dict__
assert nump in numpy.__dict__
def test_scipy_transl():
if not scipy:
skip("scipy not installed.")
from sympy.utilities.lambdify import SCIPY_TRANSLATIONS
for sym, scip in SCIPY_TRANSLATIONS.items():
assert sym in sympy.__dict__
assert scip in scipy.__dict__ or scip in scipy.special.__dict__
def test_tensorflow_transl():
if not tensorflow:
skip("tensorflow not installed")
from sympy.utilities.lambdify import TENSORFLOW_TRANSLATIONS
for sym, tens in TENSORFLOW_TRANSLATIONS.items():
assert sym in sympy.__dict__
assert tens in tensorflow.__dict__
def test_numpy_translation_abs():
if not numpy:
skip("numpy not installed.")
f = lambdify(x, Abs(x), "numpy")
assert f(-1) == 1
assert f(1) == 1
def test_numexpr_printer():
if not numexpr:
skip("numexpr not installed.")
# if translation/printing is done incorrectly then evaluating
# a lambdified numexpr expression will throw an exception
from sympy.printing.lambdarepr import NumExprPrinter
blacklist = ('where', 'complex', 'contains')
arg_tuple = (x, y, z) # some functions take more than one argument
for sym in NumExprPrinter._numexpr_functions.keys():
if sym in blacklist:
continue
ssym = S(sym)
if hasattr(ssym, '_nargs'):
nargs = ssym._nargs[0]
else:
nargs = 1
args = arg_tuple[:nargs]
f = lambdify(args, ssym(*args), modules='numexpr')
assert f(*(1, )*nargs) is not None
def test_issue_9334():
if not numexpr:
skip("numexpr not installed.")
if not numpy:
skip("numpy not installed.")
expr = S('b*a - sqrt(a**2)')
a, b = sorted(expr.free_symbols, key=lambda s: s.name)
func_numexpr = lambdify((a,b), expr, modules=[numexpr], dummify=False)
foo, bar = numpy.random.random((2, 4))
func_numexpr(foo, bar)
#================== Test some functions ============================
def test_exponentiation():
f = lambdify(x, x**2)
assert f(-1) == 1
assert f(0) == 0
assert f(1) == 1
assert f(-2) == 4
assert f(2) == 4
assert f(2.5) == 6.25
def test_sqrt():
f = lambdify(x, sqrt(x))
assert f(0) == 0.0
assert f(1) == 1.0
assert f(4) == 2.0
assert abs(f(2) - 1.414) < 0.001
assert f(6.25) == 2.5
def test_trig():
f = lambdify([x], [cos(x), sin(x)], 'math')
d = f(pi)
prec = 1e-11
assert -prec < d[0] + 1 < prec
assert -prec < d[1] < prec
d = f(3.14159)
prec = 1e-5
assert -prec < d[0] + 1 < prec
assert -prec < d[1] < prec
#================== Test vectors ===================================
def test_vector_simple():
f = lambdify((x, y, z), (z, y, x))
assert f(3, 2, 1) == (1, 2, 3)
assert f(1.0, 2.0, 3.0) == (3.0, 2.0, 1.0)
# make sure correct number of args required
raises(TypeError, lambda: f(0))
def test_vector_discontinuous():
f = lambdify(x, (-1/x, 1/x))
raises(ZeroDivisionError, lambda: f(0))
assert f(1) == (-1.0, 1.0)
assert f(2) == (-0.5, 0.5)
assert f(-2) == (0.5, -0.5)
def test_trig_symbolic():
f = lambdify([x], [cos(x), sin(x)], 'math')
d = f(pi)
assert abs(d[0] + 1) < 0.0001
assert abs(d[1] - 0) < 0.0001
def test_trig_float():
f = lambdify([x], [cos(x), sin(x)])
d = f(3.14159)
assert abs(d[0] + 1) < 0.0001
assert abs(d[1] - 0) < 0.0001
def test_docs():
f = lambdify(x, x**2)
assert f(2) == 4
f = lambdify([x, y, z], [z, y, x])
assert f(1, 2, 3) == [3, 2, 1]
f = lambdify(x, sqrt(x))
assert f(4) == 2.0
f = lambdify((x, y), sin(x*y)**2)
assert f(0, 5) == 0
def test_math():
f = lambdify((x, y), sin(x), modules="math")
assert f(0, 5) == 0
def test_sin():
f = lambdify(x, sin(x)**2)
assert isinstance(f(2), float)
f = lambdify(x, sin(x)**2, modules="math")
assert isinstance(f(2), float)
def test_matrix():
A = Matrix([[x, x*y], [sin(z) + 4, x**z]])
sol = Matrix([[1, 2], [sin(3) + 4, 1]])
f = lambdify((x, y, z), A, modules="sympy")
assert f(1, 2, 3) == sol
f = lambdify((x, y, z), (A, [A]), modules="sympy")
assert f(1, 2, 3) == (sol, [sol])
J = Matrix((x, x + y)).jacobian((x, y))
v = Matrix((x, y))
sol = Matrix([[1, 0], [1, 1]])
assert lambdify(v, J, modules='sympy')(1, 2) == sol
assert lambdify(v.T, J, modules='sympy')(1, 2) == sol
def test_numpy_matrix():
if not numpy:
skip("numpy not installed.")
A = Matrix([[x, x*y], [sin(z) + 4, x**z]])
sol_arr = numpy.array([[1, 2], [numpy.sin(3) + 4, 1]])
#Lambdify array first, to ensure return to array as default
f = lambdify((x, y, z), A, ['numpy'])
numpy.testing.assert_allclose(f(1, 2, 3), sol_arr)
#Check that the types are arrays and matrices
assert isinstance(f(1, 2, 3), numpy.ndarray)
# gh-15071
class dot(Function):
pass
x_dot_mtx = dot(x, Matrix([[2], [1], [0]]))
f_dot1 = lambdify(x, x_dot_mtx)
inp = numpy.zeros((17, 3))
assert numpy.all(f_dot1(inp) == 0)
strict_kw = dict(allow_unknown_functions=False, inline=True, fully_qualified_modules=False)
p2 = NumPyPrinter(dict(user_functions={'dot': 'dot'}, **strict_kw))
f_dot2 = lambdify(x, x_dot_mtx, printer=p2)
assert numpy.all(f_dot2(inp) == 0)
p3 = NumPyPrinter(strict_kw)
# The line below should probably fail upon construction (before calling with "(inp)"):
raises(Exception, lambda: lambdify(x, x_dot_mtx, printer=p3)(inp))
def test_numpy_transpose():
if not numpy:
skip("numpy not installed.")
A = Matrix([[1, x], [0, 1]])
f = lambdify((x), A.T, modules="numpy")
numpy.testing.assert_array_equal(f(2), numpy.array([[1, 0], [2, 1]]))
def test_numpy_dotproduct():
if not numpy:
skip("numpy not installed")
A = Matrix([x, y, z])
f1 = lambdify([x, y, z], DotProduct(A, A), modules='numpy')
f2 = lambdify([x, y, z], DotProduct(A, A.T), modules='numpy')
f3 = lambdify([x, y, z], DotProduct(A.T, A), modules='numpy')
f4 = lambdify([x, y, z], DotProduct(A, A.T), modules='numpy')
assert f1(1, 2, 3) == \
f2(1, 2, 3) == \
f3(1, 2, 3) == \
f4(1, 2, 3) == \
numpy.array([14])
def test_numpy_inverse():
if not numpy:
skip("numpy not installed.")
A = Matrix([[1, x], [0, 1]])
f = lambdify((x), A**-1, modules="numpy")
numpy.testing.assert_array_equal(f(2), numpy.array([[1, -2], [0, 1]]))
def test_numpy_old_matrix():
if not numpy:
skip("numpy not installed.")
A = Matrix([[x, x*y], [sin(z) + 4, x**z]])
sol_arr = numpy.array([[1, 2], [numpy.sin(3) + 4, 1]])
f = lambdify((x, y, z), A, [{'ImmutableDenseMatrix': numpy.matrix}, 'numpy'])
numpy.testing.assert_allclose(f(1, 2, 3), sol_arr)
assert isinstance(f(1, 2, 3), numpy.matrix)
def test_python_div_zero_issue_11306():
if not numpy:
skip("numpy not installed.")
p = Piecewise((1 / x, y < -1), (x, y < 1), (1 / x, True))
f = lambdify([x, y], p, modules='numpy')
numpy.seterr(divide='ignore')
assert float(f(numpy.array([0]),numpy.array([0.5]))) == 0
assert str(float(f(numpy.array([0]),numpy.array([1])))) == 'inf'
numpy.seterr(divide='warn')
def test_issue9474():
mods = [None, 'math']
if numpy:
mods.append('numpy')
if mpmath:
mods.append('mpmath')
for mod in mods:
f = lambdify(x, S(1)/x, modules=mod)
assert f(2) == 0.5
f = lambdify(x, floor(S(1)/x), modules=mod)
assert f(2) == 0
for absfunc, modules in product([Abs, abs], mods):
f = lambdify(x, absfunc(x), modules=modules)
assert f(-1) == 1
assert f(1) == 1
assert f(3+4j) == 5
def test_issue_9871():
if not numexpr:
skip("numexpr not installed.")
if not numpy:
skip("numpy not installed.")
r = sqrt(x**2 + y**2)
expr = diff(1/r, x)
xn = yn = numpy.linspace(1, 10, 16)
# expr(xn, xn) = -xn/(sqrt(2)*xn)^3
fv_exact = -numpy.sqrt(2.)**-3 * xn**-2
fv_numpy = lambdify((x, y), expr, modules='numpy')(xn, yn)
fv_numexpr = lambdify((x, y), expr, modules='numexpr')(xn, yn)
numpy.testing.assert_allclose(fv_numpy, fv_exact, rtol=1e-10)
numpy.testing.assert_allclose(fv_numexpr, fv_exact, rtol=1e-10)
def test_numpy_piecewise():
if not numpy:
skip("numpy not installed.")
pieces = Piecewise((x, x < 3), (x**2, x > 5), (0, True))
f = lambdify(x, pieces, modules="numpy")
numpy.testing.assert_array_equal(f(numpy.arange(10)),
numpy.array([0, 1, 2, 0, 0, 0, 36, 49, 64, 81]))
# If we evaluate somewhere all conditions are False, we should get back NaN
nodef_func = lambdify(x, Piecewise((x, x > 0), (-x, x < 0)))
numpy.testing.assert_array_equal(nodef_func(numpy.array([-1, 0, 1])),
numpy.array([1, numpy.nan, 1]))
def test_numpy_logical_ops():
if not numpy:
skip("numpy not installed.")
and_func = lambdify((x, y), And(x, y), modules="numpy")
and_func_3 = lambdify((x, y, z), And(x, y, z), modules="numpy")
or_func = lambdify((x, y), Or(x, y), modules="numpy")
or_func_3 = lambdify((x, y, z), Or(x, y, z), modules="numpy")
not_func = lambdify((x), Not(x), modules="numpy")
arr1 = numpy.array([True, True])
arr2 = numpy.array([False, True])
arr3 = numpy.array([True, False])
numpy.testing.assert_array_equal(and_func(arr1, arr2), numpy.array([False, True]))
numpy.testing.assert_array_equal(and_func_3(arr1, arr2, arr3), numpy.array([False, False]))
numpy.testing.assert_array_equal(or_func(arr1, arr2), numpy.array([True, True]))
numpy.testing.assert_array_equal(or_func_3(arr1, arr2, arr3), numpy.array([True, True]))
numpy.testing.assert_array_equal(not_func(arr2), numpy.array([True, False]))
def test_numpy_matmul():
if not numpy:
skip("numpy not installed.")
xmat = Matrix([[x, y], [z, 1+z]])
ymat = Matrix([[x**2], [Abs(x)]])
mat_func = lambdify((x, y, z), xmat*ymat, modules="numpy")
numpy.testing.assert_array_equal(mat_func(0.5, 3, 4), numpy.array([[1.625], [3.5]]))
numpy.testing.assert_array_equal(mat_func(-0.5, 3, 4), numpy.array([[1.375], [3.5]]))
# Multiple matrices chained together in multiplication
f = lambdify((x, y, z), xmat*xmat*xmat, modules="numpy")
numpy.testing.assert_array_equal(f(0.5, 3, 4), numpy.array([[72.125, 119.25],
[159, 251]]))
def test_numpy_numexpr():
if not numpy:
skip("numpy not installed.")
if not numexpr:
skip("numexpr not installed.")
a, b, c = numpy.random.randn(3, 128, 128)
# ensure that numpy and numexpr return same value for complicated expression
expr = sin(x) + cos(y) + tan(z)**2 + Abs(z-y)*acos(sin(y*z)) + \
Abs(y-z)*acosh(2+exp(y-x))- sqrt(x**2+I*y**2)
npfunc = lambdify((x, y, z), expr, modules='numpy')
nefunc = lambdify((x, y, z), expr, modules='numexpr')
assert numpy.allclose(npfunc(a, b, c), nefunc(a, b, c))
def test_numexpr_userfunctions():
if not numpy:
skip("numpy not installed.")
if not numexpr:
skip("numexpr not installed.")
a, b = numpy.random.randn(2, 10)
uf = type('uf', (Function, ),
{'eval' : classmethod(lambda x, y : y**2+1)})
func = lambdify(x, 1-uf(x), modules='numexpr')
assert numpy.allclose(func(a), -(a**2))
uf = implemented_function(Function('uf'), lambda x, y : 2*x*y+1)
func = lambdify((x, y), uf(x, y), modules='numexpr')
assert numpy.allclose(func(a, b), 2*a*b+1)
def test_tensorflow_basic_math():
if not tensorflow:
skip("tensorflow not installed.")
expr = Max(sin(x), Abs(1/(x+2)))
func = lambdify(x, expr, modules="tensorflow")
a = tensorflow.constant(0, dtype=tensorflow.float32)
s = tensorflow.Session()
assert func(a).eval(session=s) == 0.5
def test_tensorflow_placeholders():
if not tensorflow:
skip("tensorflow not installed.")
expr = Max(sin(x), Abs(1/(x+2)))
func = lambdify(x, expr, modules="tensorflow")
a = tensorflow.placeholder(dtype=tensorflow.float32)
s = tensorflow.Session()
assert func(a).eval(session=s, feed_dict={a: 0}) == 0.5
def test_tensorflow_variables():
if not tensorflow:
skip("tensorflow not installed.")
expr = Max(sin(x), Abs(1/(x+2)))
func = lambdify(x, expr, modules="tensorflow")
a = tensorflow.Variable(0, dtype=tensorflow.float32)
s = tensorflow.Session()
if V(tensorflow.__version__) < '1.0':
s.run(tensorflow.initialize_all_variables())
else:
s.run(tensorflow.global_variables_initializer())
assert func(a).eval(session=s) == 0.5
def test_tensorflow_logical_operations():
if not tensorflow:
skip("tensorflow not installed.")
expr = Not(And(Or(x, y), y))
func = lambdify([x, y], expr, modules="tensorflow")
a = tensorflow.constant(False)
b = tensorflow.constant(True)
s = tensorflow.Session()
assert func(a, b).eval(session=s) == 0
def test_tensorflow_piecewise():
if not tensorflow:
skip("tensorflow not installed.")
expr = Piecewise((0, Eq(x,0)), (-1, x < 0), (1, x > 0))
func = lambdify(x, expr, modules="tensorflow")
a = tensorflow.placeholder(dtype=tensorflow.float32)
s = tensorflow.Session()
assert func(a).eval(session=s, feed_dict={a: -1}) == -1
assert func(a).eval(session=s, feed_dict={a: 0}) == 0
assert func(a).eval(session=s, feed_dict={a: 1}) == 1
def test_tensorflow_multi_max():
if not tensorflow:
skip("tensorflow not installed.")
expr = Max(x, -x, x**2)
func = lambdify(x, expr, modules="tensorflow")
a = tensorflow.placeholder(dtype=tensorflow.float32)
s = tensorflow.Session()
assert func(a).eval(session=s, feed_dict={a: -2}) == 4
def test_tensorflow_multi_min():
if not tensorflow:
skip("tensorflow not installed.")
expr = Min(x, -x, x**2)
func = lambdify(x, expr, modules="tensorflow")
a = tensorflow.placeholder(dtype=tensorflow.float32)
s = tensorflow.Session()
assert func(a).eval(session=s, feed_dict={a: -2}) == -2
def test_tensorflow_relational():
if not tensorflow:
skip("tensorflow not installed.")
expr = x >= 0
func = lambdify(x, expr, modules="tensorflow")
a = tensorflow.placeholder(dtype=tensorflow.float32)
s = tensorflow.Session()
assert func(a).eval(session=s, feed_dict={a: 1})
def test_integral():
f = Lambda(x, exp(-x**2))
l = lambdify(x, Integral(f(x), (x, -oo, oo)), modules="sympy")
assert l(x) == Integral(exp(-x**2), (x, -oo, oo))
#================== Test symbolic ==================================
def test_sym_single_arg():
f = lambdify(x, x * y)
assert f(z) == z * y
def test_sym_list_args():
f = lambdify([x, y], x + y + z)
assert f(1, 2) == 3 + z
def test_sym_integral():
f = Lambda(x, exp(-x**2))
l = lambdify(x, Integral(f(x), (x, -oo, oo)), modules="sympy")
assert l(y).doit() == sqrt(pi)
def test_namespace_order():
# lambdify had a bug, such that module dictionaries or cached module
# dictionaries would pull earlier namespaces into themselves.
# Because the module dictionaries form the namespace of the
# generated lambda, this meant that the behavior of a previously
# generated lambda function could change as a result of later calls
# to lambdify.
n1 = {'f': lambda x: 'first f'}
n2 = {'f': lambda x: 'second f',
'g': lambda x: 'function g'}
f = sympy.Function('f')
g = sympy.Function('g')
if1 = lambdify(x, f(x), modules=(n1, "sympy"))
assert if1(1) == 'first f'
if2 = lambdify(x, g(x), modules=(n2, "sympy"))
# previously gave 'second f'
assert if1(1) == 'first f'
def test_namespace_type():
# lambdify had a bug where it would reject modules of type unicode
# on Python 2.
x = sympy.Symbol('x')
lambdify(x, x, modules=u'math')
def test_imps():
# Here we check if the default returned functions are anonymous - in
# the sense that we can have more than one function with the same name
f = implemented_function('f', lambda x: 2*x)
g = implemented_function('f', lambda x: math.sqrt(x))
l1 = lambdify(x, f(x))
l2 = lambdify(x, g(x))
assert str(f(x)) == str(g(x))
assert l1(3) == 6
assert l2(3) == math.sqrt(3)
# check that we can pass in a Function as input
func = sympy.Function('myfunc')
assert not hasattr(func, '_imp_')
my_f = implemented_function(func, lambda x: 2*x)
assert hasattr(my_f, '_imp_')
# Error for functions with same name and different implementation
f2 = implemented_function("f", lambda x: x + 101)
raises(ValueError, lambda: lambdify(x, f(f2(x))))
def test_imps_errors():
# Test errors that implemented functions can return, and still be able to
# form expressions.
# See: https://github.com/sympy/sympy/issues/10810
#
# XXX: Removed AttributeError here. This test was added due to issue 10810
# but that issue was about ValueError. It doesn't seem reasonable to
# "support" catching AttributeError in the same context...
for val, error_class in product((0, 0., 2, 2.0), (TypeError, ValueError)):
def myfunc(a):
if a == 0:
raise error_class
return 1
f = implemented_function('f', myfunc)
expr = f(val)
assert expr == f(val)
def test_imps_wrong_args():
raises(ValueError, lambda: implemented_function(sin, lambda x: x))
def test_lambdify_imps():
# Test lambdify with implemented functions
# first test basic (sympy) lambdify
f = sympy.cos
assert lambdify(x, f(x))(0) == 1
assert lambdify(x, 1 + f(x))(0) == 2
assert lambdify((x, y), y + f(x))(0, 1) == 2
# make an implemented function and test
f = implemented_function("f", lambda x: x + 100)
assert lambdify(x, f(x))(0) == 100
assert lambdify(x, 1 + f(x))(0) == 101
assert lambdify((x, y), y + f(x))(0, 1) == 101
# Can also handle tuples, lists, dicts as expressions
lam = lambdify(x, (f(x), x))
assert lam(3) == (103, 3)
lam = lambdify(x, [f(x), x])
assert lam(3) == [103, 3]
lam = lambdify(x, [f(x), (f(x), x)])
assert lam(3) == [103, (103, 3)]
lam = lambdify(x, {f(x): x})
assert lam(3) == {103: 3}
lam = lambdify(x, {f(x): x})
assert lam(3) == {103: 3}
lam = lambdify(x, {x: f(x)})
assert lam(3) == {3: 103}
# Check that imp preferred to other namespaces by default
d = {'f': lambda x: x + 99}
lam = lambdify(x, f(x), d)
assert lam(3) == 103
# Unless flag passed
lam = lambdify(x, f(x), d, use_imps=False)
assert lam(3) == 102
def test_dummification():
t = symbols('t')
F = Function('F')
G = Function('G')
#"\alpha" is not a valid python variable name
#lambdify should sub in a dummy for it, and return
#without a syntax error
alpha = symbols(r'\alpha')
some_expr = 2 * F(t)**2 / G(t)
lam = lambdify((F(t), G(t)), some_expr)
assert lam(3, 9) == 2
lam = lambdify(sin(t), 2 * sin(t)**2)
assert lam(F(t)) == 2 * F(t)**2
#Test that \alpha was properly dummified
lam = lambdify((alpha, t), 2*alpha + t)
assert lam(2, 1) == 5
raises(SyntaxError, lambda: lambdify(F(t) * G(t), F(t) * G(t) + 5))
raises(SyntaxError, lambda: lambdify(2 * F(t), 2 * F(t) + 5))
raises(SyntaxError, lambda: lambdify(2 * F(t), 4 * F(t) + 5))
def test_curly_matrix_symbol():
# Issue #15009
curlyv = sympy.MatrixSymbol("{v}", 2, 1)
lam = lambdify(curlyv, curlyv)
assert lam(1)==1
lam = lambdify(curlyv, curlyv, dummify=True)
assert lam(1)==1
def test_python_keywords():
# Test for issue 7452. The automatic dummification should ensure use of
# Python reserved keywords as symbol names will create valid lambda
# functions. This is an additional regression test.
python_if = symbols('if')
expr = python_if / 2
f = lambdify(python_if, expr)
assert f(4.0) == 2.0
def test_lambdify_docstring():
func = lambdify((w, x, y, z), w + x + y + z)
ref = (
"Created with lambdify. Signature:\n\n"
"func(w, x, y, z)\n\n"
"Expression:\n\n"
"w + x + y + z"
).splitlines()
assert func.__doc__.splitlines()[:len(ref)] == ref
syms = symbols('a1:26')
func = lambdify(syms, sum(syms))
ref = (
"Created with lambdify. Signature:\n\n"
"func(a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15,\n"
" a16, a17, a18, a19, a20, a21, a22, a23, a24, a25)\n\n"
"Expression:\n\n"
"a1 + a10 + a11 + a12 + a13 + a14 + a15 + a16 + a17 + a18 + a19 + a2 + a20 +..."
).splitlines()
assert func.__doc__.splitlines()[:len(ref)] == ref
#================== Test special printers ==========================
def test_special_printers():
class IntervalPrinter(LambdaPrinter):
"""Use ``lambda`` printer but print numbers as ``mpi`` intervals. """
def _print_Integer(self, expr):
return "mpi('%s')" % super(IntervalPrinter, self)._print_Integer(expr)
def _print_Rational(self, expr):
return "mpi('%s')" % super(IntervalPrinter, self)._print_Rational(expr)
def intervalrepr(expr):
return IntervalPrinter().doprint(expr)
expr = sqrt(sqrt(2) + sqrt(3)) + S(1)/2
func0 = lambdify((), expr, modules="mpmath", printer=intervalrepr)
func1 = lambdify((), expr, modules="mpmath", printer=IntervalPrinter)
func2 = lambdify((), expr, modules="mpmath", printer=IntervalPrinter())
mpi = type(mpmath.mpi(1, 2))
assert isinstance(func0(), mpi)
assert isinstance(func1(), mpi)
assert isinstance(func2(), mpi)
def test_true_false():
# We want exact is comparison here, not just ==
assert lambdify([], true)() is True
assert lambdify([], false)() is False
def test_issue_2790():
assert lambdify((x, (y, z)), x + y)(1, (2, 4)) == 3
assert lambdify((x, (y, (w, z))), w + x + y + z)(1, (2, (3, 4))) == 10
assert lambdify(x, x + 1, dummify=False)(1) == 2
def test_issue_12092():
f = implemented_function('f', lambda x: x**2)
assert f(f(2)).evalf() == Float(16)
def test_issue_14911():
class Variable(sympy.Symbol):
def _sympystr(self, printer):
return printer.doprint(self.name)
_lambdacode = _sympystr
_numpycode = _sympystr
x = Variable('x')
y = 2 * x
code = LambdaPrinter().doprint(y)
assert code.replace(' ', '') == '2*x'
def test_ITE():
assert lambdify((x, y, z), ITE(x, y, z))(True, 5, 3) == 5
assert lambdify((x, y, z), ITE(x, y, z))(False, 5, 3) == 3
def test_Min_Max():
# see gh-10375
assert lambdify((x, y, z), Min(x, y, z))(1, 2, 3) == 1
assert lambdify((x, y, z), Max(x, y, z))(1, 2, 3) == 3
def test_Indexed():
# Issue #10934
if not numpy:
skip("numpy not installed")
a = IndexedBase('a')
i, j = symbols('i j')
b = numpy.array([[1, 2], [3, 4]])
assert lambdify(a, Sum(a[x, y], (x, 0, 1), (y, 0, 1)))(b) == 10
def test_issue_12173():
#test for issue 12173
exp1 = lambdify((x, y), uppergamma(x, y),"mpmath")(1, 2)
exp2 = lambdify((x, y), lowergamma(x, y),"mpmath")(1, 2)
assert exp1 == uppergamma(1, 2).evalf()
assert exp2 == lowergamma(1, 2).evalf()
def test_issue_13642():
if not numpy:
skip("numpy not installed")
f = lambdify(x, sinc(x))
assert Abs(f(1) - sinc(1)).n() < 1e-15
def test_sinc_mpmath():
f = lambdify(x, sinc(x), "mpmath")
assert Abs(f(1) - sinc(1)).n() < 1e-15
def test_lambdify_dummy_arg():
d1 = Dummy()
f1 = lambdify(d1, d1 + 1, dummify=False)
assert f1(2) == 3
f1b = lambdify(d1, d1 + 1)
assert f1b(2) == 3
d2 = Dummy('x')
f2 = lambdify(d2, d2 + 1)
assert f2(2) == 3
f3 = lambdify([[d2]], d2 + 1)
assert f3([2]) == 3
def test_lambdify_mixed_symbol_dummy_args():
d = Dummy()
# Contrived example of name clash
dsym = symbols(str(d))
f = lambdify([d, dsym], d - dsym)
assert f(4, 1) == 3
def test_numpy_array_arg():
# Test for issue 14655 (numpy part)
if not numpy:
skip("numpy not installed")
f = lambdify([[x, y]], x*x + y, 'numpy')
assert f(numpy.array([2.0, 1.0])) == 5
def test_tensorflow_array_arg():
# Test for issue 14655 (tensorflow part)
if not tensorflow:
skip("tensorflow not installed.")
f = lambdify([[x, y]], x*x + y, 'tensorflow')
fcall = f(tensorflow.constant([2.0, 1.0]))
s = tensorflow.Session()
assert s.run(fcall) == 5
def test_scipy_fns():
if not scipy:
skip("scipy not installed")
single_arg_sympy_fns = [erf, erfc, factorial, gamma, loggamma, digamma]
single_arg_scipy_fns = [scipy.special.erf, scipy.special.erfc,
scipy.special.factorial, scipy.special.gamma, scipy.special.gammaln,
scipy.special.psi]
numpy.random.seed(0)
for (sympy_fn, scipy_fn) in zip(single_arg_sympy_fns, single_arg_scipy_fns):
f = lambdify(x, sympy_fn(x), modules="scipy")
for i in range(20):
tv = numpy.random.uniform(-10, 10) + 1j*numpy.random.uniform(-5, 5)
# SciPy thinks that factorial(z) is 0 when re(z) < 0.
# SymPy does not think so.
if sympy_fn == factorial and numpy.real(tv) < 0:
tv = tv + 2*numpy.abs(numpy.real(tv))
# SciPy supports gammaln for real arguments only,
# and there is also a branch cut along the negative real axis
if sympy_fn == loggamma:
tv = numpy.abs(tv)
# SymPy's digamma evaluates as polygamma(0, z)
# which SciPy supports for real arguments only
if sympy_fn == digamma:
tv = numpy.real(tv)
sympy_result = sympy_fn(tv).evalf()
assert abs(f(tv) - sympy_result) < 1e-13*(1 + abs(sympy_result))
assert abs(f(tv) - scipy_fn(tv)) < 1e-13*(1 + abs(sympy_result))
double_arg_sympy_fns = [RisingFactorial, besselj, bessely, besseli,
besselk]
double_arg_scipy_fns = [scipy.special.poch, scipy.special.jv,
scipy.special.yv, scipy.special.iv, scipy.special.kv]
for (sympy_fn, scipy_fn) in zip(double_arg_sympy_fns, double_arg_scipy_fns):
f = lambdify((x, y), sympy_fn(x, y), modules="scipy")
for i in range(20):
# SciPy supports only real orders of Bessel functions
tv1 = numpy.random.uniform(-10, 10)
tv2 = numpy.random.uniform(-10, 10) + 1j*numpy.random.uniform(-5, 5)
# SciPy supports poch for real arguments only
if sympy_fn == RisingFactorial:
tv2 = numpy.real(tv2)
sympy_result = sympy_fn(tv1, tv2).evalf()
assert abs(f(tv1, tv2) - sympy_result) < 1e-13*(1 + abs(sympy_result))
assert abs(f(tv1, tv2) - scipy_fn(tv1, tv2)) < 1e-13*(1 + abs(sympy_result))
def test_scipy_polys():
if not scipy:
skip("scipy not installed")
numpy.random.seed(0)
params = symbols('n k a b')
# list polynomials with the number of parameters
polys = [
(chebyshevt, 1),
(chebyshevu, 1),
(legendre, 1),
(hermite, 1),
(laguerre, 1),
(gegenbauer, 2),
(assoc_legendre, 2),
(assoc_laguerre, 2),
(jacobi, 3)
]
for sympy_fn, num_params in polys:
args = params[:num_params] + (x,)
f = lambdify(args, sympy_fn(*args))
for i in range(10):
tn = numpy.random.randint(3, 10)
tparams = tuple(numpy.random.uniform(0, 5, size=num_params-1))
tv = numpy.random.uniform(-10, 10) + 1j*numpy.random.uniform(-5, 5)
# SciPy supports hermite for real arguments only
if sympy_fn == hermite:
tv = numpy.real(tv)
# assoc_legendre needs x in (-1, 1) and integer param at most n
if sympy_fn == assoc_legendre:
tv = numpy.random.uniform(-1, 1)
tparams = tuple(numpy.random.randint(1, tn, size=1))
vals = (tn,) + tparams + (tv,)
sympy_result = sympy_fn(*vals).evalf()
assert abs(f(*vals) - sympy_result) < 1e-13*(1 + abs(sympy_result))
def test_lambdify_inspect():
f = lambdify(x, x**2)
# Test that inspect.getsource works but don't hard-code implementation
# details
assert 'x**2' in inspect.getsource(f)
def test_issue_14941():
x, y = Dummy(), Dummy()
# test dict
f1 = lambdify([x, y], {x: 3, y: 3}, 'sympy')
assert f1(2, 3) == {2: 3, 3: 3}
# test tuple
f2 = lambdify([x, y], (y, x), 'sympy')
assert f2(2, 3) == (3, 2)
# test list
f3 = lambdify([x, y], [y, x], 'sympy')
assert f3(2, 3) == [3, 2]
def test_lambdify_Derivative_arg_issue_16468():
f = Function('f')(x)
fx = f.diff()
assert lambdify((f, fx), f + fx)(10, 5) == 15
assert eval(lambdastr((f, fx), f/fx))(10, 5) == 2
raises(SyntaxError, lambda:
eval(lambdastr((f, fx), f/fx, dummify=False)))
assert eval(lambdastr((f, fx), f/fx, dummify=True))(10, 5) == 2
assert eval(lambdastr((fx, f), f/fx, dummify=True))(S(10), 5) == S.Half
assert lambdify(fx, 1 + fx)(41) == 42
assert eval(lambdastr(fx, 1 + fx, dummify=True))(41) == 42
def test_imag_real():
f_re = lambdify([z], sympy.re(z))
val = 3+2j
assert f_re(val) == val.real
f_im = lambdify([z], sympy.im(z)) # see #15400
assert f_im(val) == val.imag
def test_MatrixSymbol_issue_15578():
if not numpy:
skip("numpy not installed")
A = MatrixSymbol('A', 2, 2)
A0 = numpy.array([[1, 2], [3, 4]])
f = lambdify(A, A**(-1))
assert numpy.allclose(f(A0), numpy.array([[-2., 1.], [1.5, -0.5]]))
g = lambdify(A, A**3)
assert numpy.allclose(g(A0), numpy.array([[37, 54], [81, 118]]))
def test_issue_15654():
if not scipy:
skip("scipy not installed")
from sympy.abc import n, l, r, Z
from sympy.physics import hydrogen
nv, lv, rv, Zv = 1, 0, 3, 1
sympy_value = hydrogen.R_nl(nv, lv, rv, Zv).evalf()
f = lambdify((n, l, r, Z), hydrogen.R_nl(n, l, r, Z))
scipy_value = f(nv, lv, rv, Zv)
assert abs(sympy_value - scipy_value) < 1e-15
def test_issue_15827():
if not numpy:
skip("numpy not installed")
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 2, 3)
C = MatrixSymbol("C", 3, 4)
D = MatrixSymbol("D", 4, 5)
k=symbols("k")
f = lambdify(A, (2*k)*A)
g = lambdify(A, (2+k)*A)
h = lambdify(A, 2*A)
i = lambdify((B, C, D), 2*B*C*D)
assert numpy.array_equal(f(numpy.array([[1, 2, 3], [1, 2, 3], [1, 2, 3]])), \
numpy.array([[2*k, 4*k, 6*k], [2*k, 4*k, 6*k], [2*k, 4*k, 6*k]], dtype=object))
assert numpy.array_equal(g(numpy.array([[1, 2, 3], [1, 2, 3], [1, 2, 3]])), \
numpy.array([[k + 2, 2*k + 4, 3*k + 6], [k + 2, 2*k + 4, 3*k + 6], \
[k + 2, 2*k + 4, 3*k + 6]], dtype=object))
assert numpy.array_equal(h(numpy.array([[1, 2, 3], [1, 2, 3], [1, 2, 3]])), \
numpy.array([[2, 4, 6], [2, 4, 6], [2, 4, 6]]))
assert numpy.array_equal(i(numpy.array([[1, 2, 3], [1, 2, 3]]), numpy.array([[1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4]]), \
numpy.array([[1, 2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5]])), numpy.array([[ 120, 240, 360, 480, 600], \
[ 120, 240, 360, 480, 600]]))
|
49c7d86b99d2f3c53f08b05102cee4b47406339a12d37851b77774f244c7f6c8
|
""" Tests from Michael Wester's 1999 paper "Review of CAS mathematical
capabilities".
http://www.math.unm.edu/~wester/cas/book/Wester.pdf
See also http://math.unm.edu/~wester/cas_review.html for detailed output of
each tested system.
"""
from sympy import (Rational, symbols, Dummy, factorial, sqrt, log, exp, oo, zoo,
product, binomial, rf, pi, gamma, igcd, factorint, radsimp, combsimp,
npartitions, totient, primerange, factor, simplify, gcd, resultant, expand,
I, trigsimp, tan, sin, cos, cot, diff, nan, limit, EulerGamma, polygamma,
bernoulli, hyper, hyperexpand, besselj, asin, assoc_legendre, Function, re,
im, DiracDelta, chebyshevt, legendre_poly, polylog, series, O,
atan, sinh, cosh, tanh, floor, ceiling, solve, asinh, acot, csc, sec,
LambertW, N, apart, sqrtdenest, factorial2, powdenest, Mul, S, ZZ,
Poly, expand_func, E, Q, And, Or, Ne, Eq, Le, Lt, Min,
ask, refine, AlgebraicNumber, continued_fraction_iterator as cf_i,
continued_fraction_periodic as cf_p, continued_fraction_convergents as cf_c,
continued_fraction_reduce as cf_r, FiniteSet, elliptic_e, elliptic_f,
powsimp, hessian, wronskian, fibonacci, sign, Lambda, Piecewise, Subs,
residue, Derivative, logcombine, Symbol, Intersection, Union,
EmptySet, Interval, Integral, idiff, ImageSet, acos, Max)
import mpmath
from sympy.functions.combinatorial.numbers import stirling
from sympy.functions.special.delta_functions import Heaviside
from sympy.functions.special.error_functions import Ci, Si, erf
from sympy.functions.special.zeta_functions import zeta
from sympy.integrals.deltafunctions import deltaintegrate
from sympy.utilities.pytest import XFAIL, slow, SKIP, skip, ON_TRAVIS
from sympy.utilities.iterables import partitions
from mpmath import mpi, mpc
from sympy.matrices import Matrix, GramSchmidt, eye
from sympy.matrices.expressions.blockmatrix import BlockMatrix, block_collapse
from sympy.matrices.expressions import MatrixSymbol, ZeroMatrix
from sympy.physics.quantum import Commutator
from sympy.assumptions import assuming
from sympy.polys.rings import vring
from sympy.polys.fields import vfield
from sympy.polys.solvers import solve_lin_sys
from sympy.concrete import Sum
from sympy.concrete.products import Product
from sympy.integrals import integrate
from sympy.integrals.transforms import laplace_transform,\
inverse_laplace_transform, LaplaceTransform, fourier_transform,\
mellin_transform
from sympy.solvers.recurr import rsolve
from sympy.solvers.solveset import solveset, solveset_real, linsolve
from sympy.solvers.ode import dsolve
from sympy.core.relational import Equality
from sympy.core.compatibility import range, PY3
from itertools import islice, takewhile
from sympy.series.formal import fps
from sympy.series.fourier import fourier_series
R = Rational
x, y, z = symbols('x y z')
i, j, k, l, m, n = symbols('i j k l m n', integer=True)
f = Function('f')
g = Function('g')
# A. Boolean Logic and Quantifier Elimination
# Not implemented.
# B. Set Theory
def test_B1():
assert (FiniteSet(i, j, j, k, k, k) | FiniteSet(l, k, j) |
FiniteSet(j, m, j)) == FiniteSet(i, j, k, l, m)
def test_B2():
a, b, c = FiniteSet(j), FiniteSet(m), FiniteSet(j, k)
d, e = FiniteSet(i), FiniteSet(j, k, l)
assert (FiniteSet(i, j, j, k, k, k) & FiniteSet(l, k, j) &
FiniteSet(j, m, j)) == Union(a, Intersection(b, Union(c, Intersection(d, FiniteSet(l)))))
# {j} U Intersection({m}, {j, k} U Intersection({i}, {l}))
def test_B3():
assert (FiniteSet(i, j, k, l, m) - FiniteSet(j) ==
FiniteSet(i, k, l, m))
def test_B4():
assert (FiniteSet(*(FiniteSet(i, j)*FiniteSet(k, l))) ==
FiniteSet((i, k), (i, l), (j, k), (j, l)))
# C. Numbers
def test_C1():
assert (factorial(50) ==
30414093201713378043612608166064768844377641568960512000000000000)
def test_C2():
assert (factorint(factorial(50)) == {2: 47, 3: 22, 5: 12, 7: 8,
11: 4, 13: 3, 17: 2, 19: 2, 23: 2, 29: 1, 31: 1, 37: 1,
41: 1, 43: 1, 47: 1})
def test_C3():
assert (factorial2(10), factorial2(9)) == (3840, 945)
# Base conversions; not really implemented by sympy
# Whatever. Take credit!
def test_C4():
assert 0xABC == 2748
def test_C5():
assert 123 == int('234', 7)
def test_C6():
assert int('677', 8) == int('1BF', 16) == 447
def test_C7():
assert log(32768, 8) == 5
def test_C8():
# Modular multiplicative inverse. Would be nice if divmod could do this.
assert ZZ.invert(5, 7) == 3
assert ZZ.invert(5, 6) == 5
def test_C9():
assert igcd(igcd(1776, 1554), 5698) == 74
def test_C10():
x = 0
for n in range(2, 11):
x += R(1, n)
assert x == R(4861, 2520)
def test_C11():
assert R(1, 7) == S('0.[142857]')
def test_C12():
assert R(7, 11) * R(22, 7) == 2
def test_C13():
test = R(10, 7) * (1 + R(29, 1000)) ** R(1, 3)
good = 3 ** R(1, 3)
assert test == good
def test_C14():
assert sqrtdenest(sqrt(2*sqrt(3) + 4)) == 1 + sqrt(3)
def test_C15():
test = sqrtdenest(sqrt(14 + 3*sqrt(3 + 2*sqrt(5 - 12*sqrt(3 - 2*sqrt(2))))))
good = sqrt(2) + 3
assert test == good
def test_C16():
test = sqrtdenest(sqrt(10 + 2*sqrt(6) + 2*sqrt(10) + 2*sqrt(15)))
good = sqrt(2) + sqrt(3) + sqrt(5)
assert test == good
def test_C17():
test = radsimp((sqrt(3) + sqrt(2)) / (sqrt(3) - sqrt(2)))
good = 5 + 2*sqrt(6)
assert test == good
def test_C18():
assert simplify((sqrt(-2 + sqrt(-5)) * sqrt(-2 - sqrt(-5))).expand(complex=True)) == 3
@XFAIL
def test_C19():
assert radsimp(simplify((90 + 34*sqrt(7)) ** R(1, 3))) == 3 + sqrt(7)
def test_C20():
inside = (135 + 78*sqrt(3))
test = AlgebraicNumber((inside**R(2, 3) + 3) * sqrt(3) / inside**R(1, 3))
assert simplify(test) == AlgebraicNumber(12)
def test_C21():
assert simplify(AlgebraicNumber((41 + 29*sqrt(2)) ** R(1, 5))) == \
AlgebraicNumber(1 + sqrt(2))
@XFAIL
def test_C22():
test = simplify(((6 - 4*sqrt(2))*log(3 - 2*sqrt(2)) + (3 - 2*sqrt(2))*log(17
- 12*sqrt(2)) + 32 - 24*sqrt(2)) / (48*sqrt(2) - 72))
good = sqrt(2)/3 - log(sqrt(2) - 1)/3
assert test == good
def test_C23():
assert 2 * oo - 3 == oo
@XFAIL
def test_C24():
raise NotImplementedError("2**aleph_null == aleph_1")
# D. Numerical Analysis
def test_D1():
assert 0.0 / sqrt(2) == 0.0
def test_D2():
assert str(exp(-1000000).evalf()) == '3.29683147808856e-434295'
def test_D3():
assert exp(pi*sqrt(163)).evalf(50).num.ae(262537412640768744)
def test_D4():
assert floor(R(-5, 3)) == -2
assert ceiling(R(-5, 3)) == -1
@XFAIL
def test_D5():
raise NotImplementedError("cubic_spline([1, 2, 4, 5], [1, 4, 2, 3], x)(3) == 27/8")
@XFAIL
def test_D6():
raise NotImplementedError("translate sum(a[i]*x**i, (i,1,n)) to FORTRAN")
@XFAIL
def test_D7():
raise NotImplementedError("translate sum(a[i]*x**i, (i,1,n)) to C")
@XFAIL
def test_D8():
# One way is to cheat by converting the sum to a string,
# and replacing the '[' and ']' with ''.
# E.g., horner(S(str(_).replace('[','').replace(']','')))
raise NotImplementedError("apply Horner's rule to sum(a[i]*x**i, (i,1,5))")
@XFAIL
def test_D9():
raise NotImplementedError("translate D8 to FORTRAN")
@XFAIL
def test_D10():
raise NotImplementedError("translate D8 to C")
@XFAIL
def test_D11():
#Is there a way to use count_ops?
raise NotImplementedError("flops(sum(product(f[i][k], (i,1,k)), (k,1,n)))")
@XFAIL
def test_D12():
assert (mpi(-4, 2) * x + mpi(1, 3)) ** 2 == mpi(-8, 16)*x**2 + mpi(-24, 12)*x + mpi(1, 9)
@XFAIL
def test_D13():
raise NotImplementedError("discretize a PDE: diff(f(x,t),t) == diff(diff(f(x,t),x),x)")
# E. Statistics
# See scipy; all of this is numerical.
# F. Combinatorial Theory.
def test_F1():
assert rf(x, 3) == x*(1 + x)*(2 + x)
def test_F2():
assert expand_func(binomial(n, 3)) == n*(n - 1)*(n - 2)/6
@XFAIL
def test_F3():
assert combsimp(2**n * factorial(n) * factorial2(2*n - 1)) == factorial(2*n)
@XFAIL
def test_F4():
assert combsimp((2**n * factorial(n) * product(2*k - 1, (k, 1, n)))) == factorial(2*n)
@XFAIL
def test_F5():
assert gamma(n + R(1, 2)) / sqrt(pi) / factorial(n) == factorial(2*n)/2**(2*n)/factorial(n)**2
def test_F6():
partTest = [p.copy() for p in partitions(4)]
partDesired = [{4: 1}, {1: 1, 3: 1}, {2: 2}, {1: 2, 2:1}, {1: 4}]
assert partTest == partDesired
def test_F7():
assert npartitions(4) == 5
def test_F8():
assert stirling(5, 2, signed=True) == -50 # if signed, then kind=1
def test_F9():
assert totient(1776) == 576
# G. Number Theory
def test_G1():
assert list(primerange(999983, 1000004)) == [999983, 1000003]
@XFAIL
def test_G2():
raise NotImplementedError("find the primitive root of 191 == 19")
@XFAIL
def test_G3():
raise NotImplementedError("(a+b)**p mod p == a**p + b**p mod p; p prime")
# ... G14 Modular equations are not implemented.
def test_G15():
assert Rational(sqrt(3).evalf()).limit_denominator(15) == Rational(26, 15)
assert list(takewhile(lambda x: x.q <= 15, cf_c(cf_i(sqrt(3)))))[-1] == \
Rational(26, 15)
def test_G16():
assert list(islice(cf_i(pi),10)) == [3, 7, 15, 1, 292, 1, 1, 1, 2, 1]
def test_G17():
assert cf_p(0, 1, 23) == [4, [1, 3, 1, 8]]
def test_G18():
assert cf_p(1, 2, 5) == [[1]]
assert cf_r([[1]]) == S.Half + sqrt(5)/2
@XFAIL
def test_G19():
s = symbols('s', integer=True, positive=True)
it = cf_i((exp(1/s) - 1)/(exp(1/s) + 1))
assert list(islice(it, 5)) == [0, 2*s, 6*s, 10*s, 14*s]
def test_G20():
s = symbols('s', integer=True, positive=True)
# Wester erroneously has this as -s + sqrt(s**2 + 1)
assert cf_r([[2*s]]) == s + sqrt(s**2 + 1)
@XFAIL
def test_G20b():
s = symbols('s', integer=True, positive=True)
assert cf_p(s, 1, s**2 + 1) == [[2*s]]
# H. Algebra
def test_H1():
assert simplify(2*2**n) == simplify(2**(n + 1))
assert powdenest(2*2**n) == simplify(2**(n + 1))
def test_H2():
assert powsimp(4 * 2**n) == 2**(n + 2)
def test_H3():
assert (-1)**(n*(n + 1)) == 1
def test_H4():
expr = factor(6*x - 10)
assert type(expr) is Mul
assert expr.args[0] == 2
assert expr.args[1] == 3*x - 5
p1 = 64*x**34 - 21*x**47 - 126*x**8 - 46*x**5 - 16*x**60 - 81
p2 = 72*x**60 - 25*x**25 - 19*x**23 - 22*x**39 - 83*x**52 + 54*x**10 + 81
q = 34*x**19 - 25*x**16 + 70*x**7 + 20*x**3 - 91*x - 86
def test_H5():
assert gcd(p1, p2, x) == 1
def test_H6():
assert gcd(expand(p1 * q), expand(p2 * q)) == q
def test_H7():
p1 = 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5
p2 = 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z
assert gcd(p1, p2, x, y, z) == 1
def test_H8():
p1 = 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5
p2 = 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z
q = 11*x**12*y**7*z**13 - 23*x**2*y**8*z**10 + 47*x**17*y**5*z**8
assert gcd(p1 * q, p2 * q, x, y, z) == q
def test_H9():
p1 = 2*x**(n + 4) - x**(n + 2)
p2 = 4*x**(n + 1) + 3*x**n
assert gcd(p1, p2) == x**n
def test_H10():
p1 = 3*x**4 + 3*x**3 + x**2 - x - 2
p2 = x**3 - 3*x**2 + x + 5
assert resultant(p1, p2, x) == 0
def test_H11():
assert resultant(p1 * q, p2 * q, x) == 0
def test_H12():
num = x**2 - 4
den = x**2 + 4*x + 4
assert simplify(num/den) == (x - 2)/(x + 2)
@XFAIL
def test_H13():
assert simplify((exp(x) - 1) / (exp(x/2) + 1)) == exp(x/2) - 1
def test_H14():
p = (x + 1) ** 20
ep = expand(p)
assert ep == (1 + 20*x + 190*x**2 + 1140*x**3 + 4845*x**4 + 15504*x**5
+ 38760*x**6 + 77520*x**7 + 125970*x**8 + 167960*x**9 + 184756*x**10
+ 167960*x**11 + 125970*x**12 + 77520*x**13 + 38760*x**14 + 15504*x**15
+ 4845*x**16 + 1140*x**17 + 190*x**18 + 20*x**19 + x**20)
dep = diff(ep, x)
assert dep == (20 + 380*x + 3420*x**2 + 19380*x**3 + 77520*x**4
+ 232560*x**5 + 542640*x**6 + 1007760*x**7 + 1511640*x**8 + 1847560*x**9
+ 1847560*x**10 + 1511640*x**11 + 1007760*x**12 + 542640*x**13
+ 232560*x**14 + 77520*x**15 + 19380*x**16 + 3420*x**17 + 380*x**18
+ 20*x**19)
assert factor(dep) == 20*(1 + x)**19
def test_H15():
assert simplify((Mul(*[x - r for r in solveset(x**3 + x**2 - 7)]))) == x**3 + x**2 - 7
def test_H16():
assert factor(x**100 - 1) == ((x - 1)*(x + 1)*(x**2 + 1)*(x**4 - x**3
+ x**2 - x + 1)*(x**4 + x**3 + x**2 + x + 1)*(x**8 - x**6 + x**4
- x**2 + 1)*(x**20 - x**15 + x**10 - x**5 + 1)*(x**20 + x**15 + x**10
+ x**5 + 1)*(x**40 - x**30 + x**20 - x**10 + 1))
def test_H17():
assert simplify(factor(expand(p1 * p2)) - p1*p2) == 0
@XFAIL
def test_H18():
# Factor over complex rationals.
test = factor(4*x**4 + 8*x**3 + 77*x**2 + 18*x + 153)
good = (2*x + 3*I)*(2*x - 3*I)*(x + 1 - 4*I)*(x + 1 + 4*I)
assert test == good
def test_H19():
a = symbols('a')
# The idea is to let a**2 == 2, then solve 1/(a-1). Answer is a+1")
assert Poly(a - 1).invert(Poly(a**2 - 2)) == a + 1
@XFAIL
def test_H20():
raise NotImplementedError("let a**2==2; (x**3 + (a-2)*x**2 - "
+ "(2*a+3)*x - 3*a) / (x**2-2) = (x**2 - 2*x - 3) / (x-a)")
@XFAIL
def test_H21():
raise NotImplementedError("evaluate (b+c)**4 assuming b**3==2, c**2==3. \
Answer is 2*b + 8*c + 18*b**2 + 12*b*c + 9")
def test_H22():
assert factor(x**4 - 3*x**2 + 1, modulus=5) == (x - 2)**2 * (x + 2)**2
def test_H23():
f = x**11 + x + 1
g = (x**2 + x + 1) * (x**9 - x**8 + x**6 - x**5 + x**3 - x**2 + 1)
assert factor(f, modulus=65537) == g
def test_H24():
phi = AlgebraicNumber(S.GoldenRatio.expand(func=True), alias='phi')
assert factor(x**4 - 3*x**2 + 1, extension=phi) == \
(x - phi)*(x + 1 - phi)*(x - 1 + phi)*(x + phi)
def test_H25():
e = (x - 2*y**2 + 3*z**3) ** 20
assert factor(expand(e)) == e
def test_H26():
g = expand((sin(x) - 2*cos(y)**2 + 3*tan(z)**3)**20)
assert factor(g, expand=False) == (-sin(x) + 2*cos(y)**2 - 3*tan(z)**3)**20
def test_H27():
f = 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5
g = 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z
h = -2*z*y**7 \
*(6*x**9*y**9*z**3 + 10*x**7*z**6 + 17*y*x**5*z**12 + 40*y**7) \
*(3*x**22 + 47*x**17*y**5*z**8 - 6*x**15*y**9*z**2 - 24*x*y**19*z**8 - 5)
assert factor(expand(f*g)) == h
@XFAIL
def test_H28():
raise NotImplementedError("expand ((1 - c**2)**5 * (1 - s**2)**5 * "
+ "(c**2 + s**2)**10) with c**2 + s**2 = 1. Answer is c**10*s**10.")
@XFAIL
def test_H29():
assert factor(4*x**2 - 21*x*y + 20*y**2, modulus=3) == (x + y)*(x - y)
def test_H30():
test = factor(x**3 + y**3, extension=sqrt(-3))
answer = (x + y)*(x + y*(-R(1, 2) - sqrt(3)/2*I))*(x + y*(-R(1, 2) + sqrt(3)/2*I))
assert answer == test
def test_H31():
f = (x**2 + 2*x + 3)/(x**3 + 4*x**2 + 5*x + 2)
g = 2 / (x + 1)**2 - 2 / (x + 1) + 3 / (x + 2)
assert apart(f) == g
@XFAIL
def test_H32(): # issue 6558
raise NotImplementedError("[A*B*C - (A*B*C)**(-1)]*A*C*B (product \
of a non-commuting product and its inverse)")
def test_H33():
A, B, C = symbols('A, B, C', commutative=False)
assert (Commutator(A, Commutator(B, C))
+ Commutator(B, Commutator(C, A))
+ Commutator(C, Commutator(A, B))).doit().expand() == 0
# I. Trigonometry
@XFAIL
def test_I1():
assert tan(7*pi/10) == -sqrt(1 + 2/sqrt(5))
@XFAIL
def test_I2():
assert sqrt((1 + cos(6))/2) == -cos(3)
def test_I3():
assert cos(n*pi) + sin((4*n - 1)*pi/2) == (-1)**n - 1
def test_I4():
assert refine(cos(pi*cos(n*pi)) + sin(pi/2*cos(n*pi)), Q.integer(n)) == (-1)**n - 1
@XFAIL
def test_I5():
assert sin((n**5/5 + n**4/2 + n**3/3 - n/30) * pi) == 0
@XFAIL
def test_I6():
raise NotImplementedError("assuming -3*pi<x<-5*pi/2, abs(cos(x)) == -cos(x), abs(sin(x)) == -sin(x)")
@XFAIL
def test_I7():
assert cos(3*x)/cos(x) == cos(x)**2 - 3*sin(x)**2
@XFAIL
def test_I8():
assert cos(3*x)/cos(x) == 2*cos(2*x) - 1
@XFAIL
def test_I9():
# Supposed to do this with rewrite rules.
assert cos(3*x)/cos(x) == cos(x)**2 - 3*sin(x)**2
def test_I10():
assert trigsimp((tan(x)**2 + 1 - cos(x)**-2) / (sin(x)**2 + cos(x)**2 - 1)) == nan
@SKIP("hangs")
@XFAIL
def test_I11():
assert limit((tan(x)**2 + 1 - cos(x)**-2) / (sin(x)**2 + cos(x)**2 - 1), x, 0) != 0
@XFAIL
def test_I12():
try:
# This should fail or return nan or something.
diff((tan(x)**2 + 1 - cos(x)**-2) / (sin(x)**2 + cos(x)**2 - 1), x)
except:
assert True
else:
assert False, "taking the derivative with a fraction equivalent to 0/0 should fail"
# J. Special functions.
def test_J1():
assert bernoulli(16) == R(-3617, 510)
def test_J2():
assert diff(elliptic_e(x, y**2), y) == (elliptic_e(x, y**2) - elliptic_f(x, y**2))/y
@XFAIL
def test_J3():
raise NotImplementedError("Jacobi elliptic functions: diff(dn(u,k), u) == -k**2*sn(u,k)*cn(u,k)")
def test_J4():
assert gamma(R(-1, 2)) == -2*sqrt(pi)
def test_J5():
assert polygamma(0, R(1, 3)) == -log(3) - sqrt(3)*pi/6 - EulerGamma - log(sqrt(3))
def test_J6():
assert mpmath.besselj(2, 1 + 1j).ae(mpc('0.04157988694396212', '0.24739764151330632'))
def test_J7():
assert simplify(besselj(R(-5,2), pi/2)) == 12/(pi**2)
def test_J8():
p = besselj(R(3,2), z)
q = (sin(z)/z - cos(z))/sqrt(pi*z/2)
assert simplify(expand_func(p) -q) == 0
def test_J9():
assert besselj(0, z).diff(z) == - besselj(1, z)
def test_J10():
mu, nu = symbols('mu, nu', integer=True)
assert assoc_legendre(nu, mu, 0) == 2**mu*sqrt(pi)/gamma((nu - mu)/2 + 1)/gamma((-nu - mu + 1)/2)
def test_J11():
assert simplify(assoc_legendre(3, 1, x)) == simplify(-R(3, 2)*sqrt(1 - x**2)*(5*x**2 - 1))
@slow
def test_J12():
assert simplify(chebyshevt(1008, x) - 2*x*chebyshevt(1007, x) + chebyshevt(1006, x)) == 0
def test_J13():
a = symbols('a', integer=True, negative=False)
assert chebyshevt(a, -1) == (-1)**a
def test_J14():
p = hyper([S(1)/2, S(1)/2], [S(3)/2], z**2)
assert hyperexpand(p) == asin(z)/z
@XFAIL
def test_J15():
raise NotImplementedError("F((n+2)/2,-(n-2)/2,R(3,2),sin(z)**2) == sin(n*z)/(n*sin(z)*cos(z)); F(.) is hypergeometric function")
@XFAIL
def test_J16():
raise NotImplementedError("diff(zeta(x), x) @ x=0 == -log(2*pi)/2")
def test_J17():
assert integrate(f((x + 2)/5)*DiracDelta((x - 2)/3) - g(x)*diff(DiracDelta(x - 1), x), (x, 0, 3)) == 3*f(S(4)/5) + Subs(Derivative(g(x), x), x, 1)
@XFAIL
def test_J18():
raise NotImplementedError("define an antisymmetric function")
# K. The Complex Domain
def test_K1():
z1, z2 = symbols('z1, z2', complex=True)
assert re(z1 + I*z2) == -im(z2) + re(z1)
assert im(z1 + I*z2) == im(z1) + re(z2)
def test_K2():
assert abs(3 - sqrt(7) + I*sqrt(6*sqrt(7) - 15)) == 1
@XFAIL
def test_K3():
a, b = symbols('a, b', real=True)
assert simplify(abs(1/(a + I/a + I*b))) == 1/sqrt(a**2 + (I/a + b)**2)
def test_K4():
assert log(3 + 4*I).expand(complex=True) == log(5) + I*atan(R(4, 3))
def test_K5():
x, y = symbols('x, y', real=True)
assert tan(x + I*y).expand(complex=True) == (sin(2*x)/(cos(2*x) +
cosh(2*y)) + I*sinh(2*y)/(cos(2*x) + cosh(2*y)))
def test_K6():
assert sqrt(x*y*abs(z)**2)/(sqrt(x)*abs(z)) == sqrt(x*y)/sqrt(x)
assert sqrt(x*y*abs(z)**2)/(sqrt(x)*abs(z)) != sqrt(y)
def test_K7():
y = symbols('y', real=True, negative=False)
expr = sqrt(x*y*abs(z)**2)/(sqrt(x)*abs(z))
sexpr = simplify(expr)
assert sexpr == sqrt(y)
@XFAIL
def test_K8():
z = symbols('z', complex=True)
assert simplify(sqrt(1/z) - 1/sqrt(z)) != 0 # Passes
z = symbols('z', complex=True, negative=False)
assert simplify(sqrt(1/z) - 1/sqrt(z)) == 0 # Fails
def test_K9():
z = symbols('z', real=True, positive=True)
assert simplify(sqrt(1/z) - 1/sqrt(z)) == 0
def test_K10():
z = symbols('z', real=True, negative=True)
assert simplify(sqrt(1/z) + 1/sqrt(z)) == 0
# This goes up to K25
# L. Determining Zero Equivalence
def test_L1():
assert sqrt(997) - (997**3)**R(1, 6) == 0
def test_L2():
assert sqrt(999983) - (999983**3)**R(1, 6) == 0
def test_L3():
assert simplify((2**R(1, 3) + 4**R(1, 3))**3 - 6*(2**R(1, 3) + 4**R(1, 3)) - 6) == 0
def test_L4():
assert trigsimp(cos(x)**3 + cos(x)*sin(x)**2 - cos(x)) == 0
@XFAIL
def test_L5():
assert log(tan(R(1, 2)*x + pi/4)) - asinh(tan(x)) == 0
def test_L6():
assert (log(tan(x/2 + pi/4)) - asinh(tan(x))).diff(x).subs({x: 0}) == 0
@XFAIL
def test_L7():
assert simplify(log((2*sqrt(x) + 1)/(sqrt(4*x + 4*sqrt(x) + 1)))) == 0
@XFAIL
def test_L8():
assert simplify((4*x + 4*sqrt(x) + 1)**(sqrt(x)/(2*sqrt(x) + 1)) \
*(2*sqrt(x) + 1)**(1/(2*sqrt(x) + 1)) - 2*sqrt(x) - 1) == 0
@XFAIL
def test_L9():
z = symbols('z', complex=True)
assert simplify(2**(1 - z)*gamma(z)*zeta(z)*cos(z*pi/2) - pi**2*zeta(1 - z)) == 0
# M. Equations
@XFAIL
def test_M1():
assert Equality(x, 2)/2 + Equality(1, 1) == Equality(x/2 + 1, 2)
def test_M2():
# The roots of this equation should all be real. Note that this
# doesn't test that they are correct.
sol = solveset(3*x**3 - 18*x**2 + 33*x - 19, x)
assert all(s.expand(complex=True).is_real for s in sol)
@XFAIL
def test_M5():
assert solveset(x**6 - 9*x**4 - 4*x**3 + 27*x**2 - 36*x - 23, x) == FiniteSet(2**(1/3) + sqrt(3), 2**(1/3) - sqrt(3), +sqrt(3) - 1/2**(2/3) + I*sqrt(3)/2**(2/3), +sqrt(3) - 1/2**(2/3) - I*sqrt(3)/2**(2/3), -sqrt(3) - 1/2**(2/3) + I*sqrt(3)/2**(2/3), -sqrt(3) - 1/2**(2/3) - I*sqrt(3)/2**(2/3))
def test_M6():
assert set(solveset(x**7 - 1, x)) == \
{cos(n*2*pi/7) + I*sin(n*2*pi/7) for n in range(0, 7)}
# The paper asks for exp terms, but sin's and cos's may be acceptable;
# if the results are simplified, exp terms appear for all but
# -sin(pi/14) - I*cos(pi/14) and -sin(pi/14) + I*cos(pi/14) which
# will simplify if you apply the transformation foo.rewrite(exp).expand()
def test_M7():
# TODO: Replace solve with solveset, as of now test fails for solveset
sol = solve(x**8 - 8*x**7 + 34*x**6 - 92*x**5 + 175*x**4 - 236*x**3 +
226*x**2 - 140*x + 46, x)
assert [s.simplify() for s in sol] == [
1 - sqrt(-6 - 2*I*sqrt(3 + 4*sqrt(3)))/2,
1 + sqrt(-6 - 2*I*sqrt(3 + 4*sqrt(3)))/2,
1 - sqrt(-6 + 2*I*sqrt(3 + 4*sqrt(3)))/2,
1 + sqrt(-6 + 2*I*sqrt(3 + 4*sqrt (3)))/2,
1 - sqrt(-6 + 2*sqrt(-3 + 4*sqrt(3)))/2,
1 + sqrt(-6 + 2*sqrt(-3 + 4*sqrt(3)))/2,
1 - sqrt(-6 - 2*sqrt(-3 + 4*sqrt(3)))/2,
1 + sqrt(-6 - 2*sqrt(-3 + 4*sqrt(3)))/2]
@XFAIL # There are an infinite number of solutions.
def test_M8():
x = Symbol('x')
z = symbols('z', complex=True)
assert solveset(exp(2*x) + 2*exp(x) + 1 - z, x, S.Reals) == \
FiniteSet(log(1 + z - 2*sqrt(z))/2, log(1 + z + 2*sqrt(z))/2)
# This one could be simplified better (the 1/2 could be pulled into the log
# as a sqrt, and the function inside the log can be factored as a square,
# giving [log(sqrt(z) - 1), log(sqrt(z) + 1)]). Also, there should be an
# infinite number of solutions.
# x = {log(sqrt(z) - 1), log(sqrt(z) + 1) + i pi} [+ n 2 pi i, + n 2 pi i]
# where n is an arbitrary integer. See url of detailed output above.
@XFAIL
def test_M9():
x = symbols('x')
raise NotImplementedError("solveset(exp(2-x**2)-exp(-x),x) has complex solutions.")
def test_M10():
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve(exp(x) - x, x) == [-LambertW(-1)]
@XFAIL
def test_M11():
assert solveset(x**x - x, x) == FiniteSet(-1, 1)
def test_M12():
# TODO: x = [-1, 2*(+/-asinh(1)*I + n*pi}, 3*(pi/6 + n*pi/3)]
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve((x + 1)*(sin(x)**2 + 1)**2*cos(3*x)**3, x) == [
-1, pi/6, pi/2,
- I*log(1 + sqrt(2)), I*log(1 + sqrt(2)),
pi - I*log(1 + sqrt(2)), pi + I*log(1 + sqrt(2)),
]
@XFAIL
def test_M13():
n = Dummy('n')
assert solveset_real(sin(x) - cos(x), x) == ImageSet(Lambda(n, n*pi - 7*pi/4), S.Integers)
@XFAIL
def test_M14():
n = Dummy('n')
assert solveset_real(tan(x) - 1, x) == ImageSet(Lambda(n, n*pi + pi/4), S.Integers)
def test_M15():
if PY3:
n = Dummy('n')
assert solveset(sin(x) - S.Half) in (Union(ImageSet(Lambda(n, 2*n*pi + pi/6), S.Integers),
ImageSet(Lambda(n, 2*n*pi + 5*pi/6), S.Integers)),
Union(ImageSet(Lambda(n, 2*n*pi + 5*pi/6), S.Integers),
ImageSet(Lambda(n, 2*n*pi + pi/6), S.Integers)))
@XFAIL
def test_M16():
n = Dummy('n')
assert solveset(sin(x) - tan(x), x) == ImageSet(Lambda(n, n*pi), S.Integers)
@XFAIL
def test_M17():
assert solveset_real(asin(x) - atan(x), x) == FiniteSet(0)
@XFAIL
def test_M18():
assert solveset_real(acos(x) - atan(x), x) == FiniteSet(sqrt((sqrt(5) - 1)/2))
def test_M19():
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve((x - 2)/x**R(1, 3), x) == [2]
def test_M20():
assert solveset(sqrt(x**2 + 1) - x + 2, x) == EmptySet()
def test_M21():
assert solveset(x + sqrt(x) - 2) == FiniteSet(1)
def test_M22():
assert solveset(2*sqrt(x) + 3*x**R(1, 4) - 2) == FiniteSet(R(1, 16))
def test_M23():
x = symbols('x', complex=True)
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve(x - 1/sqrt(1 + x**2)) == [
-I*sqrt(S.Half + sqrt(5)/2), sqrt(-S.Half + sqrt(5)/2)]
def test_M24():
# TODO: Replace solve with solveset, as of now test fails for solveset
solution = solve(1 - binomial(m, 2)*2**k, k)
answer = log(2/(m*(m - 1)), 2)
assert solution[0].expand() == answer.expand()
def test_M25():
a, b, c, d = symbols(':d', positive=True)
x = symbols('x')
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve(a*b**x - c*d**x, x)[0].expand() == (log(c/a)/log(b/d)).expand()
def test_M26():
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve(sqrt(log(x)) - log(sqrt(x))) == [1, exp(4)]
@XFAIL
def test_M27():
x = symbols('x', real=True)
b = symbols('b', real=True)
with assuming(Q.is_true(sin(cos(1/E**2) + 1) + b > 0)):
# TODO: Replace solve with solveset
solve(log(acos(asin(x**R(2, 3) - b) - 1)) + 2, x) == [-b - sin(1 + cos(1/e**2))**R(3/2), b + sin(1 + cos(1/e**2))**R(3/2)]
@XFAIL
def test_M28():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
assert solve(5*x + exp((x - 5)/2) - 8*x**3, x, assume=Q.real(x)) == [-0.784966, -0.016291, 0.802557]
def test_M29():
x = symbols('x')
assert solveset(abs(x - 1) - 2, domain=S.Reals) == FiniteSet(-1, 3)
def test_M30():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
# assert solve(abs(2*x + 5) - abs(x - 2),x, assume=Q.real(x)) == [-1, -7]
assert solveset_real(abs(2*x + 5) - abs(x - 2), x) == FiniteSet(-1, -7)
def test_M31():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
# assert solve(1 - abs(x) - max(-x - 2, x - 2),x, assume=Q.real(x)) == [-3/2, 3/2]
assert solveset_real(1 - abs(x) - Max(-x - 2, x - 2), x) == FiniteSet(-S(3)/2, S(3)/2)
@XFAIL
def test_M32():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
assert solveset_real(Max(2 - x**2, x)- Max(-x, (x**3)/9), x) == FiniteSet(-1, 3)
@XFAIL
def test_M33():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
# Second answer can be written in another form. The second answer is the root of x**3 + 9*x**2 - 18 = 0 in the interval (-2, -1).
assert solveset_real(Max(2 - x**2, x) - x**3/9, x) == FiniteSet(-3, -1.554894, 3)
@XFAIL
def test_M34():
z = symbols('z', complex=True)
assert solveset((1 + I) * z + (2 - I) * conjugate(z) + 3*I, z) == FiniteSet(2 + 3*I)
def test_M35():
x, y = symbols('x y', real=True)
assert linsolve((3*x - 2*y - I*y + 3*I).as_real_imag(), y, x) == FiniteSet((3, 2))
def test_M36():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports solving for function
# assert solve(f**2 + f - 2, x) == [Eq(f(x), 1), Eq(f(x), -2)]
assert solveset(f(x)**2 + f(x) - 2, f(x)) == FiniteSet(-2, 1)
def test_M37():
assert linsolve([x + y + z - 6, 2*x + y + 2*z - 10, x + 3*y + z - 10 ], x, y, z) == \
FiniteSet((-z + 4, 2, z))
def test_M38():
variables = vring("k1:50", vfield("a,b,c", ZZ).to_domain())
system = [
-b*k8/a + c*k8/a, -b*k11/a + c*k11/a, -b*k10/a + c*k10/a + k2, -k3 - b*k9/a + c*k9/a,
-b*k14/a + c*k14/a, -b*k15/a + c*k15/a, -b*k18/a + c*k18/a - k2, -b*k17/a + c*k17/a,
-b*k16/a + c*k16/a + k4, -b*k13/a + c*k13/a - b*k21/a + c*k21/a + b*k5/a - c*k5/a,
b*k44/a - c*k44/a, -b*k45/a + c*k45/a, -b*k20/a + c*k20/a, -b*k44/a + c*k44/a,
b*k46/a - c*k46/a, b**2*k47/a**2 - 2*b*c*k47/a**2 + c**2*k47/a**2, k3, -k4,
-b*k12/a + c*k12/a - a*k6/b + c*k6/b, -b*k19/a + c*k19/a + a*k7/c - b*k7/c,
b*k45/a - c*k45/a, -b*k46/a + c*k46/a, -k48 + c*k48/a + c*k48/b - c**2*k48/(a*b),
-k49 + b*k49/a + b*k49/c - b**2*k49/(a*c), a*k1/b - c*k1/b, a*k4/b - c*k4/b,
a*k3/b - c*k3/b + k9, -k10 + a*k2/b - c*k2/b, a*k7/b - c*k7/b, -k9, k11,
b*k12/a - c*k12/a + a*k6/b - c*k6/b, a*k15/b - c*k15/b, k10 + a*k18/b - c*k18/b,
-k11 + a*k17/b - c*k17/b, a*k16/b - c*k16/b, -a*k13/b + c*k13/b + a*k21/b - c*k21/b + a*k5/b - c*k5/b,
-a*k44/b + c*k44/b, a*k45/b - c*k45/b, a*k14/c - b*k14/c + a*k20/b - c*k20/b,
a*k44/b - c*k44/b, -a*k46/b + c*k46/b, -k47 + c*k47/a + c*k47/b - c**2*k47/(a*b),
a*k19/b - c*k19/b, -a*k45/b + c*k45/b, a*k46/b - c*k46/b, a**2*k48/b**2 - 2*a*c*k48/b**2 + c**2*k48/b**2,
-k49 + a*k49/b + a*k49/c - a**2*k49/(b*c), k16, -k17, -a*k1/c + b*k1/c,
-k16 - a*k4/c + b*k4/c, -a*k3/c + b*k3/c, k18 - a*k2/c + b*k2/c, b*k19/a - c*k19/a - a*k7/c + b*k7/c,
-a*k6/c + b*k6/c, -a*k8/c + b*k8/c, -a*k11/c + b*k11/c + k17, -a*k10/c + b*k10/c - k18,
-a*k9/c + b*k9/c, -a*k14/c + b*k14/c - a*k20/b + c*k20/b, -a*k13/c + b*k13/c + a*k21/c - b*k21/c - a*k5/c + b*k5/c,
a*k44/c - b*k44/c, -a*k45/c + b*k45/c, -a*k44/c + b*k44/c, a*k46/c - b*k46/c,
-k47 + b*k47/a + b*k47/c - b**2*k47/(a*c), -a*k12/c + b*k12/c, a*k45/c - b*k45/c,
-a*k46/c + b*k46/c, -k48 + a*k48/b + a*k48/c - a**2*k48/(b*c),
a**2*k49/c**2 - 2*a*b*k49/c**2 + b**2*k49/c**2, k8, k11, -k15, k10 - k18,
-k17, k9, -k16, -k29, k14 - k32, -k21 + k23 - k31, -k24 - k30, -k35, k44,
-k45, k36, k13 - k23 + k39, -k20 + k38, k25 + k37, b*k26/a - c*k26/a - k34 + k42,
-2*k44, k45, k46, b*k47/a - c*k47/a, k41, k44, -k46, -b*k47/a + c*k47/a,
k12 + k24, -k19 - k25, -a*k27/b + c*k27/b - k33, k45, -k46, -a*k48/b + c*k48/b,
a*k28/c - b*k28/c + k40, -k45, k46, a*k48/b - c*k48/b, a*k49/c - b*k49/c,
-a*k49/c + b*k49/c, -k1, -k4, -k3, k15, k18 - k2, k17, k16, k22, k25 - k7,
k24 + k30, k21 + k23 - k31, k28, -k44, k45, -k30 - k6, k20 + k32, k27 + b*k33/a - c*k33/a,
k44, -k46, -b*k47/a + c*k47/a, -k36, k31 - k39 - k5, -k32 - k38, k19 - k37,
k26 - a*k34/b + c*k34/b - k42, k44, -2*k45, k46, a*k48/b - c*k48/b,
a*k35/c - b*k35/c - k41, -k44, k46, b*k47/a - c*k47/a, -a*k49/c + b*k49/c,
-k40, k45, -k46, -a*k48/b + c*k48/b, a*k49/c - b*k49/c, k1, k4, k3, -k8,
-k11, -k10 + k2, -k9, k37 + k7, -k14 - k38, -k22, -k25 - k37, -k24 + k6,
-k13 - k23 + k39, -k28 + b*k40/a - c*k40/a, k44, -k45, -k27, -k44, k46,
b*k47/a - c*k47/a, k29, k32 + k38, k31 - k39 + k5, -k12 + k30, k35 - a*k41/b + c*k41/b,
-k44, k45, -k26 + k34 + a*k42/c - b*k42/c, k44, k45, -2*k46, -b*k47/a + c*k47/a,
-a*k48/b + c*k48/b, a*k49/c - b*k49/c, k33, -k45, k46, a*k48/b - c*k48/b,
-a*k49/c + b*k49/c
]
solution = {
k49: 0, k48: 0, k47: 0, k46: 0, k45: 0, k44: 0, k41: 0, k40: 0,
k38: 0, k37: 0, k36: 0, k35: 0, k33: 0, k32: 0, k30: 0, k29: 0,
k28: 0, k27: 0, k25: 0, k24: 0, k22: 0, k21: 0, k20: 0, k19: 0,
k18: 0, k17: 0, k16: 0, k15: 0, k14: 0, k13: 0, k12: 0, k11: 0,
k10: 0, k9: 0, k8: 0, k7: 0, k6: 0, k5: 0, k4: 0, k3: 0,
k2: 0, k1: 0,
k34: b/c*k42, k31: k39, k26: a/c*k42, k23: k39
}
assert solve_lin_sys(system, variables) == solution
def test_M39():
x, y, z = symbols('x y z', complex=True)
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports non-linear multivariate
assert solve([x**2*y + 3*y*z - 4, -3*x**2*z + 2*y**2 + 1, 2*y*z**2 - z**2 - 1 ]) ==\
[{y: 1, z: 1, x: -1}, {y: 1, z: 1, x: 1},\
{y: sqrt(2)*I, z: R(1,3) - sqrt(2)*I/3, x: -sqrt(-1 - sqrt(2)*I)},\
{y: sqrt(2)*I, z: R(1,3) - sqrt(2)*I/3, x: sqrt(-1 - sqrt(2)*I)},\
{y: -sqrt(2)*I, z: R(1,3) + sqrt(2)*I/3, x: -sqrt(-1 + sqrt(2)*I)},\
{y: -sqrt(2)*I, z: R(1,3) + sqrt(2)*I/3, x: sqrt(-1 + sqrt(2)*I)}]
# N. Inequalities
def test_N1():
assert ask(Q.is_true(E**pi > pi**E))
@XFAIL
def test_N2():
x = symbols('x', real=True)
assert ask(Q.is_true(x**4 - x + 1 > 0)) is True
assert ask(Q.is_true(x**4 - x + 1 > 1)) is False
@XFAIL
def test_N3():
x = symbols('x', real=True)
assert ask(Q.is_true(And(Lt(-1, x), Lt(x, 1))), Q.is_true(abs(x) < 1 ))
@XFAIL
def test_N4():
x, y = symbols('x y', real=True)
assert ask(Q.is_true(2*x**2 > 2*y**2), Q.is_true((x > y) & (y > 0))) is True
@XFAIL
def test_N5():
x, y, k = symbols('x y k', real=True)
assert ask(Q.is_true(k*x**2 > k*y**2), Q.is_true((x > y) & (y > 0) & (k > 0))) is True
@XFAIL
def test_N6():
x, y, k, n = symbols('x y k n', real=True)
assert ask(Q.is_true(k*x**n > k*y**n), Q.is_true((x > y) & (y > 0) & (k > 0) & (n > 0))) is True
@XFAIL
def test_N7():
x, y = symbols('x y', real=True)
assert ask(Q.is_true(y > 0), Q.is_true((x > 1) & (y >= x - 1))) is True
@XFAIL
def test_N8():
x, y, z = symbols('x y z', real=True)
assert ask(Q.is_true((x == y) & (y == z)),
Q.is_true((x >= y) & (y >= z) & (z >= x)))
def test_N9():
x = Symbol('x')
assert solveset(abs(x - 1) > 2, domain=S.Reals) == Union(Interval(-oo, -1, False, True),
Interval(3, oo, True))
def test_N10():
x = Symbol('x')
p = (x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)
assert solveset(expand(p) < 0, domain=S.Reals) == Union(Interval(-oo, 1, True, True),
Interval(2, 3, True, True),
Interval(4, 5, True, True))
def test_N11():
x = Symbol('x')
assert solveset(6/(x - 3) <= 3, domain=S.Reals) == Union(Interval(-oo, 3, True, True), Interval(5, oo))
def test_N12():
x = Symbol('x')
assert solveset(sqrt(x) < 2, domain=S.Reals) == Interval(0, 4, False, True)
def test_N13():
x = Symbol('x')
assert solveset(sin(x) < 2, domain=S.Reals) == S.Reals
@XFAIL
def test_N14():
x = Symbol('x')
# Gives 'Union(Interval(Integer(0), Mul(Rational(1, 2), pi), false, true),
# Interval(Mul(Rational(1, 2), pi), Mul(Integer(2), pi), true, false))'
# which is not the correct answer, but the provided also seems wrong.
assert solveset(sin(x) < 1, x, domain=S.Reals) == Union(Interval(-oo, pi/2, True, True),
Interval(pi/2, oo, True, True))
def test_N15():
r, t = symbols('r t')
# raises NotImplementedError: only univariate inequalities are supported
solveset(abs(2*r*(cos(t) - 1) + 1) <= 1, r, S.Reals)
def test_N16():
r, t = symbols('r t')
solveset((r**2)*((cos(t) - 4)**2)*sin(t)**2 < 9, r, S.Reals)
@XFAIL
def test_N17():
# currently only univariate inequalities are supported
assert solveset((x + y > 0, x - y < 0), (x, y)) == (abs(x) < y)
def test_O1():
M = Matrix((1 + I, -2, 3*I))
assert sqrt(expand(M.dot(M.H))) == sqrt(15)
def test_O2():
assert Matrix((2, 2, -3)).cross(Matrix((1, 3, 1))) == Matrix([[11],
[-5],
[4]])
# The vector module has no way of representing vectors symbolically (without
# respect to a basis)
@XFAIL
def test_O3():
assert (va ^ vb) | (vc ^ vd) == -(va | vc)*(vb | vd) + (va | vd)*(vb | vc)
def test_O4():
from sympy.vector import CoordSys3D, Del
N = CoordSys3D("N")
delop = Del()
i, j, k = N.base_vectors()
x, y, z = N.base_scalars()
F = i*(x*y*z) + j*((x*y*z)**2) + k*((y**2)*(z**3))
assert delop.cross(F).doit() == (-2*x**2*y**2*z + 2*y*z**3)*i + x*y*j + (2*x*y**2*z**2 - x*z)*k
# The vector module has no way of representing vectors symbolically (without
# respect to a basis)
@XFAIL
def test_O5():
assert grad|(f^g)-g|(grad^f)+f|(grad^g) == 0
#testO8-O9 MISSING!!
def test_O10():
L = [Matrix([2, 3, 5]), Matrix([3, 6, 2]), Matrix([8, 3, 6])]
assert GramSchmidt(L) == [Matrix([
[2],
[3],
[5]]),
Matrix([
[S(23)/19],
[S(63)/19],
[S(-47)/19]]),
Matrix([
[S(1692)/353],
[S(-1551)/706],
[S(-423)/706]])]
@XFAIL
def test_P1():
raise NotImplementedError("Matrix property/function to extract Nth \
diagonal not implemented. See Matlab diag(A,k) \
http://www.mathworks.de/de/help/symbolic/diag.html")
def test_P2():
M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
M.row_del(1)
M.col_del(2)
assert M == Matrix([[1, 2],
[7, 8]])
@XFAIL
def test_P3():
A = Matrix([
[11, 12, 13, 14],
[21, 22, 23, 24],
[31, 32, 33, 34],
[41, 42, 43, 44]])
A11 = A[0:3, 1:4]
A12 = A[(0, 1, 3), (2, 0, 3)]
A21 = A
A221 = A[0:2, 2:4]
A222 = A[(3, 0), (2, 1)]
A22 = BlockMatrix([A221, A222])
B = BlockMatrix([[A11, A12],
[A21, A22]])
# B is a matrix consisting of several matrices
# https://github.com/sympy/sympy/issues/16278
assert B == Matrix([[12, 13, 14, 13, 11, 14],
[22, 23, 24, 23, 21, 24],
[32, 33, 34, 43, 41, 44],
[11, 12, 13, 14, 13, 14],
[21, 22, 23, 24, 23, 24],
[31, 32, 33, 34, 43, 42],
[41, 42, 43, 44, 13, 12]])
@XFAIL
def test_P4():
raise NotImplementedError("Block matrix diagonalization not supported")
@XFAIL
def test_P5():
M = Matrix([[7, 11],
[3, 8]])
# Raises exception % not supported for matrices
assert M % 2 == Matrix([[1, 1],
[1, 0]])
def test_P5_workaround():
M = Matrix([[7, 11],
[3, 8]])
assert M.applyfunc(lambda i: i % 2) == Matrix([[1, 1],
[1, 0]])
def test_P6():
M = Matrix([[cos(x), sin(x)],
[-sin(x), cos(x)]])
assert M.diff(x, 2) == Matrix([[-cos(x), -sin(x)],
[sin(x), -cos(x)]])
def test_P7():
M = Matrix([[x, y]])*(
z*Matrix([[1, 3, 5],
[2, 4, 6]]) + Matrix([[7, -9, 11],
[-8, 10, -12]]))
assert M == Matrix([[x*(z + 7) + y*(2*z - 8), x*(3*z - 9) + y*(4*z + 10),
x*(5*z + 11) + y*(6*z - 12)]])
def test_P8():
M = Matrix([[1, -2*I],
[-3*I, 4]])
assert M.norm(ord=S.Infinity) == 7
def test_P9():
a, b, c = symbols('a b c', real=True)
M = Matrix([[a/(b*c), 1/c, 1/b],
[1/c, b/(a*c), 1/a],
[1/b, 1/a, c/(a*b)]])
assert factor(M.norm('fro')) == (a**2 + b**2 + c**2)/(abs(a)*abs(b)*abs(c))
@XFAIL
def test_P10():
M = Matrix([[1, 2 + 3*I],
[f(4 - 5*I), 6]])
# conjugate(f(4 - 5*i)) is not simplified to f(4+5*I)
assert M.H == Matrix([[1, f(4 + 5*I)],
[2 + 3*I, 6]])
@XFAIL
def test_P11():
# raises NotImplementedError("Matrix([[x,y],[1,x*y]]).inv()
# not simplifying to extract common factor")
assert Matrix([[x, y],
[1, x*y]]).inv() == (1/(x**2 - 1))*Matrix([[x, -1],
[-1/y, x/y]])
def test_P12():
A11 = MatrixSymbol('A11', n, n)
A12 = MatrixSymbol('A12', n, n)
A22 = MatrixSymbol('A22', n, n)
B = BlockMatrix([[A11, A12],
[ZeroMatrix(n, n), A22]])
assert block_collapse(B.I) == BlockMatrix([[A11.I, (-1)*A11.I*A12*A22.I],
[ZeroMatrix(n, n), A22.I]])
def test_P13():
M = Matrix([[1, x - 2, x - 3],
[x - 1, x**2 - 3*x + 6, x**2 - 3*x - 2],
[x - 2, x**2 - 8, 2*(x**2) - 12*x + 14]])
L, U, _ = M.LUdecomposition()
assert simplify(L) == Matrix([[1, 0, 0],
[x - 1, 1, 0],
[x - 2, x - 3, 1]])
assert simplify(U) == Matrix([[1, x - 2, x - 3],
[0, 4, x - 5],
[0, 0, x - 7]])
def test_P14():
M = Matrix([[1, 2, 3, 1, 3],
[3, 2, 1, 1, 7],
[0, 2, 4, 1, 1],
[1, 1, 1, 1, 4]])
R, _ = M.rref()
assert R == Matrix([[1, 0, -1, 0, 2],
[0, 1, 2, 0, -1],
[0, 0, 0, 1, 3],
[0, 0, 0, 0, 0]])
def test_P15():
M = Matrix([[-1, 3, 7, -5],
[4, -2, 1, 3],
[2, 4, 15, -7]])
assert M.rank() == 2
def test_P16():
M = Matrix([[2*sqrt(2), 8],
[6*sqrt(6), 24*sqrt(3)]])
assert M.rank() == 1
def test_P17():
t = symbols('t', real=True)
M=Matrix([
[sin(2*t), cos(2*t)],
[2*(1 - (cos(t)**2))*cos(t), (1 - 2*(sin(t)**2))*sin(t)]])
assert M.rank() == 1
def test_P18():
M = Matrix([[1, 0, -2, 0],
[-2, 1, 0, 3],
[-1, 2, -6, 6]])
assert M.nullspace() == [Matrix([[2],
[4],
[1],
[0]]),
Matrix([[0],
[-3],
[0],
[1]])]
def test_P19():
w = symbols('w')
M = Matrix([[1, 1, 1, 1],
[w, x, y, z],
[w**2, x**2, y**2, z**2],
[w**3, x**3, y**3, z**3]])
assert M.det() == (w**3*x**2*y - w**3*x**2*z - w**3*x*y**2 + w**3*x*z**2
+ w**3*y**2*z - w**3*y*z**2 - w**2*x**3*y + w**2*x**3*z
+ w**2*x*y**3 - w**2*x*z**3 - w**2*y**3*z + w**2*y*z**3
+ w*x**3*y**2 - w*x**3*z**2 - w*x**2*y**3 + w*x**2*z**3
+ w*y**3*z**2 - w*y**2*z**3 - x**3*y**2*z + x**3*y*z**2
+ x**2*y**3*z - x**2*y*z**3 - x*y**3*z**2 + x*y**2*z**3
)
@XFAIL
def test_P20():
raise NotImplementedError("Matrix minimal polynomial not supported")
def test_P21():
M = Matrix([[5, -3, -7],
[-2, 1, 2],
[2, -3, -4]])
assert M.charpoly(x).as_expr() == x**3 - 2*x**2 - 5*x + 6
def test_P22():
d = 100
M = (2 - x)*eye(d)
assert M.eigenvals() == {-x + 2: d}
def test_P23():
M = Matrix([
[2, 1, 0, 0, 0],
[1, 2, 1, 0, 0],
[0, 1, 2, 1, 0],
[0, 0, 1, 2, 1],
[0, 0, 0, 1, 2]])
assert M.eigenvals() == {
S('1'): 1,
S('2'): 1,
S('3'): 1,
S('sqrt(3) + 2'): 1,
S('-sqrt(3) + 2'): 1}
def test_P24():
M = Matrix([[611, 196, -192, 407, -8, -52, -49, 29],
[196, 899, 113, -192, -71, -43, -8, -44],
[-192, 113, 899, 196, 61, 49, 8, 52],
[ 407, -192, 196, 611, 8, 44, 59, -23],
[ -8, -71, 61, 8, 411, -599, 208, 208],
[ -52, -43, 49, 44, -599, 411, 208, 208],
[ -49, -8, 8, 59, 208, 208, 99, -911],
[ 29, -44, 52, -23, 208, 208, -911, 99]])
assert M.eigenvals() == {
S('0'): 1,
S('10*sqrt(10405)'): 1,
S('100*sqrt(26) + 510'): 1,
S('1000'): 2,
S('-100*sqrt(26) + 510'): 1,
S('-10*sqrt(10405)'): 1,
S('1020'): 1}
def test_P25():
MF = N(Matrix([[ 611, 196, -192, 407, -8, -52, -49, 29],
[ 196, 899, 113, -192, -71, -43, -8, -44],
[-192, 113, 899, 196, 61, 49, 8, 52],
[ 407, -192, 196, 611, 8, 44, 59, -23],
[ -8, -71, 61, 8, 411, -599, 208, 208],
[ -52, -43, 49, 44, -599, 411, 208, 208],
[ -49, -8, 8, 59, 208, 208, 99, -911],
[ 29, -44, 52, -23, 208, 208, -911, 99]]))
assert (Matrix(sorted(MF.eigenvals())) - Matrix(
[-1020.0490184299969, 0.0, 0.09804864072151699, 1000.0,
1019.9019513592784, 1020.0, 1020.0490184299969])).norm() < 1e-13
def test_P26():
a0, a1, a2, a3, a4 = symbols('a0 a1 a2 a3 a4')
M = Matrix([[-a4, -a3, -a2, -a1, -a0, 0, 0, 0, 0],
[ 1, 0, 0, 0, 0, 0, 0, 0, 0],
[ 0, 1, 0, 0, 0, 0, 0, 0, 0],
[ 0, 0, 1, 0, 0, 0, 0, 0, 0],
[ 0, 0, 0, 1, 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0, -1, -1, 0, 0],
[ 0, 0, 0, 0, 0, 1, 0, 0, 0],
[ 0, 0, 0, 0, 0, 0, 1, -1, -1],
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]])
assert M.eigenvals(error_when_incomplete=False) == {
S('-1/2 - sqrt(3)*I/2'): 2,
S('-1/2 + sqrt(3)*I/2'): 2}
def test_P27():
a = symbols('a')
M = Matrix([[a, 0, 0, 0, 0],
[0, 0, 0, 0, 1],
[0, 0, a, 0, 0],
[0, 0, 0, a, 0],
[0, -2, 0, 0, 2]])
assert M.eigenvects() == [(a, 3, [Matrix([[1],
[0],
[0],
[0],
[0]]),
Matrix([[0],
[0],
[1],
[0],
[0]]),
Matrix([[0],
[0],
[0],
[1],
[0]])]),
(1 - I, 1, [Matrix([[ 0],
[-1/(-1 + I)],
[ 0],
[ 0],
[ 1]])]),
(1 + I, 1, [Matrix([[ 0],
[-1/(-1 - I)],
[ 0],
[ 0],
[ 1]])])]
@XFAIL
def test_P28():
raise NotImplementedError("Generalized eigenvectors not supported \
https://github.com/sympy/sympy/issues/5293")
@XFAIL
def test_P29():
raise NotImplementedError("Generalized eigenvectors not supported \
https://github.com/sympy/sympy/issues/5293")
def test_P30():
M = Matrix([[1, 0, 0, 1, -1],
[0, 1, -2, 3, -3],
[0, 0, -1, 2, -2],
[1, -1, 1, 0, 1],
[1, -1, 1, -1, 2]])
_, J = M.jordan_form()
assert J == Matrix([[-1, 0, 0, 0, 0],
[0, 1, 1, 0, 0],
[0, 0, 1, 0, 0],
[0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]])
@XFAIL
def test_P31():
raise NotImplementedError("Smith normal form not implemented")
def test_P32():
M = Matrix([[1, -2],
[2, 1]])
assert exp(M).rewrite(cos).simplify() == Matrix([[E*cos(2), -E*sin(2)],
[E*sin(2), E*cos(2)]])
def test_P33():
w, t = symbols('w t')
M = Matrix([[0, 1, 0, 0],
[0, 0, 0, 2*w],
[0, 0, 0, 1],
[0, -2*w, 3*w**2, 0]])
assert exp(M*t).rewrite(cos).expand() == Matrix([
[1, -3*t + 4*sin(t*w)/w, 6*t*w - 6*sin(t*w), -2*cos(t*w)/w + 2/w],
[0, 4*cos(t*w) - 3, -6*w*cos(t*w) + 6*w, 2*sin(t*w)],
[0, 2*cos(t*w)/w - 2/w, -3*cos(t*w) + 4, sin(t*w)/w],
[0, -2*sin(t*w), 3*w*sin(t*w), cos(t*w)]])
@XFAIL
def test_P34():
a, b, c = symbols('a b c', real=True)
M = Matrix([[a, 1, 0, 0, 0, 0],
[0, a, 0, 0, 0, 0],
[0, 0, b, 0, 0, 0],
[0, 0, 0, c, 1, 0],
[0, 0, 0, 0, c, 1],
[0, 0, 0, 0, 0, c]])
# raises exception, sin(M) not supported. exp(M*I) also not supported
# https://github.com/sympy/sympy/issues/6218
assert sin(M) == Matrix([[sin(a), cos(a), 0, 0, 0, 0],
[0, sin(a), 0, 0, 0, 0],
[0, 0, sin(b), 0, 0, 0],
[0, 0, 0, sin(c), cos(c), -sin(c)/2],
[0, 0, 0, 0, sin(c), cos(c)],
[0, 0, 0, 0, 0, sin(c)]])
@XFAIL
def test_P35():
M = pi/2*Matrix([[2, 1, 1],
[2, 3, 2],
[1, 1, 2]])
# raises exception, sin(M) not supported. exp(M*I) also not supported
# https://github.com/sympy/sympy/issues/6218
assert sin(M) == eye(3)
@XFAIL
def test_P36():
M = Matrix([[10, 7],
[7, 17]])
assert sqrt(M) == Matrix([[3, 1],
[1, 4]])
def test_P37():
M = Matrix([[1, 1, 0],
[0, 1, 0],
[0, 0, 1]])
assert M**Rational(1, 2) == Matrix([[1, R(1, 2), 0],
[0, 1, 0],
[0, 0, 1]])
@XFAIL
def test_P38():
M=Matrix([[0, 1, 0],
[0, 0, 0],
[0, 0, 0]])
#raises ValueError: Matrix det == 0; not invertible
M**Rational(1,2)
@XFAIL
def test_P39():
"""
M=Matrix([
[1, 1],
[2, 2],
[3, 3]])
M.SVD()
"""
raise NotImplementedError("Singular value decomposition not implemented")
def test_P40():
r, t = symbols('r t', real=True)
M = Matrix([r*cos(t), r*sin(t)])
assert M.jacobian(Matrix([r, t])) == Matrix([[cos(t), -r*sin(t)],
[sin(t), r*cos(t)]])
def test_P41():
r, t = symbols('r t', real=True)
assert hessian(r**2*sin(t),(r,t)) == Matrix([[ 2*sin(t), 2*r*cos(t)],
[2*r*cos(t), -r**2*sin(t)]])
def test_P42():
assert wronskian([cos(x), sin(x)], x).simplify() == 1
def test_P43():
def __my_jacobian(M, Y):
return Matrix([M.diff(v).T for v in Y]).T
r, t = symbols('r t', real=True)
M = Matrix([r*cos(t), r*sin(t)])
assert __my_jacobian(M,[r,t]) == Matrix([[cos(t), -r*sin(t)],
[sin(t), r*cos(t)]])
def test_P44():
def __my_hessian(f, Y):
V = Matrix([diff(f, v) for v in Y])
return Matrix([V.T.diff(v) for v in Y])
r, t = symbols('r t', real=True)
assert __my_hessian(r**2*sin(t), (r, t)) == Matrix([
[ 2*sin(t), 2*r*cos(t)],
[2*r*cos(t), -r**2*sin(t)]])
def test_P45():
def __my_wronskian(Y, v):
M = Matrix([Matrix(Y).T.diff(x, n) for n in range(0, len(Y))])
return M.det()
assert __my_wronskian([cos(x), sin(x)], x).simplify() == 1
# Q1-Q6 Tensor tests missing
@XFAIL
def test_R1():
i, j, n = symbols('i j n', integer=True, positive=True)
xn = MatrixSymbol('xn', n, 1)
Sm = Sum((xn[i, 0] - Sum(xn[j, 0], (j, 0, n - 1))/n)**2, (i, 0, n - 1))
# sum does not calculate
# Unknown result
Sm.doit()
raise NotImplementedError('Unknown result')
@XFAIL
def test_R2():
m, b = symbols('m b')
i, n = symbols('i n', integer=True, positive=True)
xn = MatrixSymbol('xn', n, 1)
yn = MatrixSymbol('yn', n, 1)
f = Sum((yn[i, 0] - m*xn[i, 0] - b)**2, (i, 0, n - 1))
f1 = diff(f, m)
f2 = diff(f, b)
# raises TypeError: solveset() takes at most 2 arguments (3 given)
solveset((f1, f2), (m, b), domain=S.Reals)
@XFAIL
def test_R3():
n, k = symbols('n k', integer=True, positive=True)
sk = ((-1)**k) * (binomial(2*n, k))**2
Sm = Sum(sk, (k, 1, oo))
T = Sm.doit()
T2 = T.combsimp()
# returns -((-1)**n*factorial(2*n)
# - (factorial(n))**2)*exp_polar(-I*pi)/(factorial(n))**2
assert T2 == (-1)**n*binomial(2*n, n)
@XFAIL
def test_R4():
# Macsyma indefinite sum test case:
#(c15) /* Check whether the full Gosper algorithm is implemented
# => 1/2^(n + 1) binomial(n, k - 1) */
#closedform(indefsum(binomial(n, k)/2^n - binomial(n + 1, k)/2^(n + 1), k));
#Time= 2690 msecs
# (- n + k - 1) binomial(n + 1, k)
#(d15) - --------------------------------
# n
# 2 2 (n + 1)
#
#(c16) factcomb(makefact(%));
#Time= 220 msecs
# n!
#(d16) ----------------
# n
# 2 k! 2 (n - k)!
# Might be possible after fixing https://github.com/sympy/sympy/pull/1879
raise NotImplementedError("Indefinite sum not supported")
@XFAIL
def test_R5():
a, b, c, n, k = symbols('a b c n k', integer=True, positive=True)
sk = ((-1)**k)*(binomial(a + b, a + k)
*binomial(b + c, b + k)*binomial(c + a, c + k))
Sm = Sum(sk, (k, 1, oo))
T = Sm.doit() # hypergeometric series not calculated
assert T == factorial(a+b+c)/(factorial(a)*factorial(b)*factorial(c))
def test_R6():
n, k = symbols('n k', integer=True, positive=True)
gn = MatrixSymbol('gn', n + 2, 1)
Sm = Sum(gn[k, 0] - gn[k - 1, 0], (k, 1, n + 1))
assert Sm.doit() == -gn[0, 0] + gn[n + 1, 0]
def test_R7():
n, k = symbols('n k', integer=True, positive=True)
T = Sum(k**3,(k,1,n)).doit()
assert T.factor() == n**2*(n + 1)**2/4
@XFAIL
def test_R8():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(k**2*binomial(n, k), (k, 1, n))
T = Sm.doit() #returns Piecewise function
assert T.combsimp() == n*(n + 1)*2**(n - 2)
def test_R9():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(binomial(n, k - 1)/k, (k, 1, n + 1))
assert Sm.doit().simplify() == (2**(n + 1) - 1)/(n + 1)
@XFAIL
def test_R10():
n, m, r, k = symbols('n m r k', integer=True, positive=True)
Sm = Sum(binomial(n, k)*binomial(m, r - k), (k, 0, r))
T = Sm.doit()
T2 = T.combsimp().rewrite(factorial)
assert T2 == factorial(m + n)/(factorial(r)*factorial(m + n - r))
assert T2 == binomial(m + n, r).rewrite(factorial)
# rewrite(binomial) is not working.
# https://github.com/sympy/sympy/issues/7135
T3 = T2.rewrite(binomial)
assert T3 == binomial(m + n, r)
@XFAIL
def test_R11():
n, k = symbols('n k', integer=True, positive=True)
sk = binomial(n, k)*fibonacci(k)
Sm = Sum(sk, (k, 0, n))
T = Sm.doit()
# Fibonacci simplification not implemented
# https://github.com/sympy/sympy/issues/7134
assert T == fibonacci(2*n)
@XFAIL
def test_R12():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(fibonacci(k)**2, (k, 0, n))
T = Sm.doit()
assert T == fibonacci(n)*fibonacci(n + 1)
@XFAIL
def test_R13():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(sin(k*x), (k, 1, n))
T = Sm.doit() # Sum is not calculated
assert T.simplify() == cot(x/2)/2 - cos(x*(2*n + 1)/2)/(2*sin(x/2))
@XFAIL
def test_R14():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(sin((2*k - 1)*x), (k, 1, n))
T = Sm.doit() # Sum is not calculated
assert T.simplify() == sin(n*x)**2/sin(x)
@XFAIL
def test_R15():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(binomial(n - k, k), (k, 0, floor(n/2)))
T = Sm.doit() # Sum is not calculated
assert T.simplify() == fibonacci(n + 1)
def test_R16():
k = symbols('k', integer=True, positive=True)
Sm = Sum(1/k**2 + 1/k**3, (k, 1, oo))
assert Sm.doit() == zeta(3) + pi**2/6
def test_R17():
k = symbols('k', integer=True, positive=True)
assert abs(float(Sum(1/k**2 + 1/k**3, (k, 1, oo)))
- 2.8469909700078206) < 1e-15
def test_R18():
k = symbols('k', integer=True, positive=True)
Sm = Sum(1/(2**k*k**2), (k, 1, oo))
T = Sm.doit()
assert T.simplify() == -log(2)**2/2 + pi**2/12
@slow
@XFAIL
def test_R19():
k = symbols('k', integer=True, positive=True)
Sm = Sum(1/((3*k + 1)*(3*k + 2)*(3*k + 3)), (k, 0, oo))
T = Sm.doit()
# assert fails, T not simplified
assert T.simplify() == -log(3)/4 + sqrt(3)*pi/12
@XFAIL
def test_R20():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(binomial(n, 4*k), (k, 0, oo))
T = Sm.doit()
# assert fails, T not simplified
assert T.simplify() == 2**(n/2)*cos(pi*n/4)/2 + 2**(n - 1)/2
@XFAIL
def test_R21():
k = symbols('k', integer=True, positive=True)
Sm = Sum(1/(sqrt(k*(k + 1)) * (sqrt(k) + sqrt(k + 1))), (k, 1, oo))
T = Sm.doit() # Sum not calculated
assert T.simplify() == 1
# test_R22 answer not available in Wester samples
# Sum(Sum(binomial(n, k)*binomial(n - k, n - 2*k)*x**n*y**(n - 2*k),
# (k, 0, floor(n/2))), (n, 0, oo)) with abs(x*y)<1?
@XFAIL
def test_R23():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(Sum((factorial(n)/(factorial(k)**2*factorial(n - 2*k)))*
(x/y)**k*(x*y)**(n - k), (n, 2*k, oo)), (k, 0, oo))
# Missing how to express constraint abs(x*y)<1?
T = Sm.doit() # Sum not calculated
assert T == -1/sqrt(x**2*y**2 - 4*x**2 - 2*x*y + 1)
def test_R24():
m, k = symbols('m k', integer=True, positive=True)
Sm = Sum(Product(k/(2*k - 1), (k, 1, m)), (m, 2, oo))
assert Sm.doit() == pi/2
def test_S1():
k = symbols('k', integer=True, positive=True)
Pr = Product(gamma(k/3), (k, 1, 8))
assert Pr.doit().simplify() == 640*sqrt(3)*pi**3/6561
def test_S2():
n, k = symbols('n k', integer=True, positive=True)
assert Product(k, (k, 1, n)).doit() == factorial(n)
def test_S3():
n, k = symbols('n k', integer=True, positive=True)
assert Product(x**k, (k, 1, n)).doit().simplify() == x**(n*(n + 1)/2)
def test_S4():
n, k = symbols('n k', integer=True, positive=True)
assert Product(1 + 1/k, (k, 1, n -1)).doit().simplify() == n
def test_S5():
n, k = symbols('n k', integer=True, positive=True)
assert (Product((2*k - 1)/(2*k), (k, 1, n)).doit().gammasimp() ==
gamma(n + Rational(1, 2))/(sqrt(pi)*gamma(n + 1)))
@XFAIL
def test_S6():
n, k = symbols('n k', integer=True, positive=True)
# Product does not evaluate
assert (Product(x**2 -2*x*cos(k*pi/n) + 1, (k, 1, n - 1)).doit().simplify()
== (x**(2*n) - 1)/(x**2 - 1))
@XFAIL
def test_S7():
k = symbols('k', integer=True, positive=True)
Pr = Product((k**3 - 1)/(k**3 + 1), (k, 2, oo))
T = Pr.doit() # Product does not evaluate
assert T.simplify() == Rational(2, 3)
@XFAIL
def test_S8():
k = symbols('k', integer=True, positive=True)
Pr = Product(1 - 1/(2*k)**2, (k, 1, oo))
T = Pr.doit()
# Product does not evaluate
assert T.simplify() == 2/pi
@XFAIL
def test_S9():
k = symbols('k', integer=True, positive=True)
Pr = Product(1 + (-1)**(k + 1)/(2*k - 1), (k, 1, oo))
T = Pr.doit()
# Product produces 0
# https://github.com/sympy/sympy/issues/7133
assert T.simplify() == sqrt(2)
@XFAIL
def test_S10():
k = symbols('k', integer=True, positive=True)
Pr = Product((k*(k + 1) + 1 + I)/(k*(k + 1) + 1 - I), (k, 0, oo))
T = Pr.doit()
# Product does not evaluate
assert T.simplify() == -1
def test_T1():
assert limit((1 + 1/n)**n, n, oo) == E
assert limit((1 - cos(x))/x**2, x, 0) == Rational(1, 2)
def test_T2():
assert limit((3**x + 5**x)**(1/x), x, oo) == 5
def test_T3():
assert limit(log(x)/(log(x) + sin(x)), x, oo) == 1
def test_T4():
assert limit((exp(x*exp(-x)/(exp(-x) + exp(-2*x**2/(x + 1))))
- exp(x))/x, x, oo) == -exp(2)
def test_T5():
assert limit(x*log(x)*log(x*exp(x) - x**2)**2/log(log(x**2
+ 2*exp(exp(3*x**3*log(x))))), x, oo) == Rational(1, 3)
def test_T6():
assert limit(1/n * factorial(n)**(1/n), n, oo) == exp(-1)
def test_T7():
limit(1/n * gamma(n + 1)**(1/n), n, oo)
def test_T8():
a, z = symbols('a z', real=True, positive=True)
assert limit(gamma(z + a)/gamma(z)*exp(-a*log(z)), z, oo) == 1
@XFAIL
def test_T9():
z, k = symbols('z k', real=True, positive=True)
# raises NotImplementedError:
# Don't know how to calculate the mrv of '(1, k)'
assert limit(hyper((1, k), (1,), z/k), k, oo) == exp(z)
@XFAIL
def test_T10():
# No longer raises PoleError, but should return euler-mascheroni constant
assert limit(zeta(x) - 1/(x - 1), x, 1) == integrate(-1/x + 1/floor(x), (x, 1, oo))
@XFAIL
def test_T11():
n, k = symbols('n k', integer=True, positive=True)
# evaluates to 0
assert limit(n**x/(x*product((1 + x/k), (k, 1, n))), n, oo) == gamma(x)
@XFAIL
def test_T12():
x, t = symbols('x t', real=True)
# Does not evaluate the limit but returns an expression with erf
assert limit(x * integrate(exp(-t**2), (t, 0, x))/(1 - exp(-x**2)),
x, 0) == 1
def test_T13():
x = symbols('x', real=True)
assert [limit(x/abs(x), x, 0, dir='-'),
limit(x/abs(x), x, 0, dir='+')] == [-1, 1]
def test_T14():
x = symbols('x', real=True)
assert limit(atan(-log(x)), x, 0, dir='+') == pi/2
def test_U1():
x = symbols('x', real=True)
assert diff(abs(x), x) == sign(x)
def test_U2():
f = Lambda(x, Piecewise((-x, x < 0), (x, x >= 0)))
assert diff(f(x), x) == Piecewise((-1, x < 0), (1, x >= 0))
def test_U3():
f = Lambda(x, Piecewise((x**2 - 1, x == 1), (x**3, x != 1)))
f1 = Lambda(x, diff(f(x), x))
assert f1(x) == 3*x**2
assert f1(1) == 3
@XFAIL
def test_U4():
n = symbols('n', integer=True, positive=True)
x = symbols('x', real=True)
d = diff(x**n, x, n)
assert d.rewrite(factorial) == factorial(n)
def test_U5():
# issue 6681
t = symbols('t')
ans = (
Derivative(f(g(t)), g(t))*Derivative(g(t), (t, 2)) +
Derivative(f(g(t)), (g(t), 2))*Derivative(g(t), t)**2)
assert f(g(t)).diff(t, 2) == ans
assert ans.doit() == ans
def test_U6():
h = Function('h')
T = integrate(f(y), (y, h(x), g(x)))
assert T.diff(x) == (
f(g(x))*Derivative(g(x), x) - f(h(x))*Derivative(h(x), x))
@XFAIL
def test_U7():
p, t = symbols('p t', real=True)
# Exact differential => d(V(P, T)) => dV/dP DP + dV/dT DT
# raises ValueError: Since there is more than one variable in the
# expression, the variable(s) of differentiation must be supplied to
# differentiate f(p,t)
diff(f(p, t))
def test_U8():
x, y = symbols('x y', real=True)
eq = cos(x*y) + x
# If SymPy had implicit_diff() function this hack could be avoided
# TODO: Replace solve with solveset, current test fails for solveset
assert idiff(y - eq, y, x) == (-y*sin(x*y) + 1)/(x*sin(x*y) + 1)
def test_U9():
# Wester sample case for Maple:
# O29 := diff(f(x, y), x) + diff(f(x, y), y);
# /d \ /d \
# |-- f(x, y)| + |-- f(x, y)|
# \dx / \dy /
#
# O30 := factor(subs(f(x, y) = g(x^2 + y^2), %));
# 2 2
# 2 D(g)(x + y ) (x + y)
x, y = symbols('x y', real=True)
su = diff(f(x, y), x) + diff(f(x, y), y)
s2 = su.subs(f(x, y), g(x**2 + y**2))
s3 = s2.doit().factor()
# Subs not performed, s3 = 2*(x + y)*Subs(Derivative(
# g(_xi_1), _xi_1), _xi_1, x**2 + y**2)
# Derivative(g(x*2 + y**2), x**2 + y**2) is not valid in SymPy,
# and probably will remain that way. You can take derivatives with respect
# to other expressions only if they are atomic, like a symbol or a
# function.
# D operator should be added to SymPy
# See https://github.com/sympy/sympy/issues/4719.
assert s3 == (x + y)*Subs(Derivative(g(x), x), x, x**2 + y**2)*2
def test_U10():
# see issue 2519:
assert residue((z**3 + 5)/((z**4 - 1)*(z + 1)), z, -1) == Rational(-9, 4)
@XFAIL
def test_U11():
assert (2*dx + dz) ^ (3*dx + dy + dz) ^ (dx + dy + 4*dz) == 8*dx ^ dy ^dz
@XFAIL
def test_U12():
# Wester sample case:
# (c41) /* d(3 x^5 dy /\ dz + 5 x y^2 dz /\ dx + 8 z dx /\ dy)
# => (15 x^4 + 10 x y + 8) dx /\ dy /\ dz */
# factor(ext_diff(3*x^5 * dy ~ dz + 5*x*y^2 * dz ~ dx + 8*z * dx ~ dy));
# 4
# (d41) (10 x y + 15 x + 8) dx dy dz
raise NotImplementedError(
"External diff of differential form not supported")
@XFAIL
def test_U13():
#assert minimize(x**4 - x + 1, x)== -3*2**Rational(1,3)/8 + 1
raise NotImplementedError("minimize() not supported")
@XFAIL
def test_U14():
#f = 1/(x**2 + y**2 + 1)
#assert [minimize(f), maximize(f)] == [0,1]
raise NotImplementedError("minimize(), maximize() not supported")
@XFAIL
def test_U15():
raise NotImplementedError("minimize() not supported and also solve does \
not support multivariate inequalities")
@XFAIL
def test_U16():
raise NotImplementedError("minimize() not supported in SymPy and also \
solve does not support multivariate inequalities")
@XFAIL
def test_U17():
raise NotImplementedError("Linear programming, symbolic simplex not \
supported in SymPy")
def test_V1():
x = symbols('x', real=True)
assert integrate(abs(x), x) == Piecewise((-x**2/2, x <= 0), (x**2/2, True))
def test_V2():
assert integrate(Piecewise((-x, x < 0), (x, x >= 0)), x
) == Piecewise((-x**2/2, x < 0), (x**2/2, True))
def test_V3():
assert integrate(1/(x**3 + 2),x).diff().simplify() == 1/(x**3 + 2)
def test_V4():
assert integrate(2**x/sqrt(1 + 4**x), x) == asinh(2**x)/log(2)
@XFAIL
def test_V5():
# Returns (-45*x**2 + 80*x - 41)/(5*sqrt(2*x - 1)*(4*x**2 - 4*x + 1))
assert (integrate((3*x - 5)**2/(2*x - 1)**(Rational(7, 2)), x).simplify() ==
(-41 + 80*x - 45*x**2)/(5*(2*x - 1)**Rational(5, 2)))
@XFAIL
def test_V6():
# returns RootSum(40*_z**2 - 1, Lambda(_i, _i*log(-4*_i + exp(-m*x))))/m
assert (integrate(1/(2*exp(m*x) - 5*exp(-m*x)), x) == sqrt(10)*(
log(2*exp(m*x) - sqrt(10)) - log(2*exp(m*x) + sqrt(10)))/(20*m))
def test_V7():
r1 = integrate(sinh(x)**4/cosh(x)**2)
assert r1.simplify() == -3*x/2 + sinh(x)**3/(2*cosh(x)) + 3*tanh(x)/2
@XFAIL
def test_V8_V9():
#Macsyma test case:
#(c27) /* This example involves several symbolic parameters
# => 1/sqrt(b^2 - a^2) log([sqrt(b^2 - a^2) tan(x/2) + a + b]/
# [sqrt(b^2 - a^2) tan(x/2) - a - b]) (a^2 < b^2)
# [Gradshteyn and Ryzhik 2.553(3)] */
#assume(b^2 > a^2)$
#(c28) integrate(1/(a + b*cos(x)), x);
#(c29) trigsimp(ratsimp(diff(%, x)));
# 1
#(d29) ------------
# b cos(x) + a
raise NotImplementedError(
"Integrate with assumption not supported")
def test_V10():
assert integrate(1/(3 + 3*cos(x) + 4*sin(x)), x) == log(tan(x/2) + Rational(3, 4))/4
def test_V11():
r1 = integrate(1/(4 + 3*cos(x) + 4*sin(x)), x)
r2 = factor(r1)
assert (logcombine(r2, force=True) ==
log(((tan(x/2) + 1)/(tan(x/2) + 7))**Rational(1, 3)))
@XFAIL
def test_V12():
r1 = integrate(1/(5 + 3*cos(x) + 4*sin(x)), x)
# Correct result in python2.7.4, wrong result in python3.5
# https://github.com/sympy/sympy/issues/7157
assert r1 == -1/(tan(x/2) + 2)
@XFAIL
def test_V13():
r1 = integrate(1/(6 + 3*cos(x) + 4*sin(x)), x)
# expression not simplified, returns: -sqrt(11)*I*log(tan(x/2) + 4/3
# - sqrt(11)*I/3)/11 + sqrt(11)*I*log(tan(x/2) + 4/3 + sqrt(11)*I/3)/11
assert r1.simplify() == 2*sqrt(11)*atan(sqrt(11)*(3*tan(x/2) + 4)/11)/11
@slow
@XFAIL
def test_V14():
r1 = integrate(log(abs(x**2 - y**2)), x)
# Piecewise result does not simplify to the desired result.
assert (r1.simplify() == x*log(abs(x**2 - y**2))
+ y*log(x + y) - y*log(x - y) - 2*x)
def test_V15():
r1 = integrate(x*acot(x/y), x)
assert simplify(r1 - (x*y + (x**2 + y**2)*acot(x/y))/2) == 0
@XFAIL
def test_V16():
# Integral not calculated
assert integrate(cos(5*x)*Ci(2*x), x) == Ci(2*x)*sin(5*x)/5 - (Si(3*x) + Si(7*x))/10
@XFAIL
def test_V17():
r1 = integrate((diff(f(x), x)*g(x)
- f(x)*diff(g(x), x))/(f(x)**2 - g(x)**2), x)
# integral not calculated
assert simplify(r1 - (f(x) - g(x))/(f(x) + g(x))/2) == 0
@XFAIL
def test_W1():
# The function has a pole at y.
# The integral has a Cauchy principal value of zero but SymPy returns -I*pi
# https://github.com/sympy/sympy/issues/7159
assert integrate(1/(x - y), (x, y - 1, y + 1)) == 0
@XFAIL
def test_W2():
# The function has a pole at y.
# The integral is divergent but SymPy returns -2
# https://github.com/sympy/sympy/issues/7160
# Test case in Macsyma:
# (c6) errcatch(integrate(1/(x - a)^2, x, a - 1, a + 1));
# Integral is divergent
assert integrate(1/(x - y)**2, (x, y - 1, y + 1)) == zoo
@XFAIL
def test_W3():
# integral is not calculated
# https://github.com/sympy/sympy/issues/7161
assert integrate(sqrt(x + 1/x - 2), (x, 0, 1)) == S(4)/3
@XFAIL
def test_W4():
# integral is not calculated
assert integrate(sqrt(x + 1/x - 2), (x, 1, 2)) == -2*sqrt(2)/3 + S(4)/3
@XFAIL
def test_W5():
# integral is not calculated
assert integrate(sqrt(x + 1/x - 2), (x, 0, 2)) == -2*sqrt(2)/3 + S(8)/3
@XFAIL
@slow
def test_W6():
# integral is not calculated
assert integrate(sqrt(2 - 2*cos(2*x))/2, (x, -3*pi/4, -pi/4)) == sqrt(2)
def test_W7():
a = symbols('a', real=True, positive=True)
r1 = integrate(cos(x)/(x**2 + a**2), (x, -oo, oo))
assert r1.simplify() == pi*exp(-a)/a
@XFAIL
def test_W8():
# Test case in Mathematica:
# In[19]:= Integrate[t^(a - 1)/(1 + t), {t, 0, Infinity},
# Assumptions -> 0 < a < 1]
# Out[19]= Pi Csc[a Pi]
raise NotImplementedError(
"Integrate with assumption 0 < a < 1 not supported")
@XFAIL
def test_W9():
# Integrand with a residue at infinity => -2 pi [sin(pi/5) + sin(2pi/5)]
# (principal value) [Levinson and Redheffer, p. 234] *)
r1 = integrate(5*x**3/(1 + x + x**2 + x**3 + x**4), (x, -oo, oo))
r2 = r1.doit()
assert r2 == -2*pi*(sqrt(-sqrt(5)/8 + 5/8) + sqrt(sqrt(5)/8 + 5/8))
@XFAIL
def test_W10():
# integrate(1/[1 + x + x^2 + ... + x^(2 n)], x = -infinity..infinity) =
# 2 pi/(2 n + 1) [1 + cos(pi/[2 n + 1])] csc(2 pi/[2 n + 1])
# [Levinson and Redheffer, p. 255] => 2 pi/5 [1 + cos(pi/5)] csc(2 pi/5) */
r1 = integrate(x/(1 + x + x**2 + x**4), (x, -oo, oo))
r2 = r1.doit()
assert r2 == 2*pi*(sqrt(5)/4 + 5/4)*csc(2*pi/5)/5
@XFAIL
def test_W11():
# integral not calculated
assert (integrate(sqrt(1 - x**2)/(1 + x**2), (x, -1, 1)) ==
pi*(-1 + sqrt(2)))
def test_W12():
p = symbols('p', real=True, positive=True)
q = symbols('q', real=True)
r1 = integrate(x*exp(-p*x**2 + 2*q*x), (x, -oo, oo))
assert r1.simplify() == sqrt(pi)*q*exp(q**2/p)/p**Rational(3, 2)
@XFAIL
def test_W13():
# Integral not calculated. Expected result is 2*(Euler_mascheroni_constant)
r1 = integrate(1/log(x) + 1/(1 - x) - log(log(1/x)), (x, 0, 1))
assert r1 == 2*EulerGamma
def test_W14():
assert integrate(sin(x)/x*exp(2*I*x), (x, -oo, oo)) == 0
@XFAIL
def test_W15():
# integral not calculated
assert integrate(log(gamma(x))*cos(6*pi*x), (x, 0, 1)) == S(1)/12
def test_W16():
assert integrate((1 + x)**3*legendre_poly(1, x)*legendre_poly(2, x),
(x, -1, 1)) == S(36)/35
def test_W17():
a, b = symbols('a b', real=True, positive=True)
assert integrate(exp(-a*x)*besselj(0, b*x),
(x, 0, oo)) == 1/(b*sqrt(a**2/b**2 + 1))
def test_W18():
assert integrate((besselj(1, x)/x)**2, (x, 0, oo)) == 4/(3*pi)
@XFAIL
def test_W19():
# Integral not calculated
# Expected result is (cos 7 - 1)/7 [Gradshteyn and Ryzhik 6.782(3)]
assert integrate(Ci(x)*besselj(0, 2*sqrt(7*x)), (x, 0, oo)) == (cos(7) - 1)/7
@XFAIL
def test_W20():
# integral not calculated
assert (integrate(x**2*polylog(3, 1/(x + 1)), (x, 0, 1)) ==
-pi**2/36 - S(17)/108 + zeta(3)/4 +
(-pi**2/2 - 4*log(2) + log(2)**2 + 35/3)*log(2)/9)
def test_W21():
assert abs(N(integrate(x**2*polylog(3, 1/(x + 1)), (x, 0, 1)))
- 0.210882859565594) < 1e-15
def test_W22():
t, u = symbols('t u', real=True)
s = Lambda(x, Piecewise((1, And(x >= 1, x <= 2)), (0, True)))
assert integrate(s(t)*cos(t), (t, 0, u)) == Piecewise(
(0, u < 0),
(-sin(Min(1, u)) + sin(Min(2, u)), True))
@slow
def test_W23():
a, b = symbols('a b', real=True, positive=True)
r1 = integrate(integrate(x/(x**2 + y**2), (x, a, b)), (y, -oo, oo))
assert r1.collect(pi) == pi*(-a + b)
def test_W23b():
# like W23 but limits are reversed
x = symbols('x', real=True, positive=True)
y = symbols('y', real=True)
a, b = symbols('a b', real=True, positive=True)
r2 = integrate(integrate(x/(x**2 + y**2), (y, -oo, oo)), (x, a, b))
assert r2.collect(pi) == pi*(-a + b)
@XFAIL
@slow
def test_W24():
if ON_TRAVIS:
skip("Too slow for travis.")
# Not that slow, but does not fully evaluate so simplify is slow.
# Maybe also require doit()
x, y = symbols('x y', real=True)
r1 = integrate(integrate(sqrt(x**2 + y**2), (x, 0, 1)), (y, 0, 1))
assert (r1 - (sqrt(2) + asinh(1))/3).simplify() == 0
@XFAIL
@slow
def test_W25():
if ON_TRAVIS:
skip("Too slow for travis.")
a, x, y = symbols('a x y', real=True)
i1 = integrate(
sin(a)*sin(y)/sqrt(1 - sin(a)**2*sin(x)**2*sin(y)**2),
(x, 0, pi/2))
i2 = integrate(i1, (y, 0, pi/2))
assert (i2 - pi*a/2).simplify() == 0
def test_W26():
x, y = symbols('x y', real=True)
assert integrate(integrate(abs(y - x**2), (y, 0, 2)),
(x, -1, 1)) == S(46)/15
def test_W27():
a, b, c = symbols('a b c')
assert integrate(integrate(integrate(1, (z, 0, c*(1 - x/a - y/b))),
(y, 0, b*(1 - x/a))),
(x, 0, a)) == a*b*c/6
def test_X1():
v, c = symbols('v c', real=True)
assert (series(1/sqrt(1 - (v/c)**2), v, x0=0, n=8) ==
5*v**6/(16*c**6) + 3*v**4/(8*c**4) + v**2/(2*c**2) + 1 + O(v**8))
def test_X2():
v, c = symbols('v c', real=True)
s1 = series(1/sqrt(1 - (v/c)**2), v, x0=0, n=8)
assert (1/s1**2).series(v, x0=0, n=8) == -v**2/c**2 + 1 + O(v**8)
def test_X3():
s1 = (sin(x).series()/cos(x).series()).series()
s2 = tan(x).series()
assert s2 == x + x**3/3 + 2*x**5/15 + O(x**6)
assert s1 == s2
def test_X4():
s1 = log(sin(x)/x).series()
assert s1 == -x**2/6 - x**4/180 + O(x**6)
assert log(series(sin(x)/x)).series() == s1
@XFAIL
def test_X5():
# test case in Mathematica syntax:
# In[21]:= (* => [a f'(a d) + g(b d) + integrate(h(c y), y = 0..d)]
# + [a^2 f''(a d) + b g'(b d) + h(c d)] (x - d) *)
# In[22]:= D[f[a*x], x] + g[b*x] + Integrate[h[c*y], {y, 0, x}]
# Out[22]= g[b x] + Integrate[h[c y], {y, 0, x}] + a f'[a x]
# In[23]:= Series[%, {x, d, 1}]
# Out[23]= (g[b d] + Integrate[h[c y], {y, 0, d}] + a f'[a d]) +
# 2 2
# (h[c d] + b g'[b d] + a f''[a d]) (-d + x) + O[-d + x]
h = Function('h')
a, b, c, d = symbols('a b c d', real=True)
# series() raises NotImplementedError:
# The _eval_nseries method should be added to <class
# 'sympy.core.function.Subs'> to give terms up to O(x**n) at x=0
series(diff(f(a*x), x) + g(b*x) + integrate(h(c*y), (y, 0, x)),
x, x0=d, n=2)
# assert missing, until exception is removed
def test_X6():
# Taylor series of nonscalar objects (noncommutative multiplication)
# expected result => (B A - A B) t^2/2 + O(t^3) [Stanly Steinberg]
a, b = symbols('a b', commutative=False, scalar=False)
assert (series(exp((a + b)*x) - exp(a*x) * exp(b*x), x, x0=0, n=3) ==
x**2*(-a*b/2 + b*a/2) + O(x**3))
def test_X7():
# => sum( Bernoulli[k]/k! x^(k - 2), k = 1..infinity )
# = 1/x^2 - 1/(2 x) + 1/12 - x^2/720 + x^4/30240 + O(x^6)
# [Levinson and Redheffer, p. 173]
assert (series(1/(x*(exp(x) - 1)), x, 0, 7) == x**(-2) - 1/(2*x) +
S(1)/12 - x**2/720 + x**4/30240 - x**6/1209600 + O(x**7))
def test_X8():
# Puiseux series (terms with fractional degree):
# => 1/sqrt(x - 3/2 pi) + (x - 3/2 pi)^(3/2) / 12 + O([x - 3/2 pi]^(7/2))
# see issue 7167:
x = symbols('x', real=True)
assert (series(sqrt(sec(x)), x, x0=pi*3/2, n=4) ==
1/sqrt(x - 3*pi/2) + (x - 3*pi/2)**(S(3)/2)/12 +
(x - 3*pi/2)**(S(7)/2)/160 + O((x - 3*pi/2)**4, (x, 3*pi/2)))
def test_X9():
assert (series(x**x, x, x0=0, n=4) == 1 + x*log(x) + x**2*log(x)**2/2 +
x**3*log(x)**3/6 + O(x**4*log(x)**4))
def test_X10():
z, w = symbols('z w')
assert (series(log(sinh(z)) + log(cosh(z + w)), z, x0=0, n=2) ==
log(cosh(w)) + log(z) + z*sinh(w)/cosh(w) + O(z**2))
def test_X11():
z, w = symbols('z w')
assert (series(log(sinh(z) * cosh(z + w)), z, x0=0, n=2) ==
log(cosh(w)) + log(z) + z*sinh(w)/cosh(w) + O(z**2))
@XFAIL
def test_X12():
# Look at the generalized Taylor series around x = 1
# Result => (x - 1)^a/e^b [1 - (a + 2 b) (x - 1) / 2 + O((x - 1)^2)]
a, b, x = symbols('a b x', real=True)
# series returns O(log(x-1)**2)
# https://github.com/sympy/sympy/issues/7168
assert (series(log(x)**a*exp(-b*x), x, x0=1, n=2) ==
(x - 1)**a/exp(b)*(1 - (a + 2*b)*(x - 1)/2 + O((x - 1)**2)))
def test_X13():
assert series(sqrt(2*x**2 + 1), x, x0=oo, n=1) == sqrt(2)*x + O(1/x, (x, oo))
@XFAIL
def test_X14():
# Wallis' product => 1/sqrt(pi n) + ... [Knopp, p. 385]
assert series(1/2**(2*n)*binomial(2*n, n),
n, x==oo, n=1) == 1/(sqrt(pi)*sqrt(n)) + O(1/x, (x, oo))
@SKIP("https://github.com/sympy/sympy/issues/7164")
def test_X15():
# => 0!/x - 1!/x^2 + 2!/x^3 - 3!/x^4 + O(1/x^5) [Knopp, p. 544]
x, t = symbols('x t', real=True)
# raises RuntimeError: maximum recursion depth exceeded
# https://github.com/sympy/sympy/issues/7164
# 2019-02-17: Raises
# PoleError:
# Asymptotic expansion of Ei around [-oo] is not implemented.
e1 = integrate(exp(-t)/t, (t, x, oo))
assert (series(e1, x, x0=oo, n=5) ==
6/x**4 + 2/x**3 - 1/x**2 + 1/x + O(x**(-5), (x, oo)))
def test_X16():
# Multivariate Taylor series expansion => 1 - (x^2 + 2 x y + y^2)/2 + O(x^4)
assert (series(cos(x + y), x + y, x0=0, n=4) == 1 - (x + y)**2/2 +
O(x**4 + x**3*y + x**2*y**2 + x*y**3 + y**4, x, y))
@XFAIL
def test_X17():
# Power series (compute the general formula)
# (c41) powerseries(log(sin(x)/x), x, 0);
# /aquarius/data2/opt/local/macsyma_422/library1/trgred.so being loaded.
# inf
# ==== i1 2 i1 2 i1
# \ (- 1) 2 bern(2 i1) x
# (d41) > ------------------------------
# / 2 i1 (2 i1)!
# ====
# i1 = 1
# fps does not calculate
assert fps(log(sin(x)/x)) == \
Sum((-1)**k*2**(2*k - 1)*bernoulli(2*k)*x**(2*k)/(k*factorial(2*k)), (k, 1, oo))
@XFAIL
def test_X18():
# Power series (compute the general formula). Maple FPS:
# > FormalPowerSeries(exp(-x)*sin(x), x = 0);
# infinity
# ----- (1/2 k) k
# \ 2 sin(3/4 k Pi) x
# ) -------------------------
# / k!
# -----
#
# Now, sympy returns
# oo
# _____
# \ `
# \ / k k\
# \ k |I*(-1 - I) I*(-1 + I) |
# \ x *|----------- - -----------|
# / \ 2 2 /
# / ------------------------------
# / k!
# /____,
# k = 0
k = Dummy('k')
assert fps(exp(-x)*sin(x)) == \
Sum(2**(S(1)/2*k)*sin(S(3)/4*k*pi)*x**k/factorial(k), (k, 0, oo))
@XFAIL
def test_X19():
# (c45) /* Derive an explicit Taylor series solution of y as a function of
# x from the following implicit relation:
# y = x - 1 + (x - 1)^2/2 + 2/3 (x - 1)^3 + (x - 1)^4 +
# 17/10 (x - 1)^5 + ...
# */
# x = sin(y) + cos(y);
# Time= 0 msecs
# (d45) x = sin(y) + cos(y)
#
# (c46) taylor_revert(%, y, 7);
raise NotImplementedError("Solve using series not supported. \
Inverse Taylor series expansion also not supported")
@XFAIL
def test_X20():
# Pade (rational function) approximation => (2 - x)/(2 + x)
# > numapprox[pade](exp(-x), x = 0, [1, 1]);
# bytes used=9019816, alloc=3669344, time=13.12
# 1 - 1/2 x
# ---------
# 1 + 1/2 x
# mpmath support numeric Pade approximant but there is
# no symbolic implementation in SymPy
# https://en.wikipedia.org/wiki/Pad%C3%A9_approximant
raise NotImplementedError("Symbolic Pade approximant not supported")
def test_X21():
"""
Test whether `fourier_series` of x periodical on the [-p, p] interval equals
`- (2 p / pi) sum( (-1)^n / n sin(n pi x / p), n = 1..infinity )`.
"""
p = symbols('p', positive=True)
n = symbols('n', positive=True, integer=True)
s = fourier_series(x, (x, -p, p))
# All cosine coefficients are equal to 0
assert s.an.formula == 0
# Check for sine coefficients
assert s.bn.formula.subs(s.bn.variables[0], 0) == 0
assert s.bn.formula.subs(s.bn.variables[0], n) == \
-2*p/pi * (-1)**n / n * sin(n*pi*x/p)
@XFAIL
def test_X22():
# (c52) /* => p / 2
# - (2 p / pi^2) sum( [1 - (-1)^n] cos(n pi x / p) / n^2,
# n = 1..infinity ) */
# fourier_series(abs(x), x, p);
# p
# (e52) a = -
# 0 2
#
# %nn
# (2 (- 1) - 2) p
# (e53) a = ------------------
# %nn 2 2
# %pi %nn
#
# (e54) b = 0
# %nn
#
# Time= 5290 msecs
# inf %nn %pi %nn x
# ==== (2 (- 1) - 2) cos(---------)
# \ p
# p > -------------------------------
# / 2
# ==== %nn
# %nn = 1 p
# (d54) ----------------------------------------- + -
# 2 2
# %pi
raise NotImplementedError("Fourier series not supported")
def test_Y1():
t = symbols('t', real=True, positive=True)
w = symbols('w', real=True)
s = symbols('s')
F, _, _ = laplace_transform(cos((w - 1)*t), t, s)
assert F == s/(s**2 + (w - 1)**2)
def test_Y2():
t = symbols('t', real=True, positive=True)
w = symbols('w', real=True)
s = symbols('s')
f = inverse_laplace_transform(s/(s**2 + (w - 1)**2), s, t)
assert f == cos(t*w - t)
def test_Y3():
t = symbols('t', real=True, positive=True)
w = symbols('w', real=True)
s = symbols('s')
F, _, _ = laplace_transform(sinh(w*t)*cosh(w*t), t, s)
assert F == w/(s**2 - 4*w**2)
def test_Y4():
t = symbols('t', real=True, positive=True)
s = symbols('s')
F, _, _ = laplace_transform(erf(3/sqrt(t)), t, s)
assert F == (1 - exp(-6*sqrt(s)))/s
@XFAIL
def test_Y5_Y6():
# Solve y'' + y = 4 [H(t - 1) - H(t - 2)], y(0) = 1, y'(0) = 0 where H is the
# Heaviside (unit step) function (the RHS describes a pulse of magnitude 4 and
# duration 1). See David A. Sanchez, Richard C. Allen, Jr. and Walter T.
# Kyner, _Differential Equations: An Introduction_, Addison-Wesley Publishing
# Company, 1983, p. 211. First, take the Laplace transform of the ODE
# => s^2 Y(s) - s + Y(s) = 4/s [e^(-s) - e^(-2 s)]
# where Y(s) is the Laplace transform of y(t)
t = symbols('t', real=True, positive=True)
s = symbols('s')
y = Function('y')
F, _, _ = laplace_transform(diff(y(t), t, 2)
+ y(t)
- 4*(Heaviside(t - 1)
- Heaviside(t - 2)), t, s)
# Laplace transform for diff() not calculated
# https://github.com/sympy/sympy/issues/7176
assert (F == s**2*LaplaceTransform(y(t), t, s) - s
+ LaplaceTransform(y(t), t, s) - 4*exp(-s)/s + 4*exp(-2*s)/s)
# TODO implement second part of test case
# Now, solve for Y(s) and then take the inverse Laplace transform
# => Y(s) = s/(s^2 + 1) + 4 [1/s - s/(s^2 + 1)] [e^(-s) - e^(-2 s)]
# => y(t) = cos t + 4 {[1 - cos(t - 1)] H(t - 1) - [1 - cos(t - 2)] H(t - 2)}
@XFAIL
def test_Y7():
# What is the Laplace transform of an infinite square wave?
# => 1/s + 2 sum( (-1)^n e^(- s n a)/s, n = 1..infinity )
# [Sanchez, Allen and Kyner, p. 213]
t = symbols('t', real=True, positive=True)
a = symbols('a', real=True)
s = symbols('s')
F, _, _ = laplace_transform(1 + 2*Sum((-1)**n*Heaviside(t - n*a),
(n, 1, oo)), t, s)
# returns 2*LaplaceTransform(Sum((-1)**n*Heaviside(-a*n + t),
# (n, 1, oo)), t, s) + 1/s
# https://github.com/sympy/sympy/issues/7177
assert F == 2*Sum((-1)**n*exp(-a*n*s)/s, (n, 1, oo)) + 1/s
@XFAIL
def test_Y8():
assert fourier_transform(1, x, z) == DiracDelta(z)
def test_Y9():
assert (fourier_transform(exp(-9*x**2), x, z) ==
sqrt(pi)*exp(-pi**2*z**2/9)/3)
def test_Y10():
assert (fourier_transform(abs(x)*exp(-3*abs(x)), x, z) ==
(-8*pi**2*z**2 + 18)/(16*pi**4*z**4 + 72*pi**2*z**2 + 81))
@SKIP("https://github.com/sympy/sympy/issues/7181")
@slow
def test_Y11():
# => pi cot(pi s) (0 < Re s < 1) [Gradshteyn and Ryzhik 17.43(5)]
x, s = symbols('x s')
# raises RuntimeError: maximum recursion depth exceeded
# https://github.com/sympy/sympy/issues/7181
# Update 2019-02-17 raises:
# TypeError: cannot unpack non-iterable MellinTransform object
F, _, _ = mellin_transform(1/(1 - x), x, s)
assert F == pi*cot(pi*s)
@XFAIL
def test_Y12():
# => 2^(s - 4) gamma(s/2)/gamma(4 - s/2) (0 < Re s < 1)
# [Gradshteyn and Ryzhik 17.43(16)]
x, s = symbols('x s')
# returns Wrong value -2**(s - 4)*gamma(s/2 - 3)/gamma(-s/2 + 1)
# https://github.com/sympy/sympy/issues/7182
F, _, _ = mellin_transform(besselj(3, x)/x**3, x, s)
assert F == -2**(s - 4)*gamma(s/2)/gamma(-s/2 + 4)
@XFAIL
def test_Y13():
# Z[H(t - m T)] => z/[z^m (z - 1)] (H is the Heaviside (unit step) function) z
raise NotImplementedError("z-transform not supported")
@XFAIL
def test_Y14():
# Z[H(t - m T)] => z/[z^m (z - 1)] (H is the Heaviside (unit step) function)
raise NotImplementedError("z-transform not supported")
def test_Z1():
r = Function('r')
assert (rsolve(r(n + 2) - 2*r(n + 1) + r(n) - 2, r(n),
{r(0): 1, r(1): m}).simplify() == n**2 + n*(m - 2) + 1)
def test_Z2():
r = Function('r')
assert (rsolve(r(n) - (5*r(n - 1) - 6*r(n - 2)), r(n), {r(0): 0, r(1): 1})
== -2**n + 3**n)
def test_Z3():
# => r(n) = Fibonacci[n + 1] [Cohen, p. 83]
r = Function('r')
# recurrence solution is correct, Wester expects it to be simplified to
# fibonacci(n+1), but that is quite hard
assert (rsolve(r(n) - (r(n - 1) + r(n - 2)), r(n),
{r(1): 1, r(2): 2}).simplify()
== 2**(-n)*((1 + sqrt(5))**n*(sqrt(5) + 5) +
(-sqrt(5) + 1)**n*(-sqrt(5) + 5))/10)
@XFAIL
def test_Z4():
# => [c^(n+1) [c^(n+1) - 2 c - 2] + (n+1) c^2 + 2 c - n] / [(c-1)^3 (c+1)]
# [Joan Z. Yu and Robert Israel in sci.math.symbolic]
r = Function('r')
c = symbols('c')
# raises ValueError: Polynomial or rational function expected,
# got '(c**2 - c**n)/(c - c**n)
s = rsolve(r(n) - ((1 + c - c**(n-1) - c**(n+1))/(1 - c**n)*r(n - 1)
- c*(1 - c**(n-2))/(1 - c**(n-1))*r(n - 2) + 1),
r(n), {r(1): 1, r(2): (2 + 2*c + c**2)/(1 + c)})
assert (s - (c*(n + 1)*(c*(n + 1) - 2*c - 2) +
(n + 1)*c**2 + 2*c - n)/((c-1)**3*(c+1)) == 0)
@XFAIL
def test_Z5():
# Second order ODE with initial conditions---solve directly
# transform: f(t) = sin(2 t)/8 - t cos(2 t)/4
C1, C2 = symbols('C1 C2')
# initial conditions not supported, this is a manual workaround
# https://github.com/sympy/sympy/issues/4720
eq = Derivative(f(x), x, 2) + 4*f(x) - sin(2*x)
sol = dsolve(eq, f(x))
f0 = Lambda(x, sol.rhs)
assert f0(x) == C2*sin(2*x) + (C1 - x/4)*cos(2*x)
f1 = Lambda(x, diff(f0(x), x))
# TODO: Replace solve with solveset, when it works for solveset
const_dict = solve((f0(0), f1(0)))
result = f0(x).subs(C1, const_dict[C1]).subs(C2, const_dict[C2])
assert result == -x*cos(2*x)/4 + sin(2*x)/8
# Result is OK, but ODE solving with initial conditions should be
# supported without all this manual work
raise NotImplementedError('ODE solving with initial conditions \
not supported')
@XFAIL
def test_Z6():
# Second order ODE with initial conditions---solve using Laplace
# transform: f(t) = sin(2 t)/8 - t cos(2 t)/4
t = symbols('t', real=True, positive=True)
s = symbols('s')
eq = Derivative(f(t), t, 2) + 4*f(t) - sin(2*t)
F, _, _ = laplace_transform(eq, t, s)
# Laplace transform for diff() not calculated
# https://github.com/sympy/sympy/issues/7176
assert (F == s**2*LaplaceTransform(f(t), t, s) +
4*LaplaceTransform(f(t), t, s) - 2/(s**2 + 4))
# rest of test case not implemented
|
ba5e3b4a0d3d41b92dda577b217a4f34225e0960c8c245b1a27129b4b08510c7
|
from __future__ import print_function
from textwrap import dedent
from sympy import (
symbols, Integral, Tuple, Dummy, Basic, default_sort_key, Matrix,
factorial, true)
from sympy.combinatorics import RGS_enum, RGS_unrank, Permutation
from sympy.core.compatibility import range
from sympy.utilities.iterables import (
_partition, _set_partitions, binary_partitions, bracelets, capture,
cartes, common_prefix, common_suffix, connected_components, dict_merge,
filter_symbols, flatten, generate_bell, generate_derangements,
generate_involutions, generate_oriented_forest, group, has_dups, ibin,
kbins, minlex, multiset, multiset_combinations, multiset_partitions,
multiset_permutations, necklaces, numbered_symbols, ordered, partitions,
permutations, postfixes, postorder_traversal, prefixes, reshape,
rotate_left, rotate_right, runs, sift, strongly_connected_components,
subsets, take, topological_sort, unflatten, uniq, variations,
ordered_partitions, rotations)
from sympy.utilities.enumerative import (
factoring_visitor, multiset_partitions_taocp )
from sympy.core.singleton import S
from sympy.functions.elementary.piecewise import Piecewise, ExprCondPair
from sympy.utilities.pytest import raises
w, x, y, z = symbols('w,x,y,z')
def test_postorder_traversal():
expr = z + w*(x + y)
expected = [z, w, x, y, x + y, w*(x + y), w*(x + y) + z]
assert list(postorder_traversal(expr, keys=default_sort_key)) == expected
assert list(postorder_traversal(expr, keys=True)) == expected
expr = Piecewise((x, x < 1), (x**2, True))
expected = [
x, 1, x, x < 1, ExprCondPair(x, x < 1),
2, x, x**2, true,
ExprCondPair(x**2, True), Piecewise((x, x < 1), (x**2, True))
]
assert list(postorder_traversal(expr, keys=default_sort_key)) == expected
assert list(postorder_traversal(
[expr], keys=default_sort_key)) == expected + [[expr]]
assert list(postorder_traversal(Integral(x**2, (x, 0, 1)),
keys=default_sort_key)) == [
2, x, x**2, 0, 1, x, Tuple(x, 0, 1),
Integral(x**2, Tuple(x, 0, 1))
]
assert list(postorder_traversal(('abc', ('d', 'ef')))) == [
'abc', 'd', 'ef', ('d', 'ef'), ('abc', ('d', 'ef'))]
def test_flatten():
assert flatten((1, (1,))) == [1, 1]
assert flatten((x, (x,))) == [x, x]
ls = [[(-2, -1), (1, 2)], [(0, 0)]]
assert flatten(ls, levels=0) == ls
assert flatten(ls, levels=1) == [(-2, -1), (1, 2), (0, 0)]
assert flatten(ls, levels=2) == [-2, -1, 1, 2, 0, 0]
assert flatten(ls, levels=3) == [-2, -1, 1, 2, 0, 0]
raises(ValueError, lambda: flatten(ls, levels=-1))
class MyOp(Basic):
pass
assert flatten([MyOp(x, y), z]) == [MyOp(x, y), z]
assert flatten([MyOp(x, y), z], cls=MyOp) == [x, y, z]
assert flatten({1, 11, 2}) == list({1, 11, 2})
def test_group():
assert group([]) == []
assert group([], multiple=False) == []
assert group([1]) == [[1]]
assert group([1], multiple=False) == [(1, 1)]
assert group([1, 1]) == [[1, 1]]
assert group([1, 1], multiple=False) == [(1, 2)]
assert group([1, 1, 1]) == [[1, 1, 1]]
assert group([1, 1, 1], multiple=False) == [(1, 3)]
assert group([1, 2, 1]) == [[1], [2], [1]]
assert group([1, 2, 1], multiple=False) == [(1, 1), (2, 1), (1, 1)]
assert group([1, 1, 2, 2, 2, 1, 3, 3]) == [[1, 1], [2, 2, 2], [1], [3, 3]]
assert group([1, 1, 2, 2, 2, 1, 3, 3], multiple=False) == [(1, 2),
(2, 3), (1, 1), (3, 2)]
def test_subsets():
# combinations
assert list(subsets([1, 2, 3], 0)) == [()]
assert list(subsets([1, 2, 3], 1)) == [(1,), (2,), (3,)]
assert list(subsets([1, 2, 3], 2)) == [(1, 2), (1, 3), (2, 3)]
assert list(subsets([1, 2, 3], 3)) == [(1, 2, 3)]
l = list(range(4))
assert list(subsets(l, 0, repetition=True)) == [()]
assert list(subsets(l, 1, repetition=True)) == [(0,), (1,), (2,), (3,)]
assert list(subsets(l, 2, repetition=True)) == [(0, 0), (0, 1), (0, 2),
(0, 3), (1, 1), (1, 2),
(1, 3), (2, 2), (2, 3),
(3, 3)]
assert list(subsets(l, 3, repetition=True)) == [(0, 0, 0), (0, 0, 1),
(0, 0, 2), (0, 0, 3),
(0, 1, 1), (0, 1, 2),
(0, 1, 3), (0, 2, 2),
(0, 2, 3), (0, 3, 3),
(1, 1, 1), (1, 1, 2),
(1, 1, 3), (1, 2, 2),
(1, 2, 3), (1, 3, 3),
(2, 2, 2), (2, 2, 3),
(2, 3, 3), (3, 3, 3)]
assert len(list(subsets(l, 4, repetition=True))) == 35
assert list(subsets(l[:2], 3, repetition=False)) == []
assert list(subsets(l[:2], 3, repetition=True)) == [(0, 0, 0),
(0, 0, 1),
(0, 1, 1),
(1, 1, 1)]
assert list(subsets([1, 2], repetition=True)) == \
[(), (1,), (2,), (1, 1), (1, 2), (2, 2)]
assert list(subsets([1, 2], repetition=False)) == \
[(), (1,), (2,), (1, 2)]
assert list(subsets([1, 2, 3], 2)) == \
[(1, 2), (1, 3), (2, 3)]
assert list(subsets([1, 2, 3], 2, repetition=True)) == \
[(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)]
def test_variations():
# permutations
l = list(range(4))
assert list(variations(l, 0, repetition=False)) == [()]
assert list(variations(l, 1, repetition=False)) == [(0,), (1,), (2,), (3,)]
assert list(variations(l, 2, repetition=False)) == [(0, 1), (0, 2), (0, 3), (1, 0), (1, 2), (1, 3), (2, 0), (2, 1), (2, 3), (3, 0), (3, 1), (3, 2)]
assert list(variations(l, 3, repetition=False)) == [(0, 1, 2), (0, 1, 3), (0, 2, 1), (0, 2, 3), (0, 3, 1), (0, 3, 2), (1, 0, 2), (1, 0, 3), (1, 2, 0), (1, 2, 3), (1, 3, 0), (1, 3, 2), (2, 0, 1), (2, 0, 3), (2, 1, 0), (2, 1, 3), (2, 3, 0), (2, 3, 1), (3, 0, 1), (3, 0, 2), (3, 1, 0), (3, 1, 2), (3, 2, 0), (3, 2, 1)]
assert list(variations(l, 0, repetition=True)) == [()]
assert list(variations(l, 1, repetition=True)) == [(0,), (1,), (2,), (3,)]
assert list(variations(l, 2, repetition=True)) == [(0, 0), (0, 1), (0, 2),
(0, 3), (1, 0), (1, 1),
(1, 2), (1, 3), (2, 0),
(2, 1), (2, 2), (2, 3),
(3, 0), (3, 1), (3, 2),
(3, 3)]
assert len(list(variations(l, 3, repetition=True))) == 64
assert len(list(variations(l, 4, repetition=True))) == 256
assert list(variations(l[:2], 3, repetition=False)) == []
assert list(variations(l[:2], 3, repetition=True)) == [
(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1),
(1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)
]
def test_cartes():
assert list(cartes([1, 2], [3, 4, 5])) == \
[(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)]
assert list(cartes()) == [()]
assert list(cartes('a')) == [('a',)]
assert list(cartes('a', repeat=2)) == [('a', 'a')]
assert list(cartes(list(range(2)))) == [(0,), (1,)]
def test_filter_symbols():
s = numbered_symbols()
filtered = filter_symbols(s, symbols("x0 x2 x3"))
assert take(filtered, 3) == list(symbols("x1 x4 x5"))
def test_numbered_symbols():
s = numbered_symbols(cls=Dummy)
assert isinstance(next(s), Dummy)
assert next(numbered_symbols('C', start=1, exclude=[symbols('C1')])) == \
symbols('C2')
def test_sift():
assert sift(list(range(5)), lambda _: _ % 2) == {1: [1, 3], 0: [0, 2, 4]}
assert sift([x, y], lambda _: _.has(x)) == {False: [y], True: [x]}
assert sift([S.One], lambda _: _.has(x)) == {False: [1]}
assert sift([0, 1, 2, 3], lambda x: x % 2, binary=True) == (
[1, 3], [0, 2])
assert sift([0, 1, 2, 3], lambda x: x % 3 == 1, binary=True) == (
[1], [0, 2, 3])
raises(ValueError, lambda:
sift([0, 1, 2, 3], lambda x: x % 3, binary=True))
def test_take():
X = numbered_symbols()
assert take(X, 5) == list(symbols('x0:5'))
assert take(X, 5) == list(symbols('x5:10'))
assert take([1, 2, 3, 4, 5], 5) == [1, 2, 3, 4, 5]
def test_dict_merge():
assert dict_merge({}, {1: x, y: z}) == {1: x, y: z}
assert dict_merge({1: x, y: z}, {}) == {1: x, y: z}
assert dict_merge({2: z}, {1: x, y: z}) == {1: x, 2: z, y: z}
assert dict_merge({1: x, y: z}, {2: z}) == {1: x, 2: z, y: z}
assert dict_merge({1: y, 2: z}, {1: x, y: z}) == {1: x, 2: z, y: z}
assert dict_merge({1: x, y: z}, {1: y, 2: z}) == {1: y, 2: z, y: z}
def test_prefixes():
assert list(prefixes([])) == []
assert list(prefixes([1])) == [[1]]
assert list(prefixes([1, 2])) == [[1], [1, 2]]
assert list(prefixes([1, 2, 3, 4, 5])) == \
[[1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4, 5]]
def test_postfixes():
assert list(postfixes([])) == []
assert list(postfixes([1])) == [[1]]
assert list(postfixes([1, 2])) == [[2], [1, 2]]
assert list(postfixes([1, 2, 3, 4, 5])) == \
[[5], [4, 5], [3, 4, 5], [2, 3, 4, 5], [1, 2, 3, 4, 5]]
def test_topological_sort():
V = [2, 3, 5, 7, 8, 9, 10, 11]
E = [(7, 11), (7, 8), (5, 11),
(3, 8), (3, 10), (11, 2),
(11, 9), (11, 10), (8, 9)]
assert topological_sort((V, E)) == [3, 5, 7, 8, 11, 2, 9, 10]
assert topological_sort((V, E), key=lambda v: -v) == \
[7, 5, 11, 3, 10, 8, 9, 2]
raises(ValueError, lambda: topological_sort((V, E + [(10, 7)])))
def test_strongly_connected_components():
assert strongly_connected_components(([], [])) == []
assert strongly_connected_components(([1, 2, 3], [])) == [[1], [2], [3]]
V = [1, 2, 3]
E = [(1, 2), (1, 3), (2, 1), (2, 3), (3, 1)]
assert strongly_connected_components((V, E)) == [[1, 2, 3]]
V = [1, 2, 3, 4]
E = [(1, 2), (2, 3), (3, 2), (3, 4)]
assert strongly_connected_components((V, E)) == [[4], [2, 3], [1]]
V = [1, 2, 3, 4]
E = [(1, 2), (2, 1), (3, 4), (4, 3)]
assert strongly_connected_components((V, E)) == [[1, 2], [3, 4]]
def test_connected_components():
assert connected_components(([], [])) == []
assert connected_components(([1, 2, 3], [])) == [[1], [2], [3]]
V = [1, 2, 3]
E = [(1, 2), (1, 3), (2, 1), (2, 3), (3, 1)]
assert connected_components((V, E)) == [[1, 2, 3]]
V = [1, 2, 3, 4]
E = [(1, 2), (2, 3), (3, 2), (3, 4)]
assert connected_components((V, E)) == [[1, 2, 3, 4]]
V = [1, 2, 3, 4]
E = [(1, 2), (3, 4)]
assert connected_components((V, E)) == [[1, 2], [3, 4]]
def test_rotate():
A = [0, 1, 2, 3, 4]
assert rotate_left(A, 2) == [2, 3, 4, 0, 1]
assert rotate_right(A, 1) == [4, 0, 1, 2, 3]
A = []
B = rotate_right(A, 1)
assert B == []
B.append(1)
assert A == []
B = rotate_left(A, 1)
assert B == []
B.append(1)
assert A == []
def test_multiset_partitions():
A = [0, 1, 2, 3, 4]
assert list(multiset_partitions(A, 5)) == [[[0], [1], [2], [3], [4]]]
assert len(list(multiset_partitions(A, 4))) == 10
assert len(list(multiset_partitions(A, 3))) == 25
assert list(multiset_partitions([1, 1, 1, 2, 2], 2)) == [
[[1, 1, 1, 2], [2]], [[1, 1, 1], [2, 2]], [[1, 1, 2, 2], [1]],
[[1, 1, 2], [1, 2]], [[1, 1], [1, 2, 2]]]
assert list(multiset_partitions([1, 1, 2, 2], 2)) == [
[[1, 1, 2], [2]], [[1, 1], [2, 2]], [[1, 2, 2], [1]],
[[1, 2], [1, 2]]]
assert list(multiset_partitions([1, 2, 3, 4], 2)) == [
[[1, 2, 3], [4]], [[1, 2, 4], [3]], [[1, 2], [3, 4]],
[[1, 3, 4], [2]], [[1, 3], [2, 4]], [[1, 4], [2, 3]],
[[1], [2, 3, 4]]]
assert list(multiset_partitions([1, 2, 2], 2)) == [
[[1, 2], [2]], [[1], [2, 2]]]
assert list(multiset_partitions(3)) == [
[[0, 1, 2]], [[0, 1], [2]], [[0, 2], [1]], [[0], [1, 2]],
[[0], [1], [2]]]
assert list(multiset_partitions(3, 2)) == [
[[0, 1], [2]], [[0, 2], [1]], [[0], [1, 2]]]
assert list(multiset_partitions([1] * 3, 2)) == [[[1], [1, 1]]]
assert list(multiset_partitions([1] * 3)) == [
[[1, 1, 1]], [[1], [1, 1]], [[1], [1], [1]]]
a = [3, 2, 1]
assert list(multiset_partitions(a)) == \
list(multiset_partitions(sorted(a)))
assert list(multiset_partitions(a, 5)) == []
assert list(multiset_partitions(a, 1)) == [[[1, 2, 3]]]
assert list(multiset_partitions(a + [4], 5)) == []
assert list(multiset_partitions(a + [4], 1)) == [[[1, 2, 3, 4]]]
assert list(multiset_partitions(2, 5)) == []
assert list(multiset_partitions(2, 1)) == [[[0, 1]]]
assert list(multiset_partitions('a')) == [[['a']]]
assert list(multiset_partitions('a', 2)) == []
assert list(multiset_partitions('ab')) == [[['a', 'b']], [['a'], ['b']]]
assert list(multiset_partitions('ab', 1)) == [[['a', 'b']]]
assert list(multiset_partitions('aaa', 1)) == [['aaa']]
assert list(multiset_partitions([1, 1], 1)) == [[[1, 1]]]
ans = [('mpsyy',), ('mpsy', 'y'), ('mps', 'yy'), ('mps', 'y', 'y'),
('mpyy', 's'), ('mpy', 'sy'), ('mpy', 's', 'y'), ('mp', 'syy'),
('mp', 'sy', 'y'), ('mp', 's', 'yy'), ('mp', 's', 'y', 'y'),
('msyy', 'p'), ('msy', 'py'), ('msy', 'p', 'y'), ('ms', 'pyy'),
('ms', 'py', 'y'), ('ms', 'p', 'yy'), ('ms', 'p', 'y', 'y'),
('myy', 'ps'), ('myy', 'p', 's'), ('my', 'psy'), ('my', 'ps', 'y'),
('my', 'py', 's'), ('my', 'p', 'sy'), ('my', 'p', 's', 'y'),
('m', 'psyy'), ('m', 'psy', 'y'), ('m', 'ps', 'yy'),
('m', 'ps', 'y', 'y'), ('m', 'pyy', 's'), ('m', 'py', 'sy'),
('m', 'py', 's', 'y'), ('m', 'p', 'syy'),
('m', 'p', 'sy', 'y'), ('m', 'p', 's', 'yy'),
('m', 'p', 's', 'y', 'y')]
assert list(tuple("".join(part) for part in p)
for p in multiset_partitions('sympy')) == ans
factorings = [[24], [8, 3], [12, 2], [4, 6], [4, 2, 3],
[6, 2, 2], [2, 2, 2, 3]]
assert list(factoring_visitor(p, [2,3]) for
p in multiset_partitions_taocp([3, 1])) == factorings
def test_multiset_combinations():
ans = ['iii', 'iim', 'iip', 'iis', 'imp', 'ims', 'ipp', 'ips',
'iss', 'mpp', 'mps', 'mss', 'pps', 'pss', 'sss']
assert [''.join(i) for i in
list(multiset_combinations('mississippi', 3))] == ans
M = multiset('mississippi')
assert [''.join(i) for i in
list(multiset_combinations(M, 3))] == ans
assert [''.join(i) for i in multiset_combinations(M, 30)] == []
assert list(multiset_combinations([[1], [2, 3]], 2)) == [[[1], [2, 3]]]
assert len(list(multiset_combinations('a', 3))) == 0
assert len(list(multiset_combinations('a', 0))) == 1
assert list(multiset_combinations('abc', 1)) == [['a'], ['b'], ['c']]
def test_multiset_permutations():
ans = ['abby', 'abyb', 'aybb', 'baby', 'bayb', 'bbay', 'bbya', 'byab',
'byba', 'yabb', 'ybab', 'ybba']
assert [''.join(i) for i in multiset_permutations('baby')] == ans
assert [''.join(i) for i in multiset_permutations(multiset('baby'))] == ans
assert list(multiset_permutations([0, 0, 0], 2)) == [[0, 0]]
assert list(multiset_permutations([0, 2, 1], 2)) == [
[0, 1], [0, 2], [1, 0], [1, 2], [2, 0], [2, 1]]
assert len(list(multiset_permutations('a', 0))) == 1
assert len(list(multiset_permutations('a', 3))) == 0
def test():
for i in range(1, 7):
print(i)
for p in multiset_permutations([0, 0, 1, 0, 1], i):
print(p)
assert capture(lambda: test()) == dedent('''\
1
[0]
[1]
2
[0, 0]
[0, 1]
[1, 0]
[1, 1]
3
[0, 0, 0]
[0, 0, 1]
[0, 1, 0]
[0, 1, 1]
[1, 0, 0]
[1, 0, 1]
[1, 1, 0]
4
[0, 0, 0, 1]
[0, 0, 1, 0]
[0, 0, 1, 1]
[0, 1, 0, 0]
[0, 1, 0, 1]
[0, 1, 1, 0]
[1, 0, 0, 0]
[1, 0, 0, 1]
[1, 0, 1, 0]
[1, 1, 0, 0]
5
[0, 0, 0, 1, 1]
[0, 0, 1, 0, 1]
[0, 0, 1, 1, 0]
[0, 1, 0, 0, 1]
[0, 1, 0, 1, 0]
[0, 1, 1, 0, 0]
[1, 0, 0, 0, 1]
[1, 0, 0, 1, 0]
[1, 0, 1, 0, 0]
[1, 1, 0, 0, 0]
6\n''')
def test_partitions():
ans = [[{}], [(0, {})]]
for i in range(2):
assert list(partitions(0, size=i)) == ans[i]
assert list(partitions(1, 0, size=i)) == ans[i]
assert list(partitions(6, 2, 2, size=i)) == ans[i]
assert list(partitions(6, 2, None, size=i)) != ans[i]
assert list(partitions(6, None, 2, size=i)) != ans[i]
assert list(partitions(6, 2, 0, size=i)) == ans[i]
assert [p.copy() for p in partitions(6, k=2)] == [
{2: 3}, {1: 2, 2: 2}, {1: 4, 2: 1}, {1: 6}]
assert [p.copy() for p in partitions(6, k=3)] == [
{3: 2}, {1: 1, 2: 1, 3: 1}, {1: 3, 3: 1}, {2: 3}, {1: 2, 2: 2},
{1: 4, 2: 1}, {1: 6}]
assert [p.copy() for p in partitions(8, k=4, m=3)] == [
{4: 2}, {1: 1, 3: 1, 4: 1}, {2: 2, 4: 1}, {2: 1, 3: 2}] == [
i.copy() for i in partitions(8, k=4, m=3) if all(k <= 4 for k in i)
and sum(i.values()) <=3]
assert [p.copy() for p in partitions(S(3), m=2)] == [
{3: 1}, {1: 1, 2: 1}]
assert [i.copy() for i in partitions(4, k=3)] == [
{1: 1, 3: 1}, {2: 2}, {1: 2, 2: 1}, {1: 4}] == [
i.copy() for i in partitions(4) if all(k <= 3 for k in i)]
# Consistency check on output of _partitions and RGS_unrank.
# This provides a sanity test on both routines. Also verifies that
# the total number of partitions is the same in each case.
# (from pkrathmann2)
for n in range(2, 6):
i = 0
for m, q in _set_partitions(n):
assert q == RGS_unrank(i, n)
i += 1
assert i == RGS_enum(n)
def test_binary_partitions():
assert [i[:] for i in binary_partitions(10)] == [[8, 2], [8, 1, 1],
[4, 4, 2], [4, 4, 1, 1], [4, 2, 2, 2], [4, 2, 2, 1, 1],
[4, 2, 1, 1, 1, 1], [4, 1, 1, 1, 1, 1, 1], [2, 2, 2, 2, 2],
[2, 2, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1, 1],
[2, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]]
assert len([j[:] for j in binary_partitions(16)]) == 36
def test_bell_perm():
assert [len(set(generate_bell(i))) for i in range(1, 7)] == [
factorial(i) for i in range(1, 7)]
assert list(generate_bell(3)) == [
(0, 1, 2), (0, 2, 1), (2, 0, 1), (2, 1, 0), (1, 2, 0), (1, 0, 2)]
# generate_bell and trotterjohnson are advertised to return the same
# permutations; this is not technically necessary so this test could
# be removed
for n in range(1, 5):
p = Permutation(range(n))
b = generate_bell(n)
for bi in b:
assert bi == tuple(p.array_form)
p = p.next_trotterjohnson()
raises(ValueError, lambda: list(generate_bell(0))) # XXX is this consistent with other permutation algorithms?
def test_involutions():
lengths = [1, 2, 4, 10, 26, 76]
for n, N in enumerate(lengths):
i = list(generate_involutions(n + 1))
assert len(i) == N
assert len({Permutation(j)**2 for j in i}) == 1
def test_derangements():
assert len(list(generate_derangements(list(range(6))))) == 265
assert ''.join(''.join(i) for i in generate_derangements('abcde')) == (
'badecbaecdbcaedbcdeabceadbdaecbdeacbdecabeacdbedacbedcacabedcadebcaebd'
'cdaebcdbeacdeabcdebaceabdcebadcedabcedbadabecdaebcdaecbdcaebdcbeadceab'
'dcebadeabcdeacbdebacdebcaeabcdeadbceadcbecabdecbadecdabecdbaedabcedacb'
'edbacedbca')
assert list(generate_derangements([0, 1, 2, 3])) == [
[1, 0, 3, 2], [1, 2, 3, 0], [1, 3, 0, 2], [2, 0, 3, 1],
[2, 3, 0, 1], [2, 3, 1, 0], [3, 0, 1, 2], [3, 2, 0, 1], [3, 2, 1, 0]]
assert list(generate_derangements([0, 1, 2, 2])) == [
[2, 2, 0, 1], [2, 2, 1, 0]]
def test_necklaces():
def count(n, k, f):
return len(list(necklaces(n, k, f)))
m = []
for i in range(1, 8):
m.append((
i, count(i, 2, 0), count(i, 2, 1), count(i, 3, 1)))
assert Matrix(m) == Matrix([
[1, 2, 2, 3],
[2, 3, 3, 6],
[3, 4, 4, 10],
[4, 6, 6, 21],
[5, 8, 8, 39],
[6, 14, 13, 92],
[7, 20, 18, 198]])
def test_bracelets():
bc = [i for i in bracelets(2, 4)]
assert Matrix(bc) == Matrix([
[0, 0],
[0, 1],
[0, 2],
[0, 3],
[1, 1],
[1, 2],
[1, 3],
[2, 2],
[2, 3],
[3, 3]
])
bc = [i for i in bracelets(4, 2)]
assert Matrix(bc) == Matrix([
[0, 0, 0, 0],
[0, 0, 0, 1],
[0, 0, 1, 1],
[0, 1, 0, 1],
[0, 1, 1, 1],
[1, 1, 1, 1]
])
def test_generate_oriented_forest():
assert list(generate_oriented_forest(5)) == [[0, 1, 2, 3, 4],
[0, 1, 2, 3, 3], [0, 1, 2, 3, 2], [0, 1, 2, 3, 1], [0, 1, 2, 3, 0],
[0, 1, 2, 2, 2], [0, 1, 2, 2, 1], [0, 1, 2, 2, 0], [0, 1, 2, 1, 2],
[0, 1, 2, 1, 1], [0, 1, 2, 1, 0], [0, 1, 2, 0, 1], [0, 1, 2, 0, 0],
[0, 1, 1, 1, 1], [0, 1, 1, 1, 0], [0, 1, 1, 0, 1], [0, 1, 1, 0, 0],
[0, 1, 0, 1, 0], [0, 1, 0, 0, 0], [0, 0, 0, 0, 0]]
assert len(list(generate_oriented_forest(10))) == 1842
def test_unflatten():
r = list(range(10))
assert unflatten(r) == list(zip(r[::2], r[1::2]))
assert unflatten(r, 5) == [tuple(r[:5]), tuple(r[5:])]
raises(ValueError, lambda: unflatten(list(range(10)), 3))
raises(ValueError, lambda: unflatten(list(range(10)), -2))
def test_common_prefix_suffix():
assert common_prefix([], [1]) == []
assert common_prefix(list(range(3))) == [0, 1, 2]
assert common_prefix(list(range(3)), list(range(4))) == [0, 1, 2]
assert common_prefix([1, 2, 3], [1, 2, 5]) == [1, 2]
assert common_prefix([1, 2, 3], [1, 3, 5]) == [1]
assert common_suffix([], [1]) == []
assert common_suffix(list(range(3))) == [0, 1, 2]
assert common_suffix(list(range(3)), list(range(3))) == [0, 1, 2]
assert common_suffix(list(range(3)), list(range(4))) == []
assert common_suffix([1, 2, 3], [9, 2, 3]) == [2, 3]
assert common_suffix([1, 2, 3], [9, 7, 3]) == [3]
def test_minlex():
assert minlex([1, 2, 0]) == (0, 1, 2)
assert minlex((1, 2, 0)) == (0, 1, 2)
assert minlex((1, 0, 2)) == (0, 2, 1)
assert minlex((1, 0, 2), directed=False) == (0, 1, 2)
assert minlex('aba') == 'aab'
def test_ordered():
assert list(ordered((x, y), hash, default=False)) in [[x, y], [y, x]]
assert list(ordered((x, y), hash, default=False)) == \
list(ordered((y, x), hash, default=False))
assert list(ordered((x, y))) == [x, y]
seq, keys = [[[1, 2, 1], [0, 3, 1], [1, 1, 3], [2], [1]],
(lambda x: len(x), lambda x: sum(x))]
assert list(ordered(seq, keys, default=False, warn=False)) == \
[[1], [2], [1, 2, 1], [0, 3, 1], [1, 1, 3]]
raises(ValueError, lambda:
list(ordered(seq, keys, default=False, warn=True)))
def test_runs():
assert runs([]) == []
assert runs([1]) == [[1]]
assert runs([1, 1]) == [[1], [1]]
assert runs([1, 1, 2]) == [[1], [1, 2]]
assert runs([1, 2, 1]) == [[1, 2], [1]]
assert runs([2, 1, 1]) == [[2], [1], [1]]
from operator import lt
assert runs([2, 1, 1], lt) == [[2, 1], [1]]
def test_reshape():
seq = list(range(1, 9))
assert reshape(seq, [4]) == \
[[1, 2, 3, 4], [5, 6, 7, 8]]
assert reshape(seq, (4,)) == \
[(1, 2, 3, 4), (5, 6, 7, 8)]
assert reshape(seq, (2, 2)) == \
[(1, 2, 3, 4), (5, 6, 7, 8)]
assert reshape(seq, (2, [2])) == \
[(1, 2, [3, 4]), (5, 6, [7, 8])]
assert reshape(seq, ((2,), [2])) == \
[((1, 2), [3, 4]), ((5, 6), [7, 8])]
assert reshape(seq, (1, [2], 1)) == \
[(1, [2, 3], 4), (5, [6, 7], 8)]
assert reshape(tuple(seq), ([[1], 1, (2,)],)) == \
(([[1], 2, (3, 4)],), ([[5], 6, (7, 8)],))
assert reshape(tuple(seq), ([1], 1, (2,))) == \
(([1], 2, (3, 4)), ([5], 6, (7, 8)))
assert reshape(list(range(12)), [2, [3], {2}, (1, (3,), 1)]) == \
[[0, 1, [2, 3, 4], {5, 6}, (7, (8, 9, 10), 11)]]
raises(ValueError, lambda: reshape([0, 1], [-1]))
raises(ValueError, lambda: reshape([0, 1], [3]))
def test_uniq():
assert list(uniq(p.copy() for p in partitions(4))) == \
[{4: 1}, {1: 1, 3: 1}, {2: 2}, {1: 2, 2: 1}, {1: 4}]
assert list(uniq(x % 2 for x in range(5))) == [0, 1]
assert list(uniq('a')) == ['a']
assert list(uniq('ababc')) == list('abc')
assert list(uniq([[1], [2, 1], [1]])) == [[1], [2, 1]]
assert list(uniq(permutations(i for i in [[1], 2, 2]))) == \
[([1], 2, 2), (2, [1], 2), (2, 2, [1])]
assert list(uniq([2, 3, 2, 4, [2], [1], [2], [3], [1]])) == \
[2, 3, 4, [2], [1], [3]]
def test_kbins():
assert len(list(kbins('1123', 2, ordered=1))) == 24
assert len(list(kbins('1123', 2, ordered=11))) == 36
assert len(list(kbins('1123', 2, ordered=10))) == 10
assert len(list(kbins('1123', 2, ordered=0))) == 5
assert len(list(kbins('1123', 2, ordered=None))) == 3
def test():
for ordered in [None, 0, 1, 10, 11]:
print('ordered =', ordered)
for p in kbins([0, 0, 1], 2, ordered=ordered):
print(' ', p)
assert capture(lambda : test()) == dedent('''\
ordered = None
[[0], [0, 1]]
[[0, 0], [1]]
ordered = 0
[[0, 0], [1]]
[[0, 1], [0]]
ordered = 1
[[0], [0, 1]]
[[0], [1, 0]]
[[1], [0, 0]]
ordered = 10
[[0, 0], [1]]
[[1], [0, 0]]
[[0, 1], [0]]
[[0], [0, 1]]
ordered = 11
[[0], [0, 1]]
[[0, 0], [1]]
[[0], [1, 0]]
[[0, 1], [0]]
[[1], [0, 0]]
[[1, 0], [0]]\n''')
def test():
for ordered in [None, 0, 1, 10, 11]:
print('ordered =', ordered)
for p in kbins(list(range(3)), 2, ordered=ordered):
print(' ', p)
assert capture(lambda : test()) == dedent('''\
ordered = None
[[0], [1, 2]]
[[0, 1], [2]]
ordered = 0
[[0, 1], [2]]
[[0, 2], [1]]
[[0], [1, 2]]
ordered = 1
[[0], [1, 2]]
[[0], [2, 1]]
[[1], [0, 2]]
[[1], [2, 0]]
[[2], [0, 1]]
[[2], [1, 0]]
ordered = 10
[[0, 1], [2]]
[[2], [0, 1]]
[[0, 2], [1]]
[[1], [0, 2]]
[[0], [1, 2]]
[[1, 2], [0]]
ordered = 11
[[0], [1, 2]]
[[0, 1], [2]]
[[0], [2, 1]]
[[0, 2], [1]]
[[1], [0, 2]]
[[1, 0], [2]]
[[1], [2, 0]]
[[1, 2], [0]]
[[2], [0, 1]]
[[2, 0], [1]]
[[2], [1, 0]]
[[2, 1], [0]]\n''')
def test_has_dups():
assert has_dups(set()) is False
assert has_dups(list(range(3))) is False
assert has_dups([1, 2, 1]) is True
def test__partition():
assert _partition('abcde', [1, 0, 1, 2, 0]) == [
['b', 'e'], ['a', 'c'], ['d']]
assert _partition('abcde', [1, 0, 1, 2, 0], 3) == [
['b', 'e'], ['a', 'c'], ['d']]
output = (3, [1, 0, 1, 2, 0])
assert _partition('abcde', *output) == [['b', 'e'], ['a', 'c'], ['d']]
def test_ordered_partitions():
from sympy.functions.combinatorial.numbers import nT
f = ordered_partitions
assert list(f(0, 1)) == [[]]
assert list(f(1, 0)) == [[]]
for i in range(1, 7):
for j in [None] + list(range(1, i)):
assert (
sum(1 for p in f(i, j, 1)) ==
sum(1 for p in f(i, j, 0)) ==
nT(i, j))
def test_rotations():
assert list(rotations('ab')) == [['a', 'b'], ['b', 'a']]
assert list(rotations(range(3))) == [[0, 1, 2], [1, 2, 0], [2, 0, 1]]
assert list(rotations(range(3), dir=-1)) == [[0, 1, 2], [2, 0, 1], [1, 2, 0]]
def test_ibin():
assert ibin(3) == [1, 1]
assert ibin(3, 3) == [0, 1, 1]
assert ibin(3, str=True) == '11'
assert ibin(3, 3, str=True) == '011'
assert list(ibin(2, 'all')) == [(0, 0), (0, 1), (1, 0), (1, 1)]
assert list(ibin(2, 'all', str=True)) == ['00', '01', '10', '11']
|
b870ea047d7ffcdfc6fcf1e48904837c4775ed0165ce075f796f415b767029bc
|
from sympy.core import (S, symbols, Eq, pi, Catalan, EulerGamma, Lambda,
Dummy, Function)
from sympy.core.compatibility import StringIO
from sympy import erf, Integral, Piecewise
from sympy import Equality
from sympy.matrices import Matrix, MatrixSymbol
from sympy.printing.codeprinter import Assignment
from sympy.utilities.codegen import JuliaCodeGen, codegen, make_routine
from sympy.utilities.pytest import raises
from sympy.utilities.lambdify import implemented_function
from sympy.utilities.pytest import XFAIL
import sympy
x, y, z = symbols('x,y,z')
def test_empty_jl_code():
code_gen = JuliaCodeGen()
output = StringIO()
code_gen.dump_jl([], output, "file", header=False, empty=False)
source = output.getvalue()
assert source == ""
def test_jl_simple_code():
name_expr = ("test", (x + y)*z)
result, = codegen(name_expr, "Julia", header=False, empty=False)
assert result[0] == "test.jl"
source = result[1]
expected = (
"function test(x, y, z)\n"
" out1 = z.*(x + y)\n"
" return out1\n"
"end\n"
)
assert source == expected
def test_jl_simple_code_with_header():
name_expr = ("test", (x + y)*z)
result, = codegen(name_expr, "Julia", header=True, empty=False)
assert result[0] == "test.jl"
source = result[1]
expected = (
"# Code generated with sympy " + sympy.__version__ + "\n"
"#\n"
"# See http://www.sympy.org/ for more information.\n"
"#\n"
"# This file is part of 'project'\n"
"function test(x, y, z)\n"
" out1 = z.*(x + y)\n"
" return out1\n"
"end\n"
)
assert source == expected
def test_jl_simple_code_nameout():
expr = Equality(z, (x + y))
name_expr = ("test", expr)
result, = codegen(name_expr, "Julia", header=False, empty=False)
source = result[1]
expected = (
"function test(x, y)\n"
" z = x + y\n"
" return z\n"
"end\n"
)
assert source == expected
def test_jl_numbersymbol():
name_expr = ("test", pi**Catalan)
result, = codegen(name_expr, "Julia", header=False, empty=False)
source = result[1]
expected = (
"function test()\n"
" out1 = pi^catalan\n"
" return out1\n"
"end\n"
)
assert source == expected
@XFAIL
def test_jl_numbersymbol_no_inline():
# FIXME: how to pass inline=False to the JuliaCodePrinter?
name_expr = ("test", [pi**Catalan, EulerGamma])
result, = codegen(name_expr, "Julia", header=False,
empty=False, inline=False)
source = result[1]
expected = (
"function test()\n"
" Catalan = 0.915965594177219\n"
" EulerGamma = 0.5772156649015329\n"
" out1 = pi^Catalan\n"
" out2 = EulerGamma\n"
" return out1, out2\n"
"end\n"
)
assert source == expected
def test_jl_code_argument_order():
expr = x + y
routine = make_routine("test", expr, argument_sequence=[z, x, y], language="julia")
code_gen = JuliaCodeGen()
output = StringIO()
code_gen.dump_jl([routine], output, "test", header=False, empty=False)
source = output.getvalue()
expected = (
"function test(z, x, y)\n"
" out1 = x + y\n"
" return out1\n"
"end\n"
)
assert source == expected
def test_multiple_results_m():
# Here the output order is the input order
expr1 = (x + y)*z
expr2 = (x - y)*z
name_expr = ("test", [expr1, expr2])
result, = codegen(name_expr, "Julia", header=False, empty=False)
source = result[1]
expected = (
"function test(x, y, z)\n"
" out1 = z.*(x + y)\n"
" out2 = z.*(x - y)\n"
" return out1, out2\n"
"end\n"
)
assert source == expected
def test_results_named_unordered():
# Here output order is based on name_expr
A, B, C = symbols('A,B,C')
expr1 = Equality(C, (x + y)*z)
expr2 = Equality(A, (x - y)*z)
expr3 = Equality(B, 2*x)
name_expr = ("test", [expr1, expr2, expr3])
result, = codegen(name_expr, "Julia", header=False, empty=False)
source = result[1]
expected = (
"function test(x, y, z)\n"
" C = z.*(x + y)\n"
" A = z.*(x - y)\n"
" B = 2*x\n"
" return C, A, B\n"
"end\n"
)
assert source == expected
def test_results_named_ordered():
A, B, C = symbols('A,B,C')
expr1 = Equality(C, (x + y)*z)
expr2 = Equality(A, (x - y)*z)
expr3 = Equality(B, 2*x)
name_expr = ("test", [expr1, expr2, expr3])
result = codegen(name_expr, "Julia", header=False, empty=False,
argument_sequence=(x, z, y))
assert result[0][0] == "test.jl"
source = result[0][1]
expected = (
"function test(x, z, y)\n"
" C = z.*(x + y)\n"
" A = z.*(x - y)\n"
" B = 2*x\n"
" return C, A, B\n"
"end\n"
)
assert source == expected
def test_complicated_jl_codegen():
from sympy import sin, cos, tan
name_expr = ("testlong",
[ ((sin(x) + cos(y) + tan(z))**3).expand(),
cos(cos(cos(cos(cos(cos(cos(cos(x + y + z))))))))
])
result = codegen(name_expr, "Julia", header=False, empty=False)
assert result[0][0] == "testlong.jl"
source = result[0][1]
expected = (
"function testlong(x, y, z)\n"
" out1 = sin(x).^3 + 3*sin(x).^2.*cos(y) + 3*sin(x).^2.*tan(z)"
" + 3*sin(x).*cos(y).^2 + 6*sin(x).*cos(y).*tan(z) + 3*sin(x).*tan(z).^2"
" + cos(y).^3 + 3*cos(y).^2.*tan(z) + 3*cos(y).*tan(z).^2 + tan(z).^3\n"
" out2 = cos(cos(cos(cos(cos(cos(cos(cos(x + y + z))))))))\n"
" return out1, out2\n"
"end\n"
)
assert source == expected
def test_jl_output_arg_mixed_unordered():
# named outputs are alphabetical, unnamed output appear in the given order
from sympy import sin, cos, tan
a = symbols("a")
name_expr = ("foo", [cos(2*x), Equality(y, sin(x)), cos(x), Equality(a, sin(2*x))])
result, = codegen(name_expr, "Julia", header=False, empty=False)
assert result[0] == "foo.jl"
source = result[1];
expected = (
'function foo(x)\n'
' out1 = cos(2*x)\n'
' y = sin(x)\n'
' out3 = cos(x)\n'
' a = sin(2*x)\n'
' return out1, y, out3, a\n'
'end\n'
)
assert source == expected
def test_jl_piecewise_():
pw = Piecewise((0, x < -1), (x**2, x <= 1), (-x+2, x > 1), (1, True), evaluate=False)
name_expr = ("pwtest", pw)
result, = codegen(name_expr, "Julia", header=False, empty=False)
source = result[1]
expected = (
"function pwtest(x)\n"
" out1 = ((x < -1) ? (0) :\n"
" (x <= 1) ? (x.^2) :\n"
" (x > 1) ? (2 - x) : (1))\n"
" return out1\n"
"end\n"
)
assert source == expected
@XFAIL
def test_jl_piecewise_no_inline():
# FIXME: how to pass inline=False to the JuliaCodePrinter?
pw = Piecewise((0, x < -1), (x**2, x <= 1), (-x+2, x > 1), (1, True))
name_expr = ("pwtest", pw)
result, = codegen(name_expr, "Julia", header=False, empty=False,
inline=False)
source = result[1]
expected = (
"function pwtest(x)\n"
" if (x < -1)\n"
" out1 = 0\n"
" elseif (x <= 1)\n"
" out1 = x.^2\n"
" elseif (x > 1)\n"
" out1 = -x + 2\n"
" else\n"
" out1 = 1\n"
" end\n"
" return out1\n"
"end\n"
)
assert source == expected
def test_jl_multifcns_per_file():
name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ]
result = codegen(name_expr, "Julia", header=False, empty=False)
assert result[0][0] == "foo.jl"
source = result[0][1];
expected = (
"function foo(x, y)\n"
" out1 = 2*x\n"
" out2 = 3*y\n"
" return out1, out2\n"
"end\n"
"function bar(y)\n"
" out1 = y.^2\n"
" out2 = 4*y\n"
" return out1, out2\n"
"end\n"
)
assert source == expected
def test_jl_multifcns_per_file_w_header():
name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ]
result = codegen(name_expr, "Julia", header=True, empty=False)
assert result[0][0] == "foo.jl"
source = result[0][1];
expected = (
"# Code generated with sympy " + sympy.__version__ + "\n"
"#\n"
"# See http://www.sympy.org/ for more information.\n"
"#\n"
"# This file is part of 'project'\n"
"function foo(x, y)\n"
" out1 = 2*x\n"
" out2 = 3*y\n"
" return out1, out2\n"
"end\n"
"function bar(y)\n"
" out1 = y.^2\n"
" out2 = 4*y\n"
" return out1, out2\n"
"end\n"
)
assert source == expected
def test_jl_filename_match_prefix():
name_expr = [ ("foo", [2*x, 3*y]), ("bar", [y**2, 4*y]) ]
result, = codegen(name_expr, "Julia", prefix="baz", header=False,
empty=False)
assert result[0] == "baz.jl"
def test_jl_matrix_named():
e2 = Matrix([[x, 2*y, pi*z]])
name_expr = ("test", Equality(MatrixSymbol('myout1', 1, 3), e2))
result = codegen(name_expr, "Julia", header=False, empty=False)
assert result[0][0] == "test.jl"
source = result[0][1]
expected = (
"function test(x, y, z)\n"
" myout1 = [x 2*y pi*z]\n"
" return myout1\n"
"end\n"
)
assert source == expected
def test_jl_matrix_named_matsym():
myout1 = MatrixSymbol('myout1', 1, 3)
e2 = Matrix([[x, 2*y, pi*z]])
name_expr = ("test", Equality(myout1, e2, evaluate=False))
result, = codegen(name_expr, "Julia", header=False, empty=False)
source = result[1]
expected = (
"function test(x, y, z)\n"
" myout1 = [x 2*y pi*z]\n"
" return myout1\n"
"end\n"
)
assert source == expected
def test_jl_matrix_output_autoname():
expr = Matrix([[x, x+y, 3]])
name_expr = ("test", expr)
result, = codegen(name_expr, "Julia", header=False, empty=False)
source = result[1]
expected = (
"function test(x, y)\n"
" out1 = [x x + y 3]\n"
" return out1\n"
"end\n"
)
assert source == expected
def test_jl_matrix_output_autoname_2():
e1 = (x + y)
e2 = Matrix([[2*x, 2*y, 2*z]])
e3 = Matrix([[x], [y], [z]])
e4 = Matrix([[x, y], [z, 16]])
name_expr = ("test", (e1, e2, e3, e4))
result, = codegen(name_expr, "Julia", header=False, empty=False)
source = result[1]
expected = (
"function test(x, y, z)\n"
" out1 = x + y\n"
" out2 = [2*x 2*y 2*z]\n"
" out3 = [x, y, z]\n"
" out4 = [x y;\n"
" z 16]\n"
" return out1, out2, out3, out4\n"
"end\n"
)
assert source == expected
def test_jl_results_matrix_named_ordered():
B, C = symbols('B,C')
A = MatrixSymbol('A', 1, 3)
expr1 = Equality(C, (x + y)*z)
expr2 = Equality(A, Matrix([[1, 2, x]]))
expr3 = Equality(B, 2*x)
name_expr = ("test", [expr1, expr2, expr3])
result, = codegen(name_expr, "Julia", header=False, empty=False,
argument_sequence=(x, z, y))
source = result[1]
expected = (
"function test(x, z, y)\n"
" C = z.*(x + y)\n"
" A = [1 2 x]\n"
" B = 2*x\n"
" return C, A, B\n"
"end\n"
)
assert source == expected
def test_jl_matrixsymbol_slice():
A = MatrixSymbol('A', 2, 3)
B = MatrixSymbol('B', 1, 3)
C = MatrixSymbol('C', 1, 3)
D = MatrixSymbol('D', 2, 1)
name_expr = ("test", [Equality(B, A[0, :]),
Equality(C, A[1, :]),
Equality(D, A[:, 2])])
result, = codegen(name_expr, "Julia", header=False, empty=False)
source = result[1]
expected = (
"function test(A)\n"
" B = A[1,:]\n"
" C = A[2,:]\n"
" D = A[:,3]\n"
" return B, C, D\n"
"end\n"
)
assert source == expected
def test_jl_matrixsymbol_slice2():
A = MatrixSymbol('A', 3, 4)
B = MatrixSymbol('B', 2, 2)
C = MatrixSymbol('C', 2, 2)
name_expr = ("test", [Equality(B, A[0:2, 0:2]),
Equality(C, A[0:2, 1:3])])
result, = codegen(name_expr, "Julia", header=False, empty=False)
source = result[1]
expected = (
"function test(A)\n"
" B = A[1:2,1:2]\n"
" C = A[1:2,2:3]\n"
" return B, C\n"
"end\n"
)
assert source == expected
def test_jl_matrixsymbol_slice3():
A = MatrixSymbol('A', 8, 7)
B = MatrixSymbol('B', 2, 2)
C = MatrixSymbol('C', 4, 2)
name_expr = ("test", [Equality(B, A[6:, 1::3]),
Equality(C, A[::2, ::3])])
result, = codegen(name_expr, "Julia", header=False, empty=False)
source = result[1]
expected = (
"function test(A)\n"
" B = A[7:end,2:3:end]\n"
" C = A[1:2:end,1:3:end]\n"
" return B, C\n"
"end\n"
)
assert source == expected
def test_jl_matrixsymbol_slice_autoname():
A = MatrixSymbol('A', 2, 3)
B = MatrixSymbol('B', 1, 3)
name_expr = ("test", [Equality(B, A[0,:]), A[1,:], A[:,0], A[:,1]])
result, = codegen(name_expr, "Julia", header=False, empty=False)
source = result[1]
expected = (
"function test(A)\n"
" B = A[1,:]\n"
" out2 = A[2,:]\n"
" out3 = A[:,1]\n"
" out4 = A[:,2]\n"
" return B, out2, out3, out4\n"
"end\n"
)
assert source == expected
def test_jl_loops():
# Note: an Julia programmer would probably vectorize this across one or
# more dimensions. Also, size(A) would be used rather than passing in m
# and n. Perhaps users would expect us to vectorize automatically here?
# Or is it possible to represent such things using IndexedBase?
from sympy.tensor import IndexedBase, Idx
from sympy import symbols
n, m = symbols('n m', integer=True)
A = IndexedBase('A')
x = IndexedBase('x')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
result, = codegen(('mat_vec_mult', Eq(y[i], A[i, j]*x[j])), "Julia",
header=False, empty=False)
source = result[1]
expected = (
'function mat_vec_mult(y, A, m, n, x)\n'
' for i = 1:m\n'
' y[i] = 0\n'
' end\n'
' for i = 1:m\n'
' for j = 1:n\n'
' y[i] = %(rhs)s + y[i]\n'
' end\n'
' end\n'
' return y\n'
'end\n'
)
assert (source == expected % {'rhs': 'A[%s,%s].*x[j]' % (i, j)} or
source == expected % {'rhs': 'x[j].*A[%s,%s]' % (i, j)})
def test_jl_tensor_loops_multiple_contractions():
# see comments in previous test about vectorizing
from sympy.tensor import IndexedBase, Idx
from sympy import symbols
n, m, o, p = symbols('n m o p', integer=True)
A = IndexedBase('A')
B = IndexedBase('B')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
l = Idx('l', p)
result, = codegen(('tensorthing', Eq(y[i], B[j, k, l]*A[i, j, k, l])),
"Julia", header=False, empty=False)
source = result[1]
expected = (
'function tensorthing(y, A, B, m, n, o, p)\n'
' for i = 1:m\n'
' y[i] = 0\n'
' end\n'
' for i = 1:m\n'
' for j = 1:n\n'
' for k = 1:o\n'
' for l = 1:p\n'
' y[i] = A[i,j,k,l].*B[j,k,l] + y[i]\n'
' end\n'
' end\n'
' end\n'
' end\n'
' return y\n'
'end\n'
)
assert source == expected
def test_jl_InOutArgument():
expr = Equality(x, x**2)
name_expr = ("mysqr", expr)
result, = codegen(name_expr, "Julia", header=False, empty=False)
source = result[1]
expected = (
"function mysqr(x)\n"
" x = x.^2\n"
" return x\n"
"end\n"
)
assert source == expected
def test_jl_InOutArgument_order():
# can specify the order as (x, y)
expr = Equality(x, x**2 + y)
name_expr = ("test", expr)
result, = codegen(name_expr, "Julia", header=False,
empty=False, argument_sequence=(x,y))
source = result[1]
expected = (
"function test(x, y)\n"
" x = x.^2 + y\n"
" return x\n"
"end\n"
)
assert source == expected
# make sure it gives (x, y) not (y, x)
expr = Equality(x, x**2 + y)
name_expr = ("test", expr)
result, = codegen(name_expr, "Julia", header=False, empty=False)
source = result[1]
expected = (
"function test(x, y)\n"
" x = x.^2 + y\n"
" return x\n"
"end\n"
)
assert source == expected
def test_jl_not_supported():
f = Function('f')
name_expr = ("test", [f(x).diff(x), S.ComplexInfinity])
result, = codegen(name_expr, "Julia", header=False, empty=False)
source = result[1]
expected = (
"function test(x)\n"
" # unsupported: Derivative(f(x), x)\n"
" # unsupported: zoo\n"
" out1 = Derivative(f(x), x)\n"
" out2 = zoo\n"
" return out1, out2\n"
"end\n"
)
assert source == expected
def test_global_vars_octave():
x, y, z, t = symbols("x y z t")
result = codegen(('f', x*y), "Julia", header=False, empty=False,
global_vars=(y,))
source = result[0][1]
expected = (
"function f(x)\n"
" out1 = x.*y\n"
" return out1\n"
"end\n"
)
assert source == expected
result = codegen(('f', x*y+z), "Julia", header=False, empty=False,
argument_sequence=(x, y), global_vars=(z, t))
source = result[0][1]
expected = (
"function f(x, y)\n"
" out1 = x.*y + z\n"
" return out1\n"
"end\n"
)
assert source == expected
|
e6351d6472e7ad8eb004f718459190ff0926e51e5b7bbb686364373717fa57ee
|
from __future__ import absolute_import
import shutil
from sympy.external import import_module
from sympy.utilities.pytest import skip
from sympy.utilities._compilation.compilation import compile_link_import_strings
numpy = import_module('numpy')
cython = import_module('cython')
_sources1 = [
('sigmoid.c', r"""
#include <math.h>
void sigmoid(int n, const double * const restrict in,
double * const restrict out, double lim){
for (int i=0; i<n; ++i){
const double x = in[i];
out[i] = x*pow(pow(x/lim, 8)+1, -1./8.);
}
}
"""),
('_sigmoid.pyx', r"""
import numpy as np
cimport numpy as cnp
cdef extern void c_sigmoid "sigmoid" (int, const double * const,
double * const, double)
def sigmoid(double [:] inp, double lim=350.0):
cdef cnp.ndarray[cnp.float64_t, ndim=1] out = np.empty(
inp.size, dtype=np.float64)
c_sigmoid(inp.size, &inp[0], &out[0], lim)
return out
""")
]
def npy(data, lim=350.0):
return data/((data/lim)**8+1)**(1/8.)
def test_compile_link_import_strings():
if not numpy:
skip("numpy not installed.")
if not cython:
skip("cython not installed.")
from sympy.utilities._compilation import has_c
if not has_c():
skip("No C compiler found.")
compile_kw = dict(std='c99', include_dirs=[numpy.get_include()])
info = None
try:
mod, info = compile_link_import_strings(_sources1, compile_kwargs=compile_kw)
data = numpy.random.random(1024*1024*8) # 64 MB of RAM needed..
res_mod = mod.sigmoid(data)
res_npy = npy(data)
assert numpy.allclose(res_mod, res_npy)
finally:
if info and info['build_dir']:
shutil.rmtree(info['build_dir'])
|
6f2d51017145eb2c6743236e2b6e7aea55825f719347188ba351e6209bb183d9
|
from __future__ import print_function, division
import itertools
from sympy.core import S
from sympy.core.compatibility import range, string_types
from sympy.core.containers import Tuple
from sympy.core.function import _coeff_isneg
from sympy.core.mul import Mul
from sympy.core.numbers import Rational
from sympy.core.power import Pow
from sympy.core.relational import Equality
from sympy.core.symbol import Symbol
from sympy.core.sympify import SympifyError
from sympy.printing.conventions import requires_partial
from sympy.printing.precedence import PRECEDENCE, precedence, precedence_traditional
from sympy.printing.printer import Printer
from sympy.printing.str import sstr
from sympy.utilities import default_sort_key
from sympy.utilities.iterables import has_variety
from sympy.printing.pretty.stringpict import prettyForm, stringPict
from sympy.printing.pretty.pretty_symbology import xstr, hobj, vobj, xobj, \
xsym, pretty_symbol, pretty_atom, pretty_use_unicode, greek_unicode, U, \
pretty_try_use_unicode, annotated
# rename for usage from outside
pprint_use_unicode = pretty_use_unicode
pprint_try_use_unicode = pretty_try_use_unicode
class PrettyPrinter(Printer):
"""Printer, which converts an expression into 2D ASCII-art figure."""
printmethod = "_pretty"
_default_settings = {
"order": None,
"full_prec": "auto",
"use_unicode": None,
"wrap_line": True,
"num_columns": None,
"use_unicode_sqrt_char": True,
"root_notation": True,
"mat_symbol_style": "plain",
"imaginary_unit": "i",
}
def __init__(self, settings=None):
Printer.__init__(self, settings)
if not isinstance(self._settings['imaginary_unit'], string_types):
raise TypeError("'imaginary_unit' must a string, not {}".format(self._settings['imaginary_unit']))
elif self._settings['imaginary_unit'] not in ["i", "j"]:
raise ValueError("'imaginary_unit' must be either 'i' or 'j', not '{}'".format(self._settings['imaginary_unit']))
self.emptyPrinter = lambda x: prettyForm(xstr(x))
@property
def _use_unicode(self):
if self._settings['use_unicode']:
return True
else:
return pretty_use_unicode()
def doprint(self, expr):
return self._print(expr).render(**self._settings)
# empty op so _print(stringPict) returns the same
def _print_stringPict(self, e):
return e
def _print_basestring(self, e):
return prettyForm(e)
def _print_atan2(self, e):
pform = prettyForm(*self._print_seq(e.args).parens())
pform = prettyForm(*pform.left('atan2'))
return pform
def _print_Symbol(self, e, bold_name=False):
symb = pretty_symbol(e.name, bold_name)
return prettyForm(symb)
_print_RandomSymbol = _print_Symbol
def _print_MatrixSymbol(self, e):
return self._print_Symbol(e, self._settings['mat_symbol_style'] == "bold")
def _print_Float(self, e):
# we will use StrPrinter's Float printer, but we need to handle the
# full_prec ourselves, according to the self._print_level
full_prec = self._settings["full_prec"]
if full_prec == "auto":
full_prec = self._print_level == 1
return prettyForm(sstr(e, full_prec=full_prec))
def _print_Cross(self, e):
vec1 = e._expr1
vec2 = e._expr2
pform = self._print(vec2)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('MULTIPLICATION SIGN'))))
pform = prettyForm(*pform.left(')'))
pform = prettyForm(*pform.left(self._print(vec1)))
pform = prettyForm(*pform.left('('))
return pform
def _print_Curl(self, e):
vec = e._expr
pform = self._print(vec)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('MULTIPLICATION SIGN'))))
pform = prettyForm(*pform.left(self._print(U('NABLA'))))
return pform
def _print_Divergence(self, e):
vec = e._expr
pform = self._print(vec)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('DOT OPERATOR'))))
pform = prettyForm(*pform.left(self._print(U('NABLA'))))
return pform
def _print_Dot(self, e):
vec1 = e._expr1
vec2 = e._expr2
pform = self._print(vec2)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('DOT OPERATOR'))))
pform = prettyForm(*pform.left(')'))
pform = prettyForm(*pform.left(self._print(vec1)))
pform = prettyForm(*pform.left('('))
return pform
def _print_Gradient(self, e):
func = e._expr
pform = self._print(func)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('NABLA'))))
return pform
def _print_Laplacian(self, e):
func = e._expr
pform = self._print(func)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('INCREMENT'))))
return pform
def _print_Atom(self, e):
try:
# print atoms like Exp1 or Pi
return prettyForm(pretty_atom(e.__class__.__name__, printer=self))
except KeyError:
return self.emptyPrinter(e)
# Infinity inherits from Number, so we have to override _print_XXX order
_print_Infinity = _print_Atom
_print_NegativeInfinity = _print_Atom
_print_EmptySet = _print_Atom
_print_Naturals = _print_Atom
_print_Naturals0 = _print_Atom
_print_Integers = _print_Atom
_print_Complexes = _print_Atom
def _print_Reals(self, e):
if self._use_unicode:
return self._print_Atom(e)
else:
inf_list = ['-oo', 'oo']
return self._print_seq(inf_list, '(', ')')
def _print_subfactorial(self, e):
x = e.args[0]
pform = self._print(x)
# Add parentheses if needed
if not ((x.is_Integer and x.is_nonnegative) or x.is_Symbol):
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left('!'))
return pform
def _print_factorial(self, e):
x = e.args[0]
pform = self._print(x)
# Add parentheses if needed
if not ((x.is_Integer and x.is_nonnegative) or x.is_Symbol):
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.right('!'))
return pform
def _print_factorial2(self, e):
x = e.args[0]
pform = self._print(x)
# Add parentheses if needed
if not ((x.is_Integer and x.is_nonnegative) or x.is_Symbol):
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.right('!!'))
return pform
def _print_binomial(self, e):
n, k = e.args
n_pform = self._print(n)
k_pform = self._print(k)
bar = ' '*max(n_pform.width(), k_pform.width())
pform = prettyForm(*k_pform.above(bar))
pform = prettyForm(*pform.above(n_pform))
pform = prettyForm(*pform.parens('(', ')'))
pform.baseline = (pform.baseline + 1)//2
return pform
def _print_Relational(self, e):
op = prettyForm(' ' + xsym(e.rel_op) + ' ')
l = self._print(e.lhs)
r = self._print(e.rhs)
pform = prettyForm(*stringPict.next(l, op, r))
return pform
def _print_Not(self, e):
from sympy import Equivalent, Implies
if self._use_unicode:
arg = e.args[0]
pform = self._print(arg)
if isinstance(arg, Equivalent):
return self._print_Equivalent(arg, altchar=u"\N{LEFT RIGHT DOUBLE ARROW WITH STROKE}")
if isinstance(arg, Implies):
return self._print_Implies(arg, altchar=u"\N{RIGHTWARDS ARROW WITH STROKE}")
if arg.is_Boolean and not arg.is_Not:
pform = prettyForm(*pform.parens())
return prettyForm(*pform.left(u"\N{NOT SIGN}"))
else:
return self._print_Function(e)
def __print_Boolean(self, e, char, sort=True):
args = e.args
if sort:
args = sorted(e.args, key=default_sort_key)
arg = args[0]
pform = self._print(arg)
if arg.is_Boolean and not arg.is_Not:
pform = prettyForm(*pform.parens())
for arg in args[1:]:
pform_arg = self._print(arg)
if arg.is_Boolean and not arg.is_Not:
pform_arg = prettyForm(*pform_arg.parens())
pform = prettyForm(*pform.right(u' %s ' % char))
pform = prettyForm(*pform.right(pform_arg))
return pform
def _print_And(self, e):
if self._use_unicode:
return self.__print_Boolean(e, u"\N{LOGICAL AND}")
else:
return self._print_Function(e, sort=True)
def _print_Or(self, e):
if self._use_unicode:
return self.__print_Boolean(e, u"\N{LOGICAL OR}")
else:
return self._print_Function(e, sort=True)
def _print_Xor(self, e):
if self._use_unicode:
return self.__print_Boolean(e, u"\N{XOR}")
else:
return self._print_Function(e, sort=True)
def _print_Nand(self, e):
if self._use_unicode:
return self.__print_Boolean(e, u"\N{NAND}")
else:
return self._print_Function(e, sort=True)
def _print_Nor(self, e):
if self._use_unicode:
return self.__print_Boolean(e, u"\N{NOR}")
else:
return self._print_Function(e, sort=True)
def _print_Implies(self, e, altchar=None):
if self._use_unicode:
return self.__print_Boolean(e, altchar or u"\N{RIGHTWARDS ARROW}", sort=False)
else:
return self._print_Function(e)
def _print_Equivalent(self, e, altchar=None):
if self._use_unicode:
return self.__print_Boolean(e, altchar or u"\N{LEFT RIGHT DOUBLE ARROW}")
else:
return self._print_Function(e, sort=True)
def _print_conjugate(self, e):
pform = self._print(e.args[0])
return prettyForm( *pform.above( hobj('_', pform.width())) )
def _print_Abs(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens('|', '|'))
return pform
_print_Determinant = _print_Abs
def _print_floor(self, e):
if self._use_unicode:
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens('lfloor', 'rfloor'))
return pform
else:
return self._print_Function(e)
def _print_ceiling(self, e):
if self._use_unicode:
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens('lceil', 'rceil'))
return pform
else:
return self._print_Function(e)
def _print_Derivative(self, deriv):
if requires_partial(deriv) and self._use_unicode:
deriv_symbol = U('PARTIAL DIFFERENTIAL')
else:
deriv_symbol = r'd'
x = None
count_total_deriv = 0
for sym, num in reversed(deriv.variable_count):
s = self._print(sym)
ds = prettyForm(*s.left(deriv_symbol))
count_total_deriv += num
if (not num.is_Integer) or (num > 1):
ds = ds**prettyForm(str(num))
if x is None:
x = ds
else:
x = prettyForm(*x.right(' '))
x = prettyForm(*x.right(ds))
f = prettyForm(
binding=prettyForm.FUNC, *self._print(deriv.expr).parens())
pform = prettyForm(deriv_symbol)
if (count_total_deriv > 1) != False:
pform = pform**prettyForm(str(count_total_deriv))
pform = prettyForm(*pform.below(stringPict.LINE, x))
pform.baseline = pform.baseline + 1
pform = prettyForm(*stringPict.next(pform, f))
pform.binding = prettyForm.MUL
return pform
def _print_Cycle(self, dc):
from sympy.combinatorics.permutations import Permutation, Cycle
# for Empty Cycle
if dc == Cycle():
cyc = stringPict('')
return prettyForm(*cyc.parens())
dc_list = Permutation(dc.list()).cyclic_form
# for Identity Cycle
if dc_list == []:
cyc = self._print(dc.size - 1)
return prettyForm(*cyc.parens())
cyc = stringPict('')
for i in dc_list:
l = self._print(str(tuple(i)).replace(',', ''))
cyc = prettyForm(*cyc.right(l))
return cyc
def _print_PDF(self, pdf):
lim = self._print(pdf.pdf.args[0])
lim = prettyForm(*lim.right(', '))
lim = prettyForm(*lim.right(self._print(pdf.domain[0])))
lim = prettyForm(*lim.right(', '))
lim = prettyForm(*lim.right(self._print(pdf.domain[1])))
lim = prettyForm(*lim.parens())
f = self._print(pdf.pdf.args[1])
f = prettyForm(*f.right(', '))
f = prettyForm(*f.right(lim))
f = prettyForm(*f.parens())
pform = prettyForm('PDF')
pform = prettyForm(*pform.right(f))
return pform
def _print_Integral(self, integral):
f = integral.function
# Add parentheses if arg involves addition of terms and
# create a pretty form for the argument
prettyF = self._print(f)
# XXX generalize parens
if f.is_Add:
prettyF = prettyForm(*prettyF.parens())
# dx dy dz ...
arg = prettyF
for x in integral.limits:
prettyArg = self._print(x[0])
# XXX qparens (parens if needs-parens)
if prettyArg.width() > 1:
prettyArg = prettyForm(*prettyArg.parens())
arg = prettyForm(*arg.right(' d', prettyArg))
# \int \int \int ...
firstterm = True
s = None
for lim in integral.limits:
x = lim[0]
# Create bar based on the height of the argument
h = arg.height()
H = h + 2
# XXX hack!
ascii_mode = not self._use_unicode
if ascii_mode:
H += 2
vint = vobj('int', H)
# Construct the pretty form with the integral sign and the argument
pform = prettyForm(vint)
pform.baseline = arg.baseline + (
H - h)//2 # covering the whole argument
if len(lim) > 1:
# Create pretty forms for endpoints, if definite integral.
# Do not print empty endpoints.
if len(lim) == 2:
prettyA = prettyForm("")
prettyB = self._print(lim[1])
if len(lim) == 3:
prettyA = self._print(lim[1])
prettyB = self._print(lim[2])
if ascii_mode: # XXX hack
# Add spacing so that endpoint can more easily be
# identified with the correct integral sign
spc = max(1, 3 - prettyB.width())
prettyB = prettyForm(*prettyB.left(' ' * spc))
spc = max(1, 4 - prettyA.width())
prettyA = prettyForm(*prettyA.right(' ' * spc))
pform = prettyForm(*pform.above(prettyB))
pform = prettyForm(*pform.below(prettyA))
if not ascii_mode: # XXX hack
pform = prettyForm(*pform.right(' '))
if firstterm:
s = pform # first term
firstterm = False
else:
s = prettyForm(*s.left(pform))
pform = prettyForm(*arg.left(s))
pform.binding = prettyForm.MUL
return pform
def _print_Product(self, expr):
func = expr.term
pretty_func = self._print(func)
horizontal_chr = xobj('_', 1)
corner_chr = xobj('_', 1)
vertical_chr = xobj('|', 1)
if self._use_unicode:
# use unicode corners
horizontal_chr = xobj('-', 1)
corner_chr = u'\N{BOX DRAWINGS LIGHT DOWN AND HORIZONTAL}'
func_height = pretty_func.height()
first = True
max_upper = 0
sign_height = 0
for lim in expr.limits:
width = (func_height + 2) * 5 // 3 - 2
sign_lines = [horizontal_chr + corner_chr + (horizontal_chr * (width-2)) + corner_chr + horizontal_chr]
for _ in range(func_height + 1):
sign_lines.append(' ' + vertical_chr + (' ' * (width-2)) + vertical_chr + ' ')
pretty_sign = stringPict('')
pretty_sign = prettyForm(*pretty_sign.stack(*sign_lines))
pretty_upper = self._print(lim[2])
pretty_lower = self._print(Equality(lim[0], lim[1]))
max_upper = max(max_upper, pretty_upper.height())
if first:
sign_height = pretty_sign.height()
pretty_sign = prettyForm(*pretty_sign.above(pretty_upper))
pretty_sign = prettyForm(*pretty_sign.below(pretty_lower))
if first:
pretty_func.baseline = 0
first = False
height = pretty_sign.height()
padding = stringPict('')
padding = prettyForm(*padding.stack(*[' ']*(height - 1)))
pretty_sign = prettyForm(*pretty_sign.right(padding))
pretty_func = prettyForm(*pretty_sign.right(pretty_func))
pretty_func.baseline = max_upper + sign_height//2
pretty_func.binding = prettyForm.MUL
return pretty_func
def _print_Sum(self, expr):
ascii_mode = not self._use_unicode
def asum(hrequired, lower, upper, use_ascii):
def adjust(s, wid=None, how='<^>'):
if not wid or len(s) > wid:
return s
need = wid - len(s)
if how == '<^>' or how == "<" or how not in list('<^>'):
return s + ' '*need
half = need//2
lead = ' '*half
if how == ">":
return " "*need + s
return lead + s + ' '*(need - len(lead))
h = max(hrequired, 2)
d = h//2
w = d + 1
more = hrequired % 2
lines = []
if use_ascii:
lines.append("_"*(w) + ' ')
lines.append(r"\%s`" % (' '*(w - 1)))
for i in range(1, d):
lines.append('%s\\%s' % (' '*i, ' '*(w - i)))
if more:
lines.append('%s)%s' % (' '*(d), ' '*(w - d)))
for i in reversed(range(1, d)):
lines.append('%s/%s' % (' '*i, ' '*(w - i)))
lines.append("/" + "_"*(w - 1) + ',')
return d, h + more, lines, 0
else:
w = w + more
d = d + more
vsum = vobj('sum', 4)
lines.append("_"*(w))
for i in range(0, d):
lines.append('%s%s%s' % (' '*i, vsum[2], ' '*(w - i - 1)))
for i in reversed(range(0, d)):
lines.append('%s%s%s' % (' '*i, vsum[4], ' '*(w - i - 1)))
lines.append(vsum[8]*(w))
return d, h + 2*more, lines, more
f = expr.function
prettyF = self._print(f)
if f.is_Add: # add parens
prettyF = prettyForm(*prettyF.parens())
H = prettyF.height() + 2
# \sum \sum \sum ...
first = True
max_upper = 0
sign_height = 0
for lim in expr.limits:
if len(lim) == 3:
prettyUpper = self._print(lim[2])
prettyLower = self._print(Equality(lim[0], lim[1]))
elif len(lim) == 2:
prettyUpper = self._print("")
prettyLower = self._print(Equality(lim[0], lim[1]))
elif len(lim) == 1:
prettyUpper = self._print("")
prettyLower = self._print(lim[0])
max_upper = max(max_upper, prettyUpper.height())
# Create sum sign based on the height of the argument
d, h, slines, adjustment = asum(
H, prettyLower.width(), prettyUpper.width(), ascii_mode)
prettySign = stringPict('')
prettySign = prettyForm(*prettySign.stack(*slines))
if first:
sign_height = prettySign.height()
prettySign = prettyForm(*prettySign.above(prettyUpper))
prettySign = prettyForm(*prettySign.below(prettyLower))
if first:
# change F baseline so it centers on the sign
prettyF.baseline -= d - (prettyF.height()//2 -
prettyF.baseline) - adjustment
first = False
# put padding to the right
pad = stringPict('')
pad = prettyForm(*pad.stack(*[' ']*h))
prettySign = prettyForm(*prettySign.right(pad))
# put the present prettyF to the right
prettyF = prettyForm(*prettySign.right(prettyF))
prettyF.baseline = max_upper + sign_height//2
prettyF.binding = prettyForm.MUL
return prettyF
def _print_Limit(self, l):
e, z, z0, dir = l.args
E = self._print(e)
if precedence(e) <= PRECEDENCE["Mul"]:
E = prettyForm(*E.parens('(', ')'))
Lim = prettyForm('lim')
LimArg = self._print(z)
if self._use_unicode:
LimArg = prettyForm(*LimArg.right(u'\N{BOX DRAWINGS LIGHT HORIZONTAL}\N{RIGHTWARDS ARROW}'))
else:
LimArg = prettyForm(*LimArg.right('->'))
LimArg = prettyForm(*LimArg.right(self._print(z0)))
if str(dir) == '+-' or z0 in (S.Infinity, S.NegativeInfinity):
dir = ""
else:
if self._use_unicode:
dir = u'\N{SUPERSCRIPT PLUS SIGN}' if str(dir) == "+" else u'\N{SUPERSCRIPT MINUS}'
LimArg = prettyForm(*LimArg.right(self._print(dir)))
Lim = prettyForm(*Lim.below(LimArg))
Lim = prettyForm(*Lim.right(E), binding=prettyForm.MUL)
return Lim
def _print_matrix_contents(self, e):
"""
This method factors out what is essentially grid printing.
"""
M = e # matrix
Ms = {} # i,j -> pretty(M[i,j])
for i in range(M.rows):
for j in range(M.cols):
Ms[i, j] = self._print(M[i, j])
# h- and v- spacers
hsep = 2
vsep = 1
# max width for columns
maxw = [-1] * M.cols
for j in range(M.cols):
maxw[j] = max([Ms[i, j].width() for i in range(M.rows)] or [0])
# drawing result
D = None
for i in range(M.rows):
D_row = None
for j in range(M.cols):
s = Ms[i, j]
# reshape s to maxw
# XXX this should be generalized, and go to stringPict.reshape ?
assert s.width() <= maxw[j]
# hcenter it, +0.5 to the right 2
# ( it's better to align formula starts for say 0 and r )
# XXX this is not good in all cases -- maybe introduce vbaseline?
wdelta = maxw[j] - s.width()
wleft = wdelta // 2
wright = wdelta - wleft
s = prettyForm(*s.right(' '*wright))
s = prettyForm(*s.left(' '*wleft))
# we don't need vcenter cells -- this is automatically done in
# a pretty way because when their baselines are taking into
# account in .right()
if D_row is None:
D_row = s # first box in a row
continue
D_row = prettyForm(*D_row.right(' '*hsep)) # h-spacer
D_row = prettyForm(*D_row.right(s))
if D is None:
D = D_row # first row in a picture
continue
# v-spacer
for _ in range(vsep):
D = prettyForm(*D.below(' '))
D = prettyForm(*D.below(D_row))
if D is None:
D = prettyForm('') # Empty Matrix
return D
def _print_MatrixBase(self, e):
D = self._print_matrix_contents(e)
D.baseline = D.height()//2
D = prettyForm(*D.parens('[', ']'))
return D
_print_ImmutableMatrix = _print_MatrixBase
_print_Matrix = _print_MatrixBase
def _print_TensorProduct(self, expr):
# This should somehow share the code with _print_WedgeProduct:
circled_times = "\u2297"
return self._print_seq(expr.args, None, None, circled_times,
parenthesize=lambda x: precedence_traditional(x) <= PRECEDENCE["Mul"])
def _print_WedgeProduct(self, expr):
# This should somehow share the code with _print_TensorProduct:
wedge_symbol = u"\u2227"
return self._print_seq(expr.args, None, None, wedge_symbol,
parenthesize=lambda x: precedence_traditional(x) <= PRECEDENCE["Mul"])
def _print_Trace(self, e):
D = self._print(e.arg)
D = prettyForm(*D.parens('(',')'))
D.baseline = D.height()//2
D = prettyForm(*D.left('\n'*(0) + 'tr'))
return D
def _print_MatrixElement(self, expr):
from sympy.matrices import MatrixSymbol
from sympy import Symbol
if (isinstance(expr.parent, MatrixSymbol)
and expr.i.is_number and expr.j.is_number):
return self._print(
Symbol(expr.parent.name + '_%d%d' % (expr.i, expr.j)))
else:
prettyFunc = self._print(expr.parent)
prettyFunc = prettyForm(*prettyFunc.parens())
prettyIndices = self._print_seq((expr.i, expr.j), delimiter=', '
).parens(left='[', right=']')[0]
pform = prettyForm(binding=prettyForm.FUNC,
*stringPict.next(prettyFunc, prettyIndices))
# store pform parts so it can be reassembled e.g. when powered
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyIndices
return pform
def _print_MatrixSlice(self, m):
# XXX works only for applied functions
prettyFunc = self._print(m.parent)
def ppslice(x):
x = list(x)
if x[2] == 1:
del x[2]
if x[1] == x[0] + 1:
del x[1]
if x[0] == 0:
x[0] = ''
return prettyForm(*self._print_seq(x, delimiter=':'))
prettyArgs = self._print_seq((ppslice(m.rowslice),
ppslice(m.colslice)), delimiter=', ').parens(left='[', right=']')[0]
pform = prettyForm(
binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs))
# store pform parts so it can be reassembled e.g. when powered
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyArgs
return pform
def _print_Transpose(self, expr):
pform = self._print(expr.arg)
from sympy.matrices import MatrixSymbol
if not isinstance(expr.arg, MatrixSymbol):
pform = prettyForm(*pform.parens())
pform = pform**(prettyForm('T'))
return pform
def _print_Adjoint(self, expr):
pform = self._print(expr.arg)
if self._use_unicode:
dag = prettyForm(u'\N{DAGGER}')
else:
dag = prettyForm('+')
from sympy.matrices import MatrixSymbol
if not isinstance(expr.arg, MatrixSymbol):
pform = prettyForm(*pform.parens())
pform = pform**dag
return pform
def _print_BlockMatrix(self, B):
if B.blocks.shape == (1, 1):
return self._print(B.blocks[0, 0])
return self._print(B.blocks)
def _print_MatAdd(self, expr):
s = None
for item in expr.args:
pform = self._print(item)
if s is None:
s = pform # First element
else:
coeff = item.as_coeff_mmul()[0]
if _coeff_isneg(S(coeff)):
s = prettyForm(*stringPict.next(s, ' '))
pform = self._print(item)
else:
s = prettyForm(*stringPict.next(s, ' + '))
s = prettyForm(*stringPict.next(s, pform))
return s
def _print_MatMul(self, expr):
args = list(expr.args)
from sympy import Add, MatAdd, HadamardProduct, KroneckerProduct
for i, a in enumerate(args):
if (isinstance(a, (Add, MatAdd, HadamardProduct, KroneckerProduct))
and len(expr.args) > 1):
args[i] = prettyForm(*self._print(a).parens())
else:
args[i] = self._print(a)
return prettyForm.__mul__(*args)
def _print_DotProduct(self, expr):
args = list(expr.args)
for i, a in enumerate(args):
args[i] = self._print(a)
return prettyForm.__mul__(*args)
def _print_MatPow(self, expr):
pform = self._print(expr.base)
from sympy.matrices import MatrixSymbol
if not isinstance(expr.base, MatrixSymbol):
pform = prettyForm(*pform.parens())
pform = pform**(self._print(expr.exp))
return pform
def _print_HadamardProduct(self, expr):
from sympy import MatAdd, MatMul
if self._use_unicode:
delim = pretty_atom('Ring')
else:
delim = '.*'
return self._print_seq(expr.args, None, None, delim,
parenthesize=lambda x: isinstance(x, (MatAdd, MatMul)))
def _print_KroneckerProduct(self, expr):
from sympy import MatAdd, MatMul
if self._use_unicode:
delim = u' \N{N-ARY CIRCLED TIMES OPERATOR} '
else:
delim = ' x '
return self._print_seq(expr.args, None, None, delim,
parenthesize=lambda x: isinstance(x, (MatAdd, MatMul)))
def _print_FunctionMatrix(self, X):
D = self._print(X.lamda.expr)
D = prettyForm(*D.parens('[', ']'))
return D
def _print_BasisDependent(self, expr):
from sympy.vector import Vector
if not self._use_unicode:
raise NotImplementedError("ASCII pretty printing of BasisDependent is not implemented")
if expr == expr.zero:
return prettyForm(expr.zero._pretty_form)
o1 = []
vectstrs = []
if isinstance(expr, Vector):
items = expr.separate().items()
else:
items = [(0, expr)]
for system, vect in items:
inneritems = list(vect.components.items())
inneritems.sort(key = lambda x: x[0].__str__())
for k, v in inneritems:
#if the coef of the basis vector is 1
#we skip the 1
if v == 1:
o1.append(u"" +
k._pretty_form)
#Same for -1
elif v == -1:
o1.append(u"(-1) " +
k._pretty_form)
#For a general expr
else:
#We always wrap the measure numbers in
#parentheses
arg_str = self._print(
v).parens()[0]
o1.append(arg_str + ' ' + k._pretty_form)
vectstrs.append(k._pretty_form)
#outstr = u("").join(o1)
if o1[0].startswith(u" + "):
o1[0] = o1[0][3:]
elif o1[0].startswith(" "):
o1[0] = o1[0][1:]
#Fixing the newlines
lengths = []
strs = ['']
flag = []
for i, partstr in enumerate(o1):
flag.append(0)
# XXX: What is this hack?
if '\n' in partstr:
tempstr = partstr
tempstr = tempstr.replace(vectstrs[i], '')
if u'\N{right parenthesis extension}' in tempstr: # If scalar is a fraction
for paren in range(len(tempstr)):
flag[i] = 1
if tempstr[paren] == u'\N{right parenthesis extension}':
tempstr = tempstr[:paren] + u'\N{right parenthesis extension}'\
+ ' ' + vectstrs[i] + tempstr[paren + 1:]
break
elif u'\N{RIGHT PARENTHESIS LOWER HOOK}' in tempstr:
flag[i] = 1
tempstr = tempstr.replace(u'\N{RIGHT PARENTHESIS LOWER HOOK}',
u'\N{RIGHT PARENTHESIS LOWER HOOK}'
+ ' ' + vectstrs[i])
else:
tempstr = tempstr.replace(u'\N{RIGHT PARENTHESIS UPPER HOOK}',
u'\N{RIGHT PARENTHESIS UPPER HOOK}'
+ ' ' + vectstrs[i])
o1[i] = tempstr
o1 = [x.split('\n') for x in o1]
n_newlines = max([len(x) for x in o1]) # Width of part in its pretty form
if 1 in flag: # If there was a fractional scalar
for i, parts in enumerate(o1):
if len(parts) == 1: # If part has no newline
parts.insert(0, ' ' * (len(parts[0])))
flag[i] = 1
for i, parts in enumerate(o1):
lengths.append(len(parts[flag[i]]))
for j in range(n_newlines):
if j+1 <= len(parts):
if j >= len(strs):
strs.append(' ' * (sum(lengths[:-1]) +
3*(len(lengths)-1)))
if j == flag[i]:
strs[flag[i]] += parts[flag[i]] + ' + '
else:
strs[j] += parts[j] + ' '*(lengths[-1] -
len(parts[j])+
3)
else:
if j >= len(strs):
strs.append(' ' * (sum(lengths[:-1]) +
3*(len(lengths)-1)))
strs[j] += ' '*(lengths[-1]+3)
return prettyForm(u'\n'.join([s[:-3] for s in strs]))
def _print_NDimArray(self, expr):
from sympy import ImmutableMatrix
if expr.rank() == 0:
return self._print(expr[()])
level_str = [[]] + [[] for i in range(expr.rank())]
shape_ranges = [list(range(i)) for i in expr.shape]
for outer_i in itertools.product(*shape_ranges):
level_str[-1].append(expr[outer_i])
even = True
for back_outer_i in range(expr.rank()-1, -1, -1):
if len(level_str[back_outer_i+1]) < expr.shape[back_outer_i]:
break
if even:
level_str[back_outer_i].append(level_str[back_outer_i+1])
else:
level_str[back_outer_i].append(ImmutableMatrix(level_str[back_outer_i+1]))
if len(level_str[back_outer_i + 1]) == 1:
level_str[back_outer_i][-1] = ImmutableMatrix([[level_str[back_outer_i][-1]]])
even = not even
level_str[back_outer_i+1] = []
out_expr = level_str[0][0]
if expr.rank() % 2 == 1:
out_expr = ImmutableMatrix([out_expr])
return self._print(out_expr)
_print_ImmutableDenseNDimArray = _print_NDimArray
_print_ImmutableSparseNDimArray = _print_NDimArray
_print_MutableDenseNDimArray = _print_NDimArray
_print_MutableSparseNDimArray = _print_NDimArray
def _printer_tensor_indices(self, name, indices, index_map={}):
center = stringPict(name)
top = stringPict(" "*center.width())
bot = stringPict(" "*center.width())
last_valence = None
prev_map = None
for i, index in enumerate(indices):
indpic = self._print(index.args[0])
if ((index in index_map) or prev_map) and last_valence == index.is_up:
if index.is_up:
top = prettyForm(*stringPict.next(top, ","))
else:
bot = prettyForm(*stringPict.next(bot, ","))
if index in index_map:
indpic = prettyForm(*stringPict.next(indpic, "="))
indpic = prettyForm(*stringPict.next(indpic, self._print(index_map[index])))
prev_map = True
else:
prev_map = False
if index.is_up:
top = stringPict(*top.right(indpic))
center = stringPict(*center.right(" "*indpic.width()))
bot = stringPict(*bot.right(" "*indpic.width()))
else:
bot = stringPict(*bot.right(indpic))
center = stringPict(*center.right(" "*indpic.width()))
top = stringPict(*top.right(" "*indpic.width()))
last_valence = index.is_up
pict = prettyForm(*center.above(top))
pict = prettyForm(*pict.below(bot))
return pict
def _print_Tensor(self, expr):
name = expr.args[0].name
indices = expr.get_indices()
return self._printer_tensor_indices(name, indices)
def _print_TensorElement(self, expr):
name = expr.expr.args[0].name
indices = expr.expr.get_indices()
index_map = expr.index_map
return self._printer_tensor_indices(name, indices, index_map)
def _print_TensMul(self, expr):
sign, args = expr._get_args_for_traditional_printer()
args = [
prettyForm(*self._print(i).parens()) if
precedence_traditional(i) < PRECEDENCE["Mul"] else self._print(i)
for i in args
]
pform = prettyForm.__mul__(*args)
if sign:
return prettyForm(*pform.left(sign))
else:
return pform
def _print_TensAdd(self, expr):
args = [
prettyForm(*self._print(i).parens()) if
precedence_traditional(i) < PRECEDENCE["Mul"] else self._print(i)
for i in expr.args
]
return prettyForm.__add__(*args)
def _print_TensorIndex(self, expr):
sym = expr.args[0]
if not expr.is_up:
sym = -sym
return self._print(sym)
def _print_PartialDerivative(self, deriv):
if self._use_unicode:
deriv_symbol = U('PARTIAL DIFFERENTIAL')
else:
deriv_symbol = r'd'
x = None
for variable in reversed(deriv.variables):
s = self._print(variable)
ds = prettyForm(*s.left(deriv_symbol))
if x is None:
x = ds
else:
x = prettyForm(*x.right(' '))
x = prettyForm(*x.right(ds))
f = prettyForm(
binding=prettyForm.FUNC, *self._print(deriv.expr).parens())
pform = prettyForm(deriv_symbol)
pform = prettyForm(*pform.below(stringPict.LINE, x))
pform.baseline = pform.baseline + 1
pform = prettyForm(*stringPict.next(pform, f))
pform.binding = prettyForm.MUL
return pform
def _print_Piecewise(self, pexpr):
P = {}
for n, ec in enumerate(pexpr.args):
P[n, 0] = self._print(ec.expr)
if ec.cond == True:
P[n, 1] = prettyForm('otherwise')
else:
P[n, 1] = prettyForm(
*prettyForm('for ').right(self._print(ec.cond)))
hsep = 2
vsep = 1
len_args = len(pexpr.args)
# max widths
maxw = [max([P[i, j].width() for i in range(len_args)])
for j in range(2)]
# FIXME: Refactor this code and matrix into some tabular environment.
# drawing result
D = None
for i in range(len_args):
D_row = None
for j in range(2):
p = P[i, j]
assert p.width() <= maxw[j]
wdelta = maxw[j] - p.width()
wleft = wdelta // 2
wright = wdelta - wleft
p = prettyForm(*p.right(' '*wright))
p = prettyForm(*p.left(' '*wleft))
if D_row is None:
D_row = p
continue
D_row = prettyForm(*D_row.right(' '*hsep)) # h-spacer
D_row = prettyForm(*D_row.right(p))
if D is None:
D = D_row # first row in a picture
continue
# v-spacer
for _ in range(vsep):
D = prettyForm(*D.below(' '))
D = prettyForm(*D.below(D_row))
D = prettyForm(*D.parens('{', ''))
D.baseline = D.height()//2
D.binding = prettyForm.OPEN
return D
def _print_ITE(self, ite):
from sympy.functions.elementary.piecewise import Piecewise
return self._print(ite.rewrite(Piecewise))
def _hprint_vec(self, v):
D = None
for a in v:
p = a
if D is None:
D = p
else:
D = prettyForm(*D.right(', '))
D = prettyForm(*D.right(p))
if D is None:
D = stringPict(' ')
return D
def _hprint_vseparator(self, p1, p2):
tmp = prettyForm(*p1.right(p2))
sep = stringPict(vobj('|', tmp.height()), baseline=tmp.baseline)
return prettyForm(*p1.right(sep, p2))
def _print_hyper(self, e):
# FIXME refactor Matrix, Piecewise, and this into a tabular environment
ap = [self._print(a) for a in e.ap]
bq = [self._print(b) for b in e.bq]
P = self._print(e.argument)
P.baseline = P.height()//2
# Drawing result - first create the ap, bq vectors
D = None
for v in [ap, bq]:
D_row = self._hprint_vec(v)
if D is None:
D = D_row # first row in a picture
else:
D = prettyForm(*D.below(' '))
D = prettyForm(*D.below(D_row))
# make sure that the argument `z' is centred vertically
D.baseline = D.height()//2
# insert horizontal separator
P = prettyForm(*P.left(' '))
D = prettyForm(*D.right(' '))
# insert separating `|`
D = self._hprint_vseparator(D, P)
# add parens
D = prettyForm(*D.parens('(', ')'))
# create the F symbol
above = D.height()//2 - 1
below = D.height() - above - 1
sz, t, b, add, img = annotated('F')
F = prettyForm('\n' * (above - t) + img + '\n' * (below - b),
baseline=above + sz)
add = (sz + 1)//2
F = prettyForm(*F.left(self._print(len(e.ap))))
F = prettyForm(*F.right(self._print(len(e.bq))))
F.baseline = above + add
D = prettyForm(*F.right(' ', D))
return D
def _print_meijerg(self, e):
# FIXME refactor Matrix, Piecewise, and this into a tabular environment
v = {}
v[(0, 0)] = [self._print(a) for a in e.an]
v[(0, 1)] = [self._print(a) for a in e.aother]
v[(1, 0)] = [self._print(b) for b in e.bm]
v[(1, 1)] = [self._print(b) for b in e.bother]
P = self._print(e.argument)
P.baseline = P.height()//2
vp = {}
for idx in v:
vp[idx] = self._hprint_vec(v[idx])
for i in range(2):
maxw = max(vp[(0, i)].width(), vp[(1, i)].width())
for j in range(2):
s = vp[(j, i)]
left = (maxw - s.width()) // 2
right = maxw - left - s.width()
s = prettyForm(*s.left(' ' * left))
s = prettyForm(*s.right(' ' * right))
vp[(j, i)] = s
D1 = prettyForm(*vp[(0, 0)].right(' ', vp[(0, 1)]))
D1 = prettyForm(*D1.below(' '))
D2 = prettyForm(*vp[(1, 0)].right(' ', vp[(1, 1)]))
D = prettyForm(*D1.below(D2))
# make sure that the argument `z' is centred vertically
D.baseline = D.height()//2
# insert horizontal separator
P = prettyForm(*P.left(' '))
D = prettyForm(*D.right(' '))
# insert separating `|`
D = self._hprint_vseparator(D, P)
# add parens
D = prettyForm(*D.parens('(', ')'))
# create the G symbol
above = D.height()//2 - 1
below = D.height() - above - 1
sz, t, b, add, img = annotated('G')
F = prettyForm('\n' * (above - t) + img + '\n' * (below - b),
baseline=above + sz)
pp = self._print(len(e.ap))
pq = self._print(len(e.bq))
pm = self._print(len(e.bm))
pn = self._print(len(e.an))
def adjust(p1, p2):
diff = p1.width() - p2.width()
if diff == 0:
return p1, p2
elif diff > 0:
return p1, prettyForm(*p2.left(' '*diff))
else:
return prettyForm(*p1.left(' '*-diff)), p2
pp, pm = adjust(pp, pm)
pq, pn = adjust(pq, pn)
pu = prettyForm(*pm.right(', ', pn))
pl = prettyForm(*pp.right(', ', pq))
ht = F.baseline - above - 2
if ht > 0:
pu = prettyForm(*pu.below('\n'*ht))
p = prettyForm(*pu.below(pl))
F.baseline = above
F = prettyForm(*F.right(p))
F.baseline = above + add
D = prettyForm(*F.right(' ', D))
return D
def _print_ExpBase(self, e):
# TODO should exp_polar be printed differently?
# what about exp_polar(0), exp_polar(1)?
base = prettyForm(pretty_atom('Exp1', 'e'))
return base ** self._print(e.args[0])
def _print_Function(self, e, sort=False, func_name=None):
# optional argument func_name for supplying custom names
# XXX works only for applied functions
func = e.func
args = e.args
if sort:
args = sorted(args, key=default_sort_key)
if not func_name:
func_name = func.__name__
prettyFunc = self._print(Symbol(func_name))
prettyArgs = prettyForm(*self._print_seq(args).parens())
pform = prettyForm(
binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs))
# store pform parts so it can be reassembled e.g. when powered
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyArgs
return pform
@property
def _special_function_classes(self):
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.functions.special.gamma_functions import gamma, lowergamma
from sympy.functions.special.zeta_functions import lerchphi
from sympy.functions.special.beta_functions import beta
from sympy.functions.special.delta_functions import DiracDelta
from sympy.functions.special.error_functions import Chi
return {KroneckerDelta: [greek_unicode['delta'], 'delta'],
gamma: [greek_unicode['Gamma'], 'Gamma'],
lerchphi: [greek_unicode['Phi'], 'lerchphi'],
lowergamma: [greek_unicode['gamma'], 'gamma'],
beta: [greek_unicode['Beta'], 'B'],
DiracDelta: [greek_unicode['delta'], 'delta'],
Chi: ['Chi', 'Chi']}
def _print_FunctionClass(self, expr):
for cls in self._special_function_classes:
if issubclass(expr, cls) and expr.__name__ == cls.__name__:
if self._use_unicode:
return prettyForm(self._special_function_classes[cls][0])
else:
return prettyForm(self._special_function_classes[cls][1])
func_name = expr.__name__
return prettyForm(pretty_symbol(func_name))
def _print_GeometryEntity(self, expr):
# GeometryEntity is based on Tuple but should not print like a Tuple
return self.emptyPrinter(expr)
def _print_lerchphi(self, e):
func_name = greek_unicode['Phi'] if self._use_unicode else 'lerchphi'
return self._print_Function(e, func_name=func_name)
def _print_Lambda(self, e):
vars, expr = e.args
if self._use_unicode:
arrow = u" \N{RIGHTWARDS ARROW FROM BAR} "
else:
arrow = " -> "
if len(vars) == 1:
var_form = self._print(vars[0])
else:
var_form = self._print(tuple(vars))
return prettyForm(*stringPict.next(var_form, arrow, self._print(expr)), binding=8)
def _print_Order(self, expr):
pform = self._print(expr.expr)
if (expr.point and any(p != S.Zero for p in expr.point)) or \
len(expr.variables) > 1:
pform = prettyForm(*pform.right("; "))
if len(expr.variables) > 1:
pform = prettyForm(*pform.right(self._print(expr.variables)))
elif len(expr.variables):
pform = prettyForm(*pform.right(self._print(expr.variables[0])))
if self._use_unicode:
pform = prettyForm(*pform.right(u" \N{RIGHTWARDS ARROW} "))
else:
pform = prettyForm(*pform.right(" -> "))
if len(expr.point) > 1:
pform = prettyForm(*pform.right(self._print(expr.point)))
else:
pform = prettyForm(*pform.right(self._print(expr.point[0])))
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left("O"))
return pform
def _print_SingularityFunction(self, e):
if self._use_unicode:
shift = self._print(e.args[0]-e.args[1])
n = self._print(e.args[2])
base = prettyForm("<")
base = prettyForm(*base.right(shift))
base = prettyForm(*base.right(">"))
pform = base**n
return pform
else:
n = self._print(e.args[2])
shift = self._print(e.args[0]-e.args[1])
base = self._print_seq(shift, "<", ">", ' ')
return base**n
def _print_beta(self, e):
func_name = greek_unicode['Beta'] if self._use_unicode else 'B'
return self._print_Function(e, func_name=func_name)
def _print_gamma(self, e):
func_name = greek_unicode['Gamma'] if self._use_unicode else 'Gamma'
return self._print_Function(e, func_name=func_name)
def _print_uppergamma(self, e):
func_name = greek_unicode['Gamma'] if self._use_unicode else 'Gamma'
return self._print_Function(e, func_name=func_name)
def _print_lowergamma(self, e):
func_name = greek_unicode['gamma'] if self._use_unicode else 'lowergamma'
return self._print_Function(e, func_name=func_name)
def _print_DiracDelta(self, e):
if self._use_unicode:
if len(e.args) == 2:
a = prettyForm(greek_unicode['delta'])
b = self._print(e.args[1])
b = prettyForm(*b.parens())
c = self._print(e.args[0])
c = prettyForm(*c.parens())
pform = a**b
pform = prettyForm(*pform.right(' '))
pform = prettyForm(*pform.right(c))
return pform
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left(greek_unicode['delta']))
return pform
else:
return self._print_Function(e)
def _print_expint(self, e):
from sympy import Function
if e.args[0].is_Integer and self._use_unicode:
return self._print_Function(Function('E_%s' % e.args[0])(e.args[1]))
return self._print_Function(e)
def _print_Chi(self, e):
# This needs a special case since otherwise it comes out as greek
# letter chi...
prettyFunc = prettyForm("Chi")
prettyArgs = prettyForm(*self._print_seq(e.args).parens())
pform = prettyForm(
binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs))
# store pform parts so it can be reassembled e.g. when powered
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyArgs
return pform
def _print_elliptic_e(self, e):
pforma0 = self._print(e.args[0])
if len(e.args) == 1:
pform = pforma0
else:
pforma1 = self._print(e.args[1])
pform = self._hprint_vseparator(pforma0, pforma1)
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left('E'))
return pform
def _print_elliptic_k(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left('K'))
return pform
def _print_elliptic_f(self, e):
pforma0 = self._print(e.args[0])
pforma1 = self._print(e.args[1])
pform = self._hprint_vseparator(pforma0, pforma1)
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left('F'))
return pform
def _print_elliptic_pi(self, e):
name = greek_unicode['Pi'] if self._use_unicode else 'Pi'
pforma0 = self._print(e.args[0])
pforma1 = self._print(e.args[1])
if len(e.args) == 2:
pform = self._hprint_vseparator(pforma0, pforma1)
else:
pforma2 = self._print(e.args[2])
pforma = self._hprint_vseparator(pforma1, pforma2)
pforma = prettyForm(*pforma.left('; '))
pform = prettyForm(*pforma.left(pforma0))
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left(name))
return pform
def _print_GoldenRatio(self, expr):
if self._use_unicode:
return prettyForm(pretty_symbol('phi'))
return self._print(Symbol("GoldenRatio"))
def _print_EulerGamma(self, expr):
if self._use_unicode:
return prettyForm(pretty_symbol('gamma'))
return self._print(Symbol("EulerGamma"))
def _print_Mod(self, expr):
pform = self._print(expr.args[0])
if pform.binding > prettyForm.MUL:
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.right(' mod '))
pform = prettyForm(*pform.right(self._print(expr.args[1])))
pform.binding = prettyForm.OPEN
return pform
def _print_Add(self, expr, order=None):
if self.order == 'none':
terms = list(expr.args)
else:
terms = self._as_ordered_terms(expr, order=order)
pforms, indices = [], []
def pretty_negative(pform, index):
"""Prepend a minus sign to a pretty form. """
#TODO: Move this code to prettyForm
if index == 0:
if pform.height() > 1:
pform_neg = '- '
else:
pform_neg = '-'
else:
pform_neg = ' - '
if (pform.binding > prettyForm.NEG
or pform.binding == prettyForm.ADD):
p = stringPict(*pform.parens())
else:
p = pform
p = stringPict.next(pform_neg, p)
# Lower the binding to NEG, even if it was higher. Otherwise, it
# will print as a + ( - (b)), instead of a - (b).
return prettyForm(binding=prettyForm.NEG, *p)
for i, term in enumerate(terms):
if term.is_Mul and _coeff_isneg(term):
coeff, other = term.as_coeff_mul(rational=False)
pform = self._print(Mul(-coeff, *other, evaluate=False))
pforms.append(pretty_negative(pform, i))
elif term.is_Rational and term.q > 1:
pforms.append(None)
indices.append(i)
elif term.is_Number and term < 0:
pform = self._print(-term)
pforms.append(pretty_negative(pform, i))
elif term.is_Relational:
pforms.append(prettyForm(*self._print(term).parens()))
else:
pforms.append(self._print(term))
if indices:
large = True
for pform in pforms:
if pform is not None and pform.height() > 1:
break
else:
large = False
for i in indices:
term, negative = terms[i], False
if term < 0:
term, negative = -term, True
if large:
pform = prettyForm(str(term.p))/prettyForm(str(term.q))
else:
pform = self._print(term)
if negative:
pform = pretty_negative(pform, i)
pforms[i] = pform
return prettyForm.__add__(*pforms)
def _print_Mul(self, product):
from sympy.physics.units import Quantity
a = [] # items in the numerator
b = [] # items that are in the denominator (if any)
if self.order not in ('old', 'none'):
args = product.as_ordered_factors()
else:
args = list(product.args)
# If quantities are present append them at the back
args = sorted(args, key=lambda x: isinstance(x, Quantity) or
(isinstance(x, Pow) and isinstance(x.base, Quantity)))
# Gather terms for numerator/denominator
for item in args:
if item.is_commutative and item.is_Pow and item.exp.is_Rational and item.exp.is_negative:
if item.exp != -1:
b.append(Pow(item.base, -item.exp, evaluate=False))
else:
b.append(Pow(item.base, -item.exp))
elif item.is_Rational and item is not S.Infinity:
if item.p != 1:
a.append( Rational(item.p) )
if item.q != 1:
b.append( Rational(item.q) )
else:
a.append(item)
from sympy import Integral, Piecewise, Product, Sum
# Convert to pretty forms. Add parens to Add instances if there
# is more than one term in the numer/denom
for i in range(0, len(a)):
if (a[i].is_Add and len(a) > 1) or (i != len(a) - 1 and
isinstance(a[i], (Integral, Piecewise, Product, Sum))):
a[i] = prettyForm(*self._print(a[i]).parens())
elif a[i].is_Relational:
a[i] = prettyForm(*self._print(a[i]).parens())
else:
a[i] = self._print(a[i])
for i in range(0, len(b)):
if (b[i].is_Add and len(b) > 1) or (i != len(b) - 1 and
isinstance(b[i], (Integral, Piecewise, Product, Sum))):
b[i] = prettyForm(*self._print(b[i]).parens())
else:
b[i] = self._print(b[i])
# Construct a pretty form
if len(b) == 0:
return prettyForm.__mul__(*a)
else:
if len(a) == 0:
a.append( self._print(S.One) )
return prettyForm.__mul__(*a)/prettyForm.__mul__(*b)
# A helper function for _print_Pow to print x**(1/n)
def _print_nth_root(self, base, expt):
bpretty = self._print(base)
# In very simple cases, use a single-char root sign
if (self._settings['use_unicode_sqrt_char'] and self._use_unicode
and expt is S.Half and bpretty.height() == 1
and (bpretty.width() == 1
or (base.is_Integer and base.is_nonnegative))):
return prettyForm(*bpretty.left(u'\N{SQUARE ROOT}'))
# Construct root sign, start with the \/ shape
_zZ = xobj('/', 1)
rootsign = xobj('\\', 1) + _zZ
# Make exponent number to put above it
if isinstance(expt, Rational):
exp = str(expt.q)
if exp == '2':
exp = ''
else:
exp = str(expt.args[0])
exp = exp.ljust(2)
if len(exp) > 2:
rootsign = ' '*(len(exp) - 2) + rootsign
# Stack the exponent
rootsign = stringPict(exp + '\n' + rootsign)
rootsign.baseline = 0
# Diagonal: length is one less than height of base
linelength = bpretty.height() - 1
diagonal = stringPict('\n'.join(
' '*(linelength - i - 1) + _zZ + ' '*i
for i in range(linelength)
))
# Put baseline just below lowest line: next to exp
diagonal.baseline = linelength - 1
# Make the root symbol
rootsign = prettyForm(*rootsign.right(diagonal))
# Det the baseline to match contents to fix the height
# but if the height of bpretty is one, the rootsign must be one higher
rootsign.baseline = max(1, bpretty.baseline)
#build result
s = prettyForm(hobj('_', 2 + bpretty.width()))
s = prettyForm(*bpretty.above(s))
s = prettyForm(*s.left(rootsign))
return s
def _print_Pow(self, power):
from sympy.simplify.simplify import fraction
b, e = power.as_base_exp()
if power.is_commutative:
if e is S.NegativeOne:
return prettyForm("1")/self._print(b)
n, d = fraction(e)
if n is S.One and d.is_Atom and not e.is_Integer and self._settings['root_notation']:
return self._print_nth_root(b, e)
if e.is_Rational and e < 0:
return prettyForm("1")/self._print(Pow(b, -e, evaluate=False))
if b.is_Relational:
return prettyForm(*self._print(b).parens()).__pow__(self._print(e))
return self._print(b)**self._print(e)
def _print_UnevaluatedExpr(self, expr):
return self._print(expr.args[0])
def __print_numer_denom(self, p, q):
if q == 1:
if p < 0:
return prettyForm(str(p), binding=prettyForm.NEG)
else:
return prettyForm(str(p))
elif abs(p) >= 10 and abs(q) >= 10:
# If more than one digit in numer and denom, print larger fraction
if p < 0:
return prettyForm(str(p), binding=prettyForm.NEG)/prettyForm(str(q))
# Old printing method:
#pform = prettyForm(str(-p))/prettyForm(str(q))
#return prettyForm(binding=prettyForm.NEG, *pform.left('- '))
else:
return prettyForm(str(p))/prettyForm(str(q))
else:
return None
def _print_Rational(self, expr):
result = self.__print_numer_denom(expr.p, expr.q)
if result is not None:
return result
else:
return self.emptyPrinter(expr)
def _print_Fraction(self, expr):
result = self.__print_numer_denom(expr.numerator, expr.denominator)
if result is not None:
return result
else:
return self.emptyPrinter(expr)
def _print_ProductSet(self, p):
if len(p.sets) > 1 and not has_variety(p.sets):
from sympy import Pow
return self._print(Pow(p.sets[0], len(p.sets), evaluate=False))
else:
prod_char = u"\N{MULTIPLICATION SIGN}" if self._use_unicode else 'x'
return self._print_seq(p.sets, None, None, ' %s ' % prod_char,
parenthesize=lambda set: set.is_Union or
set.is_Intersection or set.is_ProductSet)
def _print_FiniteSet(self, s):
items = sorted(s.args, key=default_sort_key)
return self._print_seq(items, '{', '}', ', ' )
def _print_Range(self, s):
if self._use_unicode:
dots = u"\N{HORIZONTAL ELLIPSIS}"
else:
dots = '...'
if s.start.is_infinite:
printset = dots, s[-1] - s.step, s[-1]
elif s.stop.is_infinite:
it = iter(s)
printset = next(it), next(it), dots
elif len(s) > 4:
it = iter(s)
printset = next(it), next(it), dots, s[-1]
else:
printset = tuple(s)
return self._print_seq(printset, '{', '}', ', ' )
def _print_Interval(self, i):
if i.start == i.end:
return self._print_seq(i.args[:1], '{', '}')
else:
if i.left_open:
left = '('
else:
left = '['
if i.right_open:
right = ')'
else:
right = ']'
return self._print_seq(i.args[:2], left, right)
def _print_AccumulationBounds(self, i):
left = '<'
right = '>'
return self._print_seq(i.args[:2], left, right)
def _print_Intersection(self, u):
delimiter = ' %s ' % pretty_atom('Intersection', 'n')
return self._print_seq(u.args, None, None, delimiter,
parenthesize=lambda set: set.is_ProductSet or
set.is_Union or set.is_Complement)
def _print_Union(self, u):
union_delimiter = ' %s ' % pretty_atom('Union', 'U')
return self._print_seq(u.args, None, None, union_delimiter,
parenthesize=lambda set: set.is_ProductSet or
set.is_Intersection or set.is_Complement)
def _print_SymmetricDifference(self, u):
if not self._use_unicode:
raise NotImplementedError("ASCII pretty printing of SymmetricDifference is not implemented")
sym_delimeter = ' %s ' % pretty_atom('SymmetricDifference')
return self._print_seq(u.args, None, None, sym_delimeter)
def _print_Complement(self, u):
delimiter = r' \ '
return self._print_seq(u.args, None, None, delimiter,
parenthesize=lambda set: set.is_ProductSet or set.is_Intersection
or set.is_Union)
def _print_ImageSet(self, ts):
if self._use_unicode:
inn = u"\N{SMALL ELEMENT OF}"
else:
inn = 'in'
variables = ts.lamda.variables
expr = self._print(ts.lamda.expr)
bar = self._print("|")
sets = [self._print(i) for i in ts.args[1:]]
if len(sets) == 1:
return self._print_seq((expr, bar, variables[0], inn, sets[0]), "{", "}", ' ')
else:
pargs = tuple(j for var, setv in zip(variables, sets) for j in (var, inn, setv, ","))
return self._print_seq((expr, bar) + pargs[:-1], "{", "}", ' ')
def _print_ConditionSet(self, ts):
if self._use_unicode:
inn = u"\N{SMALL ELEMENT OF}"
# using _and because and is a keyword and it is bad practice to
# overwrite them
_and = u"\N{LOGICAL AND}"
else:
inn = 'in'
_and = 'and'
variables = self._print_seq(Tuple(ts.sym))
as_expr = getattr(ts.condition, 'as_expr', None)
if as_expr is not None:
cond = self._print(ts.condition.as_expr())
else:
cond = self._print(ts.condition)
if self._use_unicode:
cond = self._print_seq(cond, "(", ")")
bar = self._print("|")
if ts.base_set is S.UniversalSet:
return self._print_seq((variables, bar, cond), "{", "}", ' ')
base = self._print(ts.base_set)
return self._print_seq((variables, bar, variables, inn,
base, _and, cond), "{", "}", ' ')
def _print_ComplexRegion(self, ts):
if self._use_unicode:
inn = u"\N{SMALL ELEMENT OF}"
else:
inn = 'in'
variables = self._print_seq(ts.variables)
expr = self._print(ts.expr)
bar = self._print("|")
prodsets = self._print(ts.sets)
return self._print_seq((expr, bar, variables, inn, prodsets), "{", "}", ' ')
def _print_Contains(self, e):
var, set = e.args
if self._use_unicode:
el = u" \N{ELEMENT OF} "
return prettyForm(*stringPict.next(self._print(var),
el, self._print(set)), binding=8)
else:
return prettyForm(sstr(e))
def _print_FourierSeries(self, s):
if self._use_unicode:
dots = u"\N{HORIZONTAL ELLIPSIS}"
else:
dots = '...'
return self._print_Add(s.truncate()) + self._print(dots)
def _print_FormalPowerSeries(self, s):
return self._print_Add(s.infinite)
def _print_SetExpr(self, se):
pretty_set = prettyForm(*self._print(se.set).parens())
pretty_name = self._print(Symbol("SetExpr"))
return prettyForm(*pretty_name.right(pretty_set))
def _print_SeqFormula(self, s):
if self._use_unicode:
dots = u"\N{HORIZONTAL ELLIPSIS}"
else:
dots = '...'
if len(s.start.free_symbols) > 0 or len(s.stop.free_symbols) > 0:
raise NotImplementedError("Pretty printing of sequences with symbolic bound not implemented")
if s.start is S.NegativeInfinity:
stop = s.stop
printset = (dots, s.coeff(stop - 3), s.coeff(stop - 2),
s.coeff(stop - 1), s.coeff(stop))
elif s.stop is S.Infinity or s.length > 4:
printset = s[:4]
printset.append(dots)
printset = tuple(printset)
else:
printset = tuple(s)
return self._print_list(printset)
_print_SeqPer = _print_SeqFormula
_print_SeqAdd = _print_SeqFormula
_print_SeqMul = _print_SeqFormula
def _print_seq(self, seq, left=None, right=None, delimiter=', ',
parenthesize=lambda x: False):
s = None
try:
for item in seq:
pform = self._print(item)
if parenthesize(item):
pform = prettyForm(*pform.parens())
if s is None:
# first element
s = pform
else:
# XXX: Under the tests from #15686 this raises:
# AttributeError: 'Fake' object has no attribute 'baseline'
# This is caught below but that is not the right way to
# fix it.
s = prettyForm(*stringPict.next(s, delimiter))
s = prettyForm(*stringPict.next(s, pform))
if s is None:
s = stringPict('')
except AttributeError:
s = None
for item in seq:
pform = self.doprint(item)
if parenthesize(item):
pform = prettyForm(*pform.parens())
if s is None:
# first element
s = pform
else :
s = prettyForm(*stringPict.next(s, delimiter))
s = prettyForm(*stringPict.next(s, pform))
if s is None:
s = stringPict('')
s = prettyForm(*s.parens(left, right, ifascii_nougly=True))
return s
def join(self, delimiter, args):
pform = None
for arg in args:
if pform is None:
pform = arg
else:
pform = prettyForm(*pform.right(delimiter))
pform = prettyForm(*pform.right(arg))
if pform is None:
return prettyForm("")
else:
return pform
def _print_list(self, l):
return self._print_seq(l, '[', ']')
def _print_tuple(self, t):
if len(t) == 1:
ptuple = prettyForm(*stringPict.next(self._print(t[0]), ','))
return prettyForm(*ptuple.parens('(', ')', ifascii_nougly=True))
else:
return self._print_seq(t, '(', ')')
def _print_Tuple(self, expr):
return self._print_tuple(expr)
def _print_dict(self, d):
keys = sorted(d.keys(), key=default_sort_key)
items = []
for k in keys:
K = self._print(k)
V = self._print(d[k])
s = prettyForm(*stringPict.next(K, ': ', V))
items.append(s)
return self._print_seq(items, '{', '}')
def _print_Dict(self, d):
return self._print_dict(d)
def _print_set(self, s):
if not s:
return prettyForm('set()')
items = sorted(s, key=default_sort_key)
pretty = self._print_seq(items)
pretty = prettyForm(*pretty.parens('{', '}', ifascii_nougly=True))
return pretty
def _print_frozenset(self, s):
if not s:
return prettyForm('frozenset()')
items = sorted(s, key=default_sort_key)
pretty = self._print_seq(items)
pretty = prettyForm(*pretty.parens('{', '}', ifascii_nougly=True))
pretty = prettyForm(*pretty.parens('(', ')', ifascii_nougly=True))
pretty = prettyForm(*stringPict.next(type(s).__name__, pretty))
return pretty
def _print_PolyRing(self, ring):
return prettyForm(sstr(ring))
def _print_FracField(self, field):
return prettyForm(sstr(field))
def _print_FreeGroupElement(self, elm):
return prettyForm(str(elm))
def _print_PolyElement(self, poly):
return prettyForm(sstr(poly))
def _print_FracElement(self, frac):
return prettyForm(sstr(frac))
def _print_AlgebraicNumber(self, expr):
if expr.is_aliased:
return self._print(expr.as_poly().as_expr())
else:
return self._print(expr.as_expr())
def _print_ComplexRootOf(self, expr):
args = [self._print_Add(expr.expr, order='lex'), expr.index]
pform = prettyForm(*self._print_seq(args).parens())
pform = prettyForm(*pform.left('CRootOf'))
return pform
def _print_RootSum(self, expr):
args = [self._print_Add(expr.expr, order='lex')]
if expr.fun is not S.IdentityFunction:
args.append(self._print(expr.fun))
pform = prettyForm(*self._print_seq(args).parens())
pform = prettyForm(*pform.left('RootSum'))
return pform
def _print_FiniteField(self, expr):
if self._use_unicode:
form = u'\N{DOUBLE-STRUCK CAPITAL Z}_%d'
else:
form = 'GF(%d)'
return prettyForm(pretty_symbol(form % expr.mod))
def _print_IntegerRing(self, expr):
if self._use_unicode:
return prettyForm(u'\N{DOUBLE-STRUCK CAPITAL Z}')
else:
return prettyForm('ZZ')
def _print_RationalField(self, expr):
if self._use_unicode:
return prettyForm(u'\N{DOUBLE-STRUCK CAPITAL Q}')
else:
return prettyForm('QQ')
def _print_RealField(self, domain):
if self._use_unicode:
prefix = u'\N{DOUBLE-STRUCK CAPITAL R}'
else:
prefix = 'RR'
if domain.has_default_precision:
return prettyForm(prefix)
else:
return self._print(pretty_symbol(prefix + "_" + str(domain.precision)))
def _print_ComplexField(self, domain):
if self._use_unicode:
prefix = u'\N{DOUBLE-STRUCK CAPITAL C}'
else:
prefix = 'CC'
if domain.has_default_precision:
return prettyForm(prefix)
else:
return self._print(pretty_symbol(prefix + "_" + str(domain.precision)))
def _print_PolynomialRing(self, expr):
args = list(expr.symbols)
if not expr.order.is_default:
order = prettyForm(*prettyForm("order=").right(self._print(expr.order)))
args.append(order)
pform = self._print_seq(args, '[', ']')
pform = prettyForm(*pform.left(self._print(expr.domain)))
return pform
def _print_FractionField(self, expr):
args = list(expr.symbols)
if not expr.order.is_default:
order = prettyForm(*prettyForm("order=").right(self._print(expr.order)))
args.append(order)
pform = self._print_seq(args, '(', ')')
pform = prettyForm(*pform.left(self._print(expr.domain)))
return pform
def _print_PolynomialRingBase(self, expr):
g = expr.symbols
if str(expr.order) != str(expr.default_order):
g = g + ("order=" + str(expr.order),)
pform = self._print_seq(g, '[', ']')
pform = prettyForm(*pform.left(self._print(expr.domain)))
return pform
def _print_GroebnerBasis(self, basis):
exprs = [ self._print_Add(arg, order=basis.order)
for arg in basis.exprs ]
exprs = prettyForm(*self.join(", ", exprs).parens(left="[", right="]"))
gens = [ self._print(gen) for gen in basis.gens ]
domain = prettyForm(
*prettyForm("domain=").right(self._print(basis.domain)))
order = prettyForm(
*prettyForm("order=").right(self._print(basis.order)))
pform = self.join(", ", [exprs] + gens + [domain, order])
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left(basis.__class__.__name__))
return pform
def _print_Subs(self, e):
pform = self._print(e.expr)
pform = prettyForm(*pform.parens())
h = pform.height() if pform.height() > 1 else 2
rvert = stringPict(vobj('|', h), baseline=pform.baseline)
pform = prettyForm(*pform.right(rvert))
b = pform.baseline
pform.baseline = pform.height() - 1
pform = prettyForm(*pform.right(self._print_seq([
self._print_seq((self._print(v[0]), xsym('=='), self._print(v[1])),
delimiter='') for v in zip(e.variables, e.point) ])))
pform.baseline = b
return pform
def _print_euler(self, e):
pform = prettyForm("E")
arg = self._print(e.args[0])
pform_arg = prettyForm(" "*arg.width())
pform_arg = prettyForm(*pform_arg.below(arg))
pform = prettyForm(*pform.right(pform_arg))
if len(e.args) == 1:
return pform
m, x = e.args
# TODO: copy-pasted from _print_Function: can we do better?
prettyFunc = pform
prettyArgs = prettyForm(*self._print_seq([x]).parens())
pform = prettyForm(
binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs))
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyArgs
return pform
def _print_catalan(self, e):
pform = prettyForm("C")
arg = self._print(e.args[0])
pform_arg = prettyForm(" "*arg.width())
pform_arg = prettyForm(*pform_arg.below(arg))
pform = prettyForm(*pform.right(pform_arg))
return pform
def _print_bernoulli(self, e):
pform = prettyForm("B")
arg = self._print(e.args[0])
pform_arg = prettyForm(" "*arg.width())
pform_arg = prettyForm(*pform_arg.below(arg))
pform = prettyForm(*pform.right(pform_arg))
return pform
_print_bell = _print_bernoulli
def _print_lucas(self, e):
pform = prettyForm("L")
arg = self._print(e.args[0])
pform_arg = prettyForm(" "*arg.width())
pform_arg = prettyForm(*pform_arg.below(arg))
pform = prettyForm(*pform.right(pform_arg))
return pform
def _print_fibonacci(self, e):
pform = prettyForm("F")
arg = self._print(e.args[0])
pform_arg = prettyForm(" "*arg.width())
pform_arg = prettyForm(*pform_arg.below(arg))
pform = prettyForm(*pform.right(pform_arg))
return pform
def _print_tribonacci(self, e):
pform = prettyForm("T")
arg = self._print(e.args[0])
pform_arg = prettyForm(" "*arg.width())
pform_arg = prettyForm(*pform_arg.below(arg))
pform = prettyForm(*pform.right(pform_arg))
return pform
def _print_KroneckerDelta(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.right((prettyForm(','))))
pform = prettyForm(*pform.right((self._print(e.args[1]))))
if self._use_unicode:
a = stringPict(pretty_symbol('delta'))
else:
a = stringPict('d')
b = pform
top = stringPict(*b.left(' '*a.width()))
bot = stringPict(*a.right(' '*b.width()))
return prettyForm(binding=prettyForm.POW, *bot.below(top))
def _print_RandomDomain(self, d):
if hasattr(d, 'as_boolean'):
pform = self._print('Domain: ')
pform = prettyForm(*pform.right(self._print(d.as_boolean())))
return pform
elif hasattr(d, 'set'):
pform = self._print('Domain: ')
pform = prettyForm(*pform.right(self._print(d.symbols)))
pform = prettyForm(*pform.right(self._print(' in ')))
pform = prettyForm(*pform.right(self._print(d.set)))
return pform
elif hasattr(d, 'symbols'):
pform = self._print('Domain on ')
pform = prettyForm(*pform.right(self._print(d.symbols)))
return pform
else:
return self._print(None)
def _print_DMP(self, p):
try:
if p.ring is not None:
# TODO incorporate order
return self._print(p.ring.to_sympy(p))
except SympifyError:
pass
return self._print(repr(p))
def _print_DMF(self, p):
return self._print_DMP(p)
def _print_Object(self, object):
return self._print(pretty_symbol(object.name))
def _print_Morphism(self, morphism):
arrow = xsym("-->")
domain = self._print(morphism.domain)
codomain = self._print(morphism.codomain)
tail = domain.right(arrow, codomain)[0]
return prettyForm(tail)
def _print_NamedMorphism(self, morphism):
pretty_name = self._print(pretty_symbol(morphism.name))
pretty_morphism = self._print_Morphism(morphism)
return prettyForm(pretty_name.right(":", pretty_morphism)[0])
def _print_IdentityMorphism(self, morphism):
from sympy.categories import NamedMorphism
return self._print_NamedMorphism(
NamedMorphism(morphism.domain, morphism.codomain, "id"))
def _print_CompositeMorphism(self, morphism):
circle = xsym(".")
# All components of the morphism have names and it is thus
# possible to build the name of the composite.
component_names_list = [pretty_symbol(component.name) for
component in morphism.components]
component_names_list.reverse()
component_names = circle.join(component_names_list) + ":"
pretty_name = self._print(component_names)
pretty_morphism = self._print_Morphism(morphism)
return prettyForm(pretty_name.right(pretty_morphism)[0])
def _print_Category(self, category):
return self._print(pretty_symbol(category.name))
def _print_Diagram(self, diagram):
if not diagram.premises:
# This is an empty diagram.
return self._print(S.EmptySet)
pretty_result = self._print(diagram.premises)
if diagram.conclusions:
results_arrow = " %s " % xsym("==>")
pretty_conclusions = self._print(diagram.conclusions)[0]
pretty_result = pretty_result.right(
results_arrow, pretty_conclusions)
return prettyForm(pretty_result[0])
def _print_DiagramGrid(self, grid):
from sympy.matrices import Matrix
from sympy import Symbol
matrix = Matrix([[grid[i, j] if grid[i, j] else Symbol(" ")
for j in range(grid.width)]
for i in range(grid.height)])
return self._print_matrix_contents(matrix)
def _print_FreeModuleElement(self, m):
# Print as row vector for convenience, for now.
return self._print_seq(m, '[', ']')
def _print_SubModule(self, M):
return self._print_seq(M.gens, '<', '>')
def _print_FreeModule(self, M):
return self._print(M.ring)**self._print(M.rank)
def _print_ModuleImplementedIdeal(self, M):
return self._print_seq([x for [x] in M._module.gens], '<', '>')
def _print_QuotientRing(self, R):
return self._print(R.ring) / self._print(R.base_ideal)
def _print_QuotientRingElement(self, R):
return self._print(R.data) + self._print(R.ring.base_ideal)
def _print_QuotientModuleElement(self, m):
return self._print(m.data) + self._print(m.module.killed_module)
def _print_QuotientModule(self, M):
return self._print(M.base) / self._print(M.killed_module)
def _print_MatrixHomomorphism(self, h):
matrix = self._print(h._sympy_matrix())
matrix.baseline = matrix.height() // 2
pform = prettyForm(*matrix.right(' : ', self._print(h.domain),
' %s> ' % hobj('-', 2), self._print(h.codomain)))
return pform
def _print_BaseScalarField(self, field):
string = field._coord_sys._names[field._index]
return self._print(pretty_symbol(string))
def _print_BaseVectorField(self, field):
s = U('PARTIAL DIFFERENTIAL') + '_' + field._coord_sys._names[field._index]
return self._print(pretty_symbol(s))
def _print_Differential(self, diff):
field = diff._form_field
if hasattr(field, '_coord_sys'):
string = field._coord_sys._names[field._index]
return self._print(u'\N{DOUBLE-STRUCK ITALIC SMALL D} ' + pretty_symbol(string))
else:
pform = self._print(field)
pform = prettyForm(*pform.parens())
return prettyForm(*pform.left(u"\N{DOUBLE-STRUCK ITALIC SMALL D}"))
def _print_Tr(self, p):
#TODO: Handle indices
pform = self._print(p.args[0])
pform = prettyForm(*pform.left('%s(' % (p.__class__.__name__)))
pform = prettyForm(*pform.right(')'))
return pform
def _print_primenu(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens())
if self._use_unicode:
pform = prettyForm(*pform.left(greek_unicode['nu']))
else:
pform = prettyForm(*pform.left('nu'))
return pform
def _print_primeomega(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens())
if self._use_unicode:
pform = prettyForm(*pform.left(greek_unicode['Omega']))
else:
pform = prettyForm(*pform.left('Omega'))
return pform
def _print_Quantity(self, e):
if e.name.name == 'degree':
pform = self._print(u"\N{DEGREE SIGN}")
return pform
else:
return self.emptyPrinter(e)
def _print_AssignmentBase(self, e):
op = prettyForm(' ' + xsym(e.op) + ' ')
l = self._print(e.lhs)
r = self._print(e.rhs)
pform = prettyForm(*stringPict.next(l, op, r))
return pform
def pretty(expr, **settings):
"""Returns a string containing the prettified form of expr.
For information on keyword arguments see pretty_print function.
"""
pp = PrettyPrinter(settings)
# XXX: this is an ugly hack, but at least it works
use_unicode = pp._settings['use_unicode']
uflag = pretty_use_unicode(use_unicode)
try:
return pp.doprint(expr)
finally:
pretty_use_unicode(uflag)
def pretty_print(expr, wrap_line=True, num_columns=None, use_unicode=None,
full_prec="auto", order=None, use_unicode_sqrt_char=True,
root_notation = True, mat_symbol_style="plain", imaginary_unit="i"):
"""Prints expr in pretty form.
pprint is just a shortcut for this function.
Parameters
==========
expr : expression
The expression to print.
wrap_line : bool, optional (default=True)
Line wrapping enabled/disabled.
num_columns : int or None, optional (default=None)
Number of columns before line breaking (default to None which reads
the terminal width), useful when using SymPy without terminal.
use_unicode : bool or None, optional (default=None)
Use unicode characters, such as the Greek letter pi instead of
the string pi.
full_prec : bool or string, optional (default="auto")
Use full precision.
order : bool or string, optional (default=None)
Set to 'none' for long expressions if slow; default is None.
use_unicode_sqrt_char : bool, optional (default=True)
Use compact single-character square root symbol (when unambiguous).
root_notation : bool, optional (default=True)
Set to 'False' for printing exponents of the form 1/n in fractional form.
By default exponent is printed in root form.
mat_symbol_style : string, optional (default="plain")
Set to "bold" for printing MatrixSymbols using a bold mathematical symbol face.
By default the standard face is used.
imaginary_unit : string, optional (default="i")
Letter to use for imaginary unit when use_unicode is True.
Can be "i" (default) or "j".
"""
print(pretty(expr, wrap_line=wrap_line, num_columns=num_columns,
use_unicode=use_unicode, full_prec=full_prec, order=order,
use_unicode_sqrt_char=use_unicode_sqrt_char,
root_notation=root_notation, mat_symbol_style=mat_symbol_style,
imaginary_unit=imaginary_unit))
pprint = pretty_print
def pager_print(expr, **settings):
"""Prints expr using the pager, in pretty form.
This invokes a pager command using pydoc. Lines are not wrapped
automatically. This routine is meant to be used with a pager that allows
sideways scrolling, like ``less -S``.
Parameters are the same as for ``pretty_print``. If you wish to wrap lines,
pass ``num_columns=None`` to auto-detect the width of the terminal.
"""
from pydoc import pager
from locale import getpreferredencoding
if 'num_columns' not in settings:
settings['num_columns'] = 500000 # disable line wrap
pager(pretty(expr, **settings).encode(getpreferredencoding()))
|
ec9928de3aa7eba5bf927e02f6ec203845b8ab0f5289b4ca6e3c92f505ace61c
|
"""Prettyprinter by Jurjen Bos.
(I hate spammers: mail me at pietjepuk314 at the reverse of ku.oc.oohay).
All objects have a method that create a "stringPict",
that can be used in the str method for pretty printing.
Updates by Jason Gedge (email <my last name> at cs mun ca)
- terminal_string() method
- minor fixes and changes (mostly to prettyForm)
TODO:
- Allow left/center/right alignment options for above/below and
top/center/bottom alignment options for left/right
"""
from __future__ import print_function, division
from .pretty_symbology import hobj, vobj, xsym, xobj, pretty_use_unicode
from sympy.core.compatibility import string_types, range, unicode
class stringPict(object):
"""An ASCII picture.
The pictures are represented as a list of equal length strings.
"""
#special value for stringPict.below
LINE = 'line'
def __init__(self, s, baseline=0):
"""Initialize from string.
Multiline strings are centered.
"""
self.s = s
#picture is a string that just can be printed
self.picture = stringPict.equalLengths(s.splitlines())
#baseline is the line number of the "base line"
self.baseline = baseline
self.binding = None
@staticmethod
def equalLengths(lines):
# empty lines
if not lines:
return ['']
width = max(len(line) for line in lines)
return [line.center(width) for line in lines]
def height(self):
"""The height of the picture in characters."""
return len(self.picture)
def width(self):
"""The width of the picture in characters."""
return len(self.picture[0])
@staticmethod
def next(*args):
"""Put a string of stringPicts next to each other.
Returns string, baseline arguments for stringPict.
"""
#convert everything to stringPicts
objects = []
for arg in args:
if isinstance(arg, string_types):
arg = stringPict(arg)
objects.append(arg)
#make a list of pictures, with equal height and baseline
newBaseline = max(obj.baseline for obj in objects)
newHeightBelowBaseline = max(
obj.height() - obj.baseline
for obj in objects)
newHeight = newBaseline + newHeightBelowBaseline
pictures = []
for obj in objects:
oneEmptyLine = [' '*obj.width()]
basePadding = newBaseline - obj.baseline
totalPadding = newHeight - obj.height()
pictures.append(
oneEmptyLine * basePadding +
obj.picture +
oneEmptyLine * (totalPadding - basePadding))
result = [''.join(lines) for lines in zip(*pictures)]
return '\n'.join(result), newBaseline
def right(self, *args):
r"""Put pictures next to this one.
Returns string, baseline arguments for stringPict.
(Multiline) strings are allowed, and are given a baseline of 0.
Examples
========
>>> from sympy.printing.pretty.stringpict import stringPict
>>> print(stringPict("10").right(" + ",stringPict("1\r-\r2",1))[0])
1
10 + -
2
"""
return stringPict.next(self, *args)
def left(self, *args):
"""Put pictures (left to right) at left.
Returns string, baseline arguments for stringPict.
"""
return stringPict.next(*(args + (self,)))
@staticmethod
def stack(*args):
"""Put pictures on top of each other,
from top to bottom.
Returns string, baseline arguments for stringPict.
The baseline is the baseline of the second picture.
Everything is centered.
Baseline is the baseline of the second picture.
Strings are allowed.
The special value stringPict.LINE is a row of '-' extended to the width.
"""
#convert everything to stringPicts; keep LINE
objects = []
for arg in args:
if arg is not stringPict.LINE and isinstance(arg, string_types):
arg = stringPict(arg)
objects.append(arg)
#compute new width
newWidth = max(
obj.width()
for obj in objects
if obj is not stringPict.LINE)
lineObj = stringPict(hobj('-', newWidth))
#replace LINE with proper lines
for i, obj in enumerate(objects):
if obj is stringPict.LINE:
objects[i] = lineObj
#stack the pictures, and center the result
newPicture = []
for obj in objects:
newPicture.extend(obj.picture)
newPicture = [line.center(newWidth) for line in newPicture]
newBaseline = objects[0].height() + objects[1].baseline
return '\n'.join(newPicture), newBaseline
def below(self, *args):
"""Put pictures under this picture.
Returns string, baseline arguments for stringPict.
Baseline is baseline of top picture
Examples
========
>>> from sympy.printing.pretty.stringpict import stringPict
>>> print(stringPict("x+3").below(
... stringPict.LINE, '3')[0]) #doctest: +NORMALIZE_WHITESPACE
x+3
---
3
"""
s, baseline = stringPict.stack(self, *args)
return s, self.baseline
def above(self, *args):
"""Put pictures above this picture.
Returns string, baseline arguments for stringPict.
Baseline is baseline of bottom picture.
"""
string, baseline = stringPict.stack(*(args + (self,)))
baseline = len(string.splitlines()) - self.height() + self.baseline
return string, baseline
def parens(self, left='(', right=')', ifascii_nougly=False):
"""Put parentheses around self.
Returns string, baseline arguments for stringPict.
left or right can be None or empty string which means 'no paren from
that side'
"""
h = self.height()
b = self.baseline
# XXX this is a hack -- ascii parens are ugly!
if ifascii_nougly and not pretty_use_unicode():
h = 1
b = 0
res = self
if left:
lparen = stringPict(vobj(left, h), baseline=b)
res = stringPict(*lparen.right(self))
if right:
rparen = stringPict(vobj(right, h), baseline=b)
res = stringPict(*res.right(rparen))
return ('\n'.join(res.picture), res.baseline)
def leftslash(self):
"""Precede object by a slash of the proper size.
"""
# XXX not used anywhere ?
height = max(
self.baseline,
self.height() - 1 - self.baseline)*2 + 1
slash = '\n'.join(
' '*(height - i - 1) + xobj('/', 1) + ' '*i
for i in range(height)
)
return self.left(stringPict(slash, height//2))
def root(self, n=None):
"""Produce a nice root symbol.
Produces ugly results for big n inserts.
"""
# XXX not used anywhere
# XXX duplicate of root drawing in pretty.py
#put line over expression
result = self.above('_'*self.width())
#construct right half of root symbol
height = self.height()
slash = '\n'.join(
' ' * (height - i - 1) + '/' + ' ' * i
for i in range(height)
)
slash = stringPict(slash, height - 1)
#left half of root symbol
if height > 2:
downline = stringPict('\\ \n \\', 1)
else:
downline = stringPict('\\')
#put n on top, as low as possible
if n is not None and n.width() > downline.width():
downline = downline.left(' '*(n.width() - downline.width()))
downline = downline.above(n)
#build root symbol
root = downline.right(slash)
#glue it on at the proper height
#normally, the root symbel is as high as self
#which is one less than result
#this moves the root symbol one down
#if the root became higher, the baseline has to grow too
root.baseline = result.baseline - result.height() + root.height()
return result.left(root)
def render(self, * args, **kwargs):
"""Return the string form of self.
Unless the argument line_break is set to False, it will
break the expression in a form that can be printed
on the terminal without being broken up.
"""
if kwargs["wrap_line"] is False:
return "\n".join(self.picture)
if kwargs["num_columns"] is not None:
# Read the argument num_columns if it is not None
ncols = kwargs["num_columns"]
else:
# Attempt to get a terminal width
ncols = self.terminal_width()
ncols -= 2
if ncols <= 0:
ncols = 78
# If smaller than the terminal width, no need to correct
if self.width() <= ncols:
return type(self.picture[0])(self)
# for one-line pictures we don't need v-spacers. on the other hand, for
# multiline-pictures, we need v-spacers between blocks, compare:
#
# 2 2 3 | a*c*e + a*c*f + a*d | a*c*e + a*c*f + a*d | 3.14159265358979323
# 6*x *y + 4*x*y + | | *e + a*d*f + b*c*e | 84626433832795
# | *e + a*d*f + b*c*e | + b*c*f + b*d*e + b |
# 3 4 4 | | *d*f |
# 4*y*x + x + y | + b*c*f + b*d*e + b | |
# | | |
# | *d*f
i = 0
svals = []
do_vspacers = (self.height() > 1)
while i < self.width():
svals.extend([ sval[i:i + ncols] for sval in self.picture ])
if do_vspacers:
svals.append("") # a vertical spacer
i += ncols
if svals[-1] == '':
del svals[-1] # Get rid of the last spacer
return "\n".join(svals)
def terminal_width(self):
"""Return the terminal width if possible, otherwise return 0.
"""
ncols = 0
try:
import curses
import io
try:
curses.setupterm()
ncols = curses.tigetnum('cols')
except AttributeError:
# windows curses doesn't implement setupterm or tigetnum
# code below from
# http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/440694
from ctypes import windll, create_string_buffer
# stdin handle is -10
# stdout handle is -11
# stderr handle is -12
h = windll.kernel32.GetStdHandle(-12)
csbi = create_string_buffer(22)
res = windll.kernel32.GetConsoleScreenBufferInfo(h, csbi)
if res:
import struct
(bufx, bufy, curx, cury, wattr,
left, top, right, bottom, maxx, maxy) = struct.unpack("hhhhHhhhhhh", csbi.raw)
ncols = right - left + 1
except curses.error:
pass
except io.UnsupportedOperation:
pass
except (ImportError, TypeError):
pass
return ncols
def __eq__(self, o):
if isinstance(o, string_types):
return '\n'.join(self.picture) == o
elif isinstance(o, stringPict):
return o.picture == self.picture
return False
def __hash__(self):
return super(stringPict, self).__hash__()
def __str__(self):
return str.join('\n', self.picture)
def __unicode__(self):
return unicode.join(u'\n', self.picture)
def __repr__(self):
return "stringPict(%r,%d)" % ('\n'.join(self.picture), self.baseline)
def __getitem__(self, index):
return self.picture[index]
def __len__(self):
return len(self.s)
class prettyForm(stringPict):
"""
Extension of the stringPict class that knows about basic math applications,
optimizing double minus signs.
"Binding" is interpreted as follows::
ATOM this is an atom: never needs to be parenthesized
FUNC this is a function application: parenthesize if added (?)
DIV this is a division: make wider division if divided
POW this is a power: only parenthesize if exponent
MUL this is a multiplication: parenthesize if powered
ADD this is an addition: parenthesize if multiplied or powered
NEG this is a negative number: optimize if added, parenthesize if
multiplied or powered
OPEN this is an open object: parenthesize if added, multiplied, or
powered (example: Piecewise)
"""
ATOM, FUNC, DIV, POW, MUL, ADD, NEG, OPEN = range(8)
def __init__(self, s, baseline=0, binding=0, unicode=None):
"""Initialize from stringPict and binding power."""
stringPict.__init__(self, s, baseline)
self.binding = binding
self.unicode = unicode or s
# Note: code to handle subtraction is in _print_Add
def __add__(self, *others):
"""Make a pretty addition.
Addition of negative numbers is simplified.
"""
arg = self
if arg.binding > prettyForm.NEG:
arg = stringPict(*arg.parens())
result = [arg]
for arg in others:
#add parentheses for weak binders
if arg.binding > prettyForm.NEG:
arg = stringPict(*arg.parens())
#use existing minus sign if available
if arg.binding != prettyForm.NEG:
result.append(' + ')
result.append(arg)
return prettyForm(binding=prettyForm.ADD, *stringPict.next(*result))
def __div__(self, den, slashed=False):
"""Make a pretty division; stacked or slashed.
"""
if slashed:
raise NotImplementedError("Can't do slashed fraction yet")
num = self
if num.binding == prettyForm.DIV:
num = stringPict(*num.parens())
if den.binding == prettyForm.DIV:
den = stringPict(*den.parens())
if num.binding==prettyForm.NEG:
num = num.right(" ")[0]
return prettyForm(binding=prettyForm.DIV, *stringPict.stack(
num,
stringPict.LINE,
den))
def __truediv__(self, o):
return self.__div__(o)
def __mul__(self, *others):
"""Make a pretty multiplication.
Parentheses are needed around +, - and neg.
"""
quantity = {
'degree': u"\N{DEGREE SIGN}"
}
if len(others) == 0:
return self # We aren't actually multiplying... So nothing to do here.
args = self
if args.binding > prettyForm.MUL:
arg = stringPict(*args.parens())
result = [args]
for arg in others:
if arg.picture[0] not in quantity.values():
result.append(xsym('*'))
#add parentheses for weak binders
if arg.binding > prettyForm.MUL:
arg = stringPict(*arg.parens())
result.append(arg)
len_res = len(result)
for i in range(len_res):
if i < len_res - 1 and result[i] == '-1' and result[i + 1] == xsym('*'):
# substitute -1 by -, like in -1*x -> -x
result.pop(i)
result.pop(i)
result.insert(i, '-')
if result[0][0] == '-':
# if there is a - sign in front of all
# This test was failing to catch a prettyForm.__mul__(prettyForm("-1", 0, 6)) being negative
bin = prettyForm.NEG
if result[0] == '-':
right = result[1]
if right.picture[right.baseline][0] == '-':
result[0] = '- '
else:
bin = prettyForm.MUL
return prettyForm(binding=bin, *stringPict.next(*result))
def __repr__(self):
return "prettyForm(%r,%d,%d)" % (
'\n'.join(self.picture),
self.baseline,
self.binding)
def __pow__(self, b):
"""Make a pretty power.
"""
a = self
use_inline_func_form = False
if b.binding == prettyForm.POW:
b = stringPict(*b.parens())
if a.binding > prettyForm.FUNC:
a = stringPict(*a.parens())
elif a.binding == prettyForm.FUNC:
# heuristic for when to use inline power
if b.height() > 1:
a = stringPict(*a.parens())
else:
use_inline_func_form = True
if use_inline_func_form:
# 2
# sin + + (x)
b.baseline = a.prettyFunc.baseline + b.height()
func = stringPict(*a.prettyFunc.right(b))
return prettyForm(*func.right(a.prettyArgs))
else:
# 2 <-- top
# (x+y) <-- bot
top = stringPict(*b.left(' '*a.width()))
bot = stringPict(*a.right(' '*b.width()))
return prettyForm(binding=prettyForm.POW, *bot.above(top))
simpleFunctions = ["sin", "cos", "tan"]
@staticmethod
def apply(function, *args):
"""Functions of one or more variables.
"""
if function in prettyForm.simpleFunctions:
#simple function: use only space if possible
assert len(
args) == 1, "Simple function %s must have 1 argument" % function
arg = args[0].__pretty__()
if arg.binding <= prettyForm.DIV:
#optimization: no parentheses necessary
return prettyForm(binding=prettyForm.FUNC, *arg.left(function + ' '))
argumentList = []
for arg in args:
argumentList.append(',')
argumentList.append(arg.__pretty__())
argumentList = stringPict(*stringPict.next(*argumentList[1:]))
argumentList = stringPict(*argumentList.parens())
return prettyForm(binding=prettyForm.ATOM, *argumentList.left(function))
|
b4899026b4d946801b396c68068c5586aef6bceeb38c876aceeaac92d4955178
|
from sympy import (Abs, Catalan, cos, Derivative, E, EulerGamma, exp,
factorial, factorial2, Function, GoldenRatio, TribonacciConstant, I,
Integer, Integral, Interval, Lambda, Limit, Matrix, nan, O, oo, pi, Pow,
Rational, Float, Rel, S, sin, SparseMatrix, sqrt, summation, Sum, Symbol,
symbols, Wild, WildFunction, zeta, zoo, Dummy, Dict, Tuple, FiniteSet, factor,
subfactorial, true, false, Equivalent, Xor, Complement, SymmetricDifference,
AccumBounds, UnevaluatedExpr, Eq, Ne, Quaternion, Subs)
from sympy.core import Expr, Mul
from sympy.physics.units import second, joule
from sympy.polys import Poly, rootof, RootSum, groebner, ring, field, ZZ, QQ, lex, grlex
from sympy.geometry import Point, Circle
from sympy.utilities.pytest import raises
from sympy.core.compatibility import range
from sympy.printing import sstr, sstrrepr, StrPrinter
from sympy.core.trace import Tr
from sympy import MatrixSymbol
from sympy import factorial, log, integrate
x, y, z, w, t = symbols('x,y,z,w,t')
d = Dummy('d')
def test_printmethod():
class R(Abs):
def _sympystr(self, printer):
return "foo(%s)" % printer._print(self.args[0])
assert sstr(R(x)) == "foo(x)"
class R(Abs):
def _sympystr(self, printer):
return "foo"
assert sstr(R(x)) == "foo"
def test_Abs():
assert str(Abs(x)) == "Abs(x)"
assert str(Abs(Rational(1, 6))) == "1/6"
assert str(Abs(Rational(-1, 6))) == "1/6"
def test_Add():
assert str(x + y) == "x + y"
assert str(x + 1) == "x + 1"
assert str(x + x**2) == "x**2 + x"
assert str(5 + x + y + x*y + x**2 + y**2) == "x**2 + x*y + x + y**2 + y + 5"
assert str(1 + x + x**2/2 + x**3/3) == "x**3/3 + x**2/2 + x + 1"
assert str(2*x - 7*x**2 + 2 + 3*y) == "-7*x**2 + 2*x + 3*y + 2"
assert str(x - y) == "x - y"
assert str(2 - x) == "2 - x"
assert str(x - 2) == "x - 2"
assert str(x - y - z - w) == "-w + x - y - z"
assert str(x - z*y**2*z*w) == "-w*y**2*z**2 + x"
assert str(x - 1*y*x*y) == "-x*y**2 + x"
assert str(sin(x).series(x, 0, 15)) == "x - x**3/6 + x**5/120 - x**7/5040 + x**9/362880 - x**11/39916800 + x**13/6227020800 + O(x**15)"
def test_Catalan():
assert str(Catalan) == "Catalan"
def test_ComplexInfinity():
assert str(zoo) == "zoo"
def test_Derivative():
assert str(Derivative(x, y)) == "Derivative(x, y)"
assert str(Derivative(x**2, x, evaluate=False)) == "Derivative(x**2, x)"
assert str(Derivative(
x**2/y, x, y, evaluate=False)) == "Derivative(x**2/y, x, y)"
def test_dict():
assert str({1: 1 + x}) == sstr({1: 1 + x}) == "{1: x + 1}"
assert str({1: x**2, 2: y*x}) in ("{1: x**2, 2: x*y}", "{2: x*y, 1: x**2}")
assert sstr({1: x**2, 2: y*x}) == "{1: x**2, 2: x*y}"
def test_Dict():
assert str(Dict({1: 1 + x})) == sstr({1: 1 + x}) == "{1: x + 1}"
assert str(Dict({1: x**2, 2: y*x})) in (
"{1: x**2, 2: x*y}", "{2: x*y, 1: x**2}")
assert sstr(Dict({1: x**2, 2: y*x})) == "{1: x**2, 2: x*y}"
def test_Dummy():
assert str(d) == "_d"
assert str(d + x) == "_d + x"
def test_EulerGamma():
assert str(EulerGamma) == "EulerGamma"
def test_Exp():
assert str(E) == "E"
def test_factorial():
n = Symbol('n', integer=True)
assert str(factorial(-2)) == "zoo"
assert str(factorial(0)) == "1"
assert str(factorial(7)) == "5040"
assert str(factorial(n)) == "factorial(n)"
assert str(factorial(2*n)) == "factorial(2*n)"
assert str(factorial(factorial(n))) == 'factorial(factorial(n))'
assert str(factorial(factorial2(n))) == 'factorial(factorial2(n))'
assert str(factorial2(factorial(n))) == 'factorial2(factorial(n))'
assert str(factorial2(factorial2(n))) == 'factorial2(factorial2(n))'
assert str(subfactorial(3)) == "2"
assert str(subfactorial(n)) == "subfactorial(n)"
assert str(subfactorial(2*n)) == "subfactorial(2*n)"
def test_Function():
f = Function('f')
fx = f(x)
w = WildFunction('w')
assert str(f) == "f"
assert str(fx) == "f(x)"
assert str(w) == "w_"
def test_Geometry():
assert sstr(Point(0, 0)) == 'Point2D(0, 0)'
assert sstr(Circle(Point(0, 0), 3)) == 'Circle(Point2D(0, 0), 3)'
# TODO test other Geometry entities
def test_GoldenRatio():
assert str(GoldenRatio) == "GoldenRatio"
def test_TribonacciConstant():
assert str(TribonacciConstant) == "TribonacciConstant"
def test_ImaginaryUnit():
assert str(I) == "I"
def test_Infinity():
assert str(oo) == "oo"
assert str(oo*I) == "oo*I"
def test_Integer():
assert str(Integer(-1)) == "-1"
assert str(Integer(1)) == "1"
assert str(Integer(-3)) == "-3"
assert str(Integer(0)) == "0"
assert str(Integer(25)) == "25"
def test_Integral():
assert str(Integral(sin(x), y)) == "Integral(sin(x), y)"
assert str(Integral(sin(x), (y, 0, 1))) == "Integral(sin(x), (y, 0, 1))"
def test_Interval():
n = (S.NegativeInfinity, 1, 2, S.Infinity)
for i in range(len(n)):
for j in range(i + 1, len(n)):
for l in (True, False):
for r in (True, False):
ival = Interval(n[i], n[j], l, r)
assert S(str(ival)) == ival
def test_AccumBounds():
a = Symbol('a', real=True)
assert str(AccumBounds(0, a)) == "AccumBounds(0, a)"
assert str(AccumBounds(0, 1)) == "AccumBounds(0, 1)"
def test_Lambda():
assert str(Lambda(d, d**2)) == "Lambda(_d, _d**2)"
# issue 2908
assert str(Lambda((), 1)) == "Lambda((), 1)"
assert str(Lambda((), x)) == "Lambda((), x)"
def test_Limit():
assert str(Limit(sin(x)/x, x, y)) == "Limit(sin(x)/x, x, y)"
assert str(Limit(1/x, x, 0)) == "Limit(1/x, x, 0)"
assert str(
Limit(sin(x)/x, x, y, dir="-")) == "Limit(sin(x)/x, x, y, dir='-')"
def test_list():
assert str([x]) == sstr([x]) == "[x]"
assert str([x**2, x*y + 1]) == sstr([x**2, x*y + 1]) == "[x**2, x*y + 1]"
assert str([x**2, [y + x]]) == sstr([x**2, [y + x]]) == "[x**2, [x + y]]"
def test_Matrix_str():
M = Matrix([[x**+1, 1], [y, x + y]])
assert str(M) == "Matrix([[x, 1], [y, x + y]])"
assert sstr(M) == "Matrix([\n[x, 1],\n[y, x + y]])"
M = Matrix([[1]])
assert str(M) == sstr(M) == "Matrix([[1]])"
M = Matrix([[1, 2]])
assert str(M) == sstr(M) == "Matrix([[1, 2]])"
M = Matrix()
assert str(M) == sstr(M) == "Matrix(0, 0, [])"
M = Matrix(0, 1, lambda i, j: 0)
assert str(M) == sstr(M) == "Matrix(0, 1, [])"
def test_Mul():
assert str(x/y) == "x/y"
assert str(y/x) == "y/x"
assert str(x/y/z) == "x/(y*z)"
assert str((x + 1)/(y + 2)) == "(x + 1)/(y + 2)"
assert str(2*x/3) == '2*x/3'
assert str(-2*x/3) == '-2*x/3'
assert str(-1.0*x) == '-1.0*x'
assert str(1.0*x) == '1.0*x'
# For issue 14160
assert str(Mul(-2, x, Pow(Mul(y,y,evaluate=False), -1, evaluate=False),
evaluate=False)) == '-2*x/(y*y)'
class CustomClass1(Expr):
is_commutative = True
class CustomClass2(Expr):
is_commutative = True
cc1 = CustomClass1()
cc2 = CustomClass2()
assert str(Rational(2)*cc1) == '2*CustomClass1()'
assert str(cc1*Rational(2)) == '2*CustomClass1()'
assert str(cc1*Float("1.5")) == '1.5*CustomClass1()'
assert str(cc2*Rational(2)) == '2*CustomClass2()'
assert str(cc2*Rational(2)*cc1) == '2*CustomClass1()*CustomClass2()'
assert str(cc1*Rational(2)*cc2) == '2*CustomClass1()*CustomClass2()'
def test_NaN():
assert str(nan) == "nan"
def test_NegativeInfinity():
assert str(-oo) == "-oo"
def test_Order():
assert str(O(x)) == "O(x)"
assert str(O(x**2)) == "O(x**2)"
assert str(O(x*y)) == "O(x*y, x, y)"
assert str(O(x, x)) == "O(x)"
assert str(O(x, (x, 0))) == "O(x)"
assert str(O(x, (x, oo))) == "O(x, (x, oo))"
assert str(O(x, x, y)) == "O(x, x, y)"
assert str(O(x, x, y)) == "O(x, x, y)"
assert str(O(x, (x, oo), (y, oo))) == "O(x, (x, oo), (y, oo))"
def test_Permutation_Cycle():
from sympy.combinatorics import Permutation, Cycle
# general principle: economically, canonically show all moved elements
# and the size of the permutation.
for p, s in [
(Cycle(),
'()'),
(Cycle(2),
'(2)'),
(Cycle(2, 1),
'(1 2)'),
(Cycle(1, 2)(5)(6, 7)(10),
'(1 2)(6 7)(10)'),
(Cycle(3, 4)(1, 2)(3, 4),
'(1 2)(4)'),
]:
assert str(p) == s
Permutation.print_cyclic = False
for p, s in [
(Permutation([]),
'Permutation([])'),
(Permutation([], size=1),
'Permutation([0])'),
(Permutation([], size=2),
'Permutation([0, 1])'),
(Permutation([], size=10),
'Permutation([], size=10)'),
(Permutation([1, 0, 2]),
'Permutation([1, 0, 2])'),
(Permutation([1, 0, 2, 3, 4, 5]),
'Permutation([1, 0], size=6)'),
(Permutation([1, 0, 2, 3, 4, 5], size=10),
'Permutation([1, 0], size=10)'),
]:
assert str(p) == s
Permutation.print_cyclic = True
for p, s in [
(Permutation([]),
'()'),
(Permutation([], size=1),
'(0)'),
(Permutation([], size=2),
'(1)'),
(Permutation([], size=10),
'(9)'),
(Permutation([1, 0, 2]),
'(2)(0 1)'),
(Permutation([1, 0, 2, 3, 4, 5]),
'(5)(0 1)'),
(Permutation([1, 0, 2, 3, 4, 5], size=10),
'(9)(0 1)'),
(Permutation([0, 1, 3, 2, 4, 5], size=10),
'(9)(2 3)'),
]:
assert str(p) == s
def test_Pi():
assert str(pi) == "pi"
def test_Poly():
assert str(Poly(0, x)) == "Poly(0, x, domain='ZZ')"
assert str(Poly(1, x)) == "Poly(1, x, domain='ZZ')"
assert str(Poly(x, x)) == "Poly(x, x, domain='ZZ')"
assert str(Poly(2*x + 1, x)) == "Poly(2*x + 1, x, domain='ZZ')"
assert str(Poly(2*x - 1, x)) == "Poly(2*x - 1, x, domain='ZZ')"
assert str(Poly(-1, x)) == "Poly(-1, x, domain='ZZ')"
assert str(Poly(-x, x)) == "Poly(-x, x, domain='ZZ')"
assert str(Poly(-2*x + 1, x)) == "Poly(-2*x + 1, x, domain='ZZ')"
assert str(Poly(-2*x - 1, x)) == "Poly(-2*x - 1, x, domain='ZZ')"
assert str(Poly(x - 1, x)) == "Poly(x - 1, x, domain='ZZ')"
assert str(Poly(2*x + x**5, x)) == "Poly(x**5 + 2*x, x, domain='ZZ')"
assert str(Poly(3**(2*x), 3**x)) == "Poly((3**x)**2, 3**x, domain='ZZ')"
assert str(Poly((x**2)**x)) == "Poly(((x**2)**x), (x**2)**x, domain='ZZ')"
assert str(Poly((x + y)**3, (x + y), expand=False)
) == "Poly((x + y)**3, x + y, domain='ZZ')"
assert str(Poly((x - 1)**2, (x - 1), expand=False)
) == "Poly((x - 1)**2, x - 1, domain='ZZ')"
assert str(
Poly(x**2 + 1 + y, x)) == "Poly(x**2 + y + 1, x, domain='ZZ[y]')"
assert str(
Poly(x**2 - 1 + y, x)) == "Poly(x**2 + y - 1, x, domain='ZZ[y]')"
assert str(Poly(x**2 + I*x, x)) == "Poly(x**2 + I*x, x, domain='EX')"
assert str(Poly(x**2 - I*x, x)) == "Poly(x**2 - I*x, x, domain='EX')"
assert str(Poly(-x*y*z + x*y - 1, x, y, z)
) == "Poly(-x*y*z + x*y - 1, x, y, z, domain='ZZ')"
assert str(Poly(-w*x**21*y**7*z + (1 + w)*z**3 - 2*x*z + 1, x, y, z)) == \
"Poly(-w*x**21*y**7*z - 2*x*z + (w + 1)*z**3 + 1, x, y, z, domain='ZZ[w]')"
assert str(Poly(x**2 + 1, x, modulus=2)) == "Poly(x**2 + 1, x, modulus=2)"
assert str(Poly(2*x**2 + 3*x + 4, x, modulus=17)) == "Poly(2*x**2 + 3*x + 4, x, modulus=17)"
def test_PolyRing():
assert str(ring("x", ZZ, lex)[0]) == "Polynomial ring in x over ZZ with lex order"
assert str(ring("x,y", QQ, grlex)[0]) == "Polynomial ring in x, y over QQ with grlex order"
assert str(ring("x,y,z", ZZ["t"], lex)[0]) == "Polynomial ring in x, y, z over ZZ[t] with lex order"
def test_FracField():
assert str(field("x", ZZ, lex)[0]) == "Rational function field in x over ZZ with lex order"
assert str(field("x,y", QQ, grlex)[0]) == "Rational function field in x, y over QQ with grlex order"
assert str(field("x,y,z", ZZ["t"], lex)[0]) == "Rational function field in x, y, z over ZZ[t] with lex order"
def test_PolyElement():
Ruv, u,v = ring("u,v", ZZ)
Rxyz, x,y,z = ring("x,y,z", Ruv)
assert str(x - x) == "0"
assert str(x - 1) == "x - 1"
assert str(x + 1) == "x + 1"
assert str(x**2) == "x**2"
assert str(x**(-2)) == "x**(-2)"
assert str(x**QQ(1, 2)) == "x**(1/2)"
assert str((u**2 + 3*u*v + 1)*x**2*y + u + 1) == "(u**2 + 3*u*v + 1)*x**2*y + u + 1"
assert str((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x) == "(u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x"
assert str((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x + 1) == "(u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x + 1"
assert str((-u**2 + 3*u*v - 1)*x**2*y - (u + 1)*x - 1) == "-(u**2 - 3*u*v + 1)*x**2*y - (u + 1)*x - 1"
assert str(-(v**2 + v + 1)*x + 3*u*v + 1) == "-(v**2 + v + 1)*x + 3*u*v + 1"
assert str(-(v**2 + v + 1)*x - 3*u*v + 1) == "-(v**2 + v + 1)*x - 3*u*v + 1"
def test_FracElement():
Fuv, u,v = field("u,v", ZZ)
Fxyzt, x,y,z,t = field("x,y,z,t", Fuv)
assert str(x - x) == "0"
assert str(x - 1) == "x - 1"
assert str(x + 1) == "x + 1"
assert str(x/3) == "x/3"
assert str(x/z) == "x/z"
assert str(x*y/z) == "x*y/z"
assert str(x/(z*t)) == "x/(z*t)"
assert str(x*y/(z*t)) == "x*y/(z*t)"
assert str((x - 1)/y) == "(x - 1)/y"
assert str((x + 1)/y) == "(x + 1)/y"
assert str((-x - 1)/y) == "(-x - 1)/y"
assert str((x + 1)/(y*z)) == "(x + 1)/(y*z)"
assert str(-y/(x + 1)) == "-y/(x + 1)"
assert str(y*z/(x + 1)) == "y*z/(x + 1)"
assert str(((u + 1)*x*y + 1)/((v - 1)*z - 1)) == "((u + 1)*x*y + 1)/((v - 1)*z - 1)"
assert str(((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)) == "((u + 1)*x*y + 1)/((v - 1)*z - u*v*t - 1)"
def test_Pow():
assert str(x**-1) == "1/x"
assert str(x**-2) == "x**(-2)"
assert str(x**2) == "x**2"
assert str((x + y)**-1) == "1/(x + y)"
assert str((x + y)**-2) == "(x + y)**(-2)"
assert str((x + y)**2) == "(x + y)**2"
assert str((x + y)**(1 + x)) == "(x + y)**(x + 1)"
assert str(x**Rational(1, 3)) == "x**(1/3)"
assert str(1/x**Rational(1, 3)) == "x**(-1/3)"
assert str(sqrt(sqrt(x))) == "x**(1/4)"
# not the same as x**-1
assert str(x**-1.0) == 'x**(-1.0)'
# see issue #2860
assert str(Pow(S(2), -1.0, evaluate=False)) == '2**(-1.0)'
def test_sqrt():
assert str(sqrt(x)) == "sqrt(x)"
assert str(sqrt(x**2)) == "sqrt(x**2)"
assert str(1/sqrt(x)) == "1/sqrt(x)"
assert str(1/sqrt(x**2)) == "1/sqrt(x**2)"
assert str(y/sqrt(x)) == "y/sqrt(x)"
assert str(x**0.5) == "x**0.5"
assert str(1/x**0.5) == "x**(-0.5)"
def test_Rational():
n1 = Rational(1, 4)
n2 = Rational(1, 3)
n3 = Rational(2, 4)
n4 = Rational(2, -4)
n5 = Rational(0)
n7 = Rational(3)
n8 = Rational(-3)
assert str(n1*n2) == "1/12"
assert str(n1*n2) == "1/12"
assert str(n3) == "1/2"
assert str(n1*n3) == "1/8"
assert str(n1 + n3) == "3/4"
assert str(n1 + n2) == "7/12"
assert str(n1 + n4) == "-1/4"
assert str(n4*n4) == "1/4"
assert str(n4 + n2) == "-1/6"
assert str(n4 + n5) == "-1/2"
assert str(n4*n5) == "0"
assert str(n3 + n4) == "0"
assert str(n1**n7) == "1/64"
assert str(n2**n7) == "1/27"
assert str(n2**n8) == "27"
assert str(n7**n8) == "1/27"
assert str(Rational("-25")) == "-25"
assert str(Rational("1.25")) == "5/4"
assert str(Rational("-2.6e-2")) == "-13/500"
assert str(S("25/7")) == "25/7"
assert str(S("-123/569")) == "-123/569"
assert str(S("0.1[23]", rational=1)) == "61/495"
assert str(S("5.1[666]", rational=1)) == "31/6"
assert str(S("-5.1[666]", rational=1)) == "-31/6"
assert str(S("0.[9]", rational=1)) == "1"
assert str(S("-0.[9]", rational=1)) == "-1"
assert str(sqrt(Rational(1, 4))) == "1/2"
assert str(sqrt(Rational(1, 36))) == "1/6"
assert str((123**25) ** Rational(1, 25)) == "123"
assert str((123**25 + 1)**Rational(1, 25)) != "123"
assert str((123**25 - 1)**Rational(1, 25)) != "123"
assert str((123**25 - 1)**Rational(1, 25)) != "122"
assert str(sqrt(Rational(81, 36))**3) == "27/8"
assert str(1/sqrt(Rational(81, 36))**3) == "8/27"
assert str(sqrt(-4)) == str(2*I)
assert str(2**Rational(1, 10**10)) == "2**(1/10000000000)"
assert sstr(Rational(2, 3), sympy_integers=True) == "S(2)/3"
x = Symbol("x")
assert sstr(x**Rational(2, 3), sympy_integers=True) == "x**(S(2)/3)"
assert sstr(Eq(x, Rational(2, 3)), sympy_integers=True) == "Eq(x, S(2)/3)"
assert sstr(Limit(x, x, Rational(7, 2)), sympy_integers=True) == \
"Limit(x, x, S(7)/2)"
def test_Float():
# NOTE dps is the whole number of decimal digits
assert str(Float('1.23', dps=1 + 2)) == '1.23'
assert str(Float('1.23456789', dps=1 + 8)) == '1.23456789'
assert str(
Float('1.234567890123456789', dps=1 + 18)) == '1.234567890123456789'
assert str(pi.evalf(1 + 2)) == '3.14'
assert str(pi.evalf(1 + 14)) == '3.14159265358979'
assert str(pi.evalf(1 + 64)) == ('3.141592653589793238462643383279'
'5028841971693993751058209749445923')
assert str(pi.round(-1)) == '0.'
assert str((pi**400 - (pi**400).round(1)).n(2)) == '-0.e+88'
assert str(Float(S.Infinity)) == 'inf'
assert str(Float(S.NegativeInfinity)) == '-inf'
def test_Relational():
assert str(Rel(x, y, "<")) == "x < y"
assert str(Rel(x + y, y, "==")) == "Eq(x + y, y)"
assert str(Rel(x, y, "!=")) == "Ne(x, y)"
assert str(Eq(x, 1) | Eq(x, 2)) == "Eq(x, 1) | Eq(x, 2)"
assert str(Ne(x, 1) & Ne(x, 2)) == "Ne(x, 1) & Ne(x, 2)"
def test_CRootOf():
assert str(rootof(x**5 + 2*x - 1, 0)) == "CRootOf(x**5 + 2*x - 1, 0)"
def test_RootSum():
f = x**5 + 2*x - 1
assert str(
RootSum(f, Lambda(z, z), auto=False)) == "RootSum(x**5 + 2*x - 1)"
assert str(RootSum(f, Lambda(
z, z**2), auto=False)) == "RootSum(x**5 + 2*x - 1, Lambda(z, z**2))"
def test_GroebnerBasis():
assert str(groebner(
[], x, y)) == "GroebnerBasis([], x, y, domain='ZZ', order='lex')"
F = [x**2 - 3*y - x + 1, y**2 - 2*x + y - 1]
assert str(groebner(F, order='grlex')) == \
"GroebnerBasis([x**2 - x - 3*y + 1, y**2 - 2*x + y - 1], x, y, domain='ZZ', order='grlex')"
assert str(groebner(F, order='lex')) == \
"GroebnerBasis([2*x - y**2 - y + 1, y**4 + 2*y**3 - 3*y**2 - 16*y + 7], x, y, domain='ZZ', order='lex')"
def test_set():
assert sstr(set()) == 'set()'
assert sstr(frozenset()) == 'frozenset()'
assert sstr(set([1])) == '{1}'
assert sstr(frozenset([1])) == 'frozenset({1})'
assert sstr(set([1, 2, 3])) == '{1, 2, 3}'
assert sstr(frozenset([1, 2, 3])) == 'frozenset({1, 2, 3})'
assert sstr(
set([1, x, x**2, x**3, x**4])) == '{1, x, x**2, x**3, x**4}'
assert sstr(
frozenset([1, x, x**2, x**3, x**4])) == 'frozenset({1, x, x**2, x**3, x**4})'
def test_SparseMatrix():
M = SparseMatrix([[x**+1, 1], [y, x + y]])
assert str(M) == "Matrix([[x, 1], [y, x + y]])"
assert sstr(M) == "Matrix([\n[x, 1],\n[y, x + y]])"
def test_Sum():
assert str(summation(cos(3*z), (z, x, y))) == "Sum(cos(3*z), (z, x, y))"
assert str(Sum(x*y**2, (x, -2, 2), (y, -5, 5))) == \
"Sum(x*y**2, (x, -2, 2), (y, -5, 5))"
def test_Symbol():
assert str(y) == "y"
assert str(x) == "x"
e = x
assert str(e) == "x"
def test_tuple():
assert str((x,)) == sstr((x,)) == "(x,)"
assert str((x + y, 1 + x)) == sstr((x + y, 1 + x)) == "(x + y, x + 1)"
assert str((x + y, (
1 + x, x**2))) == sstr((x + y, (1 + x, x**2))) == "(x + y, (x + 1, x**2))"
def test_Quaternion_str_printer():
q = Quaternion(x, y, z, t)
assert str(q) == "x + y*i + z*j + t*k"
q = Quaternion(x,y,z,x*t)
assert str(q) == "x + y*i + z*j + t*x*k"
q = Quaternion(x,y,z,x+t)
assert str(q) == "x + y*i + z*j + (t + x)*k"
def test_Quantity_str():
assert sstr(second, abbrev=True) == "s"
assert sstr(joule, abbrev=True) == "J"
assert str(second) == "second"
assert str(joule) == "joule"
def test_wild_str():
# Check expressions containing Wild not causing infinite recursion
w = Wild('x')
assert str(w + 1) == 'x_ + 1'
assert str(exp(2**w) + 5) == 'exp(2**x_) + 5'
assert str(3*w + 1) == '3*x_ + 1'
assert str(1/w + 1) == '1 + 1/x_'
assert str(w**2 + 1) == 'x_**2 + 1'
assert str(1/(1 - w)) == '1/(1 - x_)'
def test_zeta():
assert str(zeta(3)) == "zeta(3)"
def test_issue_3101():
e = x - y
a = str(e)
b = str(e)
assert a == b
def test_issue_3103():
e = -2*sqrt(x) - y/sqrt(x)/2
assert str(e) not in ["(-2)*x**1/2(-1/2)*x**(-1/2)*y",
"-2*x**1/2(-1/2)*x**(-1/2)*y", "-2*x**1/2-1/2*x**-1/2*w"]
assert str(e) == "-2*sqrt(x) - y/(2*sqrt(x))"
def test_issue_4021():
e = Integral(x, x) + 1
assert str(e) == 'Integral(x, x) + 1'
def test_sstrrepr():
assert sstr('abc') == 'abc'
assert sstrrepr('abc') == "'abc'"
e = ['a', 'b', 'c', x]
assert sstr(e) == "[a, b, c, x]"
assert sstrrepr(e) == "['a', 'b', 'c', x]"
def test_infinity():
assert sstr(oo*I) == "oo*I"
def test_full_prec():
assert sstr(S("0.3"), full_prec=True) == "0.300000000000000"
assert sstr(S("0.3"), full_prec="auto") == "0.300000000000000"
assert sstr(S("0.3"), full_prec=False) == "0.3"
assert sstr(S("0.3")*x, full_prec=True) in [
"0.300000000000000*x",
"x*0.300000000000000"
]
assert sstr(S("0.3")*x, full_prec="auto") in [
"0.3*x",
"x*0.3"
]
assert sstr(S("0.3")*x, full_prec=False) in [
"0.3*x",
"x*0.3"
]
def test_noncommutative():
A, B, C = symbols('A,B,C', commutative=False)
assert sstr(A*B*C**-1) == "A*B*C**(-1)"
assert sstr(C**-1*A*B) == "C**(-1)*A*B"
assert sstr(A*C**-1*B) == "A*C**(-1)*B"
assert sstr(sqrt(A)) == "sqrt(A)"
assert sstr(1/sqrt(A)) == "A**(-1/2)"
def test_empty_printer():
str_printer = StrPrinter()
assert str_printer.emptyPrinter("foo") == "foo"
assert str_printer.emptyPrinter(x*y) == "x*y"
assert str_printer.emptyPrinter(32) == "32"
def test_settings():
raises(TypeError, lambda: sstr(S(4), method="garbage"))
def test_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
X = Normal('x1', 0, 1)
assert str(where(X > 0)) == "Domain: (0 < x1) & (x1 < oo)"
D = Die('d1', 6)
assert str(where(D > 4)) == "Domain: Eq(d1, 5) | Eq(d1, 6)"
A = Exponential('a', 1)
B = Exponential('b', 1)
assert str(pspace(Tuple(A, B)).domain) == "Domain: (0 <= a) & (0 <= b) & (a < oo) & (b < oo)"
def test_FiniteSet():
assert str(FiniteSet(*range(1, 51))) == '{1, 2, 3, ..., 48, 49, 50}'
assert str(FiniteSet(*range(1, 6))) == '{1, 2, 3, 4, 5}'
def test_PrettyPoly():
from sympy.polys.domains import QQ
F = QQ.frac_field(x, y)
R = QQ[x, y]
assert sstr(F.convert(x/(x + y))) == sstr(x/(x + y))
assert sstr(R.convert(x + y)) == sstr(x + y)
def test_categories():
from sympy.categories import (Object, NamedMorphism,
IdentityMorphism, Category)
A = Object("A")
B = Object("B")
f = NamedMorphism(A, B, "f")
id_A = IdentityMorphism(A)
K = Category("K")
assert str(A) == 'Object("A")'
assert str(f) == 'NamedMorphism(Object("A"), Object("B"), "f")'
assert str(id_A) == 'IdentityMorphism(Object("A"))'
assert str(K) == 'Category("K")'
def test_Tr():
A, B = symbols('A B', commutative=False)
t = Tr(A*B)
assert str(t) == 'Tr(A*B)'
def test_issue_6387():
assert str(factor(-3.0*z + 3)) == '-3.0*(1.0*z - 1.0)'
def test_MatMul_MatAdd():
from sympy import MatrixSymbol
assert str(2*(MatrixSymbol("X", 2, 2) + MatrixSymbol("Y", 2, 2))) == \
"2*(X + Y)"
def test_MatrixSlice():
from sympy.matrices.expressions import MatrixSymbol
assert str(MatrixSymbol('X', 10, 10)[:5, 1:9:2]) == 'X[:5, 1:9:2]'
assert str(MatrixSymbol('X', 10, 10)[5, :5:2]) == 'X[5, :5:2]'
def test_true_false():
assert str(true) == repr(true) == sstr(true) == "True"
assert str(false) == repr(false) == sstr(false) == "False"
def test_Equivalent():
assert str(Equivalent(y, x)) == "Equivalent(x, y)"
def test_Xor():
assert str(Xor(y, x, evaluate=False)) == "Xor(x, y)"
def test_Complement():
assert str(Complement(S.Reals, S.Naturals)) == 'Reals \\ Naturals'
def test_SymmetricDifference():
assert str(SymmetricDifference(Interval(2, 3), Interval(3, 4),evaluate=False)) == \
'SymmetricDifference(Interval(2, 3), Interval(3, 4))'
def test_UnevaluatedExpr():
a, b = symbols("a b")
expr1 = 2*UnevaluatedExpr(a+b)
assert str(expr1) == "2*(a + b)"
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
assert(str(A[0, 0]) == "A[0, 0]")
assert(str(3 * A[0, 0]) == "3*A[0, 0]")
F = C[0, 0].subs(C, A - B)
assert str(F) == "(A - B)[0, 0]"
def test_MatrixSymbol_printing():
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
assert str(A - A*B - B) == "A - A*B - B"
assert str(A*B - (A+B)) == "-(A + B) + A*B"
assert str(A**(-1)) == "A**(-1)"
assert str(A**3) == "A**3"
def test_Subs_printing():
assert str(Subs(x, (x,), (1,))) == 'Subs(x, x, 1)'
assert str(Subs(x + y, (x, y), (1, 2))) == 'Subs(x + y, (x, y), (1, 2))'
def test_issue_15716():
x = Symbol('x')
e = -3**x*exp(-3)*log(3**x*exp(-3)/factorial(x))/factorial(x)
assert str(Integral(e, (x, -oo, oo)).doit()) == '-(Integral(-3*3**x/factorial(x), (x, -oo, oo))' \
' + Integral(3**x*log(3**x/factorial(x))/factorial(x), (x, -oo, oo)))*exp(-3)'
|
aa4c4a6d1bebb12700ded5e1bdde30cb302d422d91ce291d1238f4b3cf82d120
|
from sympy import symbols, sin, Matrix, Interval, Piecewise, Sum, lambdify,Expr
from sympy.utilities.pytest import raises
from sympy.printing.tensorflow import TensorflowPrinter
from sympy.printing.lambdarepr import lambdarepr, LambdaPrinter, NumExprPrinter
x, y, z = symbols("x,y,z")
i, a, b = symbols("i,a,b")
j, c, d = symbols("j,c,d")
def test_basic():
assert lambdarepr(x*y) == "x*y"
assert lambdarepr(x + y) in ["y + x", "x + y"]
assert lambdarepr(x**y) == "x**y"
def test_matrix():
A = Matrix([[x, y], [y*x, z**2]])
# assert lambdarepr(A) == "ImmutableDenseMatrix([[x, y], [x*y, z**2]])"
# Test printing a Matrix that has an element that is printed differently
# with the LambdaPrinter than in the StrPrinter.
p = Piecewise((x, True), evaluate=False)
A = Matrix([p])
assert lambdarepr(A) == "ImmutableDenseMatrix([[((x))]])"
def test_piecewise():
# In each case, test eval() the lambdarepr() to make sure there are a
# correct number of parentheses. It will give a SyntaxError if there aren't.
h = "lambda x: "
p = Piecewise((x, True), evaluate=False)
l = lambdarepr(p)
eval(h + l)
assert l == "((x))"
p = Piecewise((x, x < 0))
l = lambdarepr(p)
eval(h + l)
assert l == "((x) if (x < 0) else None)"
p = Piecewise(
(1, x < 1),
(2, x < 2),
(0, True)
)
l = lambdarepr(p)
eval(h + l)
assert l == "((1) if (x < 1) else (2) if (x < 2) else (0))"
p = Piecewise(
(1, x < 1),
(2, x < 2),
)
l = lambdarepr(p)
eval(h + l)
assert l == "((1) if (x < 1) else (2) if (x < 2) else None)"
p = Piecewise(
(x, x < 1),
(x**2, Interval(3, 4, True, False).contains(x)),
(0, True),
)
l = lambdarepr(p)
eval(h + l)
assert l == "((x) if (x < 1) else (x**2) if (((x <= 4)) and ((x > 3))) else (0))"
p = Piecewise(
(x**2, x < 0),
(x, x < 1),
(2 - x, x >= 1),
(0, True), evaluate=False
)
l = lambdarepr(p)
eval(h + l)
assert l == "((x**2) if (x < 0) else (x) if (x < 1)"\
" else (2 - x) if (x >= 1) else (0))"
p = Piecewise(
(x**2, x < 0),
(x, x < 1),
(2 - x, x >= 1), evaluate=False
)
l = lambdarepr(p)
eval(h + l)
assert l == "((x**2) if (x < 0) else (x) if (x < 1)"\
" else (2 - x) if (x >= 1) else None)"
p = Piecewise(
(1, x >= 1),
(2, x >= 2),
(3, x >= 3),
(4, x >= 4),
(5, x >= 5),
(6, True)
)
l = lambdarepr(p)
eval(h + l)
assert l == "((1) if (x >= 1) else (2) if (x >= 2) else (3) if (x >= 3)"\
" else (4) if (x >= 4) else (5) if (x >= 5) else (6))"
p = Piecewise(
(1, x <= 1),
(2, x <= 2),
(3, x <= 3),
(4, x <= 4),
(5, x <= 5),
(6, True)
)
l = lambdarepr(p)
eval(h + l)
assert l == "((1) if (x <= 1) else (2) if (x <= 2) else (3) if (x <= 3)"\
" else (4) if (x <= 4) else (5) if (x <= 5) else (6))"
p = Piecewise(
(1, x > 1),
(2, x > 2),
(3, x > 3),
(4, x > 4),
(5, x > 5),
(6, True)
)
l = lambdarepr(p)
eval(h + l)
assert l =="((1) if (x > 1) else (2) if (x > 2) else (3) if (x > 3)"\
" else (4) if (x > 4) else (5) if (x > 5) else (6))"
p = Piecewise(
(1, x < 1),
(2, x < 2),
(3, x < 3),
(4, x < 4),
(5, x < 5),
(6, True)
)
l = lambdarepr(p)
eval(h + l)
assert l == "((1) if (x < 1) else (2) if (x < 2) else (3) if (x < 3)"\
" else (4) if (x < 4) else (5) if (x < 5) else (6))"
p = Piecewise(
(Piecewise(
(1, x > 0),
(2, True)
), y > 0),
(3, True)
)
l = lambdarepr(p)
eval(h + l)
assert l == "((((1) if (x > 0) else (2))) if (y > 0) else (3))"
def test_sum__1():
# In each case, test eval() the lambdarepr() to make sure that
# it evaluates to the same results as the symbolic expression
s = Sum(x ** i, (i, a, b))
l = lambdarepr(s)
assert l == "(builtins.sum(x**i for i in range(a, b+1)))"
args = x, a, b
f = lambdify(args, s)
v = 2, 3, 8
assert f(*v) == s.subs(zip(args, v)).doit()
def test_sum__2():
s = Sum(i * x, (i, a, b))
l = lambdarepr(s)
assert l == "(builtins.sum(i*x for i in range(a, b+1)))"
args = x, a, b
f = lambdify(args, s)
v = 2, 3, 8
assert f(*v) == s.subs(zip(args, v)).doit()
def test_multiple_sums():
s = Sum(i * x + j, (i, a, b), (j, c, d))
l = lambdarepr(s)
assert l == "(builtins.sum(i*x + j for i in range(a, b+1) for j in range(c, d+1)))"
args = x, a, b, c, d
f = lambdify(args, s)
vals = 2, 3, 4, 5, 6
f_ref = s.subs(zip(args, vals)).doit()
f_res = f(*vals)
assert f_res == f_ref
def test_settings():
raises(TypeError, lambda: lambdarepr(sin(x), method="garbage"))
class CustomPrintedObject(Expr):
def _lambdacode(self, printer):
return 'lambda'
def _tensorflowcode(self, printer):
return 'tensorflow'
def _numpycode(self, printer):
return 'numpy'
def _numexprcode(self, printer):
return 'numexpr'
def _mpmathcode(self, printer):
return 'mpmath'
def test_printmethod():
# In each case, printmethod is called to test
# its working
obj = CustomPrintedObject()
assert LambdaPrinter().doprint(obj) == 'lambda'
assert TensorflowPrinter().doprint(obj) == 'tensorflow'
assert NumExprPrinter().doprint(obj) == "evaluate('numexpr', truediv=True)"
|
768c0de5422e0cf1686677865d22dfca5ef7c043598ad26f4285c06c0e864282
|
from sympy.core import (
S, pi, oo, symbols, Rational, Integer, Float, Mod, GoldenRatio, EulerGamma, Catalan,
Lambda, Dummy, Eq, nan, Mul, Pow
)
from sympy.functions import (
Abs, acos, acosh, asin, asinh, atan, atanh, atan2, ceiling, cos, cosh, erf,
erfc, exp, floor, gamma, log, loggamma, Max, Min, Piecewise, sign, sin, sinh,
sqrt, tan, tanh
)
from sympy.sets import Range
from sympy.logic import ITE
from sympy.codegen import For, aug_assign, Assignment
from sympy.utilities.pytest import raises, XFAIL
from sympy.printing.ccode import CCodePrinter, C89CodePrinter, C99CodePrinter, get_math_macros
from sympy.codegen.ast import (
AddAugmentedAssignment, Element, Type, FloatType, Declaration, Pointer, Variable, value_const, pointer_const,
While, Scope, Print, FunctionPrototype, FunctionDefinition, FunctionCall, Return,
real, float32, float64, float80, float128, intc, Comment, CodeBlock
)
from sympy.codegen.cfunctions import expm1, log1p, exp2, log2, fma, log10, Cbrt, hypot, Sqrt
from sympy.codegen.cnodes import restrict
from sympy.utilities.lambdify import implemented_function
from sympy.utilities.pytest import warns_deprecated_sympy
from sympy.tensor import IndexedBase, Idx
from sympy.matrices import Matrix, MatrixSymbol
from sympy import ccode
x, y, z = symbols('x,y,z')
def test_printmethod():
class fabs(Abs):
def _ccode(self, printer):
return "fabs(%s)" % printer._print(self.args[0])
assert ccode(fabs(x)) == "fabs(x)"
def test_ccode_sqrt():
assert ccode(sqrt(x)) == "sqrt(x)"
assert ccode(x**0.5) == "sqrt(x)"
assert ccode(sqrt(x)) == "sqrt(x)"
def test_ccode_Pow():
assert ccode(x**3) == "pow(x, 3)"
assert ccode(x**(y**3)) == "pow(x, pow(y, 3))"
g = implemented_function('g', Lambda(x, 2*x))
assert ccode(1/(g(x)*3.5)**(x - y**x)/(x**2 + y)) == \
"pow(3.5*2*x, -x + pow(y, x))/(pow(x, 2) + y)"
assert ccode(x**-1.0) == '1.0/x'
assert ccode(x**Rational(2, 3)) == 'pow(x, 2.0/3.0)'
assert ccode(x**Rational(2, 3), type_aliases={real: float80}) == 'powl(x, 2.0L/3.0L)'
_cond_cfunc = [(lambda base, exp: exp.is_integer, "dpowi"),
(lambda base, exp: not exp.is_integer, "pow")]
assert ccode(x**3, user_functions={'Pow': _cond_cfunc}) == 'dpowi(x, 3)'
assert ccode(x**0.5, user_functions={'Pow': _cond_cfunc}) == 'pow(x, 0.5)'
assert ccode(x**Rational(16, 5), user_functions={'Pow': _cond_cfunc}) == 'pow(x, 16.0/5.0)'
_cond_cfunc2 = [(lambda base, exp: base == 2, lambda base, exp: 'exp2(%s)' % exp),
(lambda base, exp: base != 2, 'pow')]
# Related to gh-11353
assert ccode(2**x, user_functions={'Pow': _cond_cfunc2}) == 'exp2(x)'
assert ccode(x**2, user_functions={'Pow': _cond_cfunc2}) == 'pow(x, 2)'
# For issue 14160
assert ccode(Mul(-2, x, Pow(Mul(y,y,evaluate=False), -1, evaluate=False),
evaluate=False)) == '-2*x/(y*y)'
def test_ccode_Max():
# Test for gh-11926
assert ccode(Max(x,x*x),user_functions={"Max":"my_max", "Pow":"my_pow"}) == 'my_max(x, my_pow(x, 2))'
def test_ccode_Min_performance():
#Shouldn't take more than a few seconds
big_min = Min(*symbols('a[0:50]'))
for curr_standard in ('c89', 'c99', 'c11'):
output = ccode(big_min, standard=curr_standard)
assert output.count('(') == output.count(')')
def test_ccode_constants_mathh():
assert ccode(exp(1)) == "M_E"
assert ccode(pi) == "M_PI"
assert ccode(oo, standard='c89') == "HUGE_VAL"
assert ccode(-oo, standard='c89') == "-HUGE_VAL"
assert ccode(oo) == "INFINITY"
assert ccode(-oo, standard='c99') == "-INFINITY"
assert ccode(pi, type_aliases={real: float80}) == "M_PIl"
def test_ccode_constants_other():
assert ccode(2*GoldenRatio) == "const double GoldenRatio = %s;\n2*GoldenRatio" % GoldenRatio.evalf(17)
assert ccode(
2*Catalan) == "const double Catalan = %s;\n2*Catalan" % Catalan.evalf(17)
assert ccode(2*EulerGamma) == "const double EulerGamma = %s;\n2*EulerGamma" % EulerGamma.evalf(17)
def test_ccode_Rational():
assert ccode(Rational(3, 7)) == "3.0/7.0"
assert ccode(Rational(3, 7), type_aliases={real: float80}) == "3.0L/7.0L"
assert ccode(Rational(18, 9)) == "2"
assert ccode(Rational(3, -7)) == "-3.0/7.0"
assert ccode(Rational(3, -7), type_aliases={real: float80}) == "-3.0L/7.0L"
assert ccode(Rational(-3, -7)) == "3.0/7.0"
assert ccode(Rational(-3, -7), type_aliases={real: float80}) == "3.0L/7.0L"
assert ccode(x + Rational(3, 7)) == "x + 3.0/7.0"
assert ccode(x + Rational(3, 7), type_aliases={real: float80}) == "x + 3.0L/7.0L"
assert ccode(Rational(3, 7)*x) == "(3.0/7.0)*x"
assert ccode(Rational(3, 7)*x, type_aliases={real: float80}) == "(3.0L/7.0L)*x"
def test_ccode_Integer():
assert ccode(Integer(67)) == "67"
assert ccode(Integer(-1)) == "-1"
def test_ccode_functions():
assert ccode(sin(x) ** cos(x)) == "pow(sin(x), cos(x))"
def test_ccode_inline_function():
x = symbols('x')
g = implemented_function('g', Lambda(x, 2*x))
assert ccode(g(x)) == "2*x"
g = implemented_function('g', Lambda(x, 2*x/Catalan))
assert ccode(
g(x)) == "const double Catalan = %s;\n2*x/Catalan" % Catalan.evalf(17)
A = IndexedBase('A')
i = Idx('i', symbols('n', integer=True))
g = implemented_function('g', Lambda(x, x*(1 + x)*(2 + x)))
assert ccode(g(A[i]), assign_to=A[i]) == (
"for (int i=0; i<n; i++){\n"
" A[i] = (A[i] + 1)*(A[i] + 2)*A[i];\n"
"}"
)
def test_ccode_exceptions():
assert ccode(gamma(x), standard='C99') == "tgamma(x)"
gamma_c89 = ccode(gamma(x), standard='C89')
assert 'not supported in c' in gamma_c89.lower()
gamma_c89 = ccode(gamma(x), standard='C89', allow_unknown_functions=False)
assert 'not supported in c' in gamma_c89.lower()
gamma_c89 = ccode(gamma(x), standard='C89', allow_unknown_functions=True)
assert not 'not supported in c' in gamma_c89.lower()
assert ccode(ceiling(x)) == "ceil(x)"
assert ccode(Abs(x)) == "fabs(x)"
assert ccode(gamma(x)) == "tgamma(x)"
r, s = symbols('r,s', real=True)
assert ccode(Mod(ceiling(r), ceiling(s))) == "((ceil(r)) % (ceil(s)))"
assert ccode(Mod(r, s)) == "fmod(r, s)"
def test_ccode_user_functions():
x = symbols('x', integer=False)
n = symbols('n', integer=True)
custom_functions = {
"ceiling": "ceil",
"Abs": [(lambda x: not x.is_integer, "fabs"), (lambda x: x.is_integer, "abs")],
}
assert ccode(ceiling(x), user_functions=custom_functions) == "ceil(x)"
assert ccode(Abs(x), user_functions=custom_functions) == "fabs(x)"
assert ccode(Abs(n), user_functions=custom_functions) == "abs(n)"
def test_ccode_boolean():
assert ccode(True) == "true"
assert ccode(S.true) == "true"
assert ccode(False) == "false"
assert ccode(S.false) == "false"
assert ccode(x & y) == "x && y"
assert ccode(x | y) == "x || y"
assert ccode(~x) == "!x"
assert ccode(x & y & z) == "x && y && z"
assert ccode(x | y | z) == "x || y || z"
assert ccode((x & y) | z) == "z || x && y"
assert ccode((x | y) & z) == "z && (x || y)"
def test_ccode_Relational():
from sympy import Eq, Ne, Le, Lt, Gt, Ge
assert ccode(Eq(x, y)) == "x == y"
assert ccode(Ne(x, y)) == "x != y"
assert ccode(Le(x, y)) == "x <= y"
assert ccode(Lt(x, y)) == "x < y"
assert ccode(Gt(x, y)) == "x > y"
assert ccode(Ge(x, y)) == "x >= y"
def test_ccode_Piecewise():
expr = Piecewise((x, x < 1), (x**2, True))
assert ccode(expr) == (
"((x < 1) ? (\n"
" x\n"
")\n"
": (\n"
" pow(x, 2)\n"
"))")
assert ccode(expr, assign_to="c") == (
"if (x < 1) {\n"
" c = x;\n"
"}\n"
"else {\n"
" c = pow(x, 2);\n"
"}")
expr = Piecewise((x, x < 1), (x + 1, x < 2), (x**2, True))
assert ccode(expr) == (
"((x < 1) ? (\n"
" x\n"
")\n"
": ((x < 2) ? (\n"
" x + 1\n"
")\n"
": (\n"
" pow(x, 2)\n"
")))")
assert ccode(expr, assign_to='c') == (
"if (x < 1) {\n"
" c = x;\n"
"}\n"
"else if (x < 2) {\n"
" c = x + 1;\n"
"}\n"
"else {\n"
" c = pow(x, 2);\n"
"}")
# Check that Piecewise without a True (default) condition error
expr = Piecewise((x, x < 1), (x**2, x > 1), (sin(x), x > 0))
raises(ValueError, lambda: ccode(expr))
def test_ccode_sinc():
from sympy import sinc
expr = sinc(x)
assert ccode(expr) == (
"((x != 0) ? (\n"
" sin(x)/x\n"
")\n"
": (\n"
" 1\n"
"))")
def test_ccode_Piecewise_deep():
p = ccode(2*Piecewise((x, x < 1), (x + 1, x < 2), (x**2, True)))
assert p == (
"2*((x < 1) ? (\n"
" x\n"
")\n"
": ((x < 2) ? (\n"
" x + 1\n"
")\n"
": (\n"
" pow(x, 2)\n"
")))")
expr = x*y*z + x**2 + y**2 + Piecewise((0, x < 0.5), (1, True)) + cos(z) - 1
assert ccode(expr) == (
"pow(x, 2) + x*y*z + pow(y, 2) + ((x < 0.5) ? (\n"
" 0\n"
")\n"
": (\n"
" 1\n"
")) + cos(z) - 1")
assert ccode(expr, assign_to='c') == (
"c = pow(x, 2) + x*y*z + pow(y, 2) + ((x < 0.5) ? (\n"
" 0\n"
")\n"
": (\n"
" 1\n"
")) + cos(z) - 1;")
def test_ccode_ITE():
expr = ITE(x < 1, y, z)
assert ccode(expr) == (
"((x < 1) ? (\n"
" y\n"
")\n"
": (\n"
" z\n"
"))")
def test_ccode_settings():
raises(TypeError, lambda: ccode(sin(x), method="garbage"))
def test_ccode_Indexed():
from sympy.tensor import IndexedBase, Idx
from sympy import symbols
s, n, m, o = symbols('s n m o', integer=True)
i, j, k = Idx('i', n), Idx('j', m), Idx('k', o)
x = IndexedBase('x')[j]
A = IndexedBase('A')[i, j]
B = IndexedBase('B')[i, j, k]
with warns_deprecated_sympy():
p = CCodePrinter()
p._not_c = set()
assert p._print_Indexed(x) == 'x[j]'
assert p._print_Indexed(A) == 'A[%s]' % (m*i+j)
assert p._print_Indexed(B) == 'B[%s]' % (i*o*m+j*o+k)
assert p._not_c == set()
A = IndexedBase('A', shape=(5,3))[i, j]
assert p._print_Indexed(A) == 'A[%s]' % (3*i + j)
A = IndexedBase('A', shape=(5,3), strides='F')[i, j]
assert ccode(A) == 'A[%s]' % (i + 5*j)
A = IndexedBase('A', shape=(29,29), strides=(1, s), offset=o)[i, j]
assert ccode(A) == 'A[o + s*j + i]'
Abase = IndexedBase('A', strides=(s, m, n), offset=o)
assert ccode(Abase[i, j, k]) == 'A[m*j + n*k + o + s*i]'
assert ccode(Abase[2, 3, k]) == 'A[3*m + n*k + o + 2*s]'
def test_Element():
assert ccode(Element('x', 'ij')) == 'x[i][j]'
assert ccode(Element('x', 'ij', strides='kl', offset='o')) == 'x[i*k + j*l + o]'
assert ccode(Element('x', (3,))) == 'x[3]'
assert ccode(Element('x', (3,4,5))) == 'x[3][4][5]'
def test_ccode_Indexed_without_looking_for_contraction():
len_y = 5
y = IndexedBase('y', shape=(len_y,))
x = IndexedBase('x', shape=(len_y,))
Dy = IndexedBase('Dy', shape=(len_y-1,))
i = Idx('i', len_y-1)
e=Eq(Dy[i], (y[i+1]-y[i])/(x[i+1]-x[i]))
code0 = ccode(e.rhs, assign_to=e.lhs, contract=False)
assert code0 == 'Dy[i] = (y[%s] - y[i])/(x[%s] - x[i]);' % (i + 1, i + 1)
def test_ccode_loops_matrix_vector():
n, m = symbols('n m', integer=True)
A = IndexedBase('A')
x = IndexedBase('x')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
s = (
'for (int i=0; i<m; i++){\n'
' y[i] = 0;\n'
'}\n'
'for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' y[i] = A[%s]*x[j] + y[i];\n' % (i*n + j) +\
' }\n'
'}'
)
assert ccode(A[i, j]*x[j], assign_to=y[i]) == s
def test_dummy_loops():
i, m = symbols('i m', integer=True, cls=Dummy)
x = IndexedBase('x')
y = IndexedBase('y')
i = Idx(i, m)
expected = (
'for (int i_%(icount)i=0; i_%(icount)i<m_%(mcount)i; i_%(icount)i++){\n'
' y[i_%(icount)i] = x[i_%(icount)i];\n'
'}'
) % {'icount': i.label.dummy_index, 'mcount': m.dummy_index}
assert ccode(x[i], assign_to=y[i]) == expected
def test_ccode_loops_add():
from sympy.tensor import IndexedBase, Idx
from sympy import symbols
n, m = symbols('n m', integer=True)
A = IndexedBase('A')
x = IndexedBase('x')
y = IndexedBase('y')
z = IndexedBase('z')
i = Idx('i', m)
j = Idx('j', n)
s = (
'for (int i=0; i<m; i++){\n'
' y[i] = x[i] + z[i];\n'
'}\n'
'for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' y[i] = A[%s]*x[j] + y[i];\n' % (i*n + j) +\
' }\n'
'}'
)
assert ccode(A[i, j]*x[j] + x[i] + z[i], assign_to=y[i]) == s
def test_ccode_loops_multiple_contractions():
from sympy.tensor import IndexedBase, Idx
from sympy import symbols
n, m, o, p = symbols('n m o p', integer=True)
a = IndexedBase('a')
b = IndexedBase('b')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
l = Idx('l', p)
s = (
'for (int i=0; i<m; i++){\n'
' y[i] = 0;\n'
'}\n'
'for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' for (int k=0; k<o; k++){\n'
' for (int l=0; l<p; l++){\n'
' y[i] = a[%s]*b[%s] + y[i];\n' % (i*n*o*p + j*o*p + k*p + l, j*o*p + k*p + l) +\
' }\n'
' }\n'
' }\n'
'}'
)
assert ccode(b[j, k, l]*a[i, j, k, l], assign_to=y[i]) == s
def test_ccode_loops_addfactor():
from sympy.tensor import IndexedBase, Idx
from sympy import symbols
n, m, o, p = symbols('n m o p', integer=True)
a = IndexedBase('a')
b = IndexedBase('b')
c = IndexedBase('c')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
l = Idx('l', p)
s = (
'for (int i=0; i<m; i++){\n'
' y[i] = 0;\n'
'}\n'
'for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' for (int k=0; k<o; k++){\n'
' for (int l=0; l<p; l++){\n'
' y[i] = (a[%s] + b[%s])*c[%s] + y[i];\n' % (i*n*o*p + j*o*p + k*p + l, i*n*o*p + j*o*p + k*p + l, j*o*p + k*p + l) +\
' }\n'
' }\n'
' }\n'
'}'
)
assert ccode((a[i, j, k, l] + b[i, j, k, l])*c[j, k, l], assign_to=y[i]) == s
def test_ccode_loops_multiple_terms():
from sympy.tensor import IndexedBase, Idx
from sympy import symbols
n, m, o, p = symbols('n m o p', integer=True)
a = IndexedBase('a')
b = IndexedBase('b')
c = IndexedBase('c')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
s0 = (
'for (int i=0; i<m; i++){\n'
' y[i] = 0;\n'
'}\n'
)
s1 = (
'for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' for (int k=0; k<o; k++){\n'
' y[i] = b[j]*b[k]*c[%s] + y[i];\n' % (i*n*o + j*o + k) +\
' }\n'
' }\n'
'}\n'
)
s2 = (
'for (int i=0; i<m; i++){\n'
' for (int k=0; k<o; k++){\n'
' y[i] = a[%s]*b[k] + y[i];\n' % (i*o + k) +\
' }\n'
'}\n'
)
s3 = (
'for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' y[i] = a[%s]*b[j] + y[i];\n' % (i*n + j) +\
' }\n'
'}\n'
)
c = ccode(b[j]*a[i, j] + b[k]*a[i, k] + b[j]*b[k]*c[i, j, k], assign_to=y[i])
assert (c == s0 + s1 + s2 + s3[:-1] or
c == s0 + s1 + s3 + s2[:-1] or
c == s0 + s2 + s1 + s3[:-1] or
c == s0 + s2 + s3 + s1[:-1] or
c == s0 + s3 + s1 + s2[:-1] or
c == s0 + s3 + s2 + s1[:-1])
def test_dereference_printing():
expr = x + y + sin(z) + z
assert ccode(expr, dereference=[z]) == "x + y + (*z) + sin((*z))"
def test_Matrix_printing():
# Test returning a Matrix
mat = Matrix([x*y, Piecewise((2 + x, y>0), (y, True)), sin(z)])
A = MatrixSymbol('A', 3, 1)
assert ccode(mat, A) == (
"A[0] = x*y;\n"
"if (y > 0) {\n"
" A[1] = x + 2;\n"
"}\n"
"else {\n"
" A[1] = y;\n"
"}\n"
"A[2] = sin(z);")
# Test using MatrixElements in expressions
expr = Piecewise((2*A[2, 0], x > 0), (A[2, 0], True)) + sin(A[1, 0]) + A[0, 0]
assert ccode(expr) == (
"((x > 0) ? (\n"
" 2*A[2]\n"
")\n"
": (\n"
" A[2]\n"
")) + sin(A[1]) + A[0]")
# Test using MatrixElements in a Matrix
q = MatrixSymbol('q', 5, 1)
M = MatrixSymbol('M', 3, 3)
m = Matrix([[sin(q[1,0]), 0, cos(q[2,0])],
[q[1,0] + q[2,0], q[3, 0], 5],
[2*q[4, 0]/q[1,0], sqrt(q[0,0]) + 4, 0]])
assert ccode(m, M) == (
"M[0] = sin(q[1]);\n"
"M[1] = 0;\n"
"M[2] = cos(q[2]);\n"
"M[3] = q[1] + q[2];\n"
"M[4] = q[3];\n"
"M[5] = 5;\n"
"M[6] = 2*q[4]/q[1];\n"
"M[7] = sqrt(q[0]) + 4;\n"
"M[8] = 0;")
def test_ccode_reserved_words():
x, y = symbols('x, if')
with raises(ValueError):
ccode(y**2, error_on_reserved=True, standard='C99')
assert ccode(y**2) == 'pow(if_, 2)'
assert ccode(x * y**2, dereference=[y]) == 'pow((*if_), 2)*x'
assert ccode(y**2, reserved_word_suffix='_unreserved') == 'pow(if_unreserved, 2)'
def test_ccode_sign():
expr1, ref1 = sign(x) * y, 'y*(((x) > 0) - ((x) < 0))'
expr2, ref2 = sign(cos(x)), '(((cos(x)) > 0) - ((cos(x)) < 0))'
expr3, ref3 = sign(2 * x + x**2) * x + x**2, 'pow(x, 2) + x*(((pow(x, 2) + 2*x) > 0) - ((pow(x, 2) + 2*x) < 0))'
assert ccode(expr1) == ref1
assert ccode(expr1, 'z') == 'z = %s;' % ref1
assert ccode(expr2) == ref2
assert ccode(expr3) == ref3
def test_ccode_Assignment():
assert ccode(Assignment(x, y + z)) == 'x = y + z;'
assert ccode(aug_assign(x, '+', y + z)) == 'x += y + z;'
def test_ccode_For():
f = For(x, Range(0, 10, 2), [aug_assign(y, '*', x)])
assert ccode(f) == ("for (x = 0; x < 10; x += 2) {\n"
" y *= x;\n"
"}")
def test_ccode_Max_Min():
assert ccode(Max(x, 0), standard='C89') == '((0 > x) ? 0 : x)'
assert ccode(Max(x, 0), standard='C99') == 'fmax(0, x)'
assert ccode(Min(x, 0, sqrt(x)), standard='c89') == (
'((0 < ((x < sqrt(x)) ? x : sqrt(x))) ? 0 : ((x < sqrt(x)) ? x : sqrt(x)))'
)
def test_ccode_standard():
assert ccode(expm1(x), standard='c99') == 'expm1(x)'
assert ccode(nan, standard='c99') == 'NAN'
assert ccode(float('nan'), standard='c99') == 'NAN'
def test_CCodePrinter():
with warns_deprecated_sympy():
CCodePrinter()
with warns_deprecated_sympy():
assert CCodePrinter().language == 'C'
def test_C89CodePrinter():
c89printer = C89CodePrinter()
assert c89printer.language == 'C'
assert c89printer.standard == 'C89'
assert 'void' in c89printer.reserved_words
assert 'template' not in c89printer.reserved_words
def test_C99CodePrinter():
assert C99CodePrinter().doprint(expm1(x)) == 'expm1(x)'
assert C99CodePrinter().doprint(log1p(x)) == 'log1p(x)'
assert C99CodePrinter().doprint(exp2(x)) == 'exp2(x)'
assert C99CodePrinter().doprint(log2(x)) == 'log2(x)'
assert C99CodePrinter().doprint(fma(x, y, -z)) == 'fma(x, y, -z)'
assert C99CodePrinter().doprint(log10(x)) == 'log10(x)'
assert C99CodePrinter().doprint(Cbrt(x)) == 'cbrt(x)' # note Cbrt due to cbrt already taken.
assert C99CodePrinter().doprint(hypot(x, y)) == 'hypot(x, y)'
assert C99CodePrinter().doprint(loggamma(x)) == 'lgamma(x)'
assert C99CodePrinter().doprint(Max(x, 3, x**2)) == 'fmax(3, fmax(x, pow(x, 2)))'
assert C99CodePrinter().doprint(Min(x, 3)) == 'fmin(3, x)'
c99printer = C99CodePrinter()
assert c99printer.language == 'C'
assert c99printer.standard == 'C99'
assert 'restrict' in c99printer.reserved_words
assert 'using' not in c99printer.reserved_words
@XFAIL
def test_C99CodePrinter__precision_f80():
f80_printer = C99CodePrinter(dict(type_aliases={real: float80}))
assert f80_printer.doprint(sin(x+Float('2.1'))) == 'sinl(x + 2.1L)'
def test_C99CodePrinter__precision():
n = symbols('n', integer=True)
f32_printer = C99CodePrinter(dict(type_aliases={real: float32}))
f64_printer = C99CodePrinter(dict(type_aliases={real: float64}))
f80_printer = C99CodePrinter(dict(type_aliases={real: float80}))
assert f32_printer.doprint(sin(x+2.1)) == 'sinf(x + 2.1F)'
assert f64_printer.doprint(sin(x+2.1)) == 'sin(x + 2.1000000000000001)'
assert f80_printer.doprint(sin(x+Float('2.0'))) == 'sinl(x + 2.0L)'
for printer, suffix in zip([f32_printer, f64_printer, f80_printer], ['f', '', 'l']):
def check(expr, ref):
assert printer.doprint(expr) == ref.format(s=suffix, S=suffix.upper())
check(Abs(n), 'abs(n)')
check(Abs(x + 2.0), 'fabs{s}(x + 2.0{S})')
check(sin(x + 4.0)**cos(x - 2.0), 'pow{s}(sin{s}(x + 4.0{S}), cos{s}(x - 2.0{S}))')
check(exp(x*8.0), 'exp{s}(8.0{S}*x)')
check(exp2(x), 'exp2{s}(x)')
check(expm1(x*4.0), 'expm1{s}(4.0{S}*x)')
check(Mod(n, 2), '((n) % (2))')
check(Mod(2*n + 3, 3*n + 5), '((2*n + 3) % (3*n + 5))')
check(Mod(x + 2.0, 3.0), 'fmod{s}(1.0{S}*x + 2.0{S}, 3.0{S})')
check(Mod(x, 2.0*x + 3.0), 'fmod{s}(1.0{S}*x, 2.0{S}*x + 3.0{S})')
check(log(x/2), 'log{s}((1.0{S}/2.0{S})*x)')
check(log10(3*x/2), 'log10{s}((3.0{S}/2.0{S})*x)')
check(log2(x*8.0), 'log2{s}(8.0{S}*x)')
check(log1p(x), 'log1p{s}(x)')
check(2**x, 'pow{s}(2, x)')
check(2.0**x, 'pow{s}(2.0{S}, x)')
check(x**3, 'pow{s}(x, 3)')
check(x**4.0, 'pow{s}(x, 4.0{S})')
check(sqrt(3+x), 'sqrt{s}(x + 3)')
check(Cbrt(x-2.0), 'cbrt{s}(x - 2.0{S})')
check(hypot(x, y), 'hypot{s}(x, y)')
check(sin(3.*x + 2.), 'sin{s}(3.0{S}*x + 2.0{S})')
check(cos(3.*x - 1.), 'cos{s}(3.0{S}*x - 1.0{S})')
check(tan(4.*y + 2.), 'tan{s}(4.0{S}*y + 2.0{S})')
check(asin(3.*x + 2.), 'asin{s}(3.0{S}*x + 2.0{S})')
check(acos(3.*x + 2.), 'acos{s}(3.0{S}*x + 2.0{S})')
check(atan(3.*x + 2.), 'atan{s}(3.0{S}*x + 2.0{S})')
check(atan2(3.*x, 2.*y), 'atan2{s}(3.0{S}*x, 2.0{S}*y)')
check(sinh(3.*x + 2.), 'sinh{s}(3.0{S}*x + 2.0{S})')
check(cosh(3.*x - 1.), 'cosh{s}(3.0{S}*x - 1.0{S})')
check(tanh(4.0*y + 2.), 'tanh{s}(4.0{S}*y + 2.0{S})')
check(asinh(3.*x + 2.), 'asinh{s}(3.0{S}*x + 2.0{S})')
check(acosh(3.*x + 2.), 'acosh{s}(3.0{S}*x + 2.0{S})')
check(atanh(3.*x + 2.), 'atanh{s}(3.0{S}*x + 2.0{S})')
check(erf(42.*x), 'erf{s}(42.0{S}*x)')
check(erfc(42.*x), 'erfc{s}(42.0{S}*x)')
check(gamma(x), 'tgamma{s}(x)')
check(loggamma(x), 'lgamma{s}(x)')
check(ceiling(x + 2.), "ceil{s}(x + 2.0{S})")
check(floor(x + 2.), "floor{s}(x + 2.0{S})")
check(fma(x, y, -z), 'fma{s}(x, y, -z)')
check(Max(x, 8.0, x**4.0), 'fmax{s}(8.0{S}, fmax{s}(x, pow{s}(x, 4.0{S})))')
check(Min(x, 2.0), 'fmin{s}(2.0{S}, x)')
def test_get_math_macros():
macros = get_math_macros()
assert macros[exp(1)] == 'M_E'
assert macros[1/Sqrt(2)] == 'M_SQRT1_2'
def test_ccode_Declaration():
i = symbols('i', integer=True)
var1 = Variable(i, type=Type.from_expr(i))
dcl1 = Declaration(var1)
assert ccode(dcl1) == 'int i'
var2 = Variable(x, type=float32, attrs={value_const})
dcl2a = Declaration(var2)
assert ccode(dcl2a) == 'const float x'
dcl2b = var2.as_Declaration(value=pi)
assert ccode(dcl2b) == 'const float x = M_PI'
var3 = Variable(y, type=Type('bool'))
dcl3 = Declaration(var3)
printer = C89CodePrinter()
assert 'stdbool.h' not in printer.headers
assert printer.doprint(dcl3) == 'bool y'
assert 'stdbool.h' in printer.headers
u = symbols('u', real=True)
ptr4 = Pointer.deduced(u, attrs={pointer_const, restrict})
dcl4 = Declaration(ptr4)
assert ccode(dcl4) == 'double * const restrict u'
var5 = Variable(x, Type('__float128'), attrs={value_const})
dcl5a = Declaration(var5)
assert ccode(dcl5a) == 'const __float128 x'
var5b = Variable(var5.symbol, var5.type, pi, attrs=var5.attrs)
dcl5b = Declaration(var5b)
assert ccode(dcl5b) == 'const __float128 x = M_PI'
def test_C99CodePrinter_custom_type():
# We will look at __float128 (new in glibc 2.26)
f128 = FloatType('_Float128', float128.nbits, float128.nmant, float128.nexp)
p128 = C99CodePrinter(dict(
type_aliases={real: f128},
type_literal_suffixes={f128: 'Q'},
type_func_suffixes={f128: 'f128'},
type_math_macro_suffixes={
real: 'f128',
f128: 'f128'
},
type_macros={
f128: ('__STDC_WANT_IEC_60559_TYPES_EXT__',)
}
))
assert p128.doprint(x) == 'x'
assert not p128.headers
assert not p128.libraries
assert not p128.macros
assert p128.doprint(2.0) == '2.0Q'
assert not p128.headers
assert not p128.libraries
assert p128.macros == {'__STDC_WANT_IEC_60559_TYPES_EXT__'}
assert p128.doprint(Rational(1, 2)) == '1.0Q/2.0Q'
assert p128.doprint(sin(x)) == 'sinf128(x)'
assert p128.doprint(cos(2., evaluate=False)) == 'cosf128(2.0Q)'
var5 = Variable(x, f128, attrs={value_const})
dcl5a = Declaration(var5)
assert ccode(dcl5a) == 'const _Float128 x'
var5b = Variable(x, f128, pi, attrs={value_const})
dcl5b = Declaration(var5b)
assert p128.doprint(dcl5b) == 'const _Float128 x = M_PIf128'
var5b = Variable(x, f128, value=Catalan.evalf(38), attrs={value_const})
dcl5c = Declaration(var5b)
assert p128.doprint(dcl5c) == 'const _Float128 x = %sQ' % Catalan.evalf(f128.decimal_dig)
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
assert(ccode(A[0, 0]) == "A[0]")
assert(ccode(3 * A[0, 0]) == "3*A[0]")
F = C[0, 0].subs(C, A - B)
assert(ccode(F) == "(A - B)[0]")
def test_subclass_CCodePrinter():
# issue gh-12687
class MySubClass(CCodePrinter):
pass
def test_ccode_math_macros():
assert ccode(z + exp(1)) == 'z + M_E'
assert ccode(z + log2(exp(1))) == 'z + M_LOG2E'
assert ccode(z + 1/log(2)) == 'z + M_LOG2E'
assert ccode(z + log(2)) == 'z + M_LN2'
assert ccode(z + log(10)) == 'z + M_LN10'
assert ccode(z + pi) == 'z + M_PI'
assert ccode(z + pi/2) == 'z + M_PI_2'
assert ccode(z + pi/4) == 'z + M_PI_4'
assert ccode(z + 1/pi) == 'z + M_1_PI'
assert ccode(z + 2/pi) == 'z + M_2_PI'
assert ccode(z + 2/sqrt(pi)) == 'z + M_2_SQRTPI'
assert ccode(z + 2/Sqrt(pi)) == 'z + M_2_SQRTPI'
assert ccode(z + sqrt(2)) == 'z + M_SQRT2'
assert ccode(z + Sqrt(2)) == 'z + M_SQRT2'
assert ccode(z + 1/sqrt(2)) == 'z + M_SQRT1_2'
assert ccode(z + 1/Sqrt(2)) == 'z + M_SQRT1_2'
def test_ccode_Type():
assert ccode(Type('float')) == 'float'
assert ccode(intc) == 'int'
def test_ccode_codegen_ast():
assert ccode(Comment("this is a comment")) == "// this is a comment"
assert ccode(While(abs(x) > 1, [aug_assign(x, '-', 1)])) == (
'while (fabs(x) > 1) {\n'
' x -= 1;\n'
'}'
)
assert ccode(Scope([AddAugmentedAssignment(x, 1)])) == (
'{\n'
' x += 1;\n'
'}'
)
inp_x = Declaration(Variable(x, type=real))
assert ccode(FunctionPrototype(real, 'pwer', [inp_x])) == 'double pwer(double x)'
assert ccode(FunctionDefinition(real, 'pwer', [inp_x], [Assignment(x, x**2)])) == (
'double pwer(double x){\n'
' x = pow(x, 2);\n'
'}'
)
# Elements of CodeBlock are formatted as statements:
block = CodeBlock(
x,
Print([x, y], "%d %d"),
FunctionCall('pwer', [x]),
Return(x),
)
assert ccode(block) == '\n'.join([
'x;',
'printf("%d %d", x, y);',
'pwer(x);',
'return x;',
])
|
6398fe52586f0557643a79b2a82a64a2cd429e501d91993aa01bece6b7f4583c
|
from sympy import (
Add, Abs, Chi, Ci, CosineTransform, Dict, Ei, Eq, FallingFactorial,
FiniteSet, Float, FourierTransform, Function, Indexed, IndexedBase, Integral,
Interval, InverseCosineTransform, InverseFourierTransform,
InverseLaplaceTransform, InverseMellinTransform, InverseSineTransform,
Lambda, LaplaceTransform, Limit, Matrix, Max, MellinTransform, Min, Mul,
Order, Piecewise, Poly, ring, field, ZZ, Pow, Product, Range, Rational,
RisingFactorial, rootof, RootSum, S, Shi, Si, SineTransform, Subs,
Sum, Symbol, ImageSet, Tuple, Union, Ynm, Znm, arg, asin, acsc, Mod,
assoc_laguerre, assoc_legendre, beta, binomial, catalan, ceiling, Complement,
chebyshevt, chebyshevu, conjugate, cot, coth, diff, dirichlet_eta, euler,
exp, expint, factorial, factorial2, floor, gamma, gegenbauer, hermite,
hyper, im, jacobi, laguerre, legendre, lerchphi, log,
meijerg, oo, polar_lift, polylog, re, root, sin, sqrt, symbols,
uppergamma, zeta, subfactorial, totient, elliptic_k, elliptic_f,
elliptic_e, elliptic_pi, cos, tan, Wild, true, false, Equivalent, Not,
Contains, divisor_sigma, SymmetricDifference, SeqPer, SeqFormula,
SeqAdd, SeqMul, fourier_series, pi, ConditionSet, ComplexRegion, fps,
AccumBounds, reduced_totient, primenu, primeomega, SingularityFunction,
UnevaluatedExpr, Quaternion, I, KroneckerProduct, Intersection)
from sympy.ntheory.factor_ import udivisor_sigma
from sympy.abc import mu, tau
from sympy.printing.latex import (latex, translate, greek_letters_set,
tex_greek_dictionary)
from sympy.tensor.array import (ImmutableDenseNDimArray,
ImmutableSparseNDimArray,
MutableSparseNDimArray,
MutableDenseNDimArray)
from sympy.tensor.array import tensorproduct
from sympy.utilities.pytest import XFAIL, raises
from sympy.functions import DiracDelta, Heaviside, KroneckerDelta, LeviCivita
from sympy.functions.combinatorial.numbers import bernoulli, bell, lucas, \
fibonacci, tribonacci
from sympy.logic import Implies
from sympy.logic.boolalg import And, Or, Xor
from sympy.physics.quantum import Commutator, Operator
from sympy.physics.units import degree, radian, kg, meter
from sympy.core.trace import Tr
from sympy.core.compatibility import range
from sympy.combinatorics.permutations import Cycle, Permutation
from sympy import MatrixSymbol, ln
from sympy.vector import CoordSys3D, Cross, Curl, Dot, Divergence, Gradient, Laplacian
from sympy.sets.setexpr import SetExpr
import sympy as sym
class lowergamma(sym.lowergamma):
pass # testing notation inheritance by a subclass with same name
x, y, z, t, a, b, c = symbols('x y z t a b c')
k, m, n = symbols('k m n', integer=True)
def test_printmethod():
class R(Abs):
def _latex(self, printer):
return "foo(%s)" % printer._print(self.args[0])
assert latex(R(x)) == "foo(x)"
class R(Abs):
def _latex(self, printer):
return "foo"
assert latex(R(x)) == "foo"
def test_latex_basic():
assert latex(1 + x) == "x + 1"
assert latex(x**2) == "x^{2}"
assert latex(x**(1 + x)) == "x^{x + 1}"
assert latex(x**3 + x + 1 + x**2) == "x^{3} + x^{2} + x + 1"
assert latex(2*x*y) == "2 x y"
assert latex(2*x*y, mul_symbol='dot') == r"2 \cdot x \cdot y"
assert latex(3*x**2*y, mul_symbol='\\,') == r"3\,x^{2}\,y"
assert latex(1.5*3**x, mul_symbol='\\,') == r"1.5 \cdot 3^{x}"
assert latex(1/x) == r"\frac{1}{x}"
assert latex(1/x, fold_short_frac=True) == "1 / x"
assert latex(-S(3)/2) == r"- \frac{3}{2}"
assert latex(-S(3)/2, fold_short_frac=True) == r"- 3 / 2"
assert latex(1/x**2) == r"\frac{1}{x^{2}}"
assert latex(1/(x + y)/2) == r"\frac{1}{2 \left(x + y\right)}"
assert latex(x/2) == r"\frac{x}{2}"
assert latex(x/2, fold_short_frac=True) == "x / 2"
assert latex((x + y)/(2*x)) == r"\frac{x + y}{2 x}"
assert latex((x + y)/(2*x), fold_short_frac=True) == \
r"\left(x + y\right) / 2 x"
assert latex((x + y)/(2*x), long_frac_ratio=0) == \
r"\frac{1}{2 x} \left(x + y\right)"
assert latex((x + y)/x) == r"\frac{x + y}{x}"
assert latex((x + y)/x, long_frac_ratio=3) == r"\frac{x + y}{x}"
assert latex((2*sqrt(2)*x)/3) == r"\frac{2 \sqrt{2} x}{3}"
assert latex((2*sqrt(2)*x)/3, long_frac_ratio=2) == \
r"\frac{2 x}{3} \sqrt{2}"
assert latex(2*Integral(x, x)/3) == r"\frac{2 \int x\, dx}{3}"
assert latex(2*Integral(x, x)/3, fold_short_frac=True) == \
r"\left(2 \int x\, dx\right) / 3"
assert latex(sqrt(x)) == r"\sqrt{x}"
assert latex(x**Rational(1, 3)) == r"\sqrt[3]{x}"
assert latex(x**Rational(1, 3), root_notation=False) == r"x^{\frac{1}{3}}"
assert latex(sqrt(x)**3) == r"x^{\frac{3}{2}}"
assert latex(sqrt(x), itex=True) == r"\sqrt{x}"
assert latex(x**Rational(1, 3), itex=True) == r"\root{3}{x}"
assert latex(sqrt(x)**3, itex=True) == r"x^{\frac{3}{2}}"
assert latex(x**Rational(3, 4)) == r"x^{\frac{3}{4}}"
assert latex(x**Rational(3, 4), fold_frac_powers=True) == "x^{3/4}"
assert latex((x + 1)**Rational(3, 4)) == \
r"\left(x + 1\right)^{\frac{3}{4}}"
assert latex((x + 1)**Rational(3, 4), fold_frac_powers=True) == \
r"\left(x + 1\right)^{3/4}"
assert latex(1.5e20*x) == r"1.5 \cdot 10^{20} x"
assert latex(1.5e20*x, mul_symbol='dot') == r"1.5 \cdot 10^{20} \cdot x"
assert latex(1.5e20*x, mul_symbol='times') == \
r"1.5 \times 10^{20} \times x"
assert latex(1/sin(x)) == r"\frac{1}{\sin{\left(x \right)}}"
assert latex(sin(x)**-1) == r"\frac{1}{\sin{\left(x \right)}}"
assert latex(sin(x)**Rational(3, 2)) == \
r"\sin^{\frac{3}{2}}{\left(x \right)}"
assert latex(sin(x)**Rational(3, 2), fold_frac_powers=True) == \
r"\sin^{3/2}{\left(x \right)}"
assert latex(~x) == r"\neg x"
assert latex(x & y) == r"x \wedge y"
assert latex(x & y & z) == r"x \wedge y \wedge z"
assert latex(x | y) == r"x \vee y"
assert latex(x | y | z) == r"x \vee y \vee z"
assert latex((x & y) | z) == r"z \vee \left(x \wedge y\right)"
assert latex(Implies(x, y)) == r"x \Rightarrow y"
assert latex(~(x >> ~y)) == r"x \not\Rightarrow \neg y"
assert latex(Implies(Or(x,y), z)) == r"\left(x \vee y\right) \Rightarrow z"
assert latex(Implies(z, Or(x,y))) == r"z \Rightarrow \left(x \vee y\right)"
assert latex(~x, symbol_names={x: "x_i"}) == r"\neg x_i"
assert latex(x & y, symbol_names={x: "x_i", y: "y_i"}) == \
r"x_i \wedge y_i"
assert latex(x & y & z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \
r"x_i \wedge y_i \wedge z_i"
assert latex(x | y, symbol_names={x: "x_i", y: "y_i"}) == r"x_i \vee y_i"
assert latex(x | y | z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \
r"x_i \vee y_i \vee z_i"
assert latex((x & y) | z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \
r"z_i \vee \left(x_i \wedge y_i\right)"
assert latex(Implies(x, y), symbol_names={x: "x_i", y: "y_i"}) == \
r"x_i \Rightarrow y_i"
p = Symbol('p', positive=True)
assert latex(exp(-p)*log(p)) == r"e^{- p} \log{\left(p \right)}"
def test_latex_builtins():
assert latex(True) == r"\text{True}"
assert latex(False) == r"\text{False}"
assert latex(None) == r"\text{None}"
assert latex(true) == r"\text{True}"
assert latex(false) == r'\text{False}'
def test_latex_SingularityFunction():
assert latex(SingularityFunction(x, 4, 5)) == \
r"{\left\langle x - 4 \right\rangle}^{5}"
assert latex(SingularityFunction(x, -3, 4)) == \
r"{\left\langle x + 3 \right\rangle}^{4}"
assert latex(SingularityFunction(x, 0, 4)) == \
r"{\left\langle x \right\rangle}^{4}"
assert latex(SingularityFunction(x, a, n)) == \
r"{\left\langle - a + x \right\rangle}^{n}"
assert latex(SingularityFunction(x, 4, -2)) == \
r"{\left\langle x - 4 \right\rangle}^{-2}"
assert latex(SingularityFunction(x, 4, -1)) == \
r"{\left\langle x - 4 \right\rangle}^{-1}"
def test_latex_cycle():
assert latex(Cycle(1, 2, 4)) == r"\left( 1\; 2\; 4\right)"
assert latex(Cycle(1, 2)(4, 5, 6)) == \
r"\left( 1\; 2\right)\left( 4\; 5\; 6\right)"
assert latex(Cycle()) == r"\left( \right)"
def test_latex_permutation():
assert latex(Permutation(1, 2, 4)) == r"\left( 1\; 2\; 4\right)"
assert latex(Permutation(1, 2)(4, 5, 6)) == \
r"\left( 1\; 2\right)\left( 4\; 5\; 6\right)"
assert latex(Permutation()) == r"\left( \right)"
assert latex(Permutation(2, 4)*Permutation(5)) == \
r"\left( 2\; 4\right)\left( 5\right)"
assert latex(Permutation(5)) == r"\left( 5\right)"
def test_latex_Float():
assert latex(Float(1.0e100)) == r"1.0 \cdot 10^{100}"
assert latex(Float(1.0e-100)) == r"1.0 \cdot 10^{-100}"
assert latex(Float(1.0e-100), mul_symbol="times") == \
r"1.0 \times 10^{-100}"
assert latex(1.0*oo) == r"\infty"
assert latex(-1.0*oo) == r"- \infty"
def test_latex_vector_expressions():
A = CoordSys3D('A')
assert latex(Cross(A.i, A.j*A.x*3+A.k)) == \
r"\mathbf{\hat{i}_{A}} \times \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}} + \mathbf{\hat{k}_{A}}\right)"
assert latex(Cross(A.i, A.j)) == \
r"\mathbf{\hat{i}_{A}} \times \mathbf{\hat{j}_{A}}"
assert latex(x*Cross(A.i, A.j)) == \
r"x \left(\mathbf{\hat{i}_{A}} \times \mathbf{\hat{j}_{A}}\right)"
assert latex(Cross(x*A.i, A.j)) == \
r'- \mathbf{\hat{j}_{A}} \times \left((x)\mathbf{\hat{i}_{A}}\right)'
assert latex(Curl(3*A.x*A.j)) == \
r"\nabla\times \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)"
assert latex(Curl(3*A.x*A.j+A.i)) == \
r"\nabla\times \left(\mathbf{\hat{i}_{A}} + (3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)"
assert latex(Curl(3*x*A.x*A.j)) == \
r"\nabla\times \left((3 \mathbf{{x}_{A}} x)\mathbf{\hat{j}_{A}}\right)"
assert latex(x*Curl(3*A.x*A.j)) == \
r"x \left(\nabla\times \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)\right)"
assert latex(Divergence(3*A.x*A.j+A.i)) == \
r"\nabla\cdot \left(\mathbf{\hat{i}_{A}} + (3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)"
assert latex(Divergence(3*A.x*A.j)) == \
r"\nabla\cdot \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)"
assert latex(x*Divergence(3*A.x*A.j)) == \
r"x \left(\nabla\cdot \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}}\right)\right)"
assert latex(Dot(A.i, A.j*A.x*3+A.k)) == \
r"\mathbf{\hat{i}_{A}} \cdot \left((3 \mathbf{{x}_{A}})\mathbf{\hat{j}_{A}} + \mathbf{\hat{k}_{A}}\right)"
assert latex(Dot(A.i, A.j)) == \
r"\mathbf{\hat{i}_{A}} \cdot \mathbf{\hat{j}_{A}}"
assert latex(Dot(x*A.i, A.j)) == \
r"\mathbf{\hat{j}_{A}} \cdot \left((x)\mathbf{\hat{i}_{A}}\right)"
assert latex(x*Dot(A.i, A.j)) == \
r"x \left(\mathbf{\hat{i}_{A}} \cdot \mathbf{\hat{j}_{A}}\right)"
assert latex(Gradient(A.x)) == r"\nabla \mathbf{{x}_{A}}"
assert latex(Gradient(A.x + 3*A.y)) == \
r"\nabla \left(\mathbf{{x}_{A}} + 3 \mathbf{{y}_{A}}\right)"
assert latex(x*Gradient(A.x)) == r"x \left(\nabla \mathbf{{x}_{A}}\right)"
assert latex(Gradient(x*A.x)) == r"\nabla \left(\mathbf{{x}_{A}} x\right)"
assert latex(Laplacian(A.x)) == r"\triangle \mathbf{{x}_{A}}"
assert latex(Laplacian(A.x + 3*A.y)) == \
r"\triangle \left(\mathbf{{x}_{A}} + 3 \mathbf{{y}_{A}}\right)"
assert latex(x*Laplacian(A.x)) == r"x \left(\triangle \mathbf{{x}_{A}}\right)"
assert latex(Laplacian(x*A.x)) == r"\triangle \left(\mathbf{{x}_{A}} x\right)"
def test_latex_symbols():
Gamma, lmbda, rho = symbols('Gamma, lambda, rho')
tau, Tau, TAU, taU = symbols('tau, Tau, TAU, taU')
assert latex(tau) == r"\tau"
assert latex(Tau) == "T"
assert latex(TAU) == r"\tau"
assert latex(taU) == r"\tau"
# Check that all capitalized greek letters are handled explicitly
capitalized_letters = set(l.capitalize() for l in greek_letters_set)
assert len(capitalized_letters - set(tex_greek_dictionary.keys())) == 0
assert latex(Gamma + lmbda) == r"\Gamma + \lambda"
assert latex(Gamma * lmbda) == r"\Gamma \lambda"
assert latex(Symbol('q1')) == r"q_{1}"
assert latex(Symbol('q21')) == r"q_{21}"
assert latex(Symbol('epsilon0')) == r"\epsilon_{0}"
assert latex(Symbol('omega1')) == r"\omega_{1}"
assert latex(Symbol('91')) == r"91"
assert latex(Symbol('alpha_new')) == r"\alpha_{new}"
assert latex(Symbol('C^orig')) == r"C^{orig}"
assert latex(Symbol('x^alpha')) == r"x^{\alpha}"
assert latex(Symbol('beta^alpha')) == r"\beta^{\alpha}"
assert latex(Symbol('e^Alpha')) == r"e^{A}"
assert latex(Symbol('omega_alpha^beta')) == r"\omega^{\beta}_{\alpha}"
assert latex(Symbol('omega') ** Symbol('beta')) == r"\omega^{\beta}"
@XFAIL
def test_latex_symbols_failing():
rho, mass, volume = symbols('rho, mass, volume')
assert latex(
volume * rho == mass) == r"\rho \mathrm{volume} = \mathrm{mass}"
assert latex(volume / mass * rho == 1) == \
r"\rho \mathrm{volume} {\mathrm{mass}}^{(-1)} = 1"
assert latex(mass**3 * volume**3) == \
r"{\mathrm{mass}}^{3} \cdot {\mathrm{volume}}^{3}"
def test_latex_functions():
assert latex(exp(x)) == "e^{x}"
assert latex(exp(1) + exp(2)) == "e + e^{2}"
f = Function('f')
assert latex(f(x)) == r'f{\left(x \right)}'
assert latex(f) == r'f'
g = Function('g')
assert latex(g(x, y)) == r'g{\left(x,y \right)}'
assert latex(g) == r'g'
h = Function('h')
assert latex(h(x, y, z)) == r'h{\left(x,y,z \right)}'
assert latex(h) == r'h'
Li = Function('Li')
assert latex(Li) == r'\operatorname{Li}'
assert latex(Li(x)) == r'\operatorname{Li}{\left(x \right)}'
mybeta = Function('beta')
# not to be confused with the beta function
assert latex(mybeta(x, y, z)) == r"\beta{\left(x,y,z \right)}"
assert latex(beta(x, y)) == r'\operatorname{B}\left(x, y\right)'
assert latex(beta(x, y)**2) == r'\operatorname{B}^{2}\left(x, y\right)'
assert latex(mybeta(x)) == r"\beta{\left(x \right)}"
assert latex(mybeta) == r"\beta"
g = Function('gamma')
# not to be confused with the gamma function
assert latex(g(x, y, z)) == r"\gamma{\left(x,y,z \right)}"
assert latex(g(x)) == r"\gamma{\left(x \right)}"
assert latex(g) == r"\gamma"
a1 = Function('a_1')
assert latex(a1) == r"\operatorname{a_{1}}"
assert latex(a1(x)) == r"\operatorname{a_{1}}{\left(x \right)}"
# issue 5868
omega1 = Function('omega1')
assert latex(omega1) == r"\omega_{1}"
assert latex(omega1(x)) == r"\omega_{1}{\left(x \right)}"
assert latex(sin(x)) == r"\sin{\left(x \right)}"
assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
assert latex(sin(2*x**2), fold_func_brackets=True) == \
r"\sin {2 x^{2}}"
assert latex(sin(x**2), fold_func_brackets=True) == \
r"\sin {x^{2}}"
assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left(x \right)}"
assert latex(asin(x)**2, inv_trig_style="full") == \
r"\arcsin^{2}{\left(x \right)}"
assert latex(asin(x)**2, inv_trig_style="power") == \
r"\sin^{-1}{\left(x \right)}^{2}"
assert latex(asin(x**2), inv_trig_style="power",
fold_func_brackets=True) == \
r"\sin^{-1} {x^{2}}"
assert latex(acsc(x), inv_trig_style="full") == \
r"\operatorname{arccsc}{\left(x \right)}"
assert latex(factorial(k)) == r"k!"
assert latex(factorial(-k)) == r"\left(- k\right)!"
assert latex(factorial(k)**2) == r"k!^{2}"
assert latex(subfactorial(k)) == r"!k"
assert latex(subfactorial(-k)) == r"!\left(- k\right)"
assert latex(subfactorial(k)**2) == r"\left(!k\right)^{2}"
assert latex(factorial2(k)) == r"k!!"
assert latex(factorial2(-k)) == r"\left(- k\right)!!"
assert latex(factorial2(k)**2) == r"k!!^{2}"
assert latex(binomial(2, k)) == r"{\binom{2}{k}}"
assert latex(binomial(2, k)**2) == r"{\binom{2}{k}}^{2}"
assert latex(FallingFactorial(3, k)) == r"{\left(3\right)}_{k}"
assert latex(RisingFactorial(3, k)) == r"{3}^{\left(k\right)}"
assert latex(floor(x)) == r"\left\lfloor{x}\right\rfloor"
assert latex(ceiling(x)) == r"\left\lceil{x}\right\rceil"
assert latex(floor(x)**2) == r"\left\lfloor{x}\right\rfloor^{2}"
assert latex(ceiling(x)**2) == r"\left\lceil{x}\right\rceil^{2}"
assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)"
assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}"
assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)"
assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}"
assert latex(Abs(x)) == r"\left|{x}\right|"
assert latex(Abs(x)**2) == r"\left|{x}\right|^{2}"
assert latex(re(x)) == r"\operatorname{re}{\left(x\right)}"
assert latex(re(x + y)) == \
r"\operatorname{re}{\left(x\right)} + \operatorname{re}{\left(y\right)}"
assert latex(im(x)) == r"\operatorname{im}{\left(x\right)}"
assert latex(conjugate(x)) == r"\overline{x}"
assert latex(conjugate(x)**2) == r"\overline{x}^{2}"
assert latex(conjugate(x**2)) == r"\overline{x}^{2}"
assert latex(gamma(x)) == r"\Gamma\left(x\right)"
w = Wild('w')
assert latex(gamma(w)) == r"\Gamma\left(w\right)"
assert latex(Order(x)) == r"O\left(x\right)"
assert latex(Order(x, x)) == r"O\left(x\right)"
assert latex(Order(x, (x, 0))) == r"O\left(x\right)"
assert latex(Order(x, (x, oo))) == r"O\left(x; x\rightarrow \infty\right)"
assert latex(Order(x - y, (x, y))) == \
r"O\left(x - y; x\rightarrow y\right)"
assert latex(Order(x, x, y)) == \
r"O\left(x; \left( x, \ y\right)\rightarrow \left( 0, \ 0\right)\right)"
assert latex(Order(x, x, y)) == \
r"O\left(x; \left( x, \ y\right)\rightarrow \left( 0, \ 0\right)\right)"
assert latex(Order(x, (x, oo), (y, oo))) == \
r"O\left(x; \left( x, \ y\right)\rightarrow \left( \infty, \ \infty\right)\right)"
assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
assert latex(lowergamma(x, y)**2) == r'\gamma^{2}\left(x, y\right)'
assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'
assert latex(uppergamma(x, y)**2) == r'\Gamma^{2}\left(x, y\right)'
assert latex(cot(x)) == r'\cot{\left(x \right)}'
assert latex(coth(x)) == r'\coth{\left(x \right)}'
assert latex(re(x)) == r'\operatorname{re}{\left(x\right)}'
assert latex(im(x)) == r'\operatorname{im}{\left(x\right)}'
assert latex(root(x, y)) == r'x^{\frac{1}{y}}'
assert latex(arg(x)) == r'\arg{\left(x \right)}'
assert latex(zeta(x)) == r'\zeta\left(x\right)'
assert latex(zeta(x)) == r"\zeta\left(x\right)"
assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)"
assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)"
assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)"
assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
assert latex(
polylog(x, y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)"
assert latex(elliptic_k(z)) == r"K\left(z\right)"
assert latex(elliptic_k(z)**2) == r"K^{2}\left(z\right)"
assert latex(elliptic_f(x, y)) == r"F\left(x\middle| y\right)"
assert latex(elliptic_f(x, y)**2) == r"F^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)) == r"E\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)**2) == r"E^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(z)) == r"E\left(z\right)"
assert latex(elliptic_e(z)**2) == r"E^{2}\left(z\right)"
assert latex(elliptic_pi(x, y, z)) == r"\Pi\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y, z)**2) == \
r"\Pi^{2}\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y)) == r"\Pi\left(x\middle| y\right)"
assert latex(elliptic_pi(x, y)**2) == r"\Pi^{2}\left(x\middle| y\right)"
assert latex(Ei(x)) == r'\operatorname{Ei}{\left(x \right)}'
assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left(x \right)}'
assert latex(expint(x, y)) == r'\operatorname{E}_{x}\left(y\right)'
assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)'
assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left(x \right)}'
assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left(x \right)}'
assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left(x \right)}'
assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}\left(x\right)'
assert latex(Chi(x)) == r'\operatorname{Chi}\left(x\right)'
assert latex(jacobi(n, a, b, x)) == \
r'P_{n}^{\left(a,b\right)}\left(x\right)'
assert latex(jacobi(n, a, b, x)**2) == \
r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}'
assert latex(gegenbauer(n, a, x)) == \
r'C_{n}^{\left(a\right)}\left(x\right)'
assert latex(gegenbauer(n, a, x)**2) == \
r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)'
assert latex(chebyshevt(n, x)**2) == \
r'\left(T_{n}\left(x\right)\right)^{2}'
assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)'
assert latex(chebyshevu(n, x)**2) == \
r'\left(U_{n}\left(x\right)\right)^{2}'
assert latex(legendre(n, x)) == r'P_{n}\left(x\right)'
assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}'
assert latex(assoc_legendre(n, a, x)) == \
r'P_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_legendre(n, a, x)**2) == \
r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)'
assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}'
assert latex(assoc_laguerre(n, a, x)) == \
r'L_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_laguerre(n, a, x)**2) == \
r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(hermite(n, x)) == r'H_{n}\left(x\right)'
assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}'
theta = Symbol("theta", real=True)
phi = Symbol("phi", real=True)
assert latex(Ynm(n, m, theta, phi)) == r'Y_{n}^{m}\left(\theta,\phi\right)'
assert latex(Ynm(n, m, theta, phi)**3) == \
r'\left(Y_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
assert latex(Znm(n, m, theta, phi)) == r'Z_{n}^{m}\left(\theta,\phi\right)'
assert latex(Znm(n, m, theta, phi)**3) == \
r'\left(Z_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
# Test latex printing of function names with "_"
assert latex(polar_lift(0)) == \
r"\operatorname{polar\_lift}{\left(0 \right)}"
assert latex(polar_lift(0)**3) == \
r"\operatorname{polar\_lift}^{3}{\left(0 \right)}"
assert latex(totient(n)) == r'\phi\left(n\right)'
assert latex(totient(n) ** 2) == r'\left(\phi\left(n\right)\right)^{2}'
assert latex(reduced_totient(n)) == r'\lambda\left(n\right)'
assert latex(reduced_totient(n) ** 2) == \
r'\left(\lambda\left(n\right)\right)^{2}'
assert latex(divisor_sigma(x)) == r"\sigma\left(x\right)"
assert latex(divisor_sigma(x)**2) == r"\sigma^{2}\left(x\right)"
assert latex(divisor_sigma(x, y)) == r"\sigma_y\left(x\right)"
assert latex(divisor_sigma(x, y)**2) == r"\sigma^{2}_y\left(x\right)"
assert latex(udivisor_sigma(x)) == r"\sigma^*\left(x\right)"
assert latex(udivisor_sigma(x)**2) == r"\sigma^*^{2}\left(x\right)"
assert latex(udivisor_sigma(x, y)) == r"\sigma^*_y\left(x\right)"
assert latex(udivisor_sigma(x, y)**2) == r"\sigma^*^{2}_y\left(x\right)"
assert latex(primenu(n)) == r'\nu\left(n\right)'
assert latex(primenu(n) ** 2) == r'\left(\nu\left(n\right)\right)^{2}'
assert latex(primeomega(n)) == r'\Omega\left(n\right)'
assert latex(primeomega(n) ** 2) == \
r'\left(\Omega\left(n\right)\right)^{2}'
assert latex(Mod(x, 7)) == r'x\bmod{7}'
assert latex(Mod(x + 1, 7)) == r'\left(x + 1\right)\bmod{7}'
assert latex(Mod(2 * x, 7)) == r'2 x\bmod{7}'
assert latex(Mod(x, 7) + 1) == r'\left(x\bmod{7}\right) + 1'
assert latex(2 * Mod(x, 7)) == r'2 \left(x\bmod{7}\right)'
# some unknown function name should get rendered with \operatorname
fjlkd = Function('fjlkd')
assert latex(fjlkd(x)) == r'\operatorname{fjlkd}{\left(x \right)}'
# even when it is referred to without an argument
assert latex(fjlkd) == r'\operatorname{fjlkd}'
# test that notation passes to subclasses of the same name only
def test_function_subclass_different_name():
class mygamma(gamma):
pass
assert latex(mygamma) == r"\operatorname{mygamma}"
assert latex(mygamma(x)) == r"\operatorname{mygamma}{\left(x \right)}"
def test_hyper_printing():
from sympy import pi
from sympy.abc import x, z
assert latex(meijerg(Tuple(pi, pi, x), Tuple(1),
(0, 1), Tuple(1, 2, 3/pi), z)) == \
r'{G_{4, 5}^{2, 3}\left(\begin{matrix} \pi, \pi, x & 1 \\0, 1 & 1, 2, '\
r'\frac{3}{\pi} \end{matrix} \middle| {z} \right)}'
assert latex(meijerg(Tuple(), Tuple(1), (0,), Tuple(), z)) == \
r'{G_{1, 1}^{1, 0}\left(\begin{matrix} & 1 \\0 & \end{matrix} \middle| {z} \right)}'
assert latex(hyper((x, 2), (3,), z)) == \
r'{{}_{2}F_{1}\left(\begin{matrix} x, 2 ' \
r'\\ 3 \end{matrix}\middle| {z} \right)}'
assert latex(hyper(Tuple(), Tuple(1), z)) == \
r'{{}_{0}F_{1}\left(\begin{matrix} ' \
r'\\ 1 \end{matrix}\middle| {z} \right)}'
def test_latex_bessel():
from sympy.functions.special.bessel import (besselj, bessely, besseli,
besselk, hankel1, hankel2,
jn, yn, hn1, hn2)
from sympy.abc import z
assert latex(besselj(n, z**2)**k) == r'J^{k}_{n}\left(z^{2}\right)'
assert latex(bessely(n, z)) == r'Y_{n}\left(z\right)'
assert latex(besseli(n, z)) == r'I_{n}\left(z\right)'
assert latex(besselk(n, z)) == r'K_{n}\left(z\right)'
assert latex(hankel1(n, z**2)**2) == \
r'\left(H^{(1)}_{n}\left(z^{2}\right)\right)^{2}'
assert latex(hankel2(n, z)) == r'H^{(2)}_{n}\left(z\right)'
assert latex(jn(n, z)) == r'j_{n}\left(z\right)'
assert latex(yn(n, z)) == r'y_{n}\left(z\right)'
assert latex(hn1(n, z)) == r'h^{(1)}_{n}\left(z\right)'
assert latex(hn2(n, z)) == r'h^{(2)}_{n}\left(z\right)'
def test_latex_fresnel():
from sympy.functions.special.error_functions import (fresnels, fresnelc)
from sympy.abc import z
assert latex(fresnels(z)) == r'S\left(z\right)'
assert latex(fresnelc(z)) == r'C\left(z\right)'
assert latex(fresnels(z)**2) == r'S^{2}\left(z\right)'
assert latex(fresnelc(z)**2) == r'C^{2}\left(z\right)'
def test_latex_brackets():
assert latex((-1)**x) == r"\left(-1\right)^{x}"
def test_latex_indexed():
Psi_symbol = Symbol('Psi_0', complex=True, real=False)
Psi_indexed = IndexedBase(Symbol('Psi', complex=True, real=False))
symbol_latex = latex(Psi_symbol * conjugate(Psi_symbol))
indexed_latex = latex(Psi_indexed[0] * conjugate(Psi_indexed[0]))
# \\overline{{\\Psi}_{0}} {\\Psi}_{0} vs. \\Psi_{0} \\overline{\\Psi_{0}}
assert symbol_latex == '\\Psi_{0} \\overline{\\Psi_{0}}'
assert indexed_latex == '\\overline{{\\Psi}_{0}} {\\Psi}_{0}'
# Symbol('gamma') gives r'\gamma'
assert latex(Indexed('x1', Symbol('i'))) == '{x_{1}}_{i}'
assert latex(IndexedBase('gamma')) == r'\gamma'
assert latex(IndexedBase('a b')) == 'a b'
assert latex(IndexedBase('a_b')) == 'a_{b}'
def test_latex_derivatives():
# regular "d" for ordinary derivatives
assert latex(diff(x**3, x, evaluate=False)) == \
r"\frac{d}{d x} x^{3}"
assert latex(diff(sin(x) + x**2, x, evaluate=False)) == \
r"\frac{d}{d x} \left(x^{2} + \sin{\left(x \right)}\right)"
assert latex(diff(diff(sin(x) + x**2, x, evaluate=False), evaluate=False))\
== \
r"\frac{d^{2}}{d x^{2}} \left(x^{2} + \sin{\left(x \right)}\right)"
assert latex(diff(diff(diff(sin(x) + x**2, x, evaluate=False), evaluate=False), evaluate=False)) == \
r"\frac{d^{3}}{d x^{3}} \left(x^{2} + \sin{\left(x \right)}\right)"
# \partial for partial derivatives
assert latex(diff(sin(x * y), x, evaluate=False)) == \
r"\frac{\partial}{\partial x} \sin{\left(x y \right)}"
assert latex(diff(sin(x * y) + x**2, x, evaluate=False)) == \
r"\frac{\partial}{\partial x} \left(x^{2} + \sin{\left(x y \right)}\right)"
assert latex(diff(diff(sin(x*y) + x**2, x, evaluate=False), x, evaluate=False)) == \
r"\frac{\partial^{2}}{\partial x^{2}} \left(x^{2} + \sin{\left(x y \right)}\right)"
assert latex(diff(diff(diff(sin(x*y) + x**2, x, evaluate=False), x, evaluate=False), x, evaluate=False)) == \
r"\frac{\partial^{3}}{\partial x^{3}} \left(x^{2} + \sin{\left(x y \right)}\right)"
# mixed partial derivatives
f = Function("f")
assert latex(diff(diff(f(x, y), x, evaluate=False), y, evaluate=False)) == \
r"\frac{\partial^{2}}{\partial y\partial x} " + latex(f(x, y))
assert latex(diff(diff(diff(f(x, y), x, evaluate=False), x, evaluate=False), y, evaluate=False)) == \
r"\frac{\partial^{3}}{\partial y\partial x^{2}} " + latex(f(x, y))
# use ordinary d when one of the variables has been integrated out
assert latex(diff(Integral(exp(-x*y), (x, 0, oo)), y, evaluate=False)) == \
r"\frac{d}{d y} \int\limits_{0}^{\infty} e^{- x y}\, dx"
# Derivative wrapped in power:
assert latex(diff(x, x, evaluate=False)**2) == \
r"\left(\frac{d}{d x} x\right)^{2}"
assert latex(diff(f(x), x)**2) == \
r"\left(\frac{d}{d x} f{\left(x \right)}\right)^{2}"
assert latex(diff(f(x), (x, n))) == \
r"\frac{d^{n}}{d x^{n}} f{\left(x \right)}"
def test_latex_subs():
assert latex(Subs(x*y, (
x, y), (1, 2))) == r'\left. x y \right|_{\substack{ x=1\\ y=2 }}'
def test_latex_integrals():
assert latex(Integral(log(x), x)) == r"\int \log{\left(x \right)}\, dx"
assert latex(Integral(x**2, (x, 0, 1))) == \
r"\int\limits_{0}^{1} x^{2}\, dx"
assert latex(Integral(x**2, (x, 10, 20))) == \
r"\int\limits_{10}^{20} x^{2}\, dx"
assert latex(Integral(y*x**2, (x, 0, 1), y)) == \
r"\int\int\limits_{0}^{1} x^{2} y\, dx\, dy"
assert latex(Integral(y*x**2, (x, 0, 1), y), mode='equation*') == \
r"\begin{equation*}\int\int\limits_{0}^{1} x^{2} y\, dx\, dy\end{equation*}"
assert latex(Integral(y*x**2, (x, 0, 1), y), mode='equation*', itex=True) \
== r"$$\int\int_{0}^{1} x^{2} y\, dx\, dy$$"
assert latex(Integral(x, (x, 0))) == r"\int\limits^{0} x\, dx"
assert latex(Integral(x*y, x, y)) == r"\iint x y\, dx\, dy"
assert latex(Integral(x*y*z, x, y, z)) == r"\iiint x y z\, dx\, dy\, dz"
assert latex(Integral(x*y*z*t, x, y, z, t)) == \
r"\iiiint t x y z\, dx\, dy\, dz\, dt"
assert latex(Integral(x, x, x, x, x, x, x)) == \
r"\int\int\int\int\int\int x\, dx\, dx\, dx\, dx\, dx\, dx"
assert latex(Integral(x, x, y, (z, 0, 1))) == \
r"\int\limits_{0}^{1}\int\int x\, dx\, dy\, dz"
# fix issue #10806
assert latex(Integral(z, z)**2) == r"\left(\int z\, dz\right)^{2}"
assert latex(Integral(x + z, z)) == r"\int \left(x + z\right)\, dz"
assert latex(Integral(x+z/2, z)) == \
r"\int \left(x + \frac{z}{2}\right)\, dz"
assert latex(Integral(x**y, z)) == r"\int x^{y}\, dz"
def test_latex_sets():
for s in (frozenset, set):
assert latex(s([x*y, x**2])) == r"\left\{x^{2}, x y\right\}"
assert latex(s(range(1, 6))) == r"\left\{1, 2, 3, 4, 5\right\}"
assert latex(s(range(1, 13))) == \
r"\left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right\}"
s = FiniteSet
assert latex(s(*[x*y, x**2])) == r"\left\{x^{2}, x y\right\}"
assert latex(s(*range(1, 6))) == r"\left\{1, 2, 3, 4, 5\right\}"
assert latex(s(*range(1, 13))) == \
r"\left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right\}"
def test_latex_SetExpr():
iv = Interval(1, 3)
se = SetExpr(iv)
assert latex(se) == r"SetExpr\left(\left[1, 3\right]\right)"
def test_latex_Range():
assert latex(Range(1, 51)) == \
r'\left\{1, 2, \ldots, 50\right\}'
assert latex(Range(1, 4)) == r'\left\{1, 2, 3\right\}'
assert latex(Range(0, 3, 1)) == r'\left\{0, 1, 2\right\}'
assert latex(Range(0, 30, 1)) == r'\left\{0, 1, \ldots, 29\right\}'
assert latex(Range(30, 1, -1)) == r'\left\{30, 29, \ldots, 2\right\}'
assert latex(Range(0, oo, 2)) == r'\left\{0, 2, \ldots\right\}'
assert latex(Range(oo, -2, -2)) == r'\left\{\ldots, 2, 0\right\}'
assert latex(Range(-2, -oo, -1)) == \
r'\left\{-2, -3, \ldots\right\}'
def test_latex_sequences():
s1 = SeqFormula(a**2, (0, oo))
s2 = SeqPer((1, 2))
latex_str = r'\left[0, 1, 4, 9, \ldots\right]'
assert latex(s1) == latex_str
latex_str = r'\left[1, 2, 1, 2, \ldots\right]'
assert latex(s2) == latex_str
s3 = SeqFormula(a**2, (0, 2))
s4 = SeqPer((1, 2), (0, 2))
latex_str = r'\left[0, 1, 4\right]'
assert latex(s3) == latex_str
latex_str = r'\left[1, 2, 1\right]'
assert latex(s4) == latex_str
s5 = SeqFormula(a**2, (-oo, 0))
s6 = SeqPer((1, 2), (-oo, 0))
latex_str = r'\left[\ldots, 9, 4, 1, 0\right]'
assert latex(s5) == latex_str
latex_str = r'\left[\ldots, 2, 1, 2, 1\right]'
assert latex(s6) == latex_str
latex_str = r'\left[1, 3, 5, 11, \ldots\right]'
assert latex(SeqAdd(s1, s2)) == latex_str
latex_str = r'\left[1, 3, 5\right]'
assert latex(SeqAdd(s3, s4)) == latex_str
latex_str = r'\left[\ldots, 11, 5, 3, 1\right]'
assert latex(SeqAdd(s5, s6)) == latex_str
latex_str = r'\left[0, 2, 4, 18, \ldots\right]'
assert latex(SeqMul(s1, s2)) == latex_str
latex_str = r'\left[0, 2, 4\right]'
assert latex(SeqMul(s3, s4)) == latex_str
latex_str = r'\left[\ldots, 18, 4, 2, 0\right]'
assert latex(SeqMul(s5, s6)) == latex_str
# Sequences with symbolic limits, issue 12629
s7 = SeqFormula(a**2, (a, 0, x))
latex_str = r'\left\{a^{2}\right\}_{a=0}^{x}'
assert latex(s7) == latex_str
b = Symbol('b')
s8 = SeqFormula(b*a**2, (a, 0, 2))
latex_str = r'\left[0, b, 4 b\right]'
assert latex(s8) == latex_str
def test_latex_FourierSeries():
latex_str = \
r'2 \sin{\left(x \right)} - \sin{\left(2 x \right)} + \frac{2 \sin{\left(3 x \right)}}{3} + \ldots'
assert latex(fourier_series(x, (x, -pi, pi))) == latex_str
def test_latex_FormalPowerSeries():
latex_str = r'\sum_{k=1}^{\infty} - \frac{\left(-1\right)^{- k} x^{k}}{k}'
assert latex(fps(log(1 + x))) == latex_str
def test_latex_intervals():
a = Symbol('a', real=True)
assert latex(Interval(0, 0)) == r"\left\{0\right\}"
assert latex(Interval(0, a)) == r"\left[0, a\right]"
assert latex(Interval(0, a, False, False)) == r"\left[0, a\right]"
assert latex(Interval(0, a, True, False)) == r"\left(0, a\right]"
assert latex(Interval(0, a, False, True)) == r"\left[0, a\right)"
assert latex(Interval(0, a, True, True)) == r"\left(0, a\right)"
def test_latex_AccumuBounds():
a = Symbol('a', real=True)
assert latex(AccumBounds(0, 1)) == r"\left\langle 0, 1\right\rangle"
assert latex(AccumBounds(0, a)) == r"\left\langle 0, a\right\rangle"
assert latex(AccumBounds(a + 1, a + 2)) == \
r"\left\langle a + 1, a + 2\right\rangle"
def test_latex_emptyset():
assert latex(S.EmptySet) == r"\emptyset"
def test_latex_commutator():
A = Operator('A')
B = Operator('B')
comm = Commutator(B, A)
assert latex(comm.doit()) == r"- (A B - B A)"
def test_latex_union():
assert latex(Union(Interval(0, 1), Interval(2, 3))) == \
r"\left[0, 1\right] \cup \left[2, 3\right]"
assert latex(Union(Interval(1, 1), Interval(2, 2), Interval(3, 4))) == \
r"\left\{1, 2\right\} \cup \left[3, 4\right]"
def test_latex_intersection():
assert latex(Intersection(Interval(0, 1), Interval(x, y))) == \
r"\left[0, 1\right] \cap \left[x, y\right]"
def test_latex_symmetric_difference():
assert latex(SymmetricDifference(Interval(2, 5), Interval(4, 7),
evaluate=False)) == \
r'\left[2, 5\right] \triangle \left[4, 7\right]'
def test_latex_Complement():
assert latex(Complement(S.Reals, S.Naturals)) == \
r"\mathbb{R} \setminus \mathbb{N}"
def test_latex_Complexes():
assert latex(S.Complexes) == r"\mathbb{C}"
def test_latex_productset():
line = Interval(0, 1)
bigline = Interval(0, 10)
fset = FiniteSet(1, 2, 3)
assert latex(line**2) == r"%s^{2}" % latex(line)
assert latex(line**10) == r"%s^{10}" % latex(line)
assert latex(line * bigline * fset) == r"%s \times %s \times %s" % (
latex(line), latex(bigline), latex(fset))
def test_latex_Naturals():
assert latex(S.Naturals) == r"\mathbb{N}"
def test_latex_Naturals0():
assert latex(S.Naturals0) == r"\mathbb{N}_0"
def test_latex_Integers():
assert latex(S.Integers) == r"\mathbb{Z}"
def test_latex_ImageSet():
x = Symbol('x')
assert latex(ImageSet(Lambda(x, x**2), S.Naturals)) == \
r"\left\{x^{2}\; |\; x \in \mathbb{N}\right\}"
y = Symbol('y')
imgset = ImageSet(Lambda((x, y), x + y), {1, 2, 3}, {3, 4})
assert latex(imgset) == \
r"\left\{x + y\; |\; x \in \left\{1, 2, 3\right\}, y \in \left\{3, 4\right\}\right\}"
def test_latex_ConditionSet():
x = Symbol('x')
assert latex(ConditionSet(x, Eq(x**2, 1), S.Reals)) == \
r"\left\{x \mid x \in \mathbb{R} \wedge x^{2} = 1 \right\}"
assert latex(ConditionSet(x, Eq(x**2, 1), S.UniversalSet)) == \
r"\left\{x \mid x^{2} = 1 \right\}"
def test_latex_ComplexRegion():
assert latex(ComplexRegion(Interval(3, 5)*Interval(4, 6))) == \
r"\left\{x + y i\; |\; x, y \in \left[3, 5\right] \times \left[4, 6\right] \right\}"
assert latex(ComplexRegion(Interval(0, 1)*Interval(0, 2*pi), polar=True)) == \
r"\left\{r \left(i \sin{\left(\theta \right)} + \cos{\left(\theta "\
r"\right)}\right)\; |\; r, \theta \in \left[0, 1\right] \times \left[0, 2 \pi\right) \right\}"
def test_latex_Contains():
x = Symbol('x')
assert latex(Contains(x, S.Naturals)) == r"x \in \mathbb{N}"
def test_latex_sum():
assert latex(Sum(x*y**2, (x, -2, 2), (y, -5, 5))) == \
r"\sum_{\substack{-2 \leq x \leq 2\\-5 \leq y \leq 5}} x y^{2}"
assert latex(Sum(x**2, (x, -2, 2))) == \
r"\sum_{x=-2}^{2} x^{2}"
assert latex(Sum(x**2 + y, (x, -2, 2))) == \
r"\sum_{x=-2}^{2} \left(x^{2} + y\right)"
assert latex(Sum(x**2 + y, (x, -2, 2))**2) == \
r"\left(\sum_{x=-2}^{2} \left(x^{2} + y\right)\right)^{2}"
def test_latex_product():
assert latex(Product(x*y**2, (x, -2, 2), (y, -5, 5))) == \
r"\prod_{\substack{-2 \leq x \leq 2\\-5 \leq y \leq 5}} x y^{2}"
assert latex(Product(x**2, (x, -2, 2))) == \
r"\prod_{x=-2}^{2} x^{2}"
assert latex(Product(x**2 + y, (x, -2, 2))) == \
r"\prod_{x=-2}^{2} \left(x^{2} + y\right)"
assert latex(Product(x, (x, -2, 2))**2) == \
r"\left(\prod_{x=-2}^{2} x\right)^{2}"
def test_latex_limits():
assert latex(Limit(x, x, oo)) == r"\lim_{x \to \infty} x"
# issue 8175
f = Function('f')
assert latex(Limit(f(x), x, 0)) == r"\lim_{x \to 0^+} f{\left(x \right)}"
assert latex(Limit(f(x), x, 0, "-")) == \
r"\lim_{x \to 0^-} f{\left(x \right)}"
# issue #10806
assert latex(Limit(f(x), x, 0)**2) == \
r"\left(\lim_{x \to 0^+} f{\left(x \right)}\right)^{2}"
# bi-directional limit
assert latex(Limit(f(x), x, 0, dir='+-')) == \
r"\lim_{x \to 0} f{\left(x \right)}"
def test_latex_log():
assert latex(log(x)) == r"\log{\left(x \right)}"
assert latex(ln(x)) == r"\log{\left(x \right)}"
assert latex(log(x), ln_notation=True) == r"\ln{\left(x \right)}"
assert latex(log(x)+log(y)) == \
r"\log{\left(x \right)} + \log{\left(y \right)}"
assert latex(log(x)+log(y), ln_notation=True) == \
r"\ln{\left(x \right)} + \ln{\left(y \right)}"
assert latex(pow(log(x), x)) == r"\log{\left(x \right)}^{x}"
assert latex(pow(log(x), x), ln_notation=True) == \
r"\ln{\left(x \right)}^{x}"
def test_issue_3568():
beta = Symbol(r'\beta')
y = beta + x
assert latex(y) in [r'\beta + x', r'x + \beta']
beta = Symbol(r'beta')
y = beta + x
assert latex(y) in [r'\beta + x', r'x + \beta']
def test_latex():
assert latex((2*tau)**Rational(7, 2)) == "8 \\sqrt{2} \\tau^{\\frac{7}{2}}"
assert latex((2*mu)**Rational(7, 2), mode='equation*') == \
"\\begin{equation*}8 \\sqrt{2} \\mu^{\\frac{7}{2}}\\end{equation*}"
assert latex((2*mu)**Rational(7, 2), mode='equation', itex=True) == \
"$$8 \\sqrt{2} \\mu^{\\frac{7}{2}}$$"
assert latex([2/x, y]) == r"\left[ \frac{2}{x}, \ y\right]"
def test_latex_dict():
d = {Rational(1): 1, x**2: 2, x: 3, x**3: 4}
assert latex(d) == \
r'\left\{ 1 : 1, \ x : 3, \ x^{2} : 2, \ x^{3} : 4\right\}'
D = Dict(d)
assert latex(D) == \
r'\left\{ 1 : 1, \ x : 3, \ x^{2} : 2, \ x^{3} : 4\right\}'
def test_latex_list():
ll = [Symbol('omega1'), Symbol('a'), Symbol('alpha')]
assert latex(ll) == r'\left[ \omega_{1}, \ a, \ \alpha\right]'
def test_latex_rational():
# tests issue 3973
assert latex(-Rational(1, 2)) == "- \\frac{1}{2}"
assert latex(Rational(-1, 2)) == "- \\frac{1}{2}"
assert latex(Rational(1, -2)) == "- \\frac{1}{2}"
assert latex(-Rational(-1, 2)) == "\\frac{1}{2}"
assert latex(-Rational(1, 2)*x) == "- \\frac{x}{2}"
assert latex(-Rational(1, 2)*x + Rational(-2, 3)*y) == \
"- \\frac{x}{2} - \\frac{2 y}{3}"
def test_latex_inverse():
# tests issue 4129
assert latex(1/x) == "\\frac{1}{x}"
assert latex(1/(x + y)) == "\\frac{1}{x + y}"
def test_latex_DiracDelta():
assert latex(DiracDelta(x)) == r"\delta\left(x\right)"
assert latex(DiracDelta(x)**2) == r"\left(\delta\left(x\right)\right)^{2}"
assert latex(DiracDelta(x, 0)) == r"\delta\left(x\right)"
assert latex(DiracDelta(x, 5)) == \
r"\delta^{\left( 5 \right)}\left( x \right)"
assert latex(DiracDelta(x, 5)**2) == \
r"\left(\delta^{\left( 5 \right)}\left( x \right)\right)^{2}"
def test_latex_Heaviside():
assert latex(Heaviside(x)) == r"\theta\left(x\right)"
assert latex(Heaviside(x)**2) == r"\left(\theta\left(x\right)\right)^{2}"
def test_latex_KroneckerDelta():
assert latex(KroneckerDelta(x, y)) == r"\delta_{x y}"
assert latex(KroneckerDelta(x, y + 1)) == r"\delta_{x, y + 1}"
# issue 6578
assert latex(KroneckerDelta(x + 1, y)) == r"\delta_{y, x + 1}"
assert latex(Pow(KroneckerDelta(x, y), 2, evaluate=False)) == \
r"\left(\delta_{x y}\right)^{2}"
def test_latex_LeviCivita():
assert latex(LeviCivita(x, y, z)) == r"\varepsilon_{x y z}"
assert latex(LeviCivita(x, y, z)**2) == \
r"\left(\varepsilon_{x y z}\right)^{2}"
assert latex(LeviCivita(x, y, z + 1)) == r"\varepsilon_{x, y, z + 1}"
assert latex(LeviCivita(x, y + 1, z)) == r"\varepsilon_{x, y + 1, z}"
assert latex(LeviCivita(x + 1, y, z)) == r"\varepsilon_{x + 1, y, z}"
def test_mode():
expr = x + y
assert latex(expr) == 'x + y'
assert latex(expr, mode='plain') == 'x + y'
assert latex(expr, mode='inline') == '$x + y$'
assert latex(
expr, mode='equation*') == '\\begin{equation*}x + y\\end{equation*}'
assert latex(
expr, mode='equation') == '\\begin{equation}x + y\\end{equation}'
raises(ValueError, lambda: latex(expr, mode='foo'))
def test_latex_Piecewise():
p = Piecewise((x, x < 1), (x**2, True))
assert latex(p) == "\\begin{cases} x & \\text{for}\\: x < 1 \\\\x^{2} &" \
" \\text{otherwise} \\end{cases}"
assert latex(p, itex=True) == \
"\\begin{cases} x & \\text{for}\\: x \\lt 1 \\\\x^{2} &" \
" \\text{otherwise} \\end{cases}"
p = Piecewise((x, x < 0), (0, x >= 0))
assert latex(p) == '\\begin{cases} x & \\text{for}\\: x < 0 \\\\0 &' \
' \\text{otherwise} \\end{cases}'
A, B = symbols("A B", commutative=False)
p = Piecewise((A**2, Eq(A, B)), (A*B, True))
s = r"\begin{cases} A^{2} & \text{for}\: A = B \\A B & \text{otherwise} \end{cases}"
assert latex(p) == s
assert latex(A*p) == r"A \left(%s\right)" % s
assert latex(p*A) == r"\left(%s\right) A" % s
assert latex(Piecewise((x, x < 1), (x**2, x < 2))) == \
'\\begin{cases} x & ' \
'\\text{for}\\: x < 1 \\\\x^{2} & \\text{for}\\: x < 2 \\end{cases}'
def test_latex_Matrix():
M = Matrix([[1 + x, y], [y, x - 1]])
assert latex(M) == \
r'\left[\begin{matrix}x + 1 & y\\y & x - 1\end{matrix}\right]'
assert latex(M, mode='inline') == \
r'$\left[\begin{smallmatrix}x + 1 & y\\' \
r'y & x - 1\end{smallmatrix}\right]$'
assert latex(M, mat_str='array') == \
r'\left[\begin{array}{cc}x + 1 & y\\y & x - 1\end{array}\right]'
assert latex(M, mat_str='bmatrix') == \
r'\left[\begin{bmatrix}x + 1 & y\\y & x - 1\end{bmatrix}\right]'
assert latex(M, mat_delim=None, mat_str='bmatrix') == \
r'\begin{bmatrix}x + 1 & y\\y & x - 1\end{bmatrix}'
M2 = Matrix(1, 11, range(11))
assert latex(M2) == \
r'\left[\begin{array}{ccccccccccc}' \
r'0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\end{array}\right]'
def test_latex_matrix_with_functions():
t = symbols('t')
theta1 = symbols('theta1', cls=Function)
M = Matrix([[sin(theta1(t)), cos(theta1(t))],
[cos(theta1(t).diff(t)), sin(theta1(t).diff(t))]])
expected = (r'\left[\begin{matrix}\sin{\left('
r'\theta_{1}{\left(t \right)} \right)} & '
r'\cos{\left(\theta_{1}{\left(t \right)} \right)'
r'}\\\cos{\left(\frac{d}{d t} \theta_{1}{\left(t '
r'\right)} \right)} & \sin{\left(\frac{d}{d t} '
r'\theta_{1}{\left(t \right)} \right'
r')}\end{matrix}\right]')
assert latex(M) == expected
def test_latex_NDimArray():
x, y, z, w = symbols("x y z w")
for ArrayType in (ImmutableDenseNDimArray, ImmutableSparseNDimArray,
MutableDenseNDimArray, MutableSparseNDimArray):
# Basic: scalar array
M = ArrayType(x)
assert latex(M) == "x"
M = ArrayType([[1 / x, y], [z, w]])
M1 = ArrayType([1 / x, y, z])
M2 = tensorproduct(M1, M)
M3 = tensorproduct(M, M)
assert latex(M) == \
'\\left[\\begin{matrix}\\frac{1}{x} & y\\\\z & w\\end{matrix}\\right]'
assert latex(M1) == \
"\\left[\\begin{matrix}\\frac{1}{x} & y & z\\end{matrix}\\right]"
assert latex(M2) == \
r"\left[\begin{matrix}" \
r"\left[\begin{matrix}\frac{1}{x^{2}} & \frac{y}{x}\\\frac{z}{x} & \frac{w}{x}\end{matrix}\right] & " \
r"\left[\begin{matrix}\frac{y}{x} & y^{2}\\y z & w y\end{matrix}\right] & " \
r"\left[\begin{matrix}\frac{z}{x} & y z\\z^{2} & w z\end{matrix}\right]" \
r"\end{matrix}\right]"
assert latex(M3) == \
r"""\left[\begin{matrix}"""\
r"""\left[\begin{matrix}\frac{1}{x^{2}} & \frac{y}{x}\\\frac{z}{x} & \frac{w}{x}\end{matrix}\right] & """\
r"""\left[\begin{matrix}\frac{y}{x} & y^{2}\\y z & w y\end{matrix}\right]\\"""\
r"""\left[\begin{matrix}\frac{z}{x} & y z\\z^{2} & w z\end{matrix}\right] & """\
r"""\left[\begin{matrix}\frac{w}{x} & w y\\w z & w^{2}\end{matrix}\right]"""\
r"""\end{matrix}\right]"""
Mrow = ArrayType([[x, y, 1/z]])
Mcolumn = ArrayType([[x], [y], [1/z]])
Mcol2 = ArrayType([Mcolumn.tolist()])
assert latex(Mrow) == \
r"\left[\left[\begin{matrix}x & y & \frac{1}{z}\end{matrix}\right]\right]"
assert latex(Mcolumn) == \
r"\left[\begin{matrix}x\\y\\\frac{1}{z}\end{matrix}\right]"
assert latex(Mcol2) == \
r'\left[\begin{matrix}\left[\begin{matrix}x\\y\\\frac{1}{z}\end{matrix}\right]\end{matrix}\right]'
def test_latex_mul_symbol():
assert latex(4*4**x, mul_symbol='times') == "4 \\times 4^{x}"
assert latex(4*4**x, mul_symbol='dot') == "4 \\cdot 4^{x}"
assert latex(4*4**x, mul_symbol='ldot') == r"4 \,.\, 4^{x}"
assert latex(4*x, mul_symbol='times') == "4 \\times x"
assert latex(4*x, mul_symbol='dot') == "4 \\cdot x"
assert latex(4*x, mul_symbol='ldot') == r"4 \,.\, x"
def test_latex_issue_4381():
y = 4*4**log(2)
assert latex(y) == r'4 \cdot 4^{\log{\left(2 \right)}}'
assert latex(1/y) == r'\frac{1}{4 \cdot 4^{\log{\left(2 \right)}}}'
def test_latex_issue_4576():
assert latex(Symbol("beta_13_2")) == r"\beta_{13 2}"
assert latex(Symbol("beta_132_20")) == r"\beta_{132 20}"
assert latex(Symbol("beta_13")) == r"\beta_{13}"
assert latex(Symbol("x_a_b")) == r"x_{a b}"
assert latex(Symbol("x_1_2_3")) == r"x_{1 2 3}"
assert latex(Symbol("x_a_b1")) == r"x_{a b1}"
assert latex(Symbol("x_a_1")) == r"x_{a 1}"
assert latex(Symbol("x_1_a")) == r"x_{1 a}"
assert latex(Symbol("x_1^aa")) == r"x^{aa}_{1}"
assert latex(Symbol("x_1__aa")) == r"x^{aa}_{1}"
assert latex(Symbol("x_11^a")) == r"x^{a}_{11}"
assert latex(Symbol("x_11__a")) == r"x^{a}_{11}"
assert latex(Symbol("x_a_a_a_a")) == r"x_{a a a a}"
assert latex(Symbol("x_a_a^a^a")) == r"x^{a a}_{a a}"
assert latex(Symbol("x_a_a__a__a")) == r"x^{a a}_{a a}"
assert latex(Symbol("alpha_11")) == r"\alpha_{11}"
assert latex(Symbol("alpha_11_11")) == r"\alpha_{11 11}"
assert latex(Symbol("alpha_alpha")) == r"\alpha_{\alpha}"
assert latex(Symbol("alpha^aleph")) == r"\alpha^{\aleph}"
assert latex(Symbol("alpha__aleph")) == r"\alpha^{\aleph}"
def test_latex_pow_fraction():
x = Symbol('x')
# Testing exp
assert 'e^{-x}' in latex(exp(-x)/2).replace(' ', '') # Remove Whitespace
# Testing e^{-x} in case future changes alter behavior of muls or fracs
# In particular current output is \frac{1}{2}e^{- x} but perhaps this will
# change to \frac{e^{-x}}{2}
# Testing general, non-exp, power
assert '3^{-x}' in latex(3**-x/2).replace(' ', '')
def test_noncommutative():
A, B, C = symbols('A,B,C', commutative=False)
assert latex(A*B*C**-1) == "A B C^{-1}"
assert latex(C**-1*A*B) == "C^{-1} A B"
assert latex(A*C**-1*B) == "A C^{-1} B"
def test_latex_order():
expr = x**3 + x**2*y + y**4 + 3*x*y**3
assert latex(expr, order='lex') == "x^{3} + x^{2} y + 3 x y^{3} + y^{4}"
assert latex(
expr, order='rev-lex') == "y^{4} + 3 x y^{3} + x^{2} y + x^{3}"
assert latex(expr, order='none') == "x^{3} + y^{4} + y x^{2} + 3 x y^{3}"
def test_latex_Lambda():
assert latex(Lambda(x, x + 1)) == \
r"\left( x \mapsto x + 1 \right)"
assert latex(Lambda((x, y), x + 1)) == \
r"\left( \left( x, \ y\right) \mapsto x + 1 \right)"
def test_latex_PolyElement():
Ruv, u, v = ring("u,v", ZZ)
Rxyz, x, y, z = ring("x,y,z", Ruv)
assert latex(x - x) == r"0"
assert latex(x - 1) == r"x - 1"
assert latex(x + 1) == r"x + 1"
assert latex((u**2 + 3*u*v + 1)*x**2*y + u + 1) == \
r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + u + 1"
assert latex((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x) == \
r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + \left(u + 1\right) x"
assert latex((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x + 1) == \
r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + \left(u + 1\right) x + 1"
assert latex((-u**2 + 3*u*v - 1)*x**2*y - (u + 1)*x - 1) == \
r"-\left({u}^{2} - 3 u v + 1\right) {x}^{2} y - \left(u + 1\right) x - 1"
assert latex(-(v**2 + v + 1)*x + 3*u*v + 1) == \
r"-\left({v}^{2} + v + 1\right) x + 3 u v + 1"
assert latex(-(v**2 + v + 1)*x - 3*u*v + 1) == \
r"-\left({v}^{2} + v + 1\right) x - 3 u v + 1"
def test_latex_FracElement():
Fuv, u, v = field("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Fuv)
assert latex(x - x) == r"0"
assert latex(x - 1) == r"x - 1"
assert latex(x + 1) == r"x + 1"
assert latex(x/3) == r"\frac{x}{3}"
assert latex(x/z) == r"\frac{x}{z}"
assert latex(x*y/z) == r"\frac{x y}{z}"
assert latex(x/(z*t)) == r"\frac{x}{z t}"
assert latex(x*y/(z*t)) == r"\frac{x y}{z t}"
assert latex((x - 1)/y) == r"\frac{x - 1}{y}"
assert latex((x + 1)/y) == r"\frac{x + 1}{y}"
assert latex((-x - 1)/y) == r"\frac{-x - 1}{y}"
assert latex((x + 1)/(y*z)) == r"\frac{x + 1}{y z}"
assert latex(-y/(x + 1)) == r"\frac{-y}{x + 1}"
assert latex(y*z/(x + 1)) == r"\frac{y z}{x + 1}"
assert latex(((u + 1)*x*y + 1)/((v - 1)*z - 1)) == \
r"\frac{\left(u + 1\right) x y + 1}{\left(v - 1\right) z - 1}"
assert latex(((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)) == \
r"\frac{\left(u + 1\right) x y + 1}{\left(v - 1\right) z - u v t - 1}"
def test_latex_Poly():
assert latex(Poly(x**2 + 2 * x, x)) == \
r"\operatorname{Poly}{\left( x^{2} + 2 x, x, domain=\mathbb{Z} \right)}"
assert latex(Poly(x/y, x)) == \
r"\operatorname{Poly}{\left( \frac{1}{y} x, x, domain=\mathbb{Z}\left(y\right) \right)}"
assert latex(Poly(2.0*x + y)) == \
r"\operatorname{Poly}{\left( 2.0 x + 1.0 y, x, y, domain=\mathbb{R} \right)}"
def test_latex_Poly_order():
assert latex(Poly([a, 1, b, 2, c, 3], x)) == \
'\\operatorname{Poly}{\\left( a x^{5} + x^{4} + b x^{3} + 2 x^{2} + c'\
' x + 3, x, domain=\\mathbb{Z}\\left[a, b, c\\right] \\right)}'
assert latex(Poly([a, 1, b+c, 2, 3], x)) == \
'\\operatorname{Poly}{\\left( a x^{4} + x^{3} + \\left(b + c\\right) '\
'x^{2} + 2 x + 3, x, domain=\\mathbb{Z}\\left[a, b, c\\right] \\right)}'
assert latex(Poly(a*x**3 + x**2*y - x*y - c*y**3 - b*x*y**2 + y - a*x + b,
(x, y))) == \
'\\operatorname{Poly}{\\left( a x^{3} + x^{2}y - b xy^{2} - xy - '\
'a x - c y^{3} + y + b, x, y, domain=\\mathbb{Z}\\left[a, b, c\\right] \\right)}'
def test_latex_ComplexRootOf():
assert latex(rootof(x**5 + x + 3, 0)) == \
r"\operatorname{CRootOf} {\left(x^{5} + x + 3, 0\right)}"
def test_latex_RootSum():
assert latex(RootSum(x**5 + x + 3, sin)) == \
r"\operatorname{RootSum} {\left(x^{5} + x + 3, \left( x \mapsto \sin{\left(x \right)} \right)\right)}"
def test_settings():
raises(TypeError, lambda: latex(x*y, method="garbage"))
def test_latex_numbers():
assert latex(catalan(n)) == r"C_{n}"
assert latex(catalan(n)**2) == r"C_{n}^{2}"
assert latex(bernoulli(n)) == r"B_{n}"
assert latex(bernoulli(n)**2) == r"B_{n}^{2}"
assert latex(bell(n)) == r"B_{n}"
assert latex(bell(n)**2) == r"B_{n}^{2}"
assert latex(fibonacci(n)) == r"F_{n}"
assert latex(fibonacci(n)**2) == r"F_{n}^{2}"
assert latex(lucas(n)) == r"L_{n}"
assert latex(lucas(n)**2) == r"L_{n}^{2}"
assert latex(tribonacci(n)) == r"T_{n}"
assert latex(tribonacci(n)**2) == r"T_{n}^{2}"
def test_latex_euler():
assert latex(euler(n)) == r"E_{n}"
assert latex(euler(n, x)) == r"E_{n}\left(x\right)"
assert latex(euler(n, x)**2) == r"E_{n}^{2}\left(x\right)"
def test_lamda():
assert latex(Symbol('lamda')) == r"\lambda"
assert latex(Symbol('Lamda')) == r"\Lambda"
def test_custom_symbol_names():
x = Symbol('x')
y = Symbol('y')
assert latex(x) == "x"
assert latex(x, symbol_names={x: "x_i"}) == "x_i"
assert latex(x + y, symbol_names={x: "x_i"}) == "x_i + y"
assert latex(x**2, symbol_names={x: "x_i"}) == "x_i^{2}"
assert latex(x + y, symbol_names={x: "x_i", y: "y_j"}) == "x_i + y_j"
def test_matAdd():
from sympy import MatrixSymbol
from sympy.printing.latex import LatexPrinter
C = MatrixSymbol('C', 5, 5)
B = MatrixSymbol('B', 5, 5)
l = LatexPrinter()
assert l._print(C - 2*B) in ['- 2 B + C', 'C -2 B']
assert l._print(C + 2*B) in ['2 B + C', 'C + 2 B']
assert l._print(B - 2*C) in ['B - 2 C', '- 2 C + B']
assert l._print(B + 2*C) in ['B + 2 C', '2 C + B']
def test_matMul():
from sympy import MatrixSymbol
from sympy.printing.latex import LatexPrinter
A = MatrixSymbol('A', 5, 5)
B = MatrixSymbol('B', 5, 5)
x = Symbol('x')
lp = LatexPrinter()
assert lp._print_MatMul(2*A) == '2 A'
assert lp._print_MatMul(2*x*A) == '2 x A'
assert lp._print_MatMul(-2*A) == '- 2 A'
assert lp._print_MatMul(1.5*A) == '1.5 A'
assert lp._print_MatMul(sqrt(2)*A) == r'\sqrt{2} A'
assert lp._print_MatMul(-sqrt(2)*A) == r'- \sqrt{2} A'
assert lp._print_MatMul(2*sqrt(2)*x*A) == r'2 \sqrt{2} x A'
assert lp._print_MatMul(-2*A*(A + 2*B)) in [r'- 2 A \left(A + 2 B\right)',
r'- 2 A \left(2 B + A\right)']
def test_latex_MatrixSlice():
from sympy.matrices.expressions import MatrixSymbol
assert latex(MatrixSymbol('X', 10, 10)[:5, 1:9:2]) == \
r'X\left[:5, 1:9:2\right]'
assert latex(MatrixSymbol('X', 10, 10)[5, :5:2]) == \
r'X\left[5, :5:2\right]'
def test_latex_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
from sympy.stats.rv import RandomDomain
X = Normal('x1', 0, 1)
assert latex(where(X > 0)) == r"\text{Domain: }0 < x_{1} \wedge x_{1} < \infty"
D = Die('d1', 6)
assert latex(where(D > 4)) == r"\text{Domain: }d_{1} = 5 \vee d_{1} = 6"
A = Exponential('a', 1)
B = Exponential('b', 1)
assert latex(
pspace(Tuple(A, B)).domain) == \
r"\text{Domain: }0 \leq a \wedge 0 \leq b \wedge a < \infty \wedge b < \infty"
assert latex(RandomDomain(FiniteSet(x), FiniteSet(1, 2))) == \
r'\text{Domain: }\left\{x\right\}\text{ in }\left\{1, 2\right\}'
def test_PrettyPoly():
from sympy.polys.domains import QQ
F = QQ.frac_field(x, y)
R = QQ[x, y]
assert latex(F.convert(x/(x + y))) == latex(x/(x + y))
assert latex(R.convert(x + y)) == latex(x + y)
def test_integral_transforms():
x = Symbol("x")
k = Symbol("k")
f = Function("f")
a = Symbol("a")
b = Symbol("b")
assert latex(MellinTransform(f(x), x, k)) == \
r"\mathcal{M}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseMellinTransform(f(k), k, x, a, b)) == \
r"\mathcal{M}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(LaplaceTransform(f(x), x, k)) == \
r"\mathcal{L}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseLaplaceTransform(f(k), k, x, (a, b))) == \
r"\mathcal{L}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(FourierTransform(f(x), x, k)) == \
r"\mathcal{F}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseFourierTransform(f(k), k, x)) == \
r"\mathcal{F}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(CosineTransform(f(x), x, k)) == \
r"\mathcal{COS}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseCosineTransform(f(k), k, x)) == \
r"\mathcal{COS}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(SineTransform(f(x), x, k)) == \
r"\mathcal{SIN}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseSineTransform(f(k), k, x)) == \
r"\mathcal{SIN}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
def test_PolynomialRingBase():
from sympy.polys.domains import QQ
assert latex(QQ.old_poly_ring(x, y)) == r"\mathbb{Q}\left[x, y\right]"
assert latex(QQ.old_poly_ring(x, y, order="ilex")) == \
r"S_<^{-1}\mathbb{Q}\left[x, y\right]"
def test_categories():
from sympy.categories import (Object, IdentityMorphism,
NamedMorphism, Category, Diagram,
DiagramGrid)
A1 = Object("A1")
A2 = Object("A2")
A3 = Object("A3")
f1 = NamedMorphism(A1, A2, "f1")
f2 = NamedMorphism(A2, A3, "f2")
id_A1 = IdentityMorphism(A1)
K1 = Category("K1")
assert latex(A1) == "A_{1}"
assert latex(f1) == "f_{1}:A_{1}\\rightarrow A_{2}"
assert latex(id_A1) == "id:A_{1}\\rightarrow A_{1}"
assert latex(f2*f1) == "f_{2}\\circ f_{1}:A_{1}\\rightarrow A_{3}"
assert latex(K1) == r"\mathbf{K_{1}}"
d = Diagram()
assert latex(d) == r"\emptyset"
d = Diagram({f1: "unique", f2: S.EmptySet})
assert latex(d) == r"\left\{ f_{2}\circ f_{1}:A_{1}" \
r"\rightarrow A_{3} : \emptyset, \ id:A_{1}\rightarrow " \
r"A_{1} : \emptyset, \ id:A_{2}\rightarrow A_{2} : " \
r"\emptyset, \ id:A_{3}\rightarrow A_{3} : \emptyset, " \
r"\ f_{1}:A_{1}\rightarrow A_{2} : \left\{unique\right\}, " \
r"\ f_{2}:A_{2}\rightarrow A_{3} : \emptyset\right\}"
d = Diagram({f1: "unique", f2: S.EmptySet}, {f2 * f1: "unique"})
assert latex(d) == r"\left\{ f_{2}\circ f_{1}:A_{1}" \
r"\rightarrow A_{3} : \emptyset, \ id:A_{1}\rightarrow " \
r"A_{1} : \emptyset, \ id:A_{2}\rightarrow A_{2} : " \
r"\emptyset, \ id:A_{3}\rightarrow A_{3} : \emptyset, " \
r"\ f_{1}:A_{1}\rightarrow A_{2} : \left\{unique\right\}," \
r" \ f_{2}:A_{2}\rightarrow A_{3} : \emptyset\right\}" \
r"\Longrightarrow \left\{ f_{2}\circ f_{1}:A_{1}" \
r"\rightarrow A_{3} : \left\{unique\right\}\right\}"
# A linear diagram.
A = Object("A")
B = Object("B")
C = Object("C")
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
d = Diagram([f, g])
grid = DiagramGrid(d)
assert latex(grid) == "\\begin{array}{cc}\n" \
"A & B \\\\\n" \
" & C \n" \
"\\end{array}\n"
def test_Modules():
from sympy.polys.domains import QQ
from sympy.polys.agca import homomorphism
R = QQ.old_poly_ring(x, y)
F = R.free_module(2)
M = F.submodule([x, y], [1, x**2])
assert latex(F) == r"{\mathbb{Q}\left[x, y\right]}^{2}"
assert latex(M) == \
r"\left\langle {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle"
I = R.ideal(x**2, y)
assert latex(I) == r"\left\langle {x^{2}},{y} \right\rangle"
Q = F / M
assert latex(Q) == \
r"\frac{{\mathbb{Q}\left[x, y\right]}^{2}}{\left\langle {\left[ {x},"\
r"{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle}"
assert latex(Q.submodule([1, x**3/2], [2, y])) == \
r"\left\langle {{\left[ {1},{\frac{x^{3}}{2}} \right]} + {\left"\
r"\langle {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} "\
r"\right\rangle}},{{\left[ {2},{y} \right]} + {\left\langle {\left[ "\
r"{x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle}} \right\rangle"
h = homomorphism(QQ.old_poly_ring(x).free_module(2),
QQ.old_poly_ring(x).free_module(2), [0, 0])
assert latex(h) == \
r"{\left[\begin{matrix}0 & 0\\0 & 0\end{matrix}\right]} : "\
r"{{\mathbb{Q}\left[x\right]}^{2}} \to {{\mathbb{Q}\left[x\right]}^{2}}"
def test_QuotientRing():
from sympy.polys.domains import QQ
R = QQ.old_poly_ring(x)/[x**2 + 1]
assert latex(R) == \
r"\frac{\mathbb{Q}\left[x\right]}{\left\langle {x^{2} + 1} \right\rangle}"
assert latex(R.one) == r"{1} + {\left\langle {x^{2} + 1} \right\rangle}"
def test_Tr():
#TODO: Handle indices
A, B = symbols('A B', commutative=False)
t = Tr(A*B)
assert latex(t) == r'\operatorname{tr}\left(A B\right)'
def test_Adjoint():
from sympy.matrices import MatrixSymbol, Adjoint, Inverse, Transpose
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert latex(Adjoint(X)) == r'X^{\dagger}'
assert latex(Adjoint(X + Y)) == r'\left(X + Y\right)^{\dagger}'
assert latex(Adjoint(X) + Adjoint(Y)) == r'X^{\dagger} + Y^{\dagger}'
assert latex(Adjoint(X*Y)) == r'\left(X Y\right)^{\dagger}'
assert latex(Adjoint(Y)*Adjoint(X)) == r'Y^{\dagger} X^{\dagger}'
assert latex(Adjoint(X**2)) == r'\left(X^{2}\right)^{\dagger}'
assert latex(Adjoint(X)**2) == r'\left(X^{\dagger}\right)^{2}'
assert latex(Adjoint(Inverse(X))) == r'\left(X^{-1}\right)^{\dagger}'
assert latex(Inverse(Adjoint(X))) == r'\left(X^{\dagger}\right)^{-1}'
assert latex(Adjoint(Transpose(X))) == r'\left(X^{T}\right)^{\dagger}'
assert latex(Transpose(Adjoint(X))) == r'\left(X^{\dagger}\right)^{T}'
assert latex(Transpose(Adjoint(X) + Y)) == r'\left(X^{\dagger} + Y\right)^{T}'
assert latex(Transpose(X)) == r'X^{T}'
assert latex(Transpose(X + Y)) == r'\left(X + Y\right)^{T}'
def test_Hadamard():
from sympy.matrices import MatrixSymbol, HadamardProduct
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert latex(HadamardProduct(X, Y*Y)) == r'X \circ Y^{2}'
assert latex(HadamardProduct(X, Y)*Y) == r'\left(X \circ Y\right) Y'
def test_ZeroMatrix():
from sympy import ZeroMatrix
assert latex(ZeroMatrix(1, 1)) == r"\mathbb{0}"
def test_Identity():
from sympy import Identity
assert latex(Identity(1)) == r"\mathbb{I}"
def test_boolean_args_order():
syms = symbols('a:f')
expr = And(*syms)
assert latex(expr) == 'a \\wedge b \\wedge c \\wedge d \\wedge e \\wedge f'
expr = Or(*syms)
assert latex(expr) == 'a \\vee b \\vee c \\vee d \\vee e \\vee f'
expr = Equivalent(*syms)
assert latex(expr) == \
'a \\Leftrightarrow b \\Leftrightarrow c \\Leftrightarrow d \\Leftrightarrow e \\Leftrightarrow f'
expr = Xor(*syms)
assert latex(expr) == \
'a \\veebar b \\veebar c \\veebar d \\veebar e \\veebar f'
def test_imaginary():
i = sqrt(-1)
assert latex(i) == r'i'
def test_builtins_without_args():
assert latex(sin) == r'\sin'
assert latex(cos) == r'\cos'
assert latex(tan) == r'\tan'
assert latex(log) == r'\log'
assert latex(Ei) == r'\operatorname{Ei}'
assert latex(zeta) == r'\zeta'
def test_latex_greek_functions():
# bug because capital greeks that have roman equivalents should not use
# \Alpha, \Beta, \Eta, etc.
s = Function('Alpha')
assert latex(s) == r'A'
assert latex(s(x)) == r'A{\left(x \right)}'
s = Function('Beta')
assert latex(s) == r'B'
s = Function('Eta')
assert latex(s) == r'H'
assert latex(s(x)) == r'H{\left(x \right)}'
# bug because sympy.core.numbers.Pi is special
p = Function('Pi')
# assert latex(p(x)) == r'\Pi{\left(x \right)}'
assert latex(p) == r'\Pi'
# bug because not all greeks are included
c = Function('chi')
assert latex(c(x)) == r'\chi{\left(x \right)}'
assert latex(c) == r'\chi'
def test_translate():
s = 'Alpha'
assert translate(s) == 'A'
s = 'Beta'
assert translate(s) == 'B'
s = 'Eta'
assert translate(s) == 'H'
s = 'omicron'
assert translate(s) == 'o'
s = 'Pi'
assert translate(s) == r'\Pi'
s = 'pi'
assert translate(s) == r'\pi'
s = 'LamdaHatDOT'
assert translate(s) == r'\dot{\hat{\Lambda}}'
def test_other_symbols():
from sympy.printing.latex import other_symbols
for s in other_symbols:
assert latex(symbols(s)) == "\\"+s
def test_modifiers():
# Test each modifier individually in the simplest case
# (with funny capitalizations)
assert latex(symbols("xMathring")) == r"\mathring{x}"
assert latex(symbols("xCheck")) == r"\check{x}"
assert latex(symbols("xBreve")) == r"\breve{x}"
assert latex(symbols("xAcute")) == r"\acute{x}"
assert latex(symbols("xGrave")) == r"\grave{x}"
assert latex(symbols("xTilde")) == r"\tilde{x}"
assert latex(symbols("xPrime")) == r"{x}'"
assert latex(symbols("xddDDot")) == r"\ddddot{x}"
assert latex(symbols("xDdDot")) == r"\dddot{x}"
assert latex(symbols("xDDot")) == r"\ddot{x}"
assert latex(symbols("xBold")) == r"\boldsymbol{x}"
assert latex(symbols("xnOrM")) == r"\left\|{x}\right\|"
assert latex(symbols("xAVG")) == r"\left\langle{x}\right\rangle"
assert latex(symbols("xHat")) == r"\hat{x}"
assert latex(symbols("xDot")) == r"\dot{x}"
assert latex(symbols("xBar")) == r"\bar{x}"
assert latex(symbols("xVec")) == r"\vec{x}"
assert latex(symbols("xAbs")) == r"\left|{x}\right|"
assert latex(symbols("xMag")) == r"\left|{x}\right|"
assert latex(symbols("xPrM")) == r"{x}'"
assert latex(symbols("xBM")) == r"\boldsymbol{x}"
# Test strings that are *only* the names of modifiers
assert latex(symbols("Mathring")) == r"Mathring"
assert latex(symbols("Check")) == r"Check"
assert latex(symbols("Breve")) == r"Breve"
assert latex(symbols("Acute")) == r"Acute"
assert latex(symbols("Grave")) == r"Grave"
assert latex(symbols("Tilde")) == r"Tilde"
assert latex(symbols("Prime")) == r"Prime"
assert latex(symbols("DDot")) == r"\dot{D}"
assert latex(symbols("Bold")) == r"Bold"
assert latex(symbols("NORm")) == r"NORm"
assert latex(symbols("AVG")) == r"AVG"
assert latex(symbols("Hat")) == r"Hat"
assert latex(symbols("Dot")) == r"Dot"
assert latex(symbols("Bar")) == r"Bar"
assert latex(symbols("Vec")) == r"Vec"
assert latex(symbols("Abs")) == r"Abs"
assert latex(symbols("Mag")) == r"Mag"
assert latex(symbols("PrM")) == r"PrM"
assert latex(symbols("BM")) == r"BM"
assert latex(symbols("hbar")) == r"\hbar"
# Check a few combinations
assert latex(symbols("xvecdot")) == r"\dot{\vec{x}}"
assert latex(symbols("xDotVec")) == r"\vec{\dot{x}}"
assert latex(symbols("xHATNorm")) == r"\left\|{\hat{x}}\right\|"
# Check a couple big, ugly combinations
assert latex(symbols('xMathringBm_yCheckPRM__zbreveAbs')) == \
r"\boldsymbol{\mathring{x}}^{\left|{\breve{z}}\right|}_{{\check{y}}'}"
assert latex(symbols('alphadothat_nVECDOT__tTildePrime')) == \
r"\hat{\dot{\alpha}}^{{\tilde{t}}'}_{\dot{\vec{n}}}"
def test_greek_symbols():
assert latex(Symbol('alpha')) == r'\alpha'
assert latex(Symbol('beta')) == r'\beta'
assert latex(Symbol('gamma')) == r'\gamma'
assert latex(Symbol('delta')) == r'\delta'
assert latex(Symbol('epsilon')) == r'\epsilon'
assert latex(Symbol('zeta')) == r'\zeta'
assert latex(Symbol('eta')) == r'\eta'
assert latex(Symbol('theta')) == r'\theta'
assert latex(Symbol('iota')) == r'\iota'
assert latex(Symbol('kappa')) == r'\kappa'
assert latex(Symbol('lambda')) == r'\lambda'
assert latex(Symbol('mu')) == r'\mu'
assert latex(Symbol('nu')) == r'\nu'
assert latex(Symbol('xi')) == r'\xi'
assert latex(Symbol('omicron')) == r'o'
assert latex(Symbol('pi')) == r'\pi'
assert latex(Symbol('rho')) == r'\rho'
assert latex(Symbol('sigma')) == r'\sigma'
assert latex(Symbol('tau')) == r'\tau'
assert latex(Symbol('upsilon')) == r'\upsilon'
assert latex(Symbol('phi')) == r'\phi'
assert latex(Symbol('chi')) == r'\chi'
assert latex(Symbol('psi')) == r'\psi'
assert latex(Symbol('omega')) == r'\omega'
assert latex(Symbol('Alpha')) == r'A'
assert latex(Symbol('Beta')) == r'B'
assert latex(Symbol('Gamma')) == r'\Gamma'
assert latex(Symbol('Delta')) == r'\Delta'
assert latex(Symbol('Epsilon')) == r'E'
assert latex(Symbol('Zeta')) == r'Z'
assert latex(Symbol('Eta')) == r'H'
assert latex(Symbol('Theta')) == r'\Theta'
assert latex(Symbol('Iota')) == r'I'
assert latex(Symbol('Kappa')) == r'K'
assert latex(Symbol('Lambda')) == r'\Lambda'
assert latex(Symbol('Mu')) == r'M'
assert latex(Symbol('Nu')) == r'N'
assert latex(Symbol('Xi')) == r'\Xi'
assert latex(Symbol('Omicron')) == r'O'
assert latex(Symbol('Pi')) == r'\Pi'
assert latex(Symbol('Rho')) == r'P'
assert latex(Symbol('Sigma')) == r'\Sigma'
assert latex(Symbol('Tau')) == r'T'
assert latex(Symbol('Upsilon')) == r'\Upsilon'
assert latex(Symbol('Phi')) == r'\Phi'
assert latex(Symbol('Chi')) == r'X'
assert latex(Symbol('Psi')) == r'\Psi'
assert latex(Symbol('Omega')) == r'\Omega'
assert latex(Symbol('varepsilon')) == r'\varepsilon'
assert latex(Symbol('varkappa')) == r'\varkappa'
assert latex(Symbol('varphi')) == r'\varphi'
assert latex(Symbol('varpi')) == r'\varpi'
assert latex(Symbol('varrho')) == r'\varrho'
assert latex(Symbol('varsigma')) == r'\varsigma'
assert latex(Symbol('vartheta')) == r'\vartheta'
@XFAIL
def test_builtin_without_args_mismatched_names():
assert latex(CosineTransform) == r'\mathcal{COS}'
def test_builtin_no_args():
assert latex(Chi) == r'\operatorname{Chi}'
assert latex(beta) == r'\operatorname{B}'
assert latex(gamma) == r'\Gamma'
assert latex(KroneckerDelta) == r'\delta'
assert latex(DiracDelta) == r'\delta'
assert latex(lowergamma) == r'\gamma'
def test_issue_6853():
p = Function('Pi')
assert latex(p(x)) == r"\Pi{\left(x \right)}"
def test_Mul():
e = Mul(-2, x + 1, evaluate=False)
assert latex(e) == r'- 2 \left(x + 1\right)'
e = Mul(2, x + 1, evaluate=False)
assert latex(e) == r'2 \left(x + 1\right)'
e = Mul(S.One/2, x + 1, evaluate=False)
assert latex(e) == r'\frac{x + 1}{2}'
e = Mul(y, x + 1, evaluate=False)
assert latex(e) == r'y \left(x + 1\right)'
e = Mul(-y, x + 1, evaluate=False)
assert latex(e) == r'- y \left(x + 1\right)'
e = Mul(-2, x + 1)
assert latex(e) == r'- 2 x - 2'
e = Mul(2, x + 1)
assert latex(e) == r'2 x + 2'
def test_Pow():
e = Pow(2, 2, evaluate=False)
assert latex(e) == r'2^{2}'
assert latex(x**(Rational(-1, 3))) == r'\frac{1}{\sqrt[3]{x}}'
x2 = Symbol(r'x^2')
assert latex(x2**2) == r'\left(x^{2}\right)^{2}'
def test_issue_7180():
assert latex(Equivalent(x, y)) == r"x \Leftrightarrow y"
assert latex(Not(Equivalent(x, y))) == r"x \not\Leftrightarrow y"
def test_issue_8409():
assert latex(S.Half**n) == r"\left(\frac{1}{2}\right)^{n}"
def test_issue_8470():
from sympy.parsing.sympy_parser import parse_expr
e = parse_expr("-B*A", evaluate=False)
assert latex(e) == r"A \left(- B\right)"
def test_issue_7117():
# See also issue #5031 (hence the evaluate=False in these).
e = Eq(x + 1, 2*x)
q = Mul(2, e, evaluate=False)
assert latex(q) == r"2 \left(x + 1 = 2 x\right)"
q = Add(6, e, evaluate=False)
assert latex(q) == r"6 + \left(x + 1 = 2 x\right)"
q = Pow(e, 2, evaluate=False)
assert latex(q) == r"\left(x + 1 = 2 x\right)^{2}"
def test_issue_15439():
x = MatrixSymbol('x', 2, 2)
y = MatrixSymbol('y', 2, 2)
assert latex((x * y).subs(y, -y)) == r"x \left(- y\right)"
assert latex((x * y).subs(y, -2*y)) == r"x \left(- 2 y\right)"
assert latex((x * y).subs(x, -x)) == r"- x y"
def test_issue_2934():
assert latex(Symbol(r'\frac{a_1}{b_1}')) == '\\frac{a_1}{b_1}'
def test_issue_10489():
latexSymbolWithBrace = 'C_{x_{0}}'
s = Symbol(latexSymbolWithBrace)
assert latex(s) == latexSymbolWithBrace
assert latex(cos(s)) == r'\cos{\left(C_{x_{0}} \right)}'
def test_issue_12886():
m__1, l__1 = symbols('m__1, l__1')
assert latex(m__1**2 + l__1**2) == \
r'\left(l^{1}\right)^{2} + \left(m^{1}\right)^{2}'
def test_issue_13559():
from sympy.parsing.sympy_parser import parse_expr
expr = parse_expr('5/1', evaluate=False)
assert latex(expr) == r"\frac{5}{1}"
def test_issue_13651():
expr = c + Mul(-1, a + b, evaluate=False)
assert latex(expr) == r"c - \left(a + b\right)"
def test_latex_UnevaluatedExpr():
x = symbols("x")
he = UnevaluatedExpr(1/x)
assert latex(he) == latex(1/x) == r"\frac{1}{x}"
assert latex(he**2) == r"\left(\frac{1}{x}\right)^{2}"
assert latex(he + 1) == r"1 + \frac{1}{x}"
assert latex(x*he) == r"x \frac{1}{x}"
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
assert latex(A[0, 0]) == r"A_{0, 0}"
assert latex(3 * A[0, 0]) == r"3 A_{0, 0}"
F = C[0, 0].subs(C, A - B)
assert latex(F) == r"\left(A - B\right)_{0, 0}"
i, j, k = symbols("i j k")
M = MatrixSymbol("M", k, k)
N = MatrixSymbol("N", k, k)
assert latex((M*N)[i, j]) == \
r'\sum_{i_{1}=0}^{k - 1} M_{i, i_{1}} N_{i_{1}, j}'
def test_MatrixSymbol_printing():
# test cases for issue #14237
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert latex(-A) == r"- A"
assert latex(A - A*B - B) == r"A - A B - B"
assert latex(-A*B - A*B*C - B) == r"- A B - A B C - B"
def test_KroneckerProduct_printing():
A = MatrixSymbol('A', 3, 3)
B = MatrixSymbol('B', 2, 2)
assert latex(KroneckerProduct(A, B)) == r'A \otimes B'
def test_Quaternion_latex_printing():
q = Quaternion(x, y, z, t)
assert latex(q) == "x + y i + z j + t k"
q = Quaternion(x, y, z, x*t)
assert latex(q) == "x + y i + z j + t x k"
q = Quaternion(x, y, z, x + t)
assert latex(q) == r"x + y i + z j + \left(t + x\right) k"
def test_TensorProduct_printing():
from sympy.tensor.functions import TensorProduct
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
assert latex(TensorProduct(A, B)) == r"A \otimes B"
def test_WedgeProduct_printing():
from sympy.diffgeom.rn import R2
from sympy.diffgeom import WedgeProduct
wp = WedgeProduct(R2.dx, R2.dy)
assert latex(wp) == r"\operatorname{d}x \wedge \operatorname{d}y"
def test_issue_14041():
import sympy.physics.mechanics as me
A_frame = me.ReferenceFrame('A')
thetad, phid = me.dynamicsymbols('theta, phi', 1)
L = Symbol('L')
assert latex(L*(phid + thetad)**2*A_frame.x) == \
r"L \left(\dot{\phi} + \dot{\theta}\right)^{2}\mathbf{\hat{a}_x}"
assert latex((phid + thetad)**2*A_frame.x) == \
r"\left(\dot{\phi} + \dot{\theta}\right)^{2}\mathbf{\hat{a}_x}"
assert latex((phid*thetad)**a*A_frame.x) == \
r"\left(\dot{\phi} \dot{\theta}\right)^{a}\mathbf{\hat{a}_x}"
def test_issue_9216():
expr_1 = Pow(1, -1, evaluate=False)
assert latex(expr_1) == r"1^{-1}"
expr_2 = Pow(1, Pow(1, -1, evaluate=False), evaluate=False)
assert latex(expr_2) == r"1^{1^{-1}}"
expr_3 = Pow(3, -2, evaluate=False)
assert latex(expr_3) == r"\frac{1}{9}"
expr_4 = Pow(1, -2, evaluate=False)
assert latex(expr_4) == r"1^{-2}"
def test_latex_printer_tensor():
from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
L = TensorIndexType("L")
i, j, k, l = tensor_indices("i j k l", L)
i0 = tensor_indices("i_0", L)
A, B, C, D = tensorhead("A B C D", [L], [[1]])
H = tensorhead("H", [L, L], [[1], [1]])
K = tensorhead("K", [L, L, L, L], [[1], [1], [1], [1]])
assert latex(i) == "{}^{i}"
assert latex(-i) == "{}_{i}"
expr = A(i)
assert latex(expr) == "A{}^{i}"
expr = A(i0)
assert latex(expr) == "A{}^{i_{0}}"
expr = A(-i)
assert latex(expr) == "A{}_{i}"
expr = -3*A(i)
assert latex(expr) == r"-3A{}^{i}"
expr = K(i, j, -k, -i0)
assert latex(expr) == "K{}^{ij}{}_{ki_{0}}"
expr = K(i, -j, -k, i0)
assert latex(expr) == "K{}^{i}{}_{jk}{}^{i_{0}}"
expr = K(i, -j, k, -i0)
assert latex(expr) == "K{}^{i}{}_{j}{}^{k}{}_{i_{0}}"
expr = H(i, -j)
assert latex(expr) == "H{}^{i}{}_{j}"
expr = H(i, j)
assert latex(expr) == "H{}^{ij}"
expr = H(-i, -j)
assert latex(expr) == "H{}_{ij}"
expr = (1+x)*A(i)
assert latex(expr) == r"\left(x + 1\right)A{}^{i}"
expr = H(i, -i)
assert latex(expr) == "H{}^{L_{0}}{}_{L_{0}}"
expr = H(i, -j)*A(j)*B(k)
assert latex(expr) == "H{}^{i}{}_{L_{0}}A{}^{L_{0}}B{}^{k}"
expr = A(i) + 3*B(i)
assert latex(expr) == "3B{}^{i} + A{}^{i}"
# Test ``TensorElement``:
from sympy.tensor.tensor import TensorElement
expr = TensorElement(K(i, j, k, l), {i: 3, k: 2})
assert latex(expr) == 'K{}^{i=3,j,k=2,l}'
expr = TensorElement(K(i, j, k, l), {i: 3})
assert latex(expr) == 'K{}^{i=3,jkl}'
expr = TensorElement(K(i, -j, k, l), {i: 3, k: 2})
assert latex(expr) == 'K{}^{i=3}{}_{j}{}^{k=2,l}'
expr = TensorElement(K(i, -j, k, -l), {i: 3, k: 2})
assert latex(expr) == 'K{}^{i=3}{}_{j}{}^{k=2}{}_{l}'
expr = TensorElement(K(i, j, -k, -l), {i: 3, -k: 2})
assert latex(expr) == 'K{}^{i=3,j}{}_{k=2,l}'
expr = TensorElement(K(i, j, -k, -l), {i: 3})
assert latex(expr) == 'K{}^{i=3,j}{}_{kl}'
def test_issue_15353():
from sympy import ConditionSet, Tuple, FiniteSet, S, sin, cos
a, x = symbols('a x')
# Obtained from nonlinsolve([(sin(a*x)),cos(a*x)],[x,a])
sol = ConditionSet(Tuple(x, a), FiniteSet(sin(a*x), cos(a*x)), S.Complexes)
assert latex(sol) == \
r'\left\{\left( x, \ a\right) \mid \left( x, \ a\right) \in '\
r'\mathbb{C} \wedge \left\{\sin{\left(a x \right)}, \cos{\left(a x '\
r'\right)}\right\} \right\}'
def test_trace():
# Issue 15303
from sympy import trace
A = MatrixSymbol("A", 2, 2)
assert latex(trace(A)) == r"\operatorname{tr}\left(A \right)"
assert latex(trace(A**2)) == r"\operatorname{tr}\left(A^{2} \right)"
def test_print_basic():
# Issue 15303
from sympy import Basic, Expr
# dummy class for testing printing where the function is not
# implemented in latex.py
class UnimplementedExpr(Expr):
def __new__(cls, e):
return Basic.__new__(cls, e)
# dummy function for testing
def unimplemented_expr(expr):
return UnimplementedExpr(expr).doit()
# override class name to use superscript / subscript
def unimplemented_expr_sup_sub(expr):
result = UnimplementedExpr(expr)
result.__class__.__name__ = 'UnimplementedExpr_x^1'
return result
assert latex(unimplemented_expr(x)) == r'UnimplementedExpr\left(x\right)'
assert latex(unimplemented_expr(x**2)) == \
r'UnimplementedExpr\left(x^{2}\right)'
assert latex(unimplemented_expr_sup_sub(x)) == \
r'UnimplementedExpr^{1}_{x}\left(x\right)'
def test_MatrixSymbol_bold():
# Issue #15871
from sympy import trace
A = MatrixSymbol("A", 2, 2)
assert latex(trace(A), mat_symbol_style='bold') == \
r"\operatorname{tr}\left(\mathbf{A} \right)"
assert latex(trace(A), mat_symbol_style='plain') == \
r"\operatorname{tr}\left(A \right)"
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert latex(-A, mat_symbol_style='bold') == r"- \mathbf{A}"
assert latex(A - A*B - B, mat_symbol_style='bold') == \
r"\mathbf{A} - \mathbf{A} \mathbf{B} - \mathbf{B}"
assert latex(-A*B - A*B*C - B, mat_symbol_style='bold') == \
r"- \mathbf{A} \mathbf{B} - \mathbf{A} \mathbf{B} \mathbf{C} - \mathbf{B}"
A = MatrixSymbol("A_k", 3, 3)
assert latex(A, mat_symbol_style='bold') == r"\mathbf{A_{k}}"
def test_imaginary_unit():
assert latex(1 + I) == '1 + i'
assert latex(1 + I, imaginary_unit='i') == '1 + i'
assert latex(1 + I, imaginary_unit='j') == '1 + j'
assert latex(1 + I, imaginary_unit='foo') == '1 + foo'
assert latex(I, imaginary_unit="ti") == '\\text{i}'
assert latex(I, imaginary_unit="tj") == '\\text{j}'
def test_text_re_im():
assert latex(im(x), gothic_re_im=True) == r'\Im{\left(x\right)}'
assert latex(im(x), gothic_re_im=False) == r'\operatorname{im}{\left(x\right)}'
assert latex(re(x), gothic_re_im=True) == r'\Re{\left(x\right)}'
assert latex(re(x), gothic_re_im=False) == r'\operatorname{re}{\left(x\right)}'
def test_DiffGeomMethods():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential
from sympy.diffgeom.rn import R2
m = Manifold('M', 2)
assert latex(m) == r'\text{M}'
p = Patch('P', m)
assert latex(p) == r'\text{P}_{\text{M}}'
rect = CoordSystem('rect', p)
assert latex(rect) == r'\text{rect}^{\text{P}}_{\text{M}}'
b = BaseScalarField(rect, 0)
assert latex(b) == r'\mathbf{rect_{0}}'
g = Function('g')
s_field = g(R2.x, R2.y)
assert latex(Differential(s_field)) == \
r'\operatorname{d}\left(g{\left(\mathbf{x},\mathbf{y} \right)}\right)'
|
2bb081f5cc0290798057ed77070612f618f9d634451e50b5974f0fe7d20109a2
|
from sympy import diff, Integral, Limit, sin, Symbol, Integer, Rational, cos, \
tan, asin, acos, atan, sinh, cosh, tanh, asinh, acosh, atanh, E, I, oo, \
pi, GoldenRatio, EulerGamma, Sum, Eq, Ne, Ge, Lt, Float, Matrix, Basic, \
S, MatrixSymbol, Function, Derivative, log, true, false, Range, Min, Max, \
Lambda, IndexedBase, symbols, zoo, elliptic_f, elliptic_e, elliptic_pi, Ei, \
expint, jacobi, gegenbauer, chebyshevt, chebyshevu, legendre, assoc_legendre, \
laguerre, assoc_laguerre, hermite
from sympy import elliptic_k, totient, reduced_totient, primenu, primeomega, \
fresnelc, fresnels, Heaviside
from sympy.calculus.util import AccumBounds
from sympy.core.containers import Tuple
from sympy.functions.combinatorial.factorials import factorial, factorial2, \
binomial
from sympy.functions.combinatorial.numbers import bernoulli, bell, lucas, \
fibonacci, tribonacci, catalan
from sympy.functions.elementary.complexes import re, im, Abs, conjugate
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.integers import floor, ceiling
from sympy.functions.special.gamma_functions import gamma, lowergamma, uppergamma
from sympy.functions.special.singularity_functions import SingularityFunction
from sympy.functions.special.zeta_functions import polylog, lerchphi, zeta, dirichlet_eta
from sympy.logic.boolalg import And, Or, Implies, Equivalent, Xor, Not
from sympy.matrices.expressions.determinant import Determinant
from sympy.printing.mathml import mathml, MathMLContentPrinter, \
MathMLPresentationPrinter, MathMLPrinter
from sympy.sets.sets import FiniteSet, Union, Intersection, Complement, \
SymmetricDifference, Interval, EmptySet
from sympy.stats.rv import RandomSymbol
from sympy.utilities.pytest import raises
from sympy.vector import CoordSys3D, Cross, Curl, Dot, Divergence, Gradient, Laplacian
x, y, z, a, b, c, d, e, n = symbols('x:z a:e n')
mp = MathMLContentPrinter()
mpp = MathMLPresentationPrinter()
def test_mathml_printer():
m = MathMLPrinter()
assert m.doprint(1+x) == mp.doprint(1+x)
def test_content_printmethod():
assert mp.doprint(1 + x) == '<apply><plus/><ci>x</ci><cn>1</cn></apply>'
def test_content_mathml_core():
mml_1 = mp._print(1 + x)
assert mml_1.nodeName == 'apply'
nodes = mml_1.childNodes
assert len(nodes) == 3
assert nodes[0].nodeName == 'plus'
assert nodes[0].hasChildNodes() is False
assert nodes[0].nodeValue is None
assert nodes[1].nodeName in ['cn', 'ci']
if nodes[1].nodeName == 'cn':
assert nodes[1].childNodes[0].nodeValue == '1'
assert nodes[2].childNodes[0].nodeValue == 'x'
else:
assert nodes[1].childNodes[0].nodeValue == 'x'
assert nodes[2].childNodes[0].nodeValue == '1'
mml_2 = mp._print(x**2)
assert mml_2.nodeName == 'apply'
nodes = mml_2.childNodes
assert nodes[1].childNodes[0].nodeValue == 'x'
assert nodes[2].childNodes[0].nodeValue == '2'
mml_3 = mp._print(2*x)
assert mml_3.nodeName == 'apply'
nodes = mml_3.childNodes
assert nodes[0].nodeName == 'times'
assert nodes[1].childNodes[0].nodeValue == '2'
assert nodes[2].childNodes[0].nodeValue == 'x'
mml = mp._print(Float(1.0, 2)*x)
assert mml.nodeName == 'apply'
nodes = mml.childNodes
assert nodes[0].nodeName == 'times'
assert nodes[1].childNodes[0].nodeValue == '1.0'
assert nodes[2].childNodes[0].nodeValue == 'x'
def test_content_mathml_functions():
mml_1 = mp._print(sin(x))
assert mml_1.nodeName == 'apply'
assert mml_1.childNodes[0].nodeName == 'sin'
assert mml_1.childNodes[1].nodeName == 'ci'
mml_2 = mp._print(diff(sin(x), x, evaluate=False))
assert mml_2.nodeName == 'apply'
assert mml_2.childNodes[0].nodeName == 'diff'
assert mml_2.childNodes[1].nodeName == 'bvar'
assert mml_2.childNodes[1].childNodes[
0].nodeName == 'ci' # below bvar there's <ci>x/ci>
mml_3 = mp._print(diff(cos(x*y), x, evaluate=False))
assert mml_3.nodeName == 'apply'
assert mml_3.childNodes[0].nodeName == 'partialdiff'
assert mml_3.childNodes[1].nodeName == 'bvar'
assert mml_3.childNodes[1].childNodes[
0].nodeName == 'ci' # below bvar there's <ci>x/ci>
def test_content_mathml_limits():
# XXX No unevaluated limits
lim_fun = sin(x)/x
mml_1 = mp._print(Limit(lim_fun, x, 0))
assert mml_1.childNodes[0].nodeName == 'limit'
assert mml_1.childNodes[1].nodeName == 'bvar'
assert mml_1.childNodes[2].nodeName == 'lowlimit'
assert mml_1.childNodes[3].toxml() == mp._print(lim_fun).toxml()
def test_content_mathml_integrals():
integrand = x
mml_1 = mp._print(Integral(integrand, (x, 0, 1)))
assert mml_1.childNodes[0].nodeName == 'int'
assert mml_1.childNodes[1].nodeName == 'bvar'
assert mml_1.childNodes[2].nodeName == 'lowlimit'
assert mml_1.childNodes[3].nodeName == 'uplimit'
assert mml_1.childNodes[4].toxml() == mp._print(integrand).toxml()
def test_content_mathml_matrices():
A = Matrix([1, 2, 3])
B = Matrix([[0, 5, 4], [2, 3, 1], [9, 7, 9]])
mll_1 = mp._print(A)
assert mll_1.childNodes[0].nodeName == 'matrixrow'
assert mll_1.childNodes[0].childNodes[0].nodeName == 'cn'
assert mll_1.childNodes[0].childNodes[0].childNodes[0].nodeValue == '1'
assert mll_1.childNodes[1].nodeName == 'matrixrow'
assert mll_1.childNodes[1].childNodes[0].nodeName == 'cn'
assert mll_1.childNodes[1].childNodes[0].childNodes[0].nodeValue == '2'
assert mll_1.childNodes[2].nodeName == 'matrixrow'
assert mll_1.childNodes[2].childNodes[0].nodeName == 'cn'
assert mll_1.childNodes[2].childNodes[0].childNodes[0].nodeValue == '3'
mll_2 = mp._print(B)
assert mll_2.childNodes[0].nodeName == 'matrixrow'
assert mll_2.childNodes[0].childNodes[0].nodeName == 'cn'
assert mll_2.childNodes[0].childNodes[0].childNodes[0].nodeValue == '0'
assert mll_2.childNodes[0].childNodes[1].nodeName == 'cn'
assert mll_2.childNodes[0].childNodes[1].childNodes[0].nodeValue == '5'
assert mll_2.childNodes[0].childNodes[2].nodeName == 'cn'
assert mll_2.childNodes[0].childNodes[2].childNodes[0].nodeValue == '4'
assert mll_2.childNodes[1].nodeName == 'matrixrow'
assert mll_2.childNodes[1].childNodes[0].nodeName == 'cn'
assert mll_2.childNodes[1].childNodes[0].childNodes[0].nodeValue == '2'
assert mll_2.childNodes[1].childNodes[1].nodeName == 'cn'
assert mll_2.childNodes[1].childNodes[1].childNodes[0].nodeValue == '3'
assert mll_2.childNodes[1].childNodes[2].nodeName == 'cn'
assert mll_2.childNodes[1].childNodes[2].childNodes[0].nodeValue == '1'
assert mll_2.childNodes[2].nodeName == 'matrixrow'
assert mll_2.childNodes[2].childNodes[0].nodeName == 'cn'
assert mll_2.childNodes[2].childNodes[0].childNodes[0].nodeValue == '9'
assert mll_2.childNodes[2].childNodes[1].nodeName == 'cn'
assert mll_2.childNodes[2].childNodes[1].childNodes[0].nodeValue == '7'
assert mll_2.childNodes[2].childNodes[2].nodeName == 'cn'
assert mll_2.childNodes[2].childNodes[2].childNodes[0].nodeValue == '9'
def test_content_mathml_sums():
summand = x
mml_1 = mp._print(Sum(summand, (x, 1, 10)))
assert mml_1.childNodes[0].nodeName == 'sum'
assert mml_1.childNodes[1].nodeName == 'bvar'
assert mml_1.childNodes[2].nodeName == 'lowlimit'
assert mml_1.childNodes[3].nodeName == 'uplimit'
assert mml_1.childNodes[4].toxml() == mp._print(summand).toxml()
def test_content_mathml_tuples():
mml_1 = mp._print([2])
assert mml_1.nodeName == 'list'
assert mml_1.childNodes[0].nodeName == 'cn'
assert len(mml_1.childNodes) == 1
mml_2 = mp._print([2, Integer(1)])
assert mml_2.nodeName == 'list'
assert mml_2.childNodes[0].nodeName == 'cn'
assert mml_2.childNodes[1].nodeName == 'cn'
assert len(mml_2.childNodes) == 2
def test_content_mathml_add():
mml = mp._print(x**5 - x**4 + x)
assert mml.childNodes[0].nodeName == 'plus'
assert mml.childNodes[1].childNodes[0].nodeName == 'minus'
assert mml.childNodes[1].childNodes[1].nodeName == 'apply'
def test_content_mathml_Rational():
mml_1 = mp._print(Rational(1, 1))
"""should just return a number"""
assert mml_1.nodeName == 'cn'
mml_2 = mp._print(Rational(2, 5))
assert mml_2.childNodes[0].nodeName == 'divide'
def test_content_mathml_constants():
mml = mp._print(I)
assert mml.nodeName == 'imaginaryi'
mml = mp._print(E)
assert mml.nodeName == 'exponentiale'
mml = mp._print(oo)
assert mml.nodeName == 'infinity'
mml = mp._print(pi)
assert mml.nodeName == 'pi'
assert mathml(GoldenRatio) == '<cn>φ</cn>'
mml = mathml(EulerGamma)
assert mml == '<eulergamma/>'
def test_content_mathml_trig():
mml = mp._print(sin(x))
assert mml.childNodes[0].nodeName == 'sin'
mml = mp._print(cos(x))
assert mml.childNodes[0].nodeName == 'cos'
mml = mp._print(tan(x))
assert mml.childNodes[0].nodeName == 'tan'
mml = mp._print(asin(x))
assert mml.childNodes[0].nodeName == 'arcsin'
mml = mp._print(acos(x))
assert mml.childNodes[0].nodeName == 'arccos'
mml = mp._print(atan(x))
assert mml.childNodes[0].nodeName == 'arctan'
mml = mp._print(sinh(x))
assert mml.childNodes[0].nodeName == 'sinh'
mml = mp._print(cosh(x))
assert mml.childNodes[0].nodeName == 'cosh'
mml = mp._print(tanh(x))
assert mml.childNodes[0].nodeName == 'tanh'
mml = mp._print(asinh(x))
assert mml.childNodes[0].nodeName == 'arcsinh'
mml = mp._print(atanh(x))
assert mml.childNodes[0].nodeName == 'arctanh'
mml = mp._print(acosh(x))
assert mml.childNodes[0].nodeName == 'arccosh'
def test_content_mathml_relational():
mml_1 = mp._print(Eq(x, 1))
assert mml_1.nodeName == 'apply'
assert mml_1.childNodes[0].nodeName == 'eq'
assert mml_1.childNodes[1].nodeName == 'ci'
assert mml_1.childNodes[1].childNodes[0].nodeValue == 'x'
assert mml_1.childNodes[2].nodeName == 'cn'
assert mml_1.childNodes[2].childNodes[0].nodeValue == '1'
mml_2 = mp._print(Ne(1, x))
assert mml_2.nodeName == 'apply'
assert mml_2.childNodes[0].nodeName == 'neq'
assert mml_2.childNodes[1].nodeName == 'cn'
assert mml_2.childNodes[1].childNodes[0].nodeValue == '1'
assert mml_2.childNodes[2].nodeName == 'ci'
assert mml_2.childNodes[2].childNodes[0].nodeValue == 'x'
mml_3 = mp._print(Ge(1, x))
assert mml_3.nodeName == 'apply'
assert mml_3.childNodes[0].nodeName == 'geq'
assert mml_3.childNodes[1].nodeName == 'cn'
assert mml_3.childNodes[1].childNodes[0].nodeValue == '1'
assert mml_3.childNodes[2].nodeName == 'ci'
assert mml_3.childNodes[2].childNodes[0].nodeValue == 'x'
mml_4 = mp._print(Lt(1, x))
assert mml_4.nodeName == 'apply'
assert mml_4.childNodes[0].nodeName == 'lt'
assert mml_4.childNodes[1].nodeName == 'cn'
assert mml_4.childNodes[1].childNodes[0].nodeValue == '1'
assert mml_4.childNodes[2].nodeName == 'ci'
assert mml_4.childNodes[2].childNodes[0].nodeValue == 'x'
def test_content_symbol():
mml = mp._print(x)
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeValue == 'x'
del mml
mml = mp._print(Symbol("x^2"))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeName == 'mml:msup'
assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '2'
del mml
mml = mp._print(Symbol("x__2"))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeName == 'mml:msup'
assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '2'
del mml
mml = mp._print(Symbol("x_2"))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeName == 'mml:msub'
assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '2'
del mml
mml = mp._print(Symbol("x^3_2"))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeName == 'mml:msubsup'
assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '2'
assert mml.childNodes[0].childNodes[2].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[2].childNodes[0].nodeValue == '3'
del mml
mml = mp._print(Symbol("x__3_2"))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeName == 'mml:msubsup'
assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '2'
assert mml.childNodes[0].childNodes[2].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[2].childNodes[0].nodeValue == '3'
del mml
mml = mp._print(Symbol("x_2_a"))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeName == 'mml:msub'
assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mrow'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[0].childNodes[
0].nodeValue == '2'
assert mml.childNodes[0].childNodes[1].childNodes[1].nodeName == 'mml:mo'
assert mml.childNodes[0].childNodes[1].childNodes[1].childNodes[
0].nodeValue == ' '
assert mml.childNodes[0].childNodes[1].childNodes[2].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[2].childNodes[
0].nodeValue == 'a'
del mml
mml = mp._print(Symbol("x^2^a"))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeName == 'mml:msup'
assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mrow'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[0].childNodes[
0].nodeValue == '2'
assert mml.childNodes[0].childNodes[1].childNodes[1].nodeName == 'mml:mo'
assert mml.childNodes[0].childNodes[1].childNodes[1].childNodes[
0].nodeValue == ' '
assert mml.childNodes[0].childNodes[1].childNodes[2].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[2].childNodes[
0].nodeValue == 'a'
del mml
mml = mp._print(Symbol("x__2__a"))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeName == 'mml:msup'
assert mml.childNodes[0].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].nodeName == 'mml:mrow'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[0].childNodes[
0].nodeValue == '2'
assert mml.childNodes[0].childNodes[1].childNodes[1].nodeName == 'mml:mo'
assert mml.childNodes[0].childNodes[1].childNodes[1].childNodes[
0].nodeValue == ' '
assert mml.childNodes[0].childNodes[1].childNodes[2].nodeName == 'mml:mi'
assert mml.childNodes[0].childNodes[1].childNodes[2].childNodes[
0].nodeValue == 'a'
del mml
def test_content_mathml_greek():
mml = mp._print(Symbol('alpha'))
assert mml.nodeName == 'ci'
assert mml.childNodes[0].nodeValue == u'\N{GREEK SMALL LETTER ALPHA}'
assert mp.doprint(Symbol('alpha')) == '<ci>α</ci>'
assert mp.doprint(Symbol('beta')) == '<ci>β</ci>'
assert mp.doprint(Symbol('gamma')) == '<ci>γ</ci>'
assert mp.doprint(Symbol('delta')) == '<ci>δ</ci>'
assert mp.doprint(Symbol('epsilon')) == '<ci>ε</ci>'
assert mp.doprint(Symbol('zeta')) == '<ci>ζ</ci>'
assert mp.doprint(Symbol('eta')) == '<ci>η</ci>'
assert mp.doprint(Symbol('theta')) == '<ci>θ</ci>'
assert mp.doprint(Symbol('iota')) == '<ci>ι</ci>'
assert mp.doprint(Symbol('kappa')) == '<ci>κ</ci>'
assert mp.doprint(Symbol('lambda')) == '<ci>λ</ci>'
assert mp.doprint(Symbol('mu')) == '<ci>μ</ci>'
assert mp.doprint(Symbol('nu')) == '<ci>ν</ci>'
assert mp.doprint(Symbol('xi')) == '<ci>ξ</ci>'
assert mp.doprint(Symbol('omicron')) == '<ci>ο</ci>'
assert mp.doprint(Symbol('pi')) == '<ci>π</ci>'
assert mp.doprint(Symbol('rho')) == '<ci>ρ</ci>'
assert mp.doprint(Symbol('varsigma')) == '<ci>ς</ci>'
assert mp.doprint(Symbol('sigma')) == '<ci>σ</ci>'
assert mp.doprint(Symbol('tau')) == '<ci>τ</ci>'
assert mp.doprint(Symbol('upsilon')) == '<ci>υ</ci>'
assert mp.doprint(Symbol('phi')) == '<ci>φ</ci>'
assert mp.doprint(Symbol('chi')) == '<ci>χ</ci>'
assert mp.doprint(Symbol('psi')) == '<ci>ψ</ci>'
assert mp.doprint(Symbol('omega')) == '<ci>ω</ci>'
assert mp.doprint(Symbol('Alpha')) == '<ci>Α</ci>'
assert mp.doprint(Symbol('Beta')) == '<ci>Β</ci>'
assert mp.doprint(Symbol('Gamma')) == '<ci>Γ</ci>'
assert mp.doprint(Symbol('Delta')) == '<ci>Δ</ci>'
assert mp.doprint(Symbol('Epsilon')) == '<ci>Ε</ci>'
assert mp.doprint(Symbol('Zeta')) == '<ci>Ζ</ci>'
assert mp.doprint(Symbol('Eta')) == '<ci>Η</ci>'
assert mp.doprint(Symbol('Theta')) == '<ci>Θ</ci>'
assert mp.doprint(Symbol('Iota')) == '<ci>Ι</ci>'
assert mp.doprint(Symbol('Kappa')) == '<ci>Κ</ci>'
assert mp.doprint(Symbol('Lambda')) == '<ci>Λ</ci>'
assert mp.doprint(Symbol('Mu')) == '<ci>Μ</ci>'
assert mp.doprint(Symbol('Nu')) == '<ci>Ν</ci>'
assert mp.doprint(Symbol('Xi')) == '<ci>Ξ</ci>'
assert mp.doprint(Symbol('Omicron')) == '<ci>Ο</ci>'
assert mp.doprint(Symbol('Pi')) == '<ci>Π</ci>'
assert mp.doprint(Symbol('Rho')) == '<ci>Ρ</ci>'
assert mp.doprint(Symbol('Sigma')) == '<ci>Σ</ci>'
assert mp.doprint(Symbol('Tau')) == '<ci>Τ</ci>'
assert mp.doprint(Symbol('Upsilon')) == '<ci>Υ</ci>'
assert mp.doprint(Symbol('Phi')) == '<ci>Φ</ci>'
assert mp.doprint(Symbol('Chi')) == '<ci>Χ</ci>'
assert mp.doprint(Symbol('Psi')) == '<ci>Ψ</ci>'
assert mp.doprint(Symbol('Omega')) == '<ci>Ω</ci>'
def test_content_mathml_order():
expr = x**3 + x**2*y + 3*x*y**3 + y**4
mp = MathMLContentPrinter({'order': 'lex'})
mml = mp._print(expr)
assert mml.childNodes[1].childNodes[0].nodeName == 'power'
assert mml.childNodes[1].childNodes[1].childNodes[0].data == 'x'
assert mml.childNodes[1].childNodes[2].childNodes[0].data == '3'
assert mml.childNodes[4].childNodes[0].nodeName == 'power'
assert mml.childNodes[4].childNodes[1].childNodes[0].data == 'y'
assert mml.childNodes[4].childNodes[2].childNodes[0].data == '4'
mp = MathMLContentPrinter({'order': 'rev-lex'})
mml = mp._print(expr)
assert mml.childNodes[1].childNodes[0].nodeName == 'power'
assert mml.childNodes[1].childNodes[1].childNodes[0].data == 'y'
assert mml.childNodes[1].childNodes[2].childNodes[0].data == '4'
assert mml.childNodes[4].childNodes[0].nodeName == 'power'
assert mml.childNodes[4].childNodes[1].childNodes[0].data == 'x'
assert mml.childNodes[4].childNodes[2].childNodes[0].data == '3'
def test_content_settings():
raises(TypeError, lambda: mathml(x, method="garbage"))
def test_presentation_printmethod():
assert mpp.doprint(1 + x) == '<mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow>'
assert mpp.doprint(x**2) == '<msup><mi>x</mi><mn>2</mn></msup>'
assert mpp.doprint(x**-1) == '<mfrac><mn>1</mn><mi>x</mi></mfrac>'
assert mpp.doprint(x**-2) == \
'<mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac>'
assert mpp.doprint(2*x) == \
'<mrow><mn>2</mn><mo>⁢</mo><mi>x</mi></mrow>'
def test_presentation_mathml_core():
mml_1 = mpp._print(1 + x)
assert mml_1.nodeName == 'mrow'
nodes = mml_1.childNodes
assert len(nodes) == 3
assert nodes[0].nodeName in ['mi', 'mn']
assert nodes[1].nodeName == 'mo'
if nodes[0].nodeName == 'mn':
assert nodes[0].childNodes[0].nodeValue == '1'
assert nodes[2].childNodes[0].nodeValue == 'x'
else:
assert nodes[0].childNodes[0].nodeValue == 'x'
assert nodes[2].childNodes[0].nodeValue == '1'
mml_2 = mpp._print(x**2)
assert mml_2.nodeName == 'msup'
nodes = mml_2.childNodes
assert nodes[0].childNodes[0].nodeValue == 'x'
assert nodes[1].childNodes[0].nodeValue == '2'
mml_3 = mpp._print(2*x)
assert mml_3.nodeName == 'mrow'
nodes = mml_3.childNodes
assert nodes[0].childNodes[0].nodeValue == '2'
assert nodes[1].childNodes[0].nodeValue == '⁢'
assert nodes[2].childNodes[0].nodeValue == 'x'
mml = mpp._print(Float(1.0, 2)*x)
assert mml.nodeName == 'mrow'
nodes = mml.childNodes
assert nodes[0].childNodes[0].nodeValue == '1.0'
assert nodes[1].childNodes[0].nodeValue == '⁢'
assert nodes[2].childNodes[0].nodeValue == 'x'
def test_presentation_mathml_functions():
mml_1 = mpp._print(sin(x))
assert mml_1.childNodes[0].childNodes[0
].nodeValue == 'sin'
assert mml_1.childNodes[1].childNodes[0
].childNodes[0].nodeValue == 'x'
mml_2 = mpp._print(diff(sin(x), x, evaluate=False))
assert mml_2.nodeName == 'mrow'
assert mml_2.childNodes[0].childNodes[0
].childNodes[0].childNodes[0].nodeValue == 'ⅆ'
assert mml_2.childNodes[1].childNodes[1
].nodeName == 'mfenced'
assert mml_2.childNodes[0].childNodes[1
].childNodes[0].childNodes[0].nodeValue == 'ⅆ'
mml_3 = mpp._print(diff(cos(x*y), x, evaluate=False))
assert mml_3.childNodes[0].nodeName == 'mfrac'
assert mml_3.childNodes[0].childNodes[0
].childNodes[0].childNodes[0].nodeValue == '∂'
assert mml_3.childNodes[1].childNodes[0
].childNodes[0].nodeValue == 'cos'
def test_print_derivative():
f = Function('f')
d = Derivative(f(x, y, z), x, z, x, z, z, y)
assert mathml(d) == \
'<apply><partialdiff/><bvar><ci>y</ci><ci>z</ci><degree><cn>2</cn></degree><ci>x</ci><ci>z</ci><ci>x</ci></bvar><apply><f/><ci>x</ci><ci>y</ci><ci>z</ci></apply></apply>'
assert mathml(d, printer='presentation') == \
'<mrow><mfrac><mrow><msup><mo>∂</mo><mn>6</mn></msup></mrow><mrow><mo>∂</mo><mi>y</mi><msup><mo>∂</mo><mn>2</mn></msup><mi>z</mi><mo>∂</mo><mi>x</mi><mo>∂</mo><mi>z</mi><mo>∂</mo><mi>x</mi></mrow></mfrac><mrow><mi>f</mi><mfenced><mi>x</mi><mi>y</mi><mi>z</mi></mfenced></mrow></mrow>'
def test_presentation_mathml_limits():
lim_fun = sin(x)/x
mml_1 = mpp._print(Limit(lim_fun, x, 0))
assert mml_1.childNodes[0].nodeName == 'munder'
assert mml_1.childNodes[0].childNodes[0
].childNodes[0].nodeValue == 'lim'
assert mml_1.childNodes[0].childNodes[1
].childNodes[0].childNodes[0
].nodeValue == 'x'
assert mml_1.childNodes[0].childNodes[1
].childNodes[1].childNodes[0
].nodeValue == '→'
assert mml_1.childNodes[0].childNodes[1
].childNodes[2].childNodes[0
].nodeValue == '0'
def test_presentation_mathml_integrals():
assert mpp.doprint(Integral(x, (x, 0, 1))) == \
'<mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup>'\
'<mi>x</mi><mo>ⅆ</mo><mi>x</mi></mrow>'
assert mpp.doprint(Integral(log(x), x)) == \
'<mrow><mo>∫</mo><mrow><mi>log</mi><mfenced><mi>x</mi>'\
'</mfenced></mrow><mo>ⅆ</mo><mi>x</mi></mrow>'
assert mpp.doprint(Integral(x*y, x, y)) == \
'<mrow><mo>∬</mo><mrow><mi>x</mi><mo>⁢</mo>'\
'<mi>y</mi></mrow><mo>ⅆ</mo><mi>y</mi><mo>ⅆ</mo><mi>x</mi></mrow>'
z, w = symbols('z w')
assert mpp.doprint(Integral(x*y*z, x, y, z)) == \
'<mrow><mo>∭</mo><mrow><mi>x</mi><mo>⁢</mo>'\
'<mi>y</mi><mo>⁢</mo><mi>z</mi></mrow><mo>ⅆ</mo>'\
'<mi>z</mi><mo>ⅆ</mo><mi>y</mi><mo>ⅆ</mo><mi>x</mi></mrow>'
assert mpp.doprint(Integral(x*y*z*w, x, y, z, w)) == \
'<mrow><mo>∫</mo><mo>∫</mo><mo>∫</mo>'\
'<mo>∫</mo><mrow><mi>w</mi><mo>⁢</mo>'\
'<mi>x</mi><mo>⁢</mo><mi>y</mi>'\
'<mo>⁢</mo><mi>z</mi></mrow><mo>ⅆ</mo><mi>w</mi>'\
'<mo>ⅆ</mo><mi>z</mi><mo>ⅆ</mo><mi>y</mi><mo>ⅆ</mo><mi>x</mi></mrow>'
assert mpp.doprint(Integral(x, x, y, (z, 0, 1))) == \
'<mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup>'\
'<mo>∫</mo><mo>∫</mo><mi>x</mi><mo>ⅆ</mo><mi>z</mi>'\
'<mo>ⅆ</mo><mi>y</mi><mo>ⅆ</mo><mi>x</mi></mrow>'
assert mpp.doprint(Integral(x, (x, 0))) == \
'<mrow><msup><mo>∫</mo><mn>0</mn></msup><mi>x</mi><mo>ⅆ</mo>'\
'<mi>x</mi></mrow>'
def test_presentation_mathml_matrices():
A = Matrix([1, 2, 3])
B = Matrix([[0, 5, 4], [2, 3, 1], [9, 7, 9]])
mll_1 = mpp._print(A)
assert mll_1.childNodes[0].nodeName == 'mtable'
assert mll_1.childNodes[0].childNodes[0].nodeName == 'mtr'
assert len(mll_1.childNodes[0].childNodes) == 3
assert mll_1.childNodes[0].childNodes[0].childNodes[0].nodeName == 'mtd'
assert len(mll_1.childNodes[0].childNodes[0].childNodes) == 1
assert mll_1.childNodes[0].childNodes[0].childNodes[0
].childNodes[0].childNodes[0].nodeValue == '1'
assert mll_1.childNodes[0].childNodes[1].childNodes[0
].childNodes[0].childNodes[0].nodeValue == '2'
assert mll_1.childNodes[0].childNodes[2].childNodes[0
].childNodes[0].childNodes[0].nodeValue == '3'
mll_2 = mpp._print(B)
assert mll_2.childNodes[0].nodeName == 'mtable'
assert mll_2.childNodes[0].childNodes[0].nodeName == 'mtr'
assert len(mll_2.childNodes[0].childNodes) == 3
assert mll_2.childNodes[0].childNodes[0].childNodes[0].nodeName == 'mtd'
assert len(mll_2.childNodes[0].childNodes[0].childNodes) == 3
assert mll_2.childNodes[0].childNodes[0].childNodes[0
].childNodes[0].childNodes[0].nodeValue == '0'
assert mll_2.childNodes[0].childNodes[0].childNodes[1
].childNodes[0].childNodes[0].nodeValue == '5'
assert mll_2.childNodes[0].childNodes[0].childNodes[2
].childNodes[0].childNodes[0].nodeValue == '4'
assert mll_2.childNodes[0].childNodes[1].childNodes[0
].childNodes[0].childNodes[0].nodeValue == '2'
assert mll_2.childNodes[0].childNodes[1].childNodes[1
].childNodes[0].childNodes[0].nodeValue == '3'
assert mll_2.childNodes[0].childNodes[1].childNodes[2
].childNodes[0].childNodes[0].nodeValue == '1'
assert mll_2.childNodes[0].childNodes[2].childNodes[0
].childNodes[0].childNodes[0].nodeValue == '9'
assert mll_2.childNodes[0].childNodes[2].childNodes[1
].childNodes[0].childNodes[0].nodeValue == '7'
assert mll_2.childNodes[0].childNodes[2].childNodes[2
].childNodes[0].childNodes[0].nodeValue == '9'
def test_presentation_mathml_sums():
summand = x
mml_1 = mpp._print(Sum(summand, (x, 1, 10)))
assert mml_1.childNodes[0].nodeName == 'munderover'
assert len(mml_1.childNodes[0].childNodes) == 3
assert mml_1.childNodes[0].childNodes[0].childNodes[0
].nodeValue == '∑'
assert len(mml_1.childNodes[0].childNodes[1].childNodes) == 3
assert mml_1.childNodes[0].childNodes[2].childNodes[0
].nodeValue == '10'
assert mml_1.childNodes[1].childNodes[0].nodeValue == 'x'
def test_presentation_mathml_add():
mml = mpp._print(x**5 - x**4 + x)
assert len(mml.childNodes) == 5
assert mml.childNodes[0].childNodes[0].childNodes[0
].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].childNodes[0
].nodeValue == '5'
assert mml.childNodes[1].childNodes[0].nodeValue == '-'
assert mml.childNodes[2].childNodes[0].childNodes[0
].nodeValue == 'x'
assert mml.childNodes[2].childNodes[1].childNodes[0
].nodeValue == '4'
assert mml.childNodes[3].childNodes[0].nodeValue == '+'
assert mml.childNodes[4].childNodes[0].nodeValue == 'x'
def test_presentation_mathml_Rational():
mml_1 = mpp._print(Rational(1, 1))
assert mml_1.nodeName == 'mn'
mml_2 = mpp._print(Rational(2, 5))
assert mml_2.nodeName == 'mfrac'
assert mml_2.childNodes[0].childNodes[0].nodeValue == '2'
assert mml_2.childNodes[1].childNodes[0].nodeValue == '5'
def test_presentation_mathml_constants():
mml = mpp._print(I)
assert mml.childNodes[0].nodeValue == 'ⅈ'
mml = mpp._print(E)
assert mml.childNodes[0].nodeValue == 'ⅇ'
mml = mpp._print(oo)
assert mml.childNodes[0].nodeValue == '∞'
mml = mpp._print(pi)
assert mml.childNodes[0].nodeValue == 'π'
assert mathml(GoldenRatio, printer='presentation') == '<mi>Φ</mi>'
assert mathml(zoo, printer='presentation') == \
'<mover><mo>∞</mo><mo>~</mo></mover>'
assert mathml(S.NaN, printer='presentation') == '<mi>NaN</mi>'
def test_presentation_mathml_trig():
mml = mpp._print(sin(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'sin'
mml = mpp._print(cos(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'cos'
mml = mpp._print(tan(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'tan'
mml = mpp._print(asin(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arcsin'
mml = mpp._print(acos(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arccos'
mml = mpp._print(atan(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arctan'
mml = mpp._print(sinh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'sinh'
mml = mpp._print(cosh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'cosh'
mml = mpp._print(tanh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'tanh'
mml = mpp._print(asinh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arcsinh'
mml = mpp._print(atanh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arctanh'
mml = mpp._print(acosh(x))
assert mml.childNodes[0].childNodes[0].nodeValue == 'arccosh'
def test_presentation_mathml_relational():
mml_1 = mpp._print(Eq(x, 1))
assert len(mml_1.childNodes) == 3
assert mml_1.childNodes[0].nodeName == 'mi'
assert mml_1.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml_1.childNodes[1].nodeName == 'mo'
assert mml_1.childNodes[1].childNodes[0].nodeValue == '='
assert mml_1.childNodes[2].nodeName == 'mn'
assert mml_1.childNodes[2].childNodes[0].nodeValue == '1'
mml_2 = mpp._print(Ne(1, x))
assert len(mml_2.childNodes) == 3
assert mml_2.childNodes[0].nodeName == 'mn'
assert mml_2.childNodes[0].childNodes[0].nodeValue == '1'
assert mml_2.childNodes[1].nodeName == 'mo'
assert mml_2.childNodes[1].childNodes[0].nodeValue == '≠'
assert mml_2.childNodes[2].nodeName == 'mi'
assert mml_2.childNodes[2].childNodes[0].nodeValue == 'x'
mml_3 = mpp._print(Ge(1, x))
assert len(mml_3.childNodes) == 3
assert mml_3.childNodes[0].nodeName == 'mn'
assert mml_3.childNodes[0].childNodes[0].nodeValue == '1'
assert mml_3.childNodes[1].nodeName == 'mo'
assert mml_3.childNodes[1].childNodes[0].nodeValue == '≥'
assert mml_3.childNodes[2].nodeName == 'mi'
assert mml_3.childNodes[2].childNodes[0].nodeValue == 'x'
mml_4 = mpp._print(Lt(1, x))
assert len(mml_4.childNodes) == 3
assert mml_4.childNodes[0].nodeName == 'mn'
assert mml_4.childNodes[0].childNodes[0].nodeValue == '1'
assert mml_4.childNodes[1].nodeName == 'mo'
assert mml_4.childNodes[1].childNodes[0].nodeValue == '<'
assert mml_4.childNodes[2].nodeName == 'mi'
assert mml_4.childNodes[2].childNodes[0].nodeValue == 'x'
def test_presentation_symbol():
mml = mpp._print(x)
assert mml.nodeName == 'mi'
assert mml.childNodes[0].nodeValue == 'x'
del mml
mml = mpp._print(Symbol("x^2"))
assert mml.nodeName == 'msup'
assert mml.childNodes[0].nodeName == 'mi'
assert mml.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[1].nodeName == 'mi'
assert mml.childNodes[1].childNodes[0].nodeValue == '2'
del mml
mml = mpp._print(Symbol("x__2"))
assert mml.nodeName == 'msup'
assert mml.childNodes[0].nodeName == 'mi'
assert mml.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[1].nodeName == 'mi'
assert mml.childNodes[1].childNodes[0].nodeValue == '2'
del mml
mml = mpp._print(Symbol("x_2"))
assert mml.nodeName == 'msub'
assert mml.childNodes[0].nodeName == 'mi'
assert mml.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[1].nodeName == 'mi'
assert mml.childNodes[1].childNodes[0].nodeValue == '2'
del mml
mml = mpp._print(Symbol("x^3_2"))
assert mml.nodeName == 'msubsup'
assert mml.childNodes[0].nodeName == 'mi'
assert mml.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[1].nodeName == 'mi'
assert mml.childNodes[1].childNodes[0].nodeValue == '2'
assert mml.childNodes[2].nodeName == 'mi'
assert mml.childNodes[2].childNodes[0].nodeValue == '3'
del mml
mml = mpp._print(Symbol("x__3_2"))
assert mml.nodeName == 'msubsup'
assert mml.childNodes[0].nodeName == 'mi'
assert mml.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[1].nodeName == 'mi'
assert mml.childNodes[1].childNodes[0].nodeValue == '2'
assert mml.childNodes[2].nodeName == 'mi'
assert mml.childNodes[2].childNodes[0].nodeValue == '3'
del mml
mml = mpp._print(Symbol("x_2_a"))
assert mml.nodeName == 'msub'
assert mml.childNodes[0].nodeName == 'mi'
assert mml.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[1].nodeName == 'mrow'
assert mml.childNodes[1].childNodes[0].nodeName == 'mi'
assert mml.childNodes[1].childNodes[0].childNodes[0].nodeValue == '2'
assert mml.childNodes[1].childNodes[1].nodeName == 'mo'
assert mml.childNodes[1].childNodes[1].childNodes[0].nodeValue == ' '
assert mml.childNodes[1].childNodes[2].nodeName == 'mi'
assert mml.childNodes[1].childNodes[2].childNodes[0].nodeValue == 'a'
del mml
mml = mpp._print(Symbol("x^2^a"))
assert mml.nodeName == 'msup'
assert mml.childNodes[0].nodeName == 'mi'
assert mml.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[1].nodeName == 'mrow'
assert mml.childNodes[1].childNodes[0].nodeName == 'mi'
assert mml.childNodes[1].childNodes[0].childNodes[0].nodeValue == '2'
assert mml.childNodes[1].childNodes[1].nodeName == 'mo'
assert mml.childNodes[1].childNodes[1].childNodes[0].nodeValue == ' '
assert mml.childNodes[1].childNodes[2].nodeName == 'mi'
assert mml.childNodes[1].childNodes[2].childNodes[0].nodeValue == 'a'
del mml
mml = mpp._print(Symbol("x__2__a"))
assert mml.nodeName == 'msup'
assert mml.childNodes[0].nodeName == 'mi'
assert mml.childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[1].nodeName == 'mrow'
assert mml.childNodes[1].childNodes[0].nodeName == 'mi'
assert mml.childNodes[1].childNodes[0].childNodes[0].nodeValue == '2'
assert mml.childNodes[1].childNodes[1].nodeName == 'mo'
assert mml.childNodes[1].childNodes[1].childNodes[0].nodeValue == ' '
assert mml.childNodes[1].childNodes[2].nodeName == 'mi'
assert mml.childNodes[1].childNodes[2].childNodes[0].nodeValue == 'a'
del mml
def test_presentation_mathml_greek():
mml = mpp._print(Symbol('alpha'))
assert mml.nodeName == 'mi'
assert mml.childNodes[0].nodeValue == u'\N{GREEK SMALL LETTER ALPHA}'
assert mpp.doprint(Symbol('alpha')) == '<mi>α</mi>'
assert mpp.doprint(Symbol('beta')) == '<mi>β</mi>'
assert mpp.doprint(Symbol('gamma')) == '<mi>γ</mi>'
assert mpp.doprint(Symbol('delta')) == '<mi>δ</mi>'
assert mpp.doprint(Symbol('epsilon')) == '<mi>ε</mi>'
assert mpp.doprint(Symbol('zeta')) == '<mi>ζ</mi>'
assert mpp.doprint(Symbol('eta')) == '<mi>η</mi>'
assert mpp.doprint(Symbol('theta')) == '<mi>θ</mi>'
assert mpp.doprint(Symbol('iota')) == '<mi>ι</mi>'
assert mpp.doprint(Symbol('kappa')) == '<mi>κ</mi>'
assert mpp.doprint(Symbol('lambda')) == '<mi>λ</mi>'
assert mpp.doprint(Symbol('mu')) == '<mi>μ</mi>'
assert mpp.doprint(Symbol('nu')) == '<mi>ν</mi>'
assert mpp.doprint(Symbol('xi')) == '<mi>ξ</mi>'
assert mpp.doprint(Symbol('omicron')) == '<mi>ο</mi>'
assert mpp.doprint(Symbol('pi')) == '<mi>π</mi>'
assert mpp.doprint(Symbol('rho')) == '<mi>ρ</mi>'
assert mpp.doprint(Symbol('varsigma')) == '<mi>ς</mi>'
assert mpp.doprint(Symbol('sigma')) == '<mi>σ</mi>'
assert mpp.doprint(Symbol('tau')) == '<mi>τ</mi>'
assert mpp.doprint(Symbol('upsilon')) == '<mi>υ</mi>'
assert mpp.doprint(Symbol('phi')) == '<mi>φ</mi>'
assert mpp.doprint(Symbol('chi')) == '<mi>χ</mi>'
assert mpp.doprint(Symbol('psi')) == '<mi>ψ</mi>'
assert mpp.doprint(Symbol('omega')) == '<mi>ω</mi>'
assert mpp.doprint(Symbol('Alpha')) == '<mi>Α</mi>'
assert mpp.doprint(Symbol('Beta')) == '<mi>Β</mi>'
assert mpp.doprint(Symbol('Gamma')) == '<mi>Γ</mi>'
assert mpp.doprint(Symbol('Delta')) == '<mi>Δ</mi>'
assert mpp.doprint(Symbol('Epsilon')) == '<mi>Ε</mi>'
assert mpp.doprint(Symbol('Zeta')) == '<mi>Ζ</mi>'
assert mpp.doprint(Symbol('Eta')) == '<mi>Η</mi>'
assert mpp.doprint(Symbol('Theta')) == '<mi>Θ</mi>'
assert mpp.doprint(Symbol('Iota')) == '<mi>Ι</mi>'
assert mpp.doprint(Symbol('Kappa')) == '<mi>Κ</mi>'
assert mpp.doprint(Symbol('Lambda')) == '<mi>Λ</mi>'
assert mpp.doprint(Symbol('Mu')) == '<mi>Μ</mi>'
assert mpp.doprint(Symbol('Nu')) == '<mi>Ν</mi>'
assert mpp.doprint(Symbol('Xi')) == '<mi>Ξ</mi>'
assert mpp.doprint(Symbol('Omicron')) == '<mi>Ο</mi>'
assert mpp.doprint(Symbol('Pi')) == '<mi>Π</mi>'
assert mpp.doprint(Symbol('Rho')) == '<mi>Ρ</mi>'
assert mpp.doprint(Symbol('Sigma')) == '<mi>Σ</mi>'
assert mpp.doprint(Symbol('Tau')) == '<mi>Τ</mi>'
assert mpp.doprint(Symbol('Upsilon')) == '<mi>Υ</mi>'
assert mpp.doprint(Symbol('Phi')) == '<mi>Φ</mi>'
assert mpp.doprint(Symbol('Chi')) == '<mi>Χ</mi>'
assert mpp.doprint(Symbol('Psi')) == '<mi>Ψ</mi>'
assert mpp.doprint(Symbol('Omega')) == '<mi>Ω</mi>'
def test_presentation_mathml_order():
expr = x**3 + x**2*y + 3*x*y**3 + y**4
mp = MathMLPresentationPrinter({'order': 'lex'})
mml = mp._print(expr)
assert mml.childNodes[0].nodeName == 'msup'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '3'
assert mml.childNodes[6].nodeName == 'msup'
assert mml.childNodes[6].childNodes[0].childNodes[0].nodeValue == 'y'
assert mml.childNodes[6].childNodes[1].childNodes[0].nodeValue == '4'
mp = MathMLPresentationPrinter({'order': 'rev-lex'})
mml = mp._print(expr)
assert mml.childNodes[0].nodeName == 'msup'
assert mml.childNodes[0].childNodes[0].childNodes[0].nodeValue == 'y'
assert mml.childNodes[0].childNodes[1].childNodes[0].nodeValue == '4'
assert mml.childNodes[6].nodeName == 'msup'
assert mml.childNodes[6].childNodes[0].childNodes[0].nodeValue == 'x'
assert mml.childNodes[6].childNodes[1].childNodes[0].nodeValue == '3'
def test_print_intervals():
a = Symbol('a', real=True)
assert mpp.doprint(Interval(0, a)) == \
'<mrow><mfenced close="]" open="["><mn>0</mn><mi>a</mi></mfenced></mrow>'
assert mpp.doprint(Interval(0, a, False, False)) == \
'<mrow><mfenced close="]" open="["><mn>0</mn><mi>a</mi></mfenced></mrow>'
assert mpp.doprint(Interval(0, a, True, False)) == \
'<mrow><mfenced close="]" open="("><mn>0</mn><mi>a</mi></mfenced></mrow>'
assert mpp.doprint(Interval(0, a, False, True)) == \
'<mrow><mfenced close=")" open="["><mn>0</mn><mi>a</mi></mfenced></mrow>'
assert mpp.doprint(Interval(0, a, True, True)) == \
'<mrow><mfenced close=")" open="("><mn>0</mn><mi>a</mi></mfenced></mrow>'
def test_print_tuples():
assert mpp.doprint(Tuple(0,)) == \
'<mrow><mfenced><mn>0</mn></mfenced></mrow>'
assert mpp.doprint(Tuple(0, a)) == \
'<mrow><mfenced><mn>0</mn><mi>a</mi></mfenced></mrow>'
assert mpp.doprint(Tuple(0, a, a)) == \
'<mrow><mfenced><mn>0</mn><mi>a</mi><mi>a</mi></mfenced></mrow>'
assert mpp.doprint(Tuple(0, 1, 2, 3, 4)) == \
'<mrow><mfenced><mn>0</mn><mn>1</mn><mn>2</mn><mn>3</mn><mn>4</mn></mfenced></mrow>'
assert mpp.doprint(Tuple(0, 1, Tuple(2, 3, 4))) == \
'<mrow><mfenced><mn>0</mn><mn>1</mn><mrow><mfenced><mn>2</mn><mn>3'\
'</mn><mn>4</mn></mfenced></mrow></mfenced></mrow>'
def test_print_re_im():
assert mpp.doprint(re(x)) == \
'<mrow><mi mathvariant="fraktur">R</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(im(x)) == \
'<mrow><mi mathvariant="fraktur">I</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(re(x + 1)) == \
'<mrow><mrow><mi mathvariant="fraktur">R</mi><mfenced><mi>x</mi>'\
'</mfenced></mrow><mo>+</mo><mn>1</mn></mrow>'
assert mpp.doprint(im(x + 1)) == \
'<mrow><mi mathvariant="fraktur">I</mi><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_Abs():
assert mpp.doprint(Abs(x)) == \
'<mrow><mfenced close="|" open="|"><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(Abs(x + 1)) == \
'<mrow><mfenced close="|" open="|"><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow>'
def test_print_Determinant():
assert mpp.doprint(Determinant(Matrix([[1, 2], [3, 4]]))) == \
'<mrow><mfenced close="|" open="|"><mfenced close="]" open="["><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable></mfenced></mfenced></mrow>'
def test_presentation_settings():
raises(TypeError, lambda: mathml(x, printer='presentation',
method="garbage"))
def test_toprettyxml_hooking():
# test that the patch doesn't influence the behavior of the standard
# library
import xml.dom.minidom
doc1 = xml.dom.minidom.parseString(
"<apply><plus/><ci>x</ci><cn>1</cn></apply>")
doc2 = xml.dom.minidom.parseString(
"<mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow>")
prettyxml_old1 = doc1.toprettyxml()
prettyxml_old2 = doc2.toprettyxml()
mp.apply_patch()
mp.restore_patch()
assert prettyxml_old1 == doc1.toprettyxml()
assert prettyxml_old2 == doc2.toprettyxml()
def test_print_domains():
from sympy import Complexes, Integers, Naturals, Naturals0, Reals
assert mpp.doprint(Complexes) == '<mi mathvariant="normal">ℂ</mi>'
assert mpp.doprint(Integers) == '<mi mathvariant="normal">ℤ</mi>'
assert mpp.doprint(Naturals) == '<mi mathvariant="normal">ℕ</mi>'
assert mpp.doprint(Naturals0) == \
'<msub><mi mathvariant="normal">ℕ</mi><mn>0</mn></msub>'
assert mpp.doprint(Reals) == '<mi mathvariant="normal">ℝ</mi>'
def test_print_expression_with_minus():
assert mpp.doprint(-x) == '<mrow><mo>-</mo><mi>x</mi></mrow>'
assert mpp.doprint(-x/y) == \
'<mrow><mo>-</mo><mfrac><mi>x</mi><mi>y</mi></mfrac></mrow>'
assert mpp.doprint(-Rational(1, 2)) == \
'<mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow>'
def test_print_AssocOp():
from sympy.core.operations import AssocOp
class TestAssocOp(AssocOp):
identity = 0
expr = TestAssocOp(1, 2)
mpp.doprint(expr) == \
'<mrow><mi>testassocop</mi><mn>2</mn><mn>1</mn></mrow>'
def test_print_basic():
expr = Basic(1, 2)
assert mpp.doprint(expr) == \
'<mrow><mi>basic</mi><mfenced><mn>1</mn><mn>2</mn></mfenced></mrow>'
assert mp.doprint(expr) == '<basic><cn>1</cn><cn>2</cn></basic>'
def test_mat_delim_print():
expr = Matrix([[1, 2], [3, 4]])
assert mathml(expr, printer='presentation', mat_delim='[') == \
'<mfenced close="]" open="["><mtable><mtr><mtd><mn>1</mn></mtd><mtd>'\
'<mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>4</mn>'\
'</mtd></mtr></mtable></mfenced>'
assert mathml(expr, printer='presentation', mat_delim='(') == \
'<mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd>'\
'</mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable></mfenced>'
assert mathml(expr, printer='presentation', mat_delim='') == \
'<mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr>'\
'<mtd><mn>3</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable>'
def test_ln_notation_print():
expr = log(x)
assert mathml(expr, printer='presentation') == \
'<mrow><mi>log</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mathml(expr, printer='presentation', ln_notation=False) == \
'<mrow><mi>log</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mathml(expr, printer='presentation', ln_notation=True) == \
'<mrow><mi>ln</mi><mfenced><mi>x</mi></mfenced></mrow>'
def test_mul_symbol_print():
expr = x * y
assert mathml(expr, printer='presentation') == \
'<mrow><mi>x</mi><mo>⁢</mo><mi>y</mi></mrow>'
assert mathml(expr, printer='presentation', mul_symbol=None) == \
'<mrow><mi>x</mi><mo>⁢</mo><mi>y</mi></mrow>'
assert mathml(expr, printer='presentation', mul_symbol='dot') == \
'<mrow><mi>x</mi><mo>·</mo><mi>y</mi></mrow>'
assert mathml(expr, printer='presentation', mul_symbol='ldot') == \
'<mrow><mi>x</mi><mo>․</mo><mi>y</mi></mrow>'
assert mathml(expr, printer='presentation', mul_symbol='times') == \
'<mrow><mi>x</mi><mo>×</mo><mi>y</mi></mrow>'
def test_print_lerchphi():
assert mpp.doprint(lerchphi(1, 2, 3)) == \
'<mrow><mi>Φ</mi><mfenced><mn>1</mn><mn>2</mn><mn>3</mn></mfenced></mrow>'
def test_print_polylog():
assert mp.doprint(polylog(x, y)) == \
'<apply><polylog/><ci>x</ci><ci>y</ci></apply>'
assert mpp.doprint(polylog(x, y)) == \
'<mrow><msub><mi>Li</mi><mi>x</mi></msub><mfenced><mi>y</mi></mfenced></mrow>'
def test_print_set_frozenset():
f = frozenset({1, 5, 3})
assert mpp.doprint(f) == \
'<mfenced close="}" open="{"><mn>1</mn><mn>3</mn><mn>5</mn></mfenced>'
s = set({1, 2, 3})
assert mpp.doprint(s) == \
'<mfenced close="}" open="{"><mn>1</mn><mn>2</mn><mn>3</mn></mfenced>'
def test_print_FiniteSet():
f1 = FiniteSet(x, 1, 3)
assert mpp.doprint(f1) == \
'<mfenced close="}" open="{"><mn>1</mn><mn>3</mn><mi>x</mi></mfenced>'
def test_print_EmptySet():
assert mpp.doprint(EmptySet()) == '<mo>∅</mo>'
def test_print_SetOp():
f1 = FiniteSet(x, 1, 3)
f2 = FiniteSet(y, 2, 4)
assert mpp.doprint(Union(f1, f2, evaluate=False)) == \
'<mrow><mfenced close="}" open="{"><mn>1</mn><mn>3</mn><mi>x</mi>'\
'</mfenced><mo>∪</mo><mfenced close="}" open="{"><mn>2</mn>'\
'<mn>4</mn><mi>y</mi></mfenced></mrow>'
assert mpp.doprint(Intersection(f1, f2, evaluate=False)) == \
'<mrow><mfenced close="}" open="{"><mn>1</mn><mn>3</mn><mi>x</mi>'\
'</mfenced><mo>∩</mo><mfenced close="}" open="{"><mn>2</mn>'\
'<mn>4</mn><mi>y</mi></mfenced></mrow>'
assert mpp.doprint(Complement(f1, f2, evaluate=False)) == \
'<mrow><mfenced close="}" open="{"><mn>1</mn><mn>3</mn><mi>x</mi>'\
'</mfenced><mo>∖</mo><mfenced close="}" open="{"><mn>2</mn>'\
'<mn>4</mn><mi>y</mi></mfenced></mrow>'
assert mpp.doprint(SymmetricDifference(f1, f2, evaluate=False)) == \
'<mrow><mfenced close="}" open="{"><mn>1</mn><mn>3</mn><mi>x</mi>'\
'</mfenced><mo>∆</mo><mfenced close="}" open="{"><mn>2</mn>'\
'<mn>4</mn><mi>y</mi></mfenced></mrow>'
def test_print_logic():
assert mpp.doprint(And(x, y)) == \
'<mrow><mi>x</mi><mo>∧</mo><mi>y</mi></mrow>'
assert mpp.doprint(Or(x, y)) == \
'<mrow><mi>x</mi><mo>∨</mo><mi>y</mi></mrow>'
assert mpp.doprint(Xor(x, y)) == \
'<mrow><mi>x</mi><mo>⊻</mo><mi>y</mi></mrow>'
assert mpp.doprint(Implies(x, y)) == \
'<mrow><mi>x</mi><mo>⇒</mo><mi>y</mi></mrow>'
assert mpp.doprint(Equivalent(x, y)) == \
'<mrow><mi>x</mi><mo>⇔</mo><mi>y</mi></mrow>'
assert mpp.doprint(And(Eq(x, y), x > 4)) == \
'<mrow><mrow><mi>x</mi><mo>=</mo><mi>y</mi></mrow><mo>∧</mo>'\
'<mrow><mi>x</mi><mo>></mo><mn>4</mn></mrow></mrow>'
assert mpp.doprint(And(Eq(x, 3), y < 3, x > y + 1)) == \
'<mrow><mrow><mi>x</mi><mo>=</mo><mn>3</mn></mrow><mo>∧</mo>'\
'<mrow><mi>x</mi><mo>></mo><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow>'\
'</mrow><mo>∧</mo><mrow><mi>y</mi><mo><</mo><mn>3</mn></mrow></mrow>'
assert mpp.doprint(Or(Eq(x, y), x > 4)) == \
'<mrow><mrow><mi>x</mi><mo>=</mo><mi>y</mi></mrow><mo>∨</mo>'\
'<mrow><mi>x</mi><mo>></mo><mn>4</mn></mrow></mrow>'
assert mpp.doprint(And(Eq(x, 3), Or(y < 3, x > y + 1))) == \
'<mrow><mrow><mi>x</mi><mo>=</mo><mn>3</mn></mrow><mo>∧</mo>'\
'<mfenced><mrow><mrow><mi>x</mi><mo>></mo><mrow><mi>y</mi><mo>+</mo>'\
'<mn>1</mn></mrow></mrow><mo>∨</mo><mrow><mi>y</mi><mo><</mo>'\
'<mn>3</mn></mrow></mrow></mfenced></mrow>'
assert mpp.doprint(Not(x)) == '<mrow><mo>¬</mo><mi>x</mi></mrow>'
assert mpp.doprint(Not(And(x, y))) == \
'<mrow><mo>¬</mo><mfenced><mrow><mi>x</mi><mo>∧</mo>'\
'<mi>y</mi></mrow></mfenced></mrow>'
def test_root_notation_print():
assert mathml(x**(S(1)/3), printer='presentation') == \
'<mroot><mi>x</mi><mn>3</mn></mroot>'
assert mathml(x**(S(1)/3), printer='presentation', root_notation=False) ==\
'<msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup>'
assert mathml(x**(S(1)/3), printer='content') == \
'<apply><root/><degree><ci>3</ci></degree><ci>x</ci></apply>'
assert mathml(x**(S(1)/3), printer='content', root_notation=False) == \
'<apply><power/><ci>x</ci><apply><divide/><cn>1</cn><cn>3</cn></apply></apply>'
assert mathml(x**(-S(1)/3), printer='presentation') == \
'<mfrac><mn>1</mn><mroot><mi>x</mi><mn>3</mn></mroot></mfrac>'
assert mathml(x**(-S(1)/3), printer='presentation', root_notation=False) \
== '<mfrac><mn>1</mn><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup></mfrac>'
def test_fold_frac_powers_print():
expr = x ** Rational(5, 2)
assert mathml(expr, printer='presentation') == \
'<msup><mi>x</mi><mfrac><mn>5</mn><mn>2</mn></mfrac></msup>'
assert mathml(expr, printer='presentation', fold_frac_powers=True) == \
'<msup><mi>x</mi><mfrac bevelled="true"><mn>5</mn><mn>2</mn></mfrac></msup>'
assert mathml(expr, printer='presentation', fold_frac_powers=False) == \
'<msup><mi>x</mi><mfrac><mn>5</mn><mn>2</mn></mfrac></msup>'
def test_fold_short_frac_print():
expr = Rational(2, 5)
assert mathml(expr, printer='presentation') == \
'<mfrac><mn>2</mn><mn>5</mn></mfrac>'
assert mathml(expr, printer='presentation', fold_short_frac=True) == \
'<mfrac bevelled="true"><mn>2</mn><mn>5</mn></mfrac>'
assert mathml(expr, printer='presentation', fold_short_frac=False) == \
'<mfrac><mn>2</mn><mn>5</mn></mfrac>'
def test_print_factorials():
assert mpp.doprint(factorial(x)) == '<mrow><mi>x</mi><mo>!</mo></mrow>'
assert mpp.doprint(factorial(x + 1)) == \
'<mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow>'
assert mpp.doprint(factorial2(x)) == '<mrow><mi>x</mi><mo>!!</mo></mrow>'
assert mpp.doprint(factorial2(x + 1)) == \
'<mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>!!</mo></mrow>'
assert mpp.doprint(binomial(x, y)) == \
'<mfenced><mfrac linethickness="0"><mi>x</mi><mi>y</mi></mfrac></mfenced>'
assert mpp.doprint(binomial(4, x + y)) == \
'<mfenced><mfrac linethickness="0"><mn>4</mn><mrow><mi>x</mi>'\
'<mo>+</mo><mi>y</mi></mrow></mfrac></mfenced>'
def test_print_floor():
expr = floor(x)
assert mathml(expr, printer='presentation') == \
'<mrow><mfenced close="⌋" open="⌊"><mi>x</mi></mfenced></mrow>'
def test_print_ceiling():
expr = ceiling(x)
assert mathml(expr, printer='presentation') == \
'<mrow><mfenced close="⌉" open="⌈"><mi>x</mi></mfenced></mrow>'
def test_print_Lambda():
expr = Lambda(x, x+1)
assert mathml(expr, printer='presentation') == \
'<mfenced><mrow><mi>x</mi><mo>↦</mo><mrow><mi>x</mi><mo>+</mo>'\
'<mn>1</mn></mrow></mrow></mfenced>'
expr = Lambda((x, y), x + y)
assert mathml(expr, printer='presentation') == \
'<mfenced><mrow><mrow><mfenced><mi>x</mi><mi>y</mi></mfenced></mrow>'\
'<mo>↦</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mrow></mfenced>'
def test_print_conjugate():
assert mpp.doprint(conjugate(x)) == \
'<menclose notation="top"><mi>x</mi></menclose>'
assert mpp.doprint(conjugate(x + 1)) == \
'<mrow><menclose notation="top"><mi>x</mi></menclose><mo>+</mo><mn>1</mn></mrow>'
def test_print_AccumBounds():
a = Symbol('a', real=True)
assert mpp.doprint(AccumBounds(0, 1)) == '<mfenced close="⟩" open="⟨"><mn>0</mn><mn>1</mn></mfenced>'
assert mpp.doprint(AccumBounds(0, a)) == '<mfenced close="⟩" open="⟨"><mn>0</mn><mi>a</mi></mfenced>'
assert mpp.doprint(AccumBounds(a + 1, a + 2)) == '<mfenced close="⟩" open="⟨"><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>a</mi><mo>+</mo><mn>2</mn></mrow></mfenced>'
def test_print_Float():
assert mpp.doprint(Float(1e100)) == '<mrow><mn>1.0</mn><mo>·</mo><msup><mn>10</mn><mn>100</mn></msup></mrow>'
assert mpp.doprint(Float(1e-100)) == '<mrow><mn>1.0</mn><mo>·</mo><msup><mn>10</mn><mn>-100</mn></msup></mrow>'
assert mpp.doprint(Float(-1e100)) == '<mrow><mn>-1.0</mn><mo>·</mo><msup><mn>10</mn><mn>100</mn></msup></mrow>'
assert mpp.doprint(Float(1.0*oo)) == '<mi>∞</mi>'
assert mpp.doprint(Float(-1.0*oo)) == '<mrow><mo>-</mo><mi>∞</mi></mrow>'
def test_print_different_functions():
assert mpp.doprint(gamma(x)) == '<mrow><mi>Γ</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(lowergamma(x, y)) == '<mrow><mi>γ</mi><mfenced><mi>x</mi><mi>y</mi></mfenced></mrow>'
assert mpp.doprint(uppergamma(x, y)) == '<mrow><mi>Γ</mi><mfenced><mi>x</mi><mi>y</mi></mfenced></mrow>'
assert mpp.doprint(zeta(x)) == '<mrow><mi>ζ</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(zeta(x, y)) == '<mrow><mi>ζ</mi><mfenced><mi>x</mi><mi>y</mi></mfenced></mrow>'
assert mpp.doprint(dirichlet_eta(x)) == '<mrow><mi>η</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(elliptic_k(x)) == '<mrow><mi>Κ</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(totient(x)) == '<mrow><mi>ϕ</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(reduced_totient(x)) == '<mrow><mi>λ</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(primenu(x)) == '<mrow><mi>ν</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(primeomega(x)) == '<mrow><mi>Ω</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(fresnels(x)) == '<mrow><mi>S</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(fresnelc(x)) == '<mrow><mi>C</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mpp.doprint(Heaviside(x)) == '<mrow><mi>Θ</mi><mfenced><mi>x</mi></mfenced></mrow>'
def test_mathml_builtins():
assert mpp.doprint(None) == '<mi>None</mi>'
assert mpp.doprint(true) == '<mi>True</mi>'
assert mpp.doprint(false) == '<mi>False</mi>'
def test_mathml_Range():
assert mpp.doprint(Range(1, 51)) == \
'<mfenced close="}" open="{"><mn>1</mn><mn>2</mn><mi>…</mi><mn>50</mn></mfenced>'
assert mpp.doprint(Range(1, 4)) == \
'<mfenced close="}" open="{"><mn>1</mn><mn>2</mn><mn>3</mn></mfenced>'
assert mpp.doprint(Range(0, 3, 1)) == \
'<mfenced close="}" open="{"><mn>0</mn><mn>1</mn><mn>2</mn></mfenced>'
assert mpp.doprint(Range(0, 30, 1)) == \
'<mfenced close="}" open="{"><mn>0</mn><mn>1</mn><mi>…</mi><mn>29</mn></mfenced>'
assert mpp.doprint(Range(30, 1, -1)) == \
'<mfenced close="}" open="{"><mn>30</mn><mn>29</mn><mi>…</mi>'\
'<mn>2</mn></mfenced>'
assert mpp.doprint(Range(0, oo, 2)) == \
'<mfenced close="}" open="{"><mn>0</mn><mn>2</mn><mi>…</mi></mfenced>'
assert mpp.doprint(Range(oo, -2, -2)) == \
'<mfenced close="}" open="{"><mi>…</mi><mn>2</mn><mn>0</mn></mfenced>'
assert mpp.doprint(Range(-2, -oo, -1)) == \
'<mfenced close="}" open="{"><mn>-2</mn><mn>-3</mn><mi>…</mi></mfenced>'
def test_print_exp():
assert mpp.doprint(exp(x)) == \
'<msup><mi>ⅇ</mi><mi>x</mi></msup>'
assert mpp.doprint(exp(1) + exp(2)) == \
'<mrow><mi>ⅇ</mi><mo>+</mo><msup><mi>ⅇ</mi><mn>2</mn></msup></mrow>'
def test_print_MinMax():
assert mpp.doprint(Min(x, y)) == \
'<mrow><mo>min</mo><mfenced><mi>x</mi><mi>y</mi></mfenced></mrow>'
assert mpp.doprint(Min(x, 2, x**3)) == \
'<mrow><mo>min</mo><mfenced><mn>2</mn><mi>x</mi><msup><mi>x</mi>'\
'<mn>3</mn></msup></mfenced></mrow>'
assert mpp.doprint(Max(x, y)) == \
'<mrow><mo>max</mo><mfenced><mi>x</mi><mi>y</mi></mfenced></mrow>'
assert mpp.doprint(Max(x, 2, x**3)) == \
'<mrow><mo>max</mo><mfenced><mn>2</mn><mi>x</mi><msup><mi>x</mi>'\
'<mn>3</mn></msup></mfenced></mrow>'
def test_mathml_presentation_numbers():
n = Symbol('n')
assert mathml(catalan(n), printer='presentation') == \
'<msub><mi>C</mi><mi>n</mi></msub>'
assert mathml(bernoulli(n), printer='presentation') == \
'<msub><mi>B</mi><mi>n</mi></msub>'
assert mathml(bell(n), printer='presentation') == \
'<msub><mi>B</mi><mi>n</mi></msub>'
assert mathml(fibonacci(n), printer='presentation') == \
'<msub><mi>F</mi><mi>n</mi></msub>'
assert mathml(lucas(n), printer='presentation') == \
'<msub><mi>L</mi><mi>n</mi></msub>'
assert mathml(tribonacci(n), printer='presentation') == \
'<msub><mi>T</mi><mi>n</mi></msub>'
def test_print_matrix_symbol():
A = MatrixSymbol('A', 1, 2)
assert mpp.doprint(A) == '<mi>A</mi>'
assert mp.doprint(A) == '<ci>A</ci>'
assert mathml(A, printer='presentation', mat_symbol_style="bold") == \
'<mi mathvariant="bold">A</mi>'
# No effect in content printer
assert mathml(A, mat_symbol_style="bold") == '<ci>A</ci>'
def test_print_random_symbol():
R = RandomSymbol(Symbol('R'))
assert mpp.doprint(R) == '<mi>R</mi>'
assert mp.doprint(R) == '<ci>R</ci>'
def test_print_IndexedBase():
assert mathml(IndexedBase(a)[b], printer='presentation') == \
'<msub><mi>a</mi><mi>b</mi></msub>'
assert mathml(IndexedBase(a)[b, c, d], printer='presentation') == \
'<msub><mi>a</mi><mfenced><mi>b</mi><mi>c</mi><mi>d</mi></mfenced></msub>'
assert mathml(IndexedBase(a)[b]*IndexedBase(c)[d]*IndexedBase(e),
printer='presentation') == \
'<mrow><msub><mi>a</mi><mi>b</mi></msub><mo>⁢'\
'</mo><msub><mi>c</mi><mi>d</mi></msub><mo>⁢</mo><mi>e</mi></mrow>'
def test_print_Indexed():
assert mathml(IndexedBase(a), printer='presentation') == '<mi>a</mi>'
assert mathml(IndexedBase(a/b), printer='presentation') == \
'<mrow><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow>'
assert mathml(IndexedBase((a, b)), printer='presentation') == \
'<mrow><mfenced><mi>a</mi><mi>b</mi></mfenced></mrow>'
def test_print_MatrixElement():
i, j = symbols('i j')
A = MatrixSymbol('A', i, j)
assert mathml(A[0,0],printer = 'presentation') == \
'<msub><mi>A</mi><mfenced close="" open=""><mn>0</mn><mn>0</mn></mfenced></msub>'
assert mathml(A[i,j], printer = 'presentation') == \
'<msub><mi>A</mi><mfenced close="" open=""><mi>i</mi><mi>j</mi></mfenced></msub>'
assert mathml(A[i*j,0], printer = 'presentation') == \
'<msub><mi>A</mi><mfenced close="" open=""><mrow><mi>i</mi><mo>⁢</mo><mi>j</mi></mrow><mn>0</mn></mfenced></msub>'
def test_print_Vector():
ACS = CoordSys3D('A')
assert mathml(Cross(ACS.i, ACS.j*ACS.x*3 + ACS.k), printer='presentation') == \
'<mrow><msub><mover><mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>×</mo><mfenced><mrow>'\
'<mfenced><mrow><mn>3</mn><mo>⁢</mo><msub>'\
'<mi mathvariant="bold">x</mi><mi mathvariant="bold">A</mi></msub>'\
'</mrow></mfenced><mo>⁢</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>+</mo><msub><mover>'\
'<mi mathvariant="bold">k</mi><mo>^</mo></mover><mi mathvariant="bold">'\
'A</mi></msub></mrow></mfenced></mrow>'
assert mathml(Cross(ACS.i, ACS.j), printer='presentation') == \
'<mrow><msub><mover><mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>×</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow>'
assert mathml(x*Cross(ACS.i, ACS.j), printer='presentation') == \
'<mrow><mi>x</mi><mo>⁢</mo><mrow><msub><mover>'\
'<mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>×</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mrow>'
assert mathml(Cross(x*ACS.i, ACS.j), printer='presentation') == \
'<mrow><mo>-</mo><mrow><msub><mover><mi mathvariant="bold">j</mi>'\
'<mo>^</mo></mover><mi mathvariant="bold">A</mi></msub>'\
'<mo>×</mo><mfenced><mrow><mfenced><mi>x</mi></mfenced>'\
'<mo>⁢</mo><msub><mover><mi mathvariant="bold">i</mi>'\
'<mo>^</mo></mover><mi mathvariant="bold">A</mi></msub></mrow>'\
'</mfenced></mrow></mrow>'
assert mathml(Curl(3*ACS.x*ACS.j), printer='presentation') == \
'<mrow><mo>∇</mo><mo>×</mo><mfenced><mrow><mfenced><mrow>'\
'<mn>3</mn><mo>⁢</mo><msub>'\
'<mi mathvariant="bold">x</mi><mi mathvariant="bold">A</mi></msub>'\
'</mrow></mfenced><mo>⁢</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mfenced></mrow>'
assert mathml(Curl(3*x*ACS.x*ACS.j), printer='presentation') == \
'<mrow><mo>∇</mo><mo>×</mo><mfenced><mrow><mfenced><mrow>'\
'<mn>3</mn><mo>⁢</mo><msub><mi mathvariant="bold">x'\
'</mi><mi mathvariant="bold">A</mi></msub><mo>⁢</mo>'\
'<mi>x</mi></mrow></mfenced><mo>⁢</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mfenced></mrow>'
assert mathml(x*Curl(3*ACS.x*ACS.j), printer='presentation') == \
'<mrow><mi>x</mi><mo>⁢</mo><mrow><mo>∇</mo>'\
'<mo>×</mo><mfenced><mrow><mfenced><mrow><mn>3</mn>'\
'<mo>⁢</mo><msub><mi mathvariant="bold">x</mi>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mfenced>'\
'<mo>⁢</mo><msub><mover><mi mathvariant="bold">j</mi>'\
'<mo>^</mo></mover><mi mathvariant="bold">A</mi></msub></mrow>'\
'</mfenced></mrow></mrow>'
assert mathml(Curl(3*x*ACS.x*ACS.j + ACS.i), printer='presentation') == \
'<mrow><mo>∇</mo><mo>×</mo><mfenced><mrow><msub><mover>'\
'<mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>+</mo><mfenced><mrow>'\
'<mn>3</mn><mo>⁢</mo><msub><mi mathvariant="bold">x'\
'</mi><mi mathvariant="bold">A</mi></msub><mo>⁢</mo>'\
'<mi>x</mi></mrow></mfenced><mo>⁢</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mfenced></mrow>'
assert mathml(Divergence(3*ACS.x*ACS.j), printer='presentation') == \
'<mrow><mo>∇</mo><mo>·</mo><mfenced><mrow><mfenced><mrow>'\
'<mn>3</mn><mo>⁢</mo><msub><mi mathvariant="bold">x'\
'</mi><mi mathvariant="bold">A</mi></msub></mrow></mfenced>'\
'<mo>⁢</mo><msub><mover><mi mathvariant="bold">j</mi>'\
'<mo>^</mo></mover><mi mathvariant="bold">A</mi></msub></mrow></mfenced></mrow>'
assert mathml(x*Divergence(3*ACS.x*ACS.j), printer='presentation') == \
'<mrow><mi>x</mi><mo>⁢</mo><mrow><mo>∇</mo>'\
'<mo>·</mo><mfenced><mrow><mfenced><mrow><mn>3</mn>'\
'<mo>⁢</mo><msub><mi mathvariant="bold">x</mi>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mfenced>'\
'<mo>⁢</mo><msub><mover><mi mathvariant="bold">j</mi>'\
'<mo>^</mo></mover><mi mathvariant="bold">A</mi></msub></mrow>'\
'</mfenced></mrow></mrow>'
assert mathml(Divergence(3*x*ACS.x*ACS.j + ACS.i), printer='presentation') == \
'<mrow><mo>∇</mo><mo>·</mo><mfenced><mrow><msub><mover>'\
'<mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>+</mo><mfenced><mrow>'\
'<mn>3</mn><mo>⁢</mo><msub>'\
'<mi mathvariant="bold">x</mi><mi mathvariant="bold">A</mi></msub>'\
'<mo>⁢</mo><mi>x</mi></mrow></mfenced>'\
'<mo>⁢</mo><msub><mover><mi mathvariant="bold">j</mi>'\
'<mo>^</mo></mover><mi mathvariant="bold">A</mi></msub></mrow></mfenced></mrow>'
assert mathml(Dot(ACS.i, ACS.j*ACS.x*3+ACS.k), printer='presentation') == \
'<mrow><msub><mover><mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>·</mo><mfenced><mrow>'\
'<mfenced><mrow><mn>3</mn><mo>⁢</mo><msub>'\
'<mi mathvariant="bold">x</mi><mi mathvariant="bold">A</mi></msub>'\
'</mrow></mfenced><mo>⁢</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>+</mo><msub><mover>'\
'<mi mathvariant="bold">k</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mfenced></mrow>'
assert mathml(Dot(ACS.i, ACS.j), printer='presentation') == \
'<mrow><msub><mover><mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>·</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow>'
assert mathml(Dot(x*ACS.i, ACS.j), printer='presentation') == \
'<mrow><msub><mover><mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>·</mo><mfenced><mrow>'\
'<mfenced><mi>x</mi></mfenced><mo>⁢</mo><msub><mover>'\
'<mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mfenced></mrow>'
assert mathml(x*Dot(ACS.i, ACS.j), printer='presentation') == \
'<mrow><mi>x</mi><mo>⁢</mo><mrow><msub><mover>'\
'<mi mathvariant="bold">i</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub><mo>·</mo><msub><mover>'\
'<mi mathvariant="bold">j</mi><mo>^</mo></mover>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mrow>'
assert mathml(Gradient(ACS.x), printer='presentation') == \
'<mrow><mo>∇</mo><msub><mi mathvariant="bold">x</mi>'\
'<mi mathvariant="bold">A</mi></msub></mrow>'
assert mathml(Gradient(ACS.x + 3*ACS.y), printer='presentation') == \
'<mrow><mo>∇</mo><mfenced><mrow><msub><mi mathvariant="bold">'\
'x</mi><mi mathvariant="bold">A</mi></msub><mo>+</mo><mrow><mn>3</mn>'\
'<mo>⁢</mo><msub><mi mathvariant="bold">y</mi>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mrow></mfenced></mrow>'
assert mathml(x*Gradient(ACS.x), printer='presentation') == \
'<mrow><mi>x</mi><mo>⁢</mo><mrow><mo>∇</mo>'\
'<msub><mi mathvariant="bold">x</mi><mi mathvariant="bold">A</mi>'\
'</msub></mrow></mrow>'
assert mathml(Gradient(x*ACS.x), printer='presentation') == \
'<mrow><mo>∇</mo><mfenced><mrow><msub><mi mathvariant="bold">'\
'x</mi><mi mathvariant="bold">A</mi></msub><mo>⁢</mo>'\
'<mi>x</mi></mrow></mfenced></mrow>'
assert mathml(Cross(ACS.x, ACS.z) + Cross(ACS.z, ACS.x), printer='presentation') == \
'<mover><mi mathvariant="bold">0</mi><mo>^</mo></mover>'
assert mathml(Cross(ACS.z, ACS.x), printer='presentation') == \
'<mrow><mo>-</mo><mrow><msub><mi mathvariant="bold">x</mi>'\
'<mi mathvariant="bold">A</mi></msub><mo>×</mo><msub>'\
'<mi mathvariant="bold">z</mi><mi mathvariant="bold">A</mi></msub></mrow></mrow>'
assert mathml(Laplacian(ACS.x), printer='presentation') == \
'<mrow><mo>∆</mo><msub><mi mathvariant="bold">x</mi>'\
'<mi mathvariant="bold">A</mi></msub></mrow>'
assert mathml(Laplacian(ACS.x + 3*ACS.y), printer='presentation') == \
'<mrow><mo>∆</mo><mfenced><mrow><msub><mi mathvariant="bold">'\
'x</mi><mi mathvariant="bold">A</mi></msub><mo>+</mo><mrow><mn>3</mn>'\
'<mo>⁢</mo><msub><mi mathvariant="bold">y</mi>'\
'<mi mathvariant="bold">A</mi></msub></mrow></mrow></mfenced></mrow>'
assert mathml(x*Laplacian(ACS.x), printer='presentation') == \
'<mrow><mi>x</mi><mo>⁢</mo><mrow><mo>∆</mo>'\
'<msub><mi mathvariant="bold">x</mi><mi mathvariant="bold">A</mi>'\
'</msub></mrow></mrow>'
assert mathml(Laplacian(x*ACS.x), printer='presentation') == \
'<mrow><mo>∆</mo><mfenced><mrow><msub><mi mathvariant="bold">'\
'x</mi><mi mathvariant="bold">A</mi></msub><mo>⁢</mo>'\
'<mi>x</mi></mrow></mfenced></mrow>'
def test_print_elliptic_f():
assert mathml(elliptic_f(x, y), printer = 'presentation') == \
'<mrow><mi>𝖥</mi><mfenced separators="|"><mi>x</mi><mi>y</mi></mfenced></mrow>'
assert mathml(elliptic_f(x/y, y), printer = 'presentation') == \
'<mrow><mi>𝖥</mi><mfenced separators="|"><mrow><mfrac><mi>x</mi><mi>y</mi></mfrac></mrow><mi>y</mi></mfenced></mrow>'
def test_print_elliptic_e():
assert mathml(elliptic_e(x), printer = 'presentation') == \
'<mrow><mi>𝖤</mi><mfenced separators="|"><mi>x</mi></mfenced></mrow>'
assert mathml(elliptic_e(x, y), printer = 'presentation') == \
'<mrow><mi>𝖤</mi><mfenced separators="|"><mi>x</mi><mi>y</mi></mfenced></mrow>'
def test_print_elliptic_pi():
assert mathml(elliptic_pi(x, y), printer = 'presentation') == \
'<mrow><mi>𝛱</mi><mfenced separators="|"><mi>x</mi><mi>y</mi></mfenced></mrow>'
assert mathml(elliptic_pi(x, y, z), printer = 'presentation') == \
'<mrow><mi>𝛱</mi><mfenced separators=";|"><mi>x</mi><mi>y</mi><mi>z</mi></mfenced></mrow>'
def test_print_Ei():
assert mathml(Ei(x), printer = 'presentation') == \
'<mrow><mi>Ei</mi><mfenced><mi>x</mi></mfenced></mrow>'
assert mathml(Ei(x**y), printer = 'presentation') == \
'<mrow><mi>Ei</mi><mfenced><msup><mi>x</mi><mi>y</mi></msup></mfenced></mrow>'
def test_print_expint():
assert mathml(expint(x, y), printer = 'presentation') == \
'<mrow><msub><mo>E</mo><mi>x</mi></msub><mfenced><mi>y</mi></mfenced></mrow>'
assert mathml(expint(IndexedBase(x)[1], IndexedBase(x)[2]), printer = 'presentation') == \
'<mrow><msub><mo>E</mo><msub><mi>x</mi><mn>1</mn></msub></msub><mfenced><msub><mi>x</mi><mn>2</mn></msub></mfenced></mrow>'
def test_print_jacobi():
assert mathml(jacobi(n, a, b, x), printer = 'presentation') == \
'<mrow><msubsup><mo>P</mo><mi>n</mi><mfenced><mi>a</mi><mi>b</mi></mfenced></msubsup><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_gegenbauer():
assert mathml(gegenbauer(n, a, x), printer = 'presentation') == \
'<mrow><msubsup><mo>C</mo><mi>n</mi><mfenced><mi>a</mi></mfenced></msubsup><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_chebyshevt():
assert mathml(chebyshevt(n, x), printer = 'presentation') == \
'<mrow><msub><mo>T</mo><mi>n</mi></msub><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_chebyshevu():
assert mathml(chebyshevu(n, x), printer = 'presentation') == \
'<mrow><msub><mo>U</mo><mi>n</mi></msub><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_legendre():
assert mathml(legendre(n, x), printer = 'presentation') == \
'<mrow><msub><mo>P</mo><mi>n</mi></msub><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_assoc_legendre():
assert mathml(assoc_legendre(n, a, x), printer = 'presentation') == \
'<mrow><msubsup><mo>P</mo><mi>n</mi><mfenced><mi>a</mi></mfenced></msubsup><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_laguerre():
assert mathml(laguerre(n, x), printer = 'presentation') == \
'<mrow><msub><mo>L</mo><mi>n</mi></msub><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_assoc_laguerre():
assert mathml(assoc_laguerre(n, a, x), printer = 'presentation') == \
'<mrow><msubsup><mo>L</mo><mi>n</mi><mfenced><mi>a</mi></mfenced></msubsup><mfenced><mi>x</mi></mfenced></mrow>'
def test_print_hermite():
assert mathml(hermite(n, x), printer = 'presentation') == \
'<mrow><msub><mo>H</mo><mi>n</mi></msub><mfenced><mi>x</mi></mfenced></mrow>'
def test_mathml_SingularityFunction():
assert mathml(SingularityFunction(x, 4, 5), printer='presentation') == \
'<msup><mfenced close="⟩" open="⟨"><mrow><mi>x</mi>' \
'<mo>-</mo><mn>4</mn></mrow></mfenced><mn>5</mn></msup>'
assert mathml(SingularityFunction(x, -3, 4), printer='presentation') == \
'<msup><mfenced close="⟩" open="⟨"><mrow><mi>x</mi>' \
'<mo>+</mo><mn>3</mn></mrow></mfenced><mn>4</mn></msup>'
assert mathml(SingularityFunction(x, 0, 4), printer='presentation') == \
'<msup><mfenced close="⟩" open="⟨"><mi>x</mi></mfenced>' \
'<mn>4</mn></msup>'
assert mathml(SingularityFunction(x, a, n), printer='presentation') == \
'<msup><mfenced close="⟩" open="⟨"><mrow><mrow>' \
'<mo>-</mo><mi>a</mi></mrow><mo>+</mo><mi>x</mi></mrow></mfenced>' \
'<mi>n</mi></msup>'
assert mathml(SingularityFunction(x, 4, -2), printer='presentation') == \
'<msup><mfenced close="⟩" open="⟨"><mrow><mi>x</mi>' \
'<mo>-</mo><mn>4</mn></mrow></mfenced><mn>-2</mn></msup>'
assert mathml(SingularityFunction(x, 4, -1), printer='presentation') == \
'<msup><mfenced close="⟩" open="⟨"><mrow><mi>x</mi>' \
'<mo>-</mo><mn>4</mn></mrow></mfenced><mn>-1</mn></msup>'
def test_mathml_matrix_functions():
from sympy.matrices import MatrixSymbol, Adjoint, Inverse, Transpose
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert mathml(Adjoint(X), printer='presentation') == \
'<msup><mi>X</mi><mo>†</mo></msup>'
assert mathml(Adjoint(X + Y), printer='presentation') == \
'<msup><mfenced><mrow><mi>X</mi><mo>+</mo><mi>Y</mi></mrow></mfenced><mo>†</mo></msup>'
assert mathml(Adjoint(X) + Adjoint(Y), printer='presentation') == \
'<mrow><msup><mi>X</mi><mo>†</mo></msup><mo>+</mo><msup>' \
'<mi>Y</mi><mo>†</mo></msup></mrow>'
assert mathml(Adjoint(X*Y), printer='presentation') == \
'<msup><mfenced><mrow><mi>X</mi><mo>⁢</mo>' \
'<mi>Y</mi></mrow></mfenced><mo>†</mo></msup>'
assert mathml(Adjoint(Y)*Adjoint(X), printer='presentation') == \
'<mrow><msup><mi>Y</mi><mo>†</mo></msup><mo>⁢' \
'</mo><msup><mi>X</mi><mo>†</mo></msup></mrow>'
assert mathml(Adjoint(X**2), printer='presentation') == \
'<msup><mfenced><msup><mi>X</mi><mn>2</mn></msup></mfenced><mo>†</mo></msup>'
assert mathml(Adjoint(X)**2, printer='presentation') == \
'<msup><mfenced><msup><mi>X</mi><mo>†</mo></msup></mfenced><mn>2</mn></msup>'
assert mathml(Adjoint(Inverse(X)), printer='presentation') == \
'<msup><mfenced><msup><mi>X</mi><mn>-1</mn></msup></mfenced><mo>†</mo></msup>'
assert mathml(Inverse(Adjoint(X)), printer='presentation') == \
'<msup><mfenced><msup><mi>X</mi><mo>†</mo></msup></mfenced><mn>-1</mn></msup>'
assert mathml(Adjoint(Transpose(X)), printer='presentation') == \
'<msup><mfenced><msup><mi>X</mi><mo>T</mo></msup></mfenced><mo>†</mo></msup>'
assert mathml(Transpose(Adjoint(X)), printer='presentation') == \
'<msup><mfenced><msup><mi>X</mi><mo>†</mo></msup></mfenced><mo>T</mo></msup>'
assert mathml(Transpose(Adjoint(X) + Y), printer='presentation') == \
'<msup><mfenced><mrow><msup><mi>X</mi><mo>†</mo></msup>' \
'<mo>+</mo><mi>Y</mi></mrow></mfenced><mo>T</mo></msup>'
assert mathml(Transpose(X), printer='presentation') == \
'<msup><mi>X</mi><mo>T</mo></msup>'
assert mathml(Transpose(X + Y), printer='presentation') == \
'<msup><mfenced><mrow><mi>X</mi><mo>+</mo><mi>Y</mi></mrow></mfenced><mo>T</mo></msup>'
def test_mathml_special_matrices():
from sympy.matrices import Identity, ZeroMatrix
assert mathml(Identity(4), printer='presentation') == '<mi>𝕀</mi>'
assert mathml(ZeroMatrix(2, 2), printer='presentation') == '<mn>𝟘</mn>'
|
4af30aa1786dfcc5162f4bf7f938a4996278d6d72eaa56851cb37c5cd686cebb
|
from sympy.core import (S, pi, oo, symbols, Function,
Rational, Integer, Tuple, Derivative)
from sympy.integrals import Integral
from sympy.concrete import Sum
from sympy.functions import exp, sin, cos, conjugate, Max, Min
from sympy import mathematica_code as mcode
x, y, z = symbols('x,y,z')
f = Function('f')
def test_Integer():
assert mcode(Integer(67)) == "67"
assert mcode(Integer(-1)) == "-1"
def test_Rational():
assert mcode(Rational(3, 7)) == "3/7"
assert mcode(Rational(18, 9)) == "2"
assert mcode(Rational(3, -7)) == "-3/7"
assert mcode(Rational(-3, -7)) == "3/7"
assert mcode(x + Rational(3, 7)) == "x + 3/7"
assert mcode(Rational(3, 7)*x) == "(3/7)*x"
def test_Function():
assert mcode(f(x, y, z)) == "f[x, y, z]"
assert mcode(sin(x) ** cos(x)) == "Sin[x]^Cos[x]"
assert mcode(conjugate(x)) == "Conjugate[x]"
assert mcode(Max(x,y,z)*Min(y,z)) == "Max[x, y, z]*Min[y, z]"
def test_Pow():
assert mcode(x**3) == "x^3"
assert mcode(x**(y**3)) == "x^(y^3)"
assert mcode(1/(f(x)*3.5)**(x - y**x)/(x**2 + y)) == \
"(3.5*f[x])^(-x + y^x)/(x^2 + y)"
assert mcode(x**-1.0) == 'x^(-1.0)'
assert mcode(x**Rational(2, 3)) == 'x^(2/3)'
def test_Mul():
A, B, C, D = symbols('A B C D', commutative=False)
assert mcode(x*y*z) == "x*y*z"
assert mcode(x*y*A) == "x*y*A"
assert mcode(x*y*A*B) == "x*y*A**B"
assert mcode(x*y*A*B*C) == "x*y*A**B**C"
assert mcode(x*A*B*(C + D)*A*y) == "x*y*A**B**(C + D)**A"
def test_constants():
assert mcode(S.Zero) == "0"
assert mcode(S.One) == "1"
assert mcode(S.NegativeOne) == "-1"
assert mcode(S.Half) == "1/2"
assert mcode(S.ImaginaryUnit) == "I"
assert mcode(oo) == "Infinity"
assert mcode(S.NegativeInfinity) == "-Infinity"
assert mcode(S.ComplexInfinity) == "ComplexInfinity"
assert mcode(S.NaN) == "Indeterminate"
assert mcode(S.Exp1) == "E"
assert mcode(pi) == "Pi"
assert mcode(S.GoldenRatio) == "GoldenRatio"
assert mcode(S.TribonacciConstant) == \
"1/3 + (1/3)*(19 - 3*33^(1/2))^(1/3) + " \
"(1/3)*(3*33^(1/2) + 19)^(1/3)"
assert mcode(S.EulerGamma) == "EulerGamma"
assert mcode(S.Catalan) == "Catalan"
def test_containers():
assert mcode([1, 2, 3, [4, 5, [6, 7]], 8, [9, 10], 11]) == \
"{1, 2, 3, {4, 5, {6, 7}}, 8, {9, 10}, 11}"
assert mcode((1, 2, (3, 4))) == "{1, 2, {3, 4}}"
assert mcode([1]) == "{1}"
assert mcode((1,)) == "{1}"
assert mcode(Tuple(*[1, 2, 3])) == "{1, 2, 3}"
def test_matrices():
from sympy.matrices import MutableDenseMatrix, MutableSparseMatrix, \
ImmutableDenseMatrix, ImmutableSparseMatrix
A = MutableDenseMatrix(
[[1, -1, 0, 0],
[0, 1, -1, 0],
[0, 0, 1, -1],
[0, 0, 0, 1]]
)
B = MutableSparseMatrix(A)
C = ImmutableDenseMatrix(A)
D = ImmutableSparseMatrix(A)
assert mcode(C) == mcode(A) == \
"{{1, -1, 0, 0}, " \
"{0, 1, -1, 0}, " \
"{0, 0, 1, -1}, " \
"{0, 0, 0, 1}}"
assert mcode(D) == mcode(B) == \
"SparseArray[{" \
"{1, 1} -> 1, {1, 2} -> -1, {2, 2} -> 1, {2, 3} -> -1, " \
"{3, 3} -> 1, {3, 4} -> -1, {4, 4} -> 1" \
"}, {4, 4}]"
# Trivial cases of matrices
assert mcode(MutableDenseMatrix(0, 0, [])) == '{}'
assert mcode(MutableSparseMatrix(0, 0, [])) == 'SparseArray[{}, {0, 0}]'
assert mcode(MutableDenseMatrix(0, 3, [])) == '{}'
assert mcode(MutableSparseMatrix(0, 3, [])) == 'SparseArray[{}, {0, 3}]'
assert mcode(MutableDenseMatrix(3, 0, [])) == '{{}, {}, {}}'
assert mcode(MutableSparseMatrix(3, 0, [])) == 'SparseArray[{}, {3, 0}]'
def test_NDArray():
from sympy.tensor.array import (
MutableDenseNDimArray, ImmutableDenseNDimArray,
MutableSparseNDimArray, ImmutableSparseNDimArray)
example = MutableDenseNDimArray(
[[[1, 2, 3, 4],
[5, 6, 7, 8],
[9, 10, 11, 12]],
[[13, 14, 15, 16],
[17, 18, 19, 20],
[21, 22, 23, 24]]]
)
assert mcode(example) == \
"{{{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}, " \
"{{13, 14, 15, 16}, {17, 18, 19, 20}, {21, 22, 23, 24}}}"
example = ImmutableDenseNDimArray(example)
assert mcode(example) == \
"{{{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}, " \
"{{13, 14, 15, 16}, {17, 18, 19, 20}, {21, 22, 23, 24}}}"
example = MutableSparseNDimArray(example)
assert mcode(example) == \
"SparseArray[{" \
"{1, 1, 1} -> 1, {1, 1, 2} -> 2, {1, 1, 3} -> 3, " \
"{1, 1, 4} -> 4, {1, 2, 1} -> 5, {1, 2, 2} -> 6, " \
"{1, 2, 3} -> 7, {1, 2, 4} -> 8, {1, 3, 1} -> 9, " \
"{1, 3, 2} -> 10, {1, 3, 3} -> 11, {1, 3, 4} -> 12, " \
"{2, 1, 1} -> 13, {2, 1, 2} -> 14, {2, 1, 3} -> 15, " \
"{2, 1, 4} -> 16, {2, 2, 1} -> 17, {2, 2, 2} -> 18, " \
"{2, 2, 3} -> 19, {2, 2, 4} -> 20, {2, 3, 1} -> 21, " \
"{2, 3, 2} -> 22, {2, 3, 3} -> 23, {2, 3, 4} -> 24" \
"}, {2, 3, 4}]"
example = ImmutableSparseNDimArray(example)
assert mcode(example) == \
"SparseArray[{" \
"{1, 1, 1} -> 1, {1, 1, 2} -> 2, {1, 1, 3} -> 3, " \
"{1, 1, 4} -> 4, {1, 2, 1} -> 5, {1, 2, 2} -> 6, " \
"{1, 2, 3} -> 7, {1, 2, 4} -> 8, {1, 3, 1} -> 9, " \
"{1, 3, 2} -> 10, {1, 3, 3} -> 11, {1, 3, 4} -> 12, " \
"{2, 1, 1} -> 13, {2, 1, 2} -> 14, {2, 1, 3} -> 15, " \
"{2, 1, 4} -> 16, {2, 2, 1} -> 17, {2, 2, 2} -> 18, " \
"{2, 2, 3} -> 19, {2, 2, 4} -> 20, {2, 3, 1} -> 21, " \
"{2, 3, 2} -> 22, {2, 3, 3} -> 23, {2, 3, 4} -> 24" \
"}, {2, 3, 4}]"
def test_Integral():
assert mcode(Integral(sin(sin(x)), x)) == "Hold[Integrate[Sin[Sin[x]], x]]"
assert mcode(Integral(exp(-x**2 - y**2),
(x, -oo, oo),
(y, -oo, oo))) == \
"Hold[Integrate[Exp[-x^2 - y^2], {x, -Infinity, Infinity}, " \
"{y, -Infinity, Infinity}]]"
def test_Derivative():
assert mcode(Derivative(sin(x), x)) == "Hold[D[Sin[x], x]]"
assert mcode(Derivative(x, x)) == "Hold[D[x, x]]"
assert mcode(Derivative(sin(x)*y**4, x, 2)) == "Hold[D[y^4*Sin[x], {x, 2}]]"
assert mcode(Derivative(sin(x)*y**4, x, y, x)) == "Hold[D[y^4*Sin[x], x, y, x]]"
assert mcode(Derivative(sin(x)*y**4, x, y, 3, x)) == "Hold[D[y^4*Sin[x], x, {y, 3}, x]]"
def test_Sum():
assert mcode(Sum(sin(x), (x, 0, 10))) == "Hold[Sum[Sin[x], {x, 0, 10}]]"
assert mcode(Sum(exp(-x**2 - y**2),
(x, -oo, oo),
(y, -oo, oo))) == \
"Hold[Sum[Exp[-x^2 - y^2], {x, -Infinity, Infinity}, " \
"{y, -Infinity, Infinity}]]"
def test_comment():
from sympy.printing.mathematica import MCodePrinter
assert MCodePrinter()._get_comment("Hello World") == \
"(* Hello World *)"
def test_userfuncs():
# Dictionary mutation test
some_function = symbols("some_function", cls=Function)
my_user_functions = {"some_function": "SomeFunction"}
assert mcode(
some_function(z),
user_functions=my_user_functions) == \
'SomeFunction[z]'
assert mcode(
some_function(z),
user_functions=my_user_functions) == \
'SomeFunction[z]'
# List argument test
my_user_functions = \
{"some_function": [(lambda x: True, "SomeOtherFunction")]}
assert mcode(
some_function(z),
user_functions=my_user_functions) == \
'SomeOtherFunction[z]'
|
b5828b4fd2eea144686d20d5c7bc80ef84754a60164f31662df4a88d736bd457
|
# -*- coding: utf-8 -*-
from sympy import (
Add, And, Basic, Derivative, Dict, Eq, Equivalent, FF,
FiniteSet, Function, Ge, Gt, I, Implies, Integral, SingularityFunction,
Lambda, Le, Limit, Lt, Matrix, Mul, Nand, Ne, Nor, Not, O, Or,
Pow, Product, QQ, RR, Rational, Ray, rootof, RootSum, S,
Segment, Subs, Sum, Symbol, Tuple, Trace, Xor, ZZ, conjugate,
groebner, oo, pi, symbols, ilex, grlex, Range, Contains,
SeqPer, SeqFormula, SeqAdd, SeqMul, fourier_series, fps, ITE,
Complement, Interval, Intersection, Union, EulerGamma, GoldenRatio)
from sympy.codegen.ast import (Assignment, AddAugmentedAssignment,
SubAugmentedAssignment, MulAugmentedAssignment, DivAugmentedAssignment, ModAugmentedAssignment)
from sympy.core.compatibility import range, u_decode as u, PY3
from sympy.core.expr import UnevaluatedExpr
from sympy.core.trace import Tr
from sympy.functions import (Abs, Chi, Ci, Ei, KroneckerDelta,
Piecewise, Shi, Si, atan2, beta, binomial, catalan, ceiling, cos,
euler, exp, expint, factorial, factorial2, floor, gamma, hyper, log,
meijerg, sin, sqrt, subfactorial, tan, uppergamma, lerchphi,
elliptic_k, elliptic_f, elliptic_e, elliptic_pi, DiracDelta, bell,
bernoulli, fibonacci, tribonacci, lucas)
from sympy.matrices import Adjoint, Inverse, MatrixSymbol, Transpose, KroneckerProduct
from sympy.physics import mechanics
from sympy.physics.units import joule, degree
from sympy.printing.pretty import pprint, pretty as xpretty
from sympy.printing.pretty.pretty_symbology import center_accent
from sympy.sets import ImageSet
from sympy.sets.setexpr import SetExpr
from sympy.tensor.array import (ImmutableDenseNDimArray, ImmutableSparseNDimArray,
MutableDenseNDimArray, MutableSparseNDimArray, tensorproduct)
from sympy.tensor.functions import TensorProduct
from sympy.tensor.tensor import (TensorIndexType, tensor_indices, tensorhead,
TensorElement)
from sympy.utilities.pytest import raises, XFAIL
from sympy.vector import CoordSys3D, Gradient, Curl, Divergence, Dot, Cross, Laplacian
import sympy as sym
class lowergamma(sym.lowergamma):
pass # testing notation inheritance by a subclass with same name
a, b, c, d, x, y, z, k, n = symbols('a,b,c,d,x,y,z,k,n')
f = Function("f")
th = Symbol('theta')
ph = Symbol('phi')
"""
Expressions whose pretty-printing is tested here:
(A '#' to the right of an expression indicates that its various acceptable
orderings are accounted for by the tests.)
BASIC EXPRESSIONS:
oo
(x**2)
1/x
y*x**-2
x**Rational(-5,2)
(-2)**x
Pow(3, 1, evaluate=False)
(x**2 + x + 1) #
1-x #
1-2*x #
x/y
-x/y
(x+2)/y #
(1+x)*y #3
-5*x/(x+10) # correct placement of negative sign
1 - Rational(3,2)*(x+1)
-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5) # issue 5524
ORDERING:
x**2 + x + 1
1 - x
1 - 2*x
2*x**4 + y**2 - x**2 + y**3
RELATIONAL:
Eq(x, y)
Lt(x, y)
Gt(x, y)
Le(x, y)
Ge(x, y)
Ne(x/(y+1), y**2) #
RATIONAL NUMBERS:
y*x**-2
y**Rational(3,2) * x**Rational(-5,2)
sin(x)**3/tan(x)**2
FUNCTIONS (ABS, CONJ, EXP, FUNCTION BRACES, FACTORIAL, FLOOR, CEILING):
(2*x + exp(x)) #
Abs(x)
Abs(x/(x**2+1)) #
Abs(1 / (y - Abs(x)))
factorial(n)
factorial(2*n)
subfactorial(n)
subfactorial(2*n)
factorial(factorial(factorial(n)))
factorial(n+1) #
conjugate(x)
conjugate(f(x+1)) #
f(x)
f(x, y)
f(x/(y+1), y) #
f(x**x**x**x**x**x)
sin(x)**2
conjugate(a+b*I)
conjugate(exp(a+b*I))
conjugate( f(1 + conjugate(f(x))) ) #
f(x/(y+1), y) # denom of first arg
floor(1 / (y - floor(x)))
ceiling(1 / (y - ceiling(x)))
SQRT:
sqrt(2)
2**Rational(1,3)
2**Rational(1,1000)
sqrt(x**2 + 1)
(1 + sqrt(5))**Rational(1,3)
2**(1/x)
sqrt(2+pi)
(2+(1+x**2)/(2+x))**Rational(1,4)+(1+x**Rational(1,1000))/sqrt(3+x**2)
DERIVATIVES:
Derivative(log(x), x, evaluate=False)
Derivative(log(x), x, evaluate=False) + x #
Derivative(log(x) + x**2, x, y, evaluate=False)
Derivative(2*x*y, y, x, evaluate=False) + x**2 #
beta(alpha).diff(alpha)
INTEGRALS:
Integral(log(x), x)
Integral(x**2, x)
Integral((sin(x))**2 / (tan(x))**2)
Integral(x**(2**x), x)
Integral(x**2, (x,1,2))
Integral(x**2, (x,Rational(1,2),10))
Integral(x**2*y**2, x,y)
Integral(x**2, (x, None, 1))
Integral(x**2, (x, 1, None))
Integral(sin(th)/cos(ph), (th,0,pi), (ph, 0, 2*pi))
MATRICES:
Matrix([[x**2+1, 1], [y, x+y]]) #
Matrix([[x/y, y, th], [0, exp(I*k*ph), 1]])
PIECEWISE:
Piecewise((x,x<1),(x**2,True))
ITE:
ITE(x, y, z)
SEQUENCES (TUPLES, LISTS, DICTIONARIES):
()
[]
{}
(1/x,)
[x**2, 1/x, x, y, sin(th)**2/cos(ph)**2]
(x**2, 1/x, x, y, sin(th)**2/cos(ph)**2)
{x: sin(x)}
{1/x: 1/y, x: sin(x)**2} #
[x**2]
(x**2,)
{x**2: 1}
LIMITS:
Limit(x, x, oo)
Limit(x**2, x, 0)
Limit(1/x, x, 0)
Limit(sin(x)/x, x, 0)
UNITS:
joule => kg*m**2/s
SUBS:
Subs(f(x), x, ph**2)
Subs(f(x).diff(x), x, 0)
Subs(f(x).diff(x)/y, (x, y), (0, Rational(1, 2)))
ORDER:
O(1)
O(1/x)
O(x**2 + y**2)
"""
def pretty(expr, order=None):
"""ASCII pretty-printing"""
return xpretty(expr, order=order, use_unicode=False, wrap_line=False)
def upretty(expr, order=None):
"""Unicode pretty-printing"""
return xpretty(expr, order=order, use_unicode=True, wrap_line=False)
def test_pretty_ascii_str():
assert pretty( 'xxx' ) == 'xxx'
assert pretty( "xxx" ) == 'xxx'
assert pretty( 'xxx\'xxx' ) == 'xxx\'xxx'
assert pretty( 'xxx"xxx' ) == 'xxx\"xxx'
assert pretty( 'xxx\"xxx' ) == 'xxx\"xxx'
assert pretty( "xxx'xxx" ) == 'xxx\'xxx'
assert pretty( "xxx\'xxx" ) == 'xxx\'xxx'
assert pretty( "xxx\"xxx" ) == 'xxx\"xxx'
assert pretty( "xxx\"xxx\'xxx" ) == 'xxx"xxx\'xxx'
assert pretty( "xxx\nxxx" ) == 'xxx\nxxx'
def test_pretty_unicode_str():
assert pretty( u'xxx' ) == u'xxx'
assert pretty( u'xxx' ) == u'xxx'
assert pretty( u'xxx\'xxx' ) == u'xxx\'xxx'
assert pretty( u'xxx"xxx' ) == u'xxx\"xxx'
assert pretty( u'xxx\"xxx' ) == u'xxx\"xxx'
assert pretty( u"xxx'xxx" ) == u'xxx\'xxx'
assert pretty( u"xxx\'xxx" ) == u'xxx\'xxx'
assert pretty( u"xxx\"xxx" ) == u'xxx\"xxx'
assert pretty( u"xxx\"xxx\'xxx" ) == u'xxx"xxx\'xxx'
assert pretty( u"xxx\nxxx" ) == u'xxx\nxxx'
def test_upretty_greek():
assert upretty( oo ) == u'∞'
assert upretty( Symbol('alpha^+_1') ) == u'α⁺₁'
assert upretty( Symbol('beta') ) == u'β'
assert upretty(Symbol('lambda')) == u'λ'
def test_upretty_multiindex():
assert upretty( Symbol('beta12') ) == u'β₁₂'
assert upretty( Symbol('Y00') ) == u'Y₀₀'
assert upretty( Symbol('Y_00') ) == u'Y₀₀'
assert upretty( Symbol('F^+-') ) == u'F⁺⁻'
def test_upretty_sub_super():
assert upretty( Symbol('beta_1_2') ) == u'β₁ ₂'
assert upretty( Symbol('beta^1^2') ) == u'β¹ ²'
assert upretty( Symbol('beta_1^2') ) == u'β²₁'
assert upretty( Symbol('beta_10_20') ) == u'β₁₀ ₂₀'
assert upretty( Symbol('beta_ax_gamma^i') ) == u'βⁱₐₓ ᵧ'
assert upretty( Symbol("F^1^2_3_4") ) == u'F¹ ²₃ ₄'
assert upretty( Symbol("F_1_2^3^4") ) == u'F³ ⁴₁ ₂'
assert upretty( Symbol("F_1_2_3_4") ) == u'F₁ ₂ ₃ ₄'
assert upretty( Symbol("F^1^2^3^4") ) == u'F¹ ² ³ ⁴'
def test_upretty_subs_missing_in_24():
assert upretty( Symbol('F_beta') ) == u'Fᵦ'
assert upretty( Symbol('F_gamma') ) == u'Fᵧ'
assert upretty( Symbol('F_rho') ) == u'Fᵨ'
assert upretty( Symbol('F_phi') ) == u'Fᵩ'
assert upretty( Symbol('F_chi') ) == u'Fᵪ'
assert upretty( Symbol('F_a') ) == u'Fₐ'
assert upretty( Symbol('F_e') ) == u'Fₑ'
assert upretty( Symbol('F_i') ) == u'Fᵢ'
assert upretty( Symbol('F_o') ) == u'Fₒ'
assert upretty( Symbol('F_u') ) == u'Fᵤ'
assert upretty( Symbol('F_r') ) == u'Fᵣ'
assert upretty( Symbol('F_v') ) == u'Fᵥ'
assert upretty( Symbol('F_x') ) == u'Fₓ'
def test_missing_in_2X_issue_9047():
if PY3:
assert upretty( Symbol('F_h') ) == u'Fₕ'
assert upretty( Symbol('F_k') ) == u'Fₖ'
assert upretty( Symbol('F_l') ) == u'Fₗ'
assert upretty( Symbol('F_m') ) == u'Fₘ'
assert upretty( Symbol('F_n') ) == u'Fₙ'
assert upretty( Symbol('F_p') ) == u'Fₚ'
assert upretty( Symbol('F_s') ) == u'Fₛ'
assert upretty( Symbol('F_t') ) == u'Fₜ'
def test_upretty_modifiers():
# Accents
assert upretty( Symbol('Fmathring') ) == u'F̊'
assert upretty( Symbol('Fddddot') ) == u'F⃜'
assert upretty( Symbol('Fdddot') ) == u'F⃛'
assert upretty( Symbol('Fddot') ) == u'F̈'
assert upretty( Symbol('Fdot') ) == u'Ḟ'
assert upretty( Symbol('Fcheck') ) == u'F̌'
assert upretty( Symbol('Fbreve') ) == u'F̆'
assert upretty( Symbol('Facute') ) == u'F́'
assert upretty( Symbol('Fgrave') ) == u'F̀'
assert upretty( Symbol('Ftilde') ) == u'F̃'
assert upretty( Symbol('Fhat') ) == u'F̂'
assert upretty( Symbol('Fbar') ) == u'F̅'
assert upretty( Symbol('Fvec') ) == u'F⃗'
assert upretty( Symbol('Fprime') ) == u'F′'
assert upretty( Symbol('Fprm') ) == u'F′'
# No faces are actually implemented, but test to make sure the modifiers are stripped
assert upretty( Symbol('Fbold') ) == u'Fbold'
assert upretty( Symbol('Fbm') ) == u'Fbm'
assert upretty( Symbol('Fcal') ) == u'Fcal'
assert upretty( Symbol('Fscr') ) == u'Fscr'
assert upretty( Symbol('Ffrak') ) == u'Ffrak'
# Brackets
assert upretty( Symbol('Fnorm') ) == u'‖F‖'
assert upretty( Symbol('Favg') ) == u'⟨F⟩'
assert upretty( Symbol('Fabs') ) == u'|F|'
assert upretty( Symbol('Fmag') ) == u'|F|'
# Combinations
assert upretty( Symbol('xvecdot') ) == u'x⃗̇'
assert upretty( Symbol('xDotVec') ) == u'ẋ⃗'
assert upretty( Symbol('xHATNorm') ) == u'‖x̂‖'
assert upretty( Symbol('xMathring_yCheckPRM__zbreveAbs') ) == u'x̊_y̌′__|z̆|'
assert upretty( Symbol('alphadothat_nVECDOT__tTildePrime') ) == u'α̇̂_n⃗̇__t̃′'
assert upretty( Symbol('x_dot') ) == u'x_dot'
assert upretty( Symbol('x__dot') ) == u'x__dot'
def test_pretty_Cycle():
from sympy.combinatorics.permutations import Cycle
assert pretty(Cycle(1, 2)) == '(1 2)'
assert pretty(Cycle(2)) == '(2)'
assert pretty(Cycle(1, 3)(4, 5)) == '(1 3)(4 5)'
assert pretty(Cycle()) == '()'
def test_pretty_basic():
assert pretty( -Rational(1)/2 ) == '-1/2'
assert pretty( -Rational(13)/22 ) == \
"""\
-13 \n\
----\n\
22 \
"""
expr = oo
ascii_str = \
"""\
oo\
"""
ucode_str = \
u("""\
∞\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2)
ascii_str = \
"""\
2\n\
x \
"""
ucode_str = \
u("""\
2\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 1/x
ascii_str = \
"""\
1\n\
-\n\
x\
"""
ucode_str = \
u("""\
1\n\
─\n\
x\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# not the same as 1/x
expr = x**-1.0
ascii_str = \
"""\
-1.0\n\
x \
"""
ucode_str = \
("""\
-1.0\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# see issue #2860
expr = Pow(S(2), -1.0, evaluate=False)
ascii_str = \
"""\
-1.0\n\
2 \
"""
ucode_str = \
("""\
-1.0\n\
2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = y*x**-2
ascii_str = \
"""\
y \n\
--\n\
2\n\
x \
"""
ucode_str = \
u("""\
y \n\
──\n\
2\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
#see issue #14033
expr = x**Rational(1, 3)
ascii_str = \
"""\
1/3\n\
x \
"""
ucode_str = \
u("""\
1/3\n\
x \
""")
assert xpretty(expr, use_unicode=False, wrap_line=False,\
root_notation = False) == ascii_str
assert xpretty(expr, use_unicode=True, wrap_line=False,\
root_notation = False) == ucode_str
expr = x**Rational(-5, 2)
ascii_str = \
"""\
1 \n\
----\n\
5/2\n\
x \
"""
ucode_str = \
u("""\
1 \n\
────\n\
5/2\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (-2)**x
ascii_str = \
"""\
x\n\
(-2) \
"""
ucode_str = \
u("""\
x\n\
(-2) \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# See issue 4923
expr = Pow(3, 1, evaluate=False)
ascii_str = \
"""\
1\n\
3 \
"""
ucode_str = \
u("""\
1\n\
3 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2 + x + 1)
ascii_str_1 = \
"""\
2\n\
1 + x + x \
"""
ascii_str_2 = \
"""\
2 \n\
x + x + 1\
"""
ascii_str_3 = \
"""\
2 \n\
x + 1 + x\
"""
ucode_str_1 = \
u("""\
2\n\
1 + x + x \
""")
ucode_str_2 = \
u("""\
2 \n\
x + x + 1\
""")
ucode_str_3 = \
u("""\
2 \n\
x + 1 + x\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2, ascii_str_3]
assert upretty(expr) in [ucode_str_1, ucode_str_2, ucode_str_3]
expr = 1 - x
ascii_str_1 = \
"""\
1 - x\
"""
ascii_str_2 = \
"""\
-x + 1\
"""
ucode_str_1 = \
u("""\
1 - x\
""")
ucode_str_2 = \
u("""\
-x + 1\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = 1 - 2*x
ascii_str_1 = \
"""\
1 - 2*x\
"""
ascii_str_2 = \
"""\
-2*x + 1\
"""
ucode_str_1 = \
u("""\
1 - 2⋅x\
""")
ucode_str_2 = \
u("""\
-2⋅x + 1\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = x/y
ascii_str = \
"""\
x\n\
-\n\
y\
"""
ucode_str = \
u("""\
x\n\
─\n\
y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x/y
ascii_str = \
"""\
-x \n\
---\n\
y \
"""
ucode_str = \
u("""\
-x \n\
───\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x + 2)/y
ascii_str_1 = \
"""\
2 + x\n\
-----\n\
y \
"""
ascii_str_2 = \
"""\
x + 2\n\
-----\n\
y \
"""
ucode_str_1 = \
u("""\
2 + x\n\
─────\n\
y \
""")
ucode_str_2 = \
u("""\
x + 2\n\
─────\n\
y \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = (1 + x)*y
ascii_str_1 = \
"""\
y*(1 + x)\
"""
ascii_str_2 = \
"""\
(1 + x)*y\
"""
ascii_str_3 = \
"""\
y*(x + 1)\
"""
ucode_str_1 = \
u("""\
y⋅(1 + x)\
""")
ucode_str_2 = \
u("""\
(1 + x)⋅y\
""")
ucode_str_3 = \
u("""\
y⋅(x + 1)\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2, ascii_str_3]
assert upretty(expr) in [ucode_str_1, ucode_str_2, ucode_str_3]
# Test for correct placement of the negative sign
expr = -5*x/(x + 10)
ascii_str_1 = \
"""\
-5*x \n\
------\n\
10 + x\
"""
ascii_str_2 = \
"""\
-5*x \n\
------\n\
x + 10\
"""
ucode_str_1 = \
u("""\
-5⋅x \n\
──────\n\
10 + x\
""")
ucode_str_2 = \
u("""\
-5⋅x \n\
──────\n\
x + 10\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = -S(1)/2 - 3*x
ascii_str = \
"""\
-3*x - 1/2\
"""
ucode_str = \
u("""\
-3⋅x - 1/2\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = S(1)/2 - 3*x
ascii_str = \
"""\
1/2 - 3*x\
"""
ucode_str = \
u("""\
1/2 - 3⋅x\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -S(1)/2 - 3*x/2
ascii_str = \
"""\
3*x 1\n\
- --- - -\n\
2 2\
"""
ucode_str = \
u("""\
3⋅x 1\n\
- ─── - ─\n\
2 2\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = S(1)/2 - 3*x/2
ascii_str = \
"""\
1 3*x\n\
- - ---\n\
2 2 \
"""
ucode_str = \
u("""\
1 3⋅x\n\
─ - ───\n\
2 2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_negative_fractions():
expr = -x/y
ascii_str =\
"""\
-x \n\
---\n\
y \
"""
ucode_str =\
u("""\
-x \n\
───\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x*z/y
ascii_str =\
"""\
-x*z \n\
-----\n\
y \
"""
ucode_str =\
u("""\
-x⋅z \n\
─────\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x**2/y
ascii_str =\
"""\
2\n\
x \n\
--\n\
y \
"""
ucode_str =\
u("""\
2\n\
x \n\
──\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x**2/y
ascii_str =\
"""\
2 \n\
-x \n\
----\n\
y \
"""
ucode_str =\
u("""\
2 \n\
-x \n\
────\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x/(y*z)
ascii_str =\
"""\
-x \n\
---\n\
y*z\
"""
ucode_str =\
u("""\
-x \n\
───\n\
y⋅z\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -a/y**2
ascii_str =\
"""\
-a \n\
---\n\
2\n\
y \
"""
ucode_str =\
u("""\
-a \n\
───\n\
2\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = y**(-a/b)
ascii_str =\
"""\
-a \n\
---\n\
b \n\
y \
"""
ucode_str =\
u("""\
-a \n\
───\n\
b \n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -1/y**2
ascii_str =\
"""\
-1 \n\
---\n\
2\n\
y \
"""
ucode_str =\
u("""\
-1 \n\
───\n\
2\n\
y \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -10/b**2
ascii_str =\
"""\
-10 \n\
----\n\
2 \n\
b \
"""
ucode_str =\
u("""\
-10 \n\
────\n\
2 \n\
b \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Rational(-200, 37)
ascii_str =\
"""\
-200 \n\
-----\n\
37 \
"""
ucode_str =\
u("""\
-200 \n\
─────\n\
37 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_issue_5524():
assert pretty(-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5)) == \
"""\
2 / ___ \\\n\
- (5 - y) + (x - 5)*\\-x - 2*\\/ 2 + 5/\
"""
assert upretty(-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5)) == \
u("""\
2 \n\
- (5 - y) + (x - 5)⋅(-x - 2⋅√2 + 5)\
""")
def test_pretty_ordering():
assert pretty(x**2 + x + 1, order='lex') == \
"""\
2 \n\
x + x + 1\
"""
assert pretty(x**2 + x + 1, order='rev-lex') == \
"""\
2\n\
1 + x + x \
"""
assert pretty(1 - x, order='lex') == '-x + 1'
assert pretty(1 - x, order='rev-lex') == '1 - x'
assert pretty(1 - 2*x, order='lex') == '-2*x + 1'
assert pretty(1 - 2*x, order='rev-lex') == '1 - 2*x'
f = 2*x**4 + y**2 - x**2 + y**3
assert pretty(f, order=None) == \
"""\
4 2 3 2\n\
2*x - x + y + y \
"""
assert pretty(f, order='lex') == \
"""\
4 2 3 2\n\
2*x - x + y + y \
"""
assert pretty(f, order='rev-lex') == \
"""\
2 3 2 4\n\
y + y - x + 2*x \
"""
expr = x - x**3/6 + x**5/120 + O(x**6)
ascii_str = \
"""\
3 5 \n\
x x / 6\\\n\
x - -- + --- + O\\x /\n\
6 120 \
"""
ucode_str = \
u("""\
3 5 \n\
x x ⎛ 6⎞\n\
x - ── + ─── + O⎝x ⎠\n\
6 120 \
""")
assert pretty(expr, order=None) == ascii_str
assert upretty(expr, order=None) == ucode_str
assert pretty(expr, order='lex') == ascii_str
assert upretty(expr, order='lex') == ucode_str
assert pretty(expr, order='rev-lex') == ascii_str
assert upretty(expr, order='rev-lex') == ucode_str
def test_EulerGamma():
assert pretty(EulerGamma) == str(EulerGamma) == "EulerGamma"
assert upretty(EulerGamma) == u"γ"
def test_GoldenRatio():
assert pretty(GoldenRatio) == str(GoldenRatio) == "GoldenRatio"
assert upretty(GoldenRatio) == u"φ"
def test_pretty_relational():
expr = Eq(x, y)
ascii_str = \
"""\
x = y\
"""
ucode_str = \
u("""\
x = y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lt(x, y)
ascii_str = \
"""\
x < y\
"""
ucode_str = \
u("""\
x < y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Gt(x, y)
ascii_str = \
"""\
x > y\
"""
ucode_str = \
u("""\
x > y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Le(x, y)
ascii_str = \
"""\
x <= y\
"""
ucode_str = \
u("""\
x ≤ y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Ge(x, y)
ascii_str = \
"""\
x >= y\
"""
ucode_str = \
u("""\
x ≥ y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Ne(x/(y + 1), y**2)
ascii_str_1 = \
"""\
x 2\n\
----- != y \n\
1 + y \
"""
ascii_str_2 = \
"""\
x 2\n\
----- != y \n\
y + 1 \
"""
ucode_str_1 = \
u("""\
x 2\n\
───── ≠ y \n\
1 + y \
""")
ucode_str_2 = \
u("""\
x 2\n\
───── ≠ y \n\
y + 1 \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
def test_Assignment():
expr = Assignment(x, y)
ascii_str = \
"""\
x := y\
"""
ucode_str = \
u("""\
x := y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_AugmentedAssignment():
expr = AddAugmentedAssignment(x, y)
ascii_str = \
"""\
x += y\
"""
ucode_str = \
u("""\
x += y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = SubAugmentedAssignment(x, y)
ascii_str = \
"""\
x -= y\
"""
ucode_str = \
u("""\
x -= y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = MulAugmentedAssignment(x, y)
ascii_str = \
"""\
x *= y\
"""
ucode_str = \
u("""\
x *= y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = DivAugmentedAssignment(x, y)
ascii_str = \
"""\
x /= y\
"""
ucode_str = \
u("""\
x /= y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = ModAugmentedAssignment(x, y)
ascii_str = \
"""\
x %= y\
"""
ucode_str = \
u("""\
x %= y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_issue_7117():
# See also issue #5031 (hence the evaluate=False in these).
e = Eq(x + 1, x/2)
q = Mul(2, e, evaluate=False)
assert upretty(q) == u("""\
⎛ x⎞\n\
2⋅⎜x + 1 = ─⎟\n\
⎝ 2⎠\
""")
q = Add(e, 6, evaluate=False)
assert upretty(q) == u("""\
⎛ x⎞\n\
6 + ⎜x + 1 = ─⎟\n\
⎝ 2⎠\
""")
q = Pow(e, 2, evaluate=False)
assert upretty(q) == u("""\
2\n\
⎛ x⎞ \n\
⎜x + 1 = ─⎟ \n\
⎝ 2⎠ \
""")
e2 = Eq(x, 2)
q = Mul(e, e2, evaluate=False)
assert upretty(q) == u("""\
⎛ x⎞ \n\
⎜x + 1 = ─⎟⋅(x = 2)\n\
⎝ 2⎠ \
""")
def test_pretty_rational():
expr = y*x**-2
ascii_str = \
"""\
y \n\
--\n\
2\n\
x \
"""
ucode_str = \
u("""\
y \n\
──\n\
2\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = y**Rational(3, 2) * x**Rational(-5, 2)
ascii_str = \
"""\
3/2\n\
y \n\
----\n\
5/2\n\
x \
"""
ucode_str = \
u("""\
3/2\n\
y \n\
────\n\
5/2\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sin(x)**3/tan(x)**2
ascii_str = \
"""\
3 \n\
sin (x)\n\
-------\n\
2 \n\
tan (x)\
"""
ucode_str = \
u("""\
3 \n\
sin (x)\n\
───────\n\
2 \n\
tan (x)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_functions():
"""Tests for Abs, conjugate, exp, function braces, and factorial."""
expr = (2*x + exp(x))
ascii_str_1 = \
"""\
x\n\
2*x + e \
"""
ascii_str_2 = \
"""\
x \n\
e + 2*x\
"""
ucode_str_1 = \
u("""\
x\n\
2⋅x + ℯ \
""")
ucode_str_2 = \
u("""\
x \n\
ℯ + 2⋅x\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Abs(x)
ascii_str = \
"""\
|x|\
"""
ucode_str = \
u("""\
│x│\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Abs(x/(x**2 + 1))
ascii_str_1 = \
"""\
| x |\n\
|------|\n\
| 2|\n\
|1 + x |\
"""
ascii_str_2 = \
"""\
| x |\n\
|------|\n\
| 2 |\n\
|x + 1|\
"""
ucode_str_1 = \
u("""\
│ x │\n\
│──────│\n\
│ 2│\n\
│1 + x │\
""")
ucode_str_2 = \
u("""\
│ x │\n\
│──────│\n\
│ 2 │\n\
│x + 1│\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Abs(1 / (y - Abs(x)))
ascii_str = \
"""\
| 1 |\n\
|-------|\n\
|y - |x||\
"""
ucode_str = \
u("""\
│ 1 │\n\
│───────│\n\
│y - │x││\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
n = Symbol('n', integer=True)
expr = factorial(n)
ascii_str = \
"""\
n!\
"""
ucode_str = \
u("""\
n!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial(2*n)
ascii_str = \
"""\
(2*n)!\
"""
ucode_str = \
u("""\
(2⋅n)!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial(factorial(factorial(n)))
ascii_str = \
"""\
((n!)!)!\
"""
ucode_str = \
u("""\
((n!)!)!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial(n + 1)
ascii_str_1 = \
"""\
(1 + n)!\
"""
ascii_str_2 = \
"""\
(n + 1)!\
"""
ucode_str_1 = \
u("""\
(1 + n)!\
""")
ucode_str_2 = \
u("""\
(n + 1)!\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = subfactorial(n)
ascii_str = \
"""\
!n\
"""
ucode_str = \
u("""\
!n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = subfactorial(2*n)
ascii_str = \
"""\
!(2*n)\
"""
ucode_str = \
u("""\
!(2⋅n)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
n = Symbol('n', integer=True)
expr = factorial2(n)
ascii_str = \
"""\
n!!\
"""
ucode_str = \
u("""\
n!!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial2(2*n)
ascii_str = \
"""\
(2*n)!!\
"""
ucode_str = \
u("""\
(2⋅n)!!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial2(factorial2(factorial2(n)))
ascii_str = \
"""\
((n!!)!!)!!\
"""
ucode_str = \
u("""\
((n!!)!!)!!\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial2(n + 1)
ascii_str_1 = \
"""\
(1 + n)!!\
"""
ascii_str_2 = \
"""\
(n + 1)!!\
"""
ucode_str_1 = \
u("""\
(1 + n)!!\
""")
ucode_str_2 = \
u("""\
(n + 1)!!\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = 2*binomial(n, k)
ascii_str = \
"""\
/n\\\n\
2*| |\n\
\\k/\
"""
ucode_str = \
u("""\
⎛n⎞\n\
2⋅⎜ ⎟\n\
⎝k⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2*binomial(2*n, k)
ascii_str = \
"""\
/2*n\\\n\
2*| |\n\
\\ k /\
"""
ucode_str = \
u("""\
⎛2⋅n⎞\n\
2⋅⎜ ⎟\n\
⎝ k ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2*binomial(n**2, k)
ascii_str = \
"""\
/ 2\\\n\
|n |\n\
2*| |\n\
\\k /\
"""
ucode_str = \
u("""\
⎛ 2⎞\n\
⎜n ⎟\n\
2⋅⎜ ⎟\n\
⎝k ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = catalan(n)
ascii_str = \
"""\
C \n\
n\
"""
ucode_str = \
u("""\
C \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = catalan(n)
ascii_str = \
"""\
C \n\
n\
"""
ucode_str = \
u("""\
C \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = bell(n)
ascii_str = \
"""\
B \n\
n\
"""
ucode_str = \
u("""\
B \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = bernoulli(n)
ascii_str = \
"""\
B \n\
n\
"""
ucode_str = \
u("""\
B \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = fibonacci(n)
ascii_str = \
"""\
F \n\
n\
"""
ucode_str = \
u("""\
F \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = lucas(n)
ascii_str = \
"""\
L \n\
n\
"""
ucode_str = \
u("""\
L \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = tribonacci(n)
ascii_str = \
"""\
T \n\
n\
"""
ucode_str = \
u("""\
T \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate(x)
ascii_str = \
"""\
_\n\
x\
"""
ucode_str = \
u("""\
_\n\
x\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
f = Function('f')
expr = conjugate(f(x + 1))
ascii_str_1 = \
"""\
________\n\
f(1 + x)\
"""
ascii_str_2 = \
"""\
________\n\
f(x + 1)\
"""
ucode_str_1 = \
u("""\
________\n\
f(1 + x)\
""")
ucode_str_2 = \
u("""\
________\n\
f(x + 1)\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = f(x)
ascii_str = \
"""\
f(x)\
"""
ucode_str = \
u("""\
f(x)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = f(x, y)
ascii_str = \
"""\
f(x, y)\
"""
ucode_str = \
u("""\
f(x, y)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = f(x/(y + 1), y)
ascii_str_1 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\1 + y /\
"""
ascii_str_2 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\y + 1 /\
"""
ucode_str_1 = \
u("""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝1 + y ⎠\
""")
ucode_str_2 = \
u("""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝y + 1 ⎠\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = f(x**x**x**x**x**x)
ascii_str = \
"""\
/ / / / / x\\\\\\\\\\
| | | | \\x /||||
| | | \\x /|||
| | \\x /||
| \\x /|
f\\x /\
"""
ucode_str = \
u("""\
⎛ ⎛ ⎛ ⎛ ⎛ x⎞⎞⎞⎞⎞
⎜ ⎜ ⎜ ⎜ ⎝x ⎠⎟⎟⎟⎟
⎜ ⎜ ⎜ ⎝x ⎠⎟⎟⎟
⎜ ⎜ ⎝x ⎠⎟⎟
⎜ ⎝x ⎠⎟
f⎝x ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sin(x)**2
ascii_str = \
"""\
2 \n\
sin (x)\
"""
ucode_str = \
u("""\
2 \n\
sin (x)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate(a + b*I)
ascii_str = \
"""\
_ _\n\
a - I*b\
"""
ucode_str = \
u("""\
_ _\n\
a - ⅈ⋅b\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate(exp(a + b*I))
ascii_str = \
"""\
_ _\n\
a - I*b\n\
e \
"""
ucode_str = \
u("""\
_ _\n\
a - ⅈ⋅b\n\
ℯ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate( f(1 + conjugate(f(x))) )
ascii_str_1 = \
"""\
___________\n\
/ ____\\\n\
f\\1 + f(x)/\
"""
ascii_str_2 = \
"""\
___________\n\
/____ \\\n\
f\\f(x) + 1/\
"""
ucode_str_1 = \
u("""\
___________\n\
⎛ ____⎞\n\
f⎝1 + f(x)⎠\
""")
ucode_str_2 = \
u("""\
___________\n\
⎛____ ⎞\n\
f⎝f(x) + 1⎠\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = f(x/(y + 1), y)
ascii_str_1 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\1 + y /\
"""
ascii_str_2 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\y + 1 /\
"""
ucode_str_1 = \
u("""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝1 + y ⎠\
""")
ucode_str_2 = \
u("""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝y + 1 ⎠\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = floor(1 / (y - floor(x)))
ascii_str = \
"""\
/ 1 \\\n\
floor|------------|\n\
\\y - floor(x)/\
"""
ucode_str = \
u("""\
⎢ 1 ⎥\n\
⎢───────⎥\n\
⎣y - ⌊x⌋⎦\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = ceiling(1 / (y - ceiling(x)))
ascii_str = \
"""\
/ 1 \\\n\
ceiling|--------------|\n\
\\y - ceiling(x)/\
"""
ucode_str = \
u("""\
⎡ 1 ⎤\n\
⎢───────⎥\n\
⎢y - ⌈x⌉⎥\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(n)
ascii_str = \
"""\
E \n\
n\
"""
ucode_str = \
u("""\
E \n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(1/(1 + 1/(1 + 1/n)))
ascii_str = \
"""\
E \n\
1 \n\
---------\n\
1 \n\
1 + -----\n\
1\n\
1 + -\n\
n\
"""
ucode_str = \
u("""\
E \n\
1 \n\
─────────\n\
1 \n\
1 + ─────\n\
1\n\
1 + ─\n\
n\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(n, x)
ascii_str = \
"""\
E (x)\n\
n \
"""
ucode_str = \
u("""\
E (x)\n\
n \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(n, x/2)
ascii_str = \
"""\
/x\\\n\
E |-|\n\
n\\2/\
"""
ucode_str = \
u("""\
⎛x⎞\n\
E ⎜─⎟\n\
n⎝2⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_sqrt():
expr = sqrt(2)
ascii_str = \
"""\
___\n\
\\/ 2 \
"""
ucode_str = \
u"√2"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2**Rational(1, 3)
ascii_str = \
"""\
3 ___\n\
\\/ 2 \
"""
ucode_str = \
u("""\
3 ___\n\
╲╱ 2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2**Rational(1, 1000)
ascii_str = \
"""\
1000___\n\
\\/ 2 \
"""
ucode_str = \
u("""\
1000___\n\
╲╱ 2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sqrt(x**2 + 1)
ascii_str = \
"""\
________\n\
/ 2 \n\
\\/ x + 1 \
"""
ucode_str = \
u("""\
________\n\
╱ 2 \n\
╲╱ x + 1 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (1 + sqrt(5))**Rational(1, 3)
ascii_str = \
"""\
___________\n\
3 / ___ \n\
\\/ 1 + \\/ 5 \
"""
ucode_str = \
u("""\
3 ________\n\
╲╱ 1 + √5 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2**(1/x)
ascii_str = \
"""\
x ___\n\
\\/ 2 \
"""
ucode_str = \
u("""\
x ___\n\
╲╱ 2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sqrt(2 + pi)
ascii_str = \
"""\
________\n\
\\/ 2 + pi \
"""
ucode_str = \
u("""\
_______\n\
╲╱ 2 + π \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (2 + (
1 + x**2)/(2 + x))**Rational(1, 4) + (1 + x**Rational(1, 1000))/sqrt(3 + x**2)
ascii_str = \
"""\
____________ \n\
/ 2 1000___ \n\
/ x + 1 \\/ x + 1\n\
4 / 2 + ------ + -----------\n\
\\/ x + 2 ________\n\
/ 2 \n\
\\/ x + 3 \
"""
ucode_str = \
u("""\
____________ \n\
╱ 2 1000___ \n\
╱ x + 1 ╲╱ x + 1\n\
4 ╱ 2 + ────── + ───────────\n\
╲╱ x + 2 ________\n\
╱ 2 \n\
╲╱ x + 3 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_sqrt_char_knob():
# See PR #9234.
expr = sqrt(2)
ucode_str1 = \
u("""\
___\n\
╲╱ 2 \
""")
ucode_str2 = \
u"√2"
assert xpretty(expr, use_unicode=True,
use_unicode_sqrt_char=False) == ucode_str1
assert xpretty(expr, use_unicode=True,
use_unicode_sqrt_char=True) == ucode_str2
def test_pretty_sqrt_longsymbol_no_sqrt_char():
# Do not use unicode sqrt char for long symbols (see PR #9234).
expr = sqrt(Symbol('C1'))
ucode_str = \
u("""\
____\n\
╲╱ C₁ \
""")
assert upretty(expr) == ucode_str
def test_pretty_KroneckerDelta():
x, y = symbols("x, y")
expr = KroneckerDelta(x, y)
ascii_str = \
"""\
d \n\
x,y\
"""
ucode_str = \
u("""\
δ \n\
x,y\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_product():
n, m, k, l = symbols('n m k l')
f = symbols('f', cls=Function)
expr = Product(f((n/3)**2), (n, k**2, l))
unicode_str = \
u("""\
l \n\
─┬──────┬─ \n\
│ │ ⎛ 2⎞\n\
│ │ ⎜n ⎟\n\
│ │ f⎜──⎟\n\
│ │ ⎝9 ⎠\n\
│ │ \n\
2 \n\
n = k """)
ascii_str = \
"""\
l \n\
__________ \n\
| | / 2\\\n\
| | |n |\n\
| | f|--|\n\
| | \\9 /\n\
| | \n\
2 \n\
n = k """
expr = Product(f((n/3)**2), (n, k**2, l), (l, 1, m))
unicode_str = \
u("""\
m l \n\
─┬──────┬─ ─┬──────┬─ \n\
│ │ │ │ ⎛ 2⎞\n\
│ │ │ │ ⎜n ⎟\n\
│ │ │ │ f⎜──⎟\n\
│ │ │ │ ⎝9 ⎠\n\
│ │ │ │ \n\
l = 1 2 \n\
n = k """)
ascii_str = \
"""\
m l \n\
__________ __________ \n\
| | | | / 2\\\n\
| | | | |n |\n\
| | | | f|--|\n\
| | | | \\9 /\n\
| | | | \n\
l = 1 2 \n\
n = k """
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
def test_pretty_lambda():
# S.IdentityFunction is a special case
expr = Lambda(y, y)
assert pretty(expr) == "x -> x"
assert upretty(expr) == u"x ↦ x"
expr = Lambda(x, x+1)
assert pretty(expr) == "x -> x + 1"
assert upretty(expr) == u"x ↦ x + 1"
expr = Lambda(x, x**2)
ascii_str = \
"""\
2\n\
x -> x \
"""
ucode_str = \
u("""\
2\n\
x ↦ x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda(x, x**2)**2
ascii_str = \
"""\
2
/ 2\\ \n\
\\x -> x / \
"""
ucode_str = \
u("""\
2
⎛ 2⎞ \n\
⎝x ↦ x ⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda((x, y), x)
ascii_str = "(x, y) -> x"
ucode_str = u"(x, y) ↦ x"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda((x, y), x**2)
ascii_str = \
"""\
2\n\
(x, y) -> x \
"""
ucode_str = \
u("""\
2\n\
(x, y) ↦ x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_order():
expr = O(1)
ascii_str = \
"""\
O(1)\
"""
ucode_str = \
u("""\
O(1)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(1/x)
ascii_str = \
"""\
/1\\\n\
O|-|\n\
\\x/\
"""
ucode_str = \
u("""\
⎛1⎞\n\
O⎜─⎟\n\
⎝x⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(x**2 + y**2)
ascii_str = \
"""\
/ 2 2 \\\n\
O\\x + y ; (x, y) -> (0, 0)/\
"""
ucode_str = \
u("""\
⎛ 2 2 ⎞\n\
O⎝x + y ; (x, y) → (0, 0)⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(1, (x, oo))
ascii_str = \
"""\
O(1; x -> oo)\
"""
ucode_str = \
u("""\
O(1; x → ∞)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(1/x, (x, oo))
ascii_str = \
"""\
/1 \\\n\
O|-; x -> oo|\n\
\\x /\
"""
ucode_str = \
u("""\
⎛1 ⎞\n\
O⎜─; x → ∞⎟\n\
⎝x ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(x**2 + y**2, (x, oo), (y, oo))
ascii_str = \
"""\
/ 2 2 \\\n\
O\\x + y ; (x, y) -> (oo, oo)/\
"""
ucode_str = \
u("""\
⎛ 2 2 ⎞\n\
O⎝x + y ; (x, y) → (∞, ∞)⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_derivatives():
# Simple
expr = Derivative(log(x), x, evaluate=False)
ascii_str = \
"""\
d \n\
--(log(x))\n\
dx \
"""
ucode_str = \
u("""\
d \n\
──(log(x))\n\
dx \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(log(x), x, evaluate=False) + x
ascii_str_1 = \
"""\
d \n\
x + --(log(x))\n\
dx \
"""
ascii_str_2 = \
"""\
d \n\
--(log(x)) + x\n\
dx \
"""
ucode_str_1 = \
u("""\
d \n\
x + ──(log(x))\n\
dx \
""")
ucode_str_2 = \
u("""\
d \n\
──(log(x)) + x\n\
dx \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
# basic partial derivatives
expr = Derivative(log(x + y) + x, x)
ascii_str_1 = \
"""\
d \n\
--(log(x + y) + x)\n\
dx \
"""
ascii_str_2 = \
"""\
d \n\
--(x + log(x + y))\n\
dx \
"""
ucode_str_1 = \
u("""\
∂ \n\
──(log(x + y) + x)\n\
∂x \
""")
ucode_str_2 = \
u("""\
∂ \n\
──(x + log(x + y))\n\
∂x \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2], upretty(expr)
# Multiple symbols
expr = Derivative(log(x) + x**2, x, y)
ascii_str_1 = \
"""\
2 \n\
d / 2\\\n\
-----\\log(x) + x /\n\
dy dx \
"""
ascii_str_2 = \
"""\
2 \n\
d / 2 \\\n\
-----\\x + log(x)/\n\
dy dx \
"""
ucode_str_1 = \
u("""\
2 \n\
d ⎛ 2⎞\n\
─────⎝log(x) + x ⎠\n\
dy dx \
""")
ucode_str_2 = \
u("""\
2 \n\
d ⎛ 2 ⎞\n\
─────⎝x + log(x)⎠\n\
dy dx \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Derivative(2*x*y, y, x) + x**2
ascii_str_1 = \
"""\
2 \n\
d 2\n\
-----(2*x*y) + x \n\
dx dy \
"""
ascii_str_2 = \
"""\
2 \n\
2 d \n\
x + -----(2*x*y)\n\
dx dy \
"""
ucode_str_1 = \
u("""\
2 \n\
∂ 2\n\
─────(2⋅x⋅y) + x \n\
∂x ∂y \
""")
ucode_str_2 = \
u("""\
2 \n\
2 ∂ \n\
x + ─────(2⋅x⋅y)\n\
∂x ∂y \
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Derivative(2*x*y, x, x)
ascii_str = \
"""\
2 \n\
d \n\
---(2*x*y)\n\
2 \n\
dx \
"""
ucode_str = \
u("""\
2 \n\
∂ \n\
───(2⋅x⋅y)\n\
2 \n\
∂x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(2*x*y, x, 17)
ascii_str = \
"""\
17 \n\
d \n\
----(2*x*y)\n\
17 \n\
dx \
"""
ucode_str = \
u("""\
17 \n\
∂ \n\
────(2⋅x⋅y)\n\
17 \n\
∂x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(2*x*y, x, x, y)
ascii_str = \
"""\
3 \n\
d \n\
------(2*x*y)\n\
2 \n\
dy dx \
"""
ucode_str = \
u("""\
3 \n\
∂ \n\
──────(2⋅x⋅y)\n\
2 \n\
∂y ∂x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# Greek letters
alpha = Symbol('alpha')
beta = Function('beta')
expr = beta(alpha).diff(alpha)
ascii_str = \
"""\
d \n\
------(beta(alpha))\n\
dalpha \
"""
ucode_str = \
u("""\
d \n\
──(β(α))\n\
dα \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(f(x), (x, n))
ascii_str = \
"""\
n \n\
d \n\
---(f(x))\n\
n \n\
dx \
"""
ucode_str = \
u("""\
n \n\
d \n\
───(f(x))\n\
n \n\
dx \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_integrals():
expr = Integral(log(x), x)
ascii_str = \
"""\
/ \n\
| \n\
| log(x) dx\n\
| \n\
/ \
"""
ucode_str = \
u("""\
⌠ \n\
⎮ log(x) dx\n\
⌡ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2, x)
ascii_str = \
"""\
/ \n\
| \n\
| 2 \n\
| x dx\n\
| \n\
/ \
"""
ucode_str = \
u("""\
⌠ \n\
⎮ 2 \n\
⎮ x dx\n\
⌡ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral((sin(x))**2 / (tan(x))**2)
ascii_str = \
"""\
/ \n\
| \n\
| 2 \n\
| sin (x) \n\
| ------- dx\n\
| 2 \n\
| tan (x) \n\
| \n\
/ \
"""
ucode_str = \
u("""\
⌠ \n\
⎮ 2 \n\
⎮ sin (x) \n\
⎮ ─────── dx\n\
⎮ 2 \n\
⎮ tan (x) \n\
⌡ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**(2**x), x)
ascii_str = \
"""\
/ \n\
| \n\
| / x\\ \n\
| \\2 / \n\
| x dx\n\
| \n\
/ \
"""
ucode_str = \
u("""\
⌠ \n\
⎮ ⎛ x⎞ \n\
⎮ ⎝2 ⎠ \n\
⎮ x dx\n\
⌡ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2, (x, 1, 2))
ascii_str = \
"""\
2 \n\
/ \n\
| \n\
| 2 \n\
| x dx\n\
| \n\
/ \n\
1 \
"""
ucode_str = \
u("""\
2 \n\
⌠ \n\
⎮ 2 \n\
⎮ x dx\n\
⌡ \n\
1 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2, (x, Rational(1, 2), 10))
ascii_str = \
"""\
10 \n\
/ \n\
| \n\
| 2 \n\
| x dx\n\
| \n\
/ \n\
1/2 \
"""
ucode_str = \
u("""\
10 \n\
⌠ \n\
⎮ 2 \n\
⎮ x dx\n\
⌡ \n\
1/2 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2*y**2, x, y)
ascii_str = \
"""\
/ / \n\
| | \n\
| | 2 2 \n\
| | x *y dx dy\n\
| | \n\
/ / \
"""
ucode_str = \
u("""\
⌠ ⌠ \n\
⎮ ⎮ 2 2 \n\
⎮ ⎮ x ⋅y dx dy\n\
⌡ ⌡ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(sin(th)/cos(ph), (th, 0, pi), (ph, 0, 2*pi))
ascii_str = \
"""\
2*pi pi \n\
/ / \n\
| | \n\
| | sin(theta) \n\
| | ---------- d(theta) d(phi)\n\
| | cos(phi) \n\
| | \n\
/ / \n\
0 0 \
"""
ucode_str = \
u("""\
2⋅π π \n\
⌠ ⌠ \n\
⎮ ⎮ sin(θ) \n\
⎮ ⎮ ────── dθ dφ\n\
⎮ ⎮ cos(φ) \n\
⌡ ⌡ \n\
0 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_matrix():
# Empty Matrix
expr = Matrix()
ascii_str = "[]"
unicode_str = "[]"
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Matrix(2, 0, lambda i, j: 0)
ascii_str = "[]"
unicode_str = "[]"
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Matrix(0, 2, lambda i, j: 0)
ascii_str = "[]"
unicode_str = "[]"
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Matrix([[x**2 + 1, 1], [y, x + y]])
ascii_str_1 = \
"""\
[ 2 ]
[1 + x 1 ]
[ ]
[ y x + y]\
"""
ascii_str_2 = \
"""\
[ 2 ]
[x + 1 1 ]
[ ]
[ y x + y]\
"""
ucode_str_1 = \
u("""\
⎡ 2 ⎤
⎢1 + x 1 ⎥
⎢ ⎥
⎣ y x + y⎦\
""")
ucode_str_2 = \
u("""\
⎡ 2 ⎤
⎢x + 1 1 ⎥
⎢ ⎥
⎣ y x + y⎦\
""")
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Matrix([[x/y, y, th], [0, exp(I*k*ph), 1]])
ascii_str = \
"""\
[x ]
[- y theta]
[y ]
[ ]
[ I*k*phi ]
[0 e 1 ]\
"""
ucode_str = \
u("""\
⎡x ⎤
⎢─ y θ⎥
⎢y ⎥
⎢ ⎥
⎢ ⅈ⋅k⋅φ ⎥
⎣0 ℯ 1⎦\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_ndim_arrays():
x, y, z, w = symbols("x y z w")
for ArrayType in (ImmutableDenseNDimArray, ImmutableSparseNDimArray, MutableDenseNDimArray, MutableSparseNDimArray):
# Basic: scalar array
M = ArrayType(x)
assert pretty(M) == "x"
assert upretty(M) == "x"
M = ArrayType([[1/x, y], [z, w]])
M1 = ArrayType([1/x, y, z])
M2 = tensorproduct(M1, M)
M3 = tensorproduct(M, M)
ascii_str = \
"""\
[1 ]\n\
[- y]\n\
[x ]\n\
[ ]\n\
[z w]\
"""
ucode_str = \
u("""\
⎡1 ⎤\n\
⎢─ y⎥\n\
⎢x ⎥\n\
⎢ ⎥\n\
⎣z w⎦\
""")
assert pretty(M) == ascii_str
assert upretty(M) == ucode_str
ascii_str = \
"""\
[1 ]\n\
[- y z]\n\
[x ]\
"""
ucode_str = \
u("""\
⎡1 ⎤\n\
⎢─ y z⎥\n\
⎣x ⎦\
""")
assert pretty(M1) == ascii_str
assert upretty(M1) == ucode_str
ascii_str = \
"""\
[[1 y] ]\n\
[[-- -] [z ]]\n\
[[ 2 x] [ y 2 ] [- y*z]]\n\
[[x ] [ - y ] [x ]]\n\
[[ ] [ x ] [ ]]\n\
[[z w] [ ] [ 2 ]]\n\
[[- -] [y*z w*y] [z w*z]]\n\
[[x x] ]\
"""
ucode_str = \
u("""\
⎡⎡1 y⎤ ⎤\n\
⎢⎢── ─⎥ ⎡z ⎤⎥\n\
⎢⎢ 2 x⎥ ⎡ y 2 ⎤ ⎢─ y⋅z⎥⎥\n\
⎢⎢x ⎥ ⎢ ─ y ⎥ ⎢x ⎥⎥\n\
⎢⎢ ⎥ ⎢ x ⎥ ⎢ ⎥⎥\n\
⎢⎢z w⎥ ⎢ ⎥ ⎢ 2 ⎥⎥\n\
⎢⎢─ ─⎥ ⎣y⋅z w⋅y⎦ ⎣z w⋅z⎦⎥\n\
⎣⎣x x⎦ ⎦\
""")
assert pretty(M2) == ascii_str
assert upretty(M2) == ucode_str
ascii_str = \
"""\
[ [1 y] ]\n\
[ [-- -] ]\n\
[ [ 2 x] [ y 2 ]]\n\
[ [x ] [ - y ]]\n\
[ [ ] [ x ]]\n\
[ [z w] [ ]]\n\
[ [- -] [y*z w*y]]\n\
[ [x x] ]\n\
[ ]\n\
[[z ] [ w ]]\n\
[[- y*z] [ - w*y]]\n\
[[x ] [ x ]]\n\
[[ ] [ ]]\n\
[[ 2 ] [ 2 ]]\n\
[[z w*z] [w*z w ]]\
"""
ucode_str = \
u("""\
⎡ ⎡1 y⎤ ⎤\n\
⎢ ⎢── ─⎥ ⎥\n\
⎢ ⎢ 2 x⎥ ⎡ y 2 ⎤⎥\n\
⎢ ⎢x ⎥ ⎢ ─ y ⎥⎥\n\
⎢ ⎢ ⎥ ⎢ x ⎥⎥\n\
⎢ ⎢z w⎥ ⎢ ⎥⎥\n\
⎢ ⎢─ ─⎥ ⎣y⋅z w⋅y⎦⎥\n\
⎢ ⎣x x⎦ ⎥\n\
⎢ ⎥\n\
⎢⎡z ⎤ ⎡ w ⎤⎥\n\
⎢⎢─ y⋅z⎥ ⎢ ─ w⋅y⎥⎥\n\
⎢⎢x ⎥ ⎢ x ⎥⎥\n\
⎢⎢ ⎥ ⎢ ⎥⎥\n\
⎢⎢ 2 ⎥ ⎢ 2 ⎥⎥\n\
⎣⎣z w⋅z⎦ ⎣w⋅z w ⎦⎦\
""")
assert pretty(M3) == ascii_str
assert upretty(M3) == ucode_str
Mrow = ArrayType([[x, y, 1 / z]])
Mcolumn = ArrayType([[x], [y], [1 / z]])
Mcol2 = ArrayType([Mcolumn.tolist()])
ascii_str = \
"""\
[[ 1]]\n\
[[x y -]]\n\
[[ z]]\
"""
ucode_str = \
u("""\
⎡⎡ 1⎤⎤\n\
⎢⎢x y ─⎥⎥\n\
⎣⎣ z⎦⎦\
""")
assert pretty(Mrow) == ascii_str
assert upretty(Mrow) == ucode_str
ascii_str = \
"""\
[x]\n\
[ ]\n\
[y]\n\
[ ]\n\
[1]\n\
[-]\n\
[z]\
"""
ucode_str = \
u("""\
⎡x⎤\n\
⎢ ⎥\n\
⎢y⎥\n\
⎢ ⎥\n\
⎢1⎥\n\
⎢─⎥\n\
⎣z⎦\
""")
assert pretty(Mcolumn) == ascii_str
assert upretty(Mcolumn) == ucode_str
ascii_str = \
"""\
[[x]]\n\
[[ ]]\n\
[[y]]\n\
[[ ]]\n\
[[1]]\n\
[[-]]\n\
[[z]]\
"""
ucode_str = \
u("""\
⎡⎡x⎤⎤\n\
⎢⎢ ⎥⎥\n\
⎢⎢y⎥⎥\n\
⎢⎢ ⎥⎥\n\
⎢⎢1⎥⎥\n\
⎢⎢─⎥⎥\n\
⎣⎣z⎦⎦\
""")
assert pretty(Mcol2) == ascii_str
assert upretty(Mcol2) == ucode_str
def test_tensor_TensorProduct():
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
assert upretty(TensorProduct(A, B)) == "A\u2297B"
assert upretty(TensorProduct(A, B, A)) == "A\u2297B\u2297A"
def test_diffgeom_print_WedgeProduct():
from sympy.diffgeom.rn import R2
from sympy.diffgeom import WedgeProduct
wp = WedgeProduct(R2.dx, R2.dy)
assert upretty(wp) == u("ⅆ x∧ⅆ y")
def test_Adjoint():
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert pretty(Adjoint(X)) == " +\nX "
assert pretty(Adjoint(X + Y)) == " +\n(X + Y) "
assert pretty(Adjoint(X) + Adjoint(Y)) == " + +\nX + Y "
assert pretty(Adjoint(X*Y)) == " +\n(X*Y) "
assert pretty(Adjoint(Y)*Adjoint(X)) == " + +\nY *X "
assert pretty(Adjoint(X**2)) == " +\n/ 2\\ \n\\X / "
assert pretty(Adjoint(X)**2) == " 2\n/ +\\ \n\\X / "
assert pretty(Adjoint(Inverse(X))) == " +\n/ -1\\ \n\\X / "
assert pretty(Inverse(Adjoint(X))) == " -1\n/ +\\ \n\\X / "
assert pretty(Adjoint(Transpose(X))) == " +\n/ T\\ \n\\X / "
assert pretty(Transpose(Adjoint(X))) == " T\n/ +\\ \n\\X / "
assert upretty(Adjoint(X)) == u" †\nX "
assert upretty(Adjoint(X + Y)) == u" †\n(X + Y) "
assert upretty(Adjoint(X) + Adjoint(Y)) == u" † †\nX + Y "
assert upretty(Adjoint(X*Y)) == u" †\n(X⋅Y) "
assert upretty(Adjoint(Y)*Adjoint(X)) == u" † †\nY ⋅X "
assert upretty(Adjoint(X**2)) == \
u" †\n⎛ 2⎞ \n⎝X ⎠ "
assert upretty(Adjoint(X)**2) == \
u" 2\n⎛ †⎞ \n⎝X ⎠ "
assert upretty(Adjoint(Inverse(X))) == \
u" †\n⎛ -1⎞ \n⎝X ⎠ "
assert upretty(Inverse(Adjoint(X))) == \
u" -1\n⎛ †⎞ \n⎝X ⎠ "
assert upretty(Adjoint(Transpose(X))) == \
u" †\n⎛ T⎞ \n⎝X ⎠ "
assert upretty(Transpose(Adjoint(X))) == \
u" T\n⎛ †⎞ \n⎝X ⎠ "
def test_pretty_Trace_issue_9044():
X = Matrix([[1, 2], [3, 4]])
Y = Matrix([[2, 4], [6, 8]])
ascii_str_1 = \
"""\
/[1 2]\\
tr|[ ]|
\\[3 4]/\
"""
ucode_str_1 = \
u("""\
⎛⎡1 2⎤⎞
tr⎜⎢ ⎥⎟
⎝⎣3 4⎦⎠\
""")
ascii_str_2 = \
"""\
/[1 2]\\ /[2 4]\\
tr|[ ]| + tr|[ ]|
\\[3 4]/ \\[6 8]/\
"""
ucode_str_2 = \
u("""\
⎛⎡1 2⎤⎞ ⎛⎡2 4⎤⎞
tr⎜⎢ ⎥⎟ + tr⎜⎢ ⎥⎟
⎝⎣3 4⎦⎠ ⎝⎣6 8⎦⎠\
""")
assert pretty(Trace(X)) == ascii_str_1
assert upretty(Trace(X)) == ucode_str_1
assert pretty(Trace(X) + Trace(Y)) == ascii_str_2
assert upretty(Trace(X) + Trace(Y)) == ucode_str_2
def test_MatrixExpressions():
n = Symbol('n', integer=True)
X = MatrixSymbol('X', n, n)
assert pretty(X) == upretty(X) == "X"
Y = X[1:2:3, 4:5:6]
ascii_str = ucode_str = "X[1:3, 4:6]"
assert pretty(Y) == ascii_str
assert upretty(Y) == ucode_str
Z = X[1:10:2]
ascii_str = ucode_str = "X[1:10:2, :n]"
assert pretty(Z) == ascii_str
assert upretty(Z) == ucode_str
def test_pretty_dotproduct():
from sympy.matrices import Matrix, MatrixSymbol
from sympy.matrices.expressions.dotproduct import DotProduct
n = symbols("n", integer=True)
A = MatrixSymbol('A', n, 1)
B = MatrixSymbol('B', n, 1)
C = Matrix(1, 3, [1, 2, 3])
D = Matrix(1, 3, [1, 3, 4])
assert pretty(DotProduct(A, B)) == u"A*B"
assert pretty(DotProduct(C, D)) == u"[1 2 3]*[1 3 4]"
assert upretty(DotProduct(A, B)) == u"A⋅B"
assert upretty(DotProduct(C, D)) == u"[1 2 3]⋅[1 3 4]"
def test_pretty_piecewise():
expr = Piecewise((x, x < 1), (x**2, True))
ascii_str = \
"""\
/x for x < 1\n\
| \n\
< 2 \n\
|x otherwise\n\
\\ \
"""
ucode_str = \
u("""\
⎧x for x < 1\n\
⎪ \n\
⎨ 2 \n\
⎪x otherwise\n\
⎩ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -Piecewise((x, x < 1), (x**2, True))
ascii_str = \
"""\
//x for x < 1\\\n\
|| |\n\
-|< 2 |\n\
||x otherwise|\n\
\\\\ /\
"""
ucode_str = \
u("""\
⎛⎧x for x < 1⎞\n\
⎜⎪ ⎟\n\
-⎜⎨ 2 ⎟\n\
⎜⎪x otherwise⎟\n\
⎝⎩ ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x + Piecewise((x, x > 0), (y, True)) + Piecewise((x/y, x < 2),
(y**2, x > 2), (1, True)) + 1
ascii_str = \
"""\
//x \\ \n\
||- for x < 2| \n\
||y | \n\
//x for x > 0\\ || | \n\
x + |< | + |< 2 | + 1\n\
\\\\y otherwise/ ||y for x > 2| \n\
|| | \n\
||1 otherwise| \n\
\\\\ / \
"""
ucode_str = \
u("""\
⎛⎧x ⎞ \n\
⎜⎪─ for x < 2⎟ \n\
⎜⎪y ⎟ \n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟ \n\
x + ⎜⎨ ⎟ + ⎜⎨ 2 ⎟ + 1\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟ \n\
⎜⎪ ⎟ \n\
⎜⎪1 otherwise⎟ \n\
⎝⎩ ⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x - Piecewise((x, x > 0), (y, True)) + Piecewise((x/y, x < 2),
(y**2, x > 2), (1, True)) + 1
ascii_str = \
"""\
//x \\ \n\
||- for x < 2| \n\
||y | \n\
//x for x > 0\\ || | \n\
x - |< | + |< 2 | + 1\n\
\\\\y otherwise/ ||y for x > 2| \n\
|| | \n\
||1 otherwise| \n\
\\\\ / \
"""
ucode_str = \
u("""\
⎛⎧x ⎞ \n\
⎜⎪─ for x < 2⎟ \n\
⎜⎪y ⎟ \n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟ \n\
x - ⎜⎨ ⎟ + ⎜⎨ 2 ⎟ + 1\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟ \n\
⎜⎪ ⎟ \n\
⎜⎪1 otherwise⎟ \n\
⎝⎩ ⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x*Piecewise((x, x > 0), (y, True))
ascii_str = \
"""\
//x for x > 0\\\n\
x*|< |\n\
\\\\y otherwise/\
"""
ucode_str = \
u("""\
⎛⎧x for x > 0⎞\n\
x⋅⎜⎨ ⎟\n\
⎝⎩y otherwise⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Piecewise((x, x > 0), (y, True))*Piecewise((x/y, x < 2), (y**2, x >
2), (1, True))
ascii_str = \
"""\
//x \\\n\
||- for x < 2|\n\
||y |\n\
//x for x > 0\\ || |\n\
|< |*|< 2 |\n\
\\\\y otherwise/ ||y for x > 2|\n\
|| |\n\
||1 otherwise|\n\
\\\\ /\
"""
ucode_str = \
u("""\
⎛⎧x ⎞\n\
⎜⎪─ for x < 2⎟\n\
⎜⎪y ⎟\n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟\n\
⎜⎨ ⎟⋅⎜⎨ 2 ⎟\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟\n\
⎜⎪ ⎟\n\
⎜⎪1 otherwise⎟\n\
⎝⎩ ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -Piecewise((x, x > 0), (y, True))*Piecewise((x/y, x < 2), (y**2, x
> 2), (1, True))
ascii_str = \
"""\
//x \\\n\
||- for x < 2|\n\
||y |\n\
//x for x > 0\\ || |\n\
-|< |*|< 2 |\n\
\\\\y otherwise/ ||y for x > 2|\n\
|| |\n\
||1 otherwise|\n\
\\\\ /\
"""
ucode_str = \
u("""\
⎛⎧x ⎞\n\
⎜⎪─ for x < 2⎟\n\
⎜⎪y ⎟\n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟\n\
-⎜⎨ ⎟⋅⎜⎨ 2 ⎟\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟\n\
⎜⎪ ⎟\n\
⎜⎪1 otherwise⎟\n\
⎝⎩ ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Piecewise((0, Abs(1/y) < 1), (1, Abs(y) < 1), (y*meijerg(((2, 1),
()), ((), (1, 0)), 1/y), True))
ascii_str = \
"""\
/ |1| \n\
| 0 for |-| < 1\n\
| |y| \n\
| \n\
< 1 for |y| < 1\n\
| \n\
| __0, 2 /2, 1 | 1\\ \n\
|y*/__ | | -| otherwise \n\
\\ \\_|2, 2 \\ 1, 0 | y/ \
"""
ucode_str = \
u("""\
⎧ │1│ \n\
⎪ 0 for │─│ < 1\n\
⎪ │y│ \n\
⎪ \n\
⎨ 1 for │y│ < 1\n\
⎪ \n\
⎪ ╭─╮0, 2 ⎛2, 1 │ 1⎞ \n\
⎪y⋅│╶┐ ⎜ │ ─⎟ otherwise \n\
⎩ ╰─╯2, 2 ⎝ 1, 0 │ y⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# XXX: We have to use evaluate=False here because Piecewise._eval_power
# denests the power.
expr = Pow(Piecewise((x, x > 0), (y, True)), 2, evaluate=False)
ascii_str = \
"""\
2\n\
//x for x > 0\\ \n\
|< | \n\
\\\\y otherwise/ \
"""
ucode_str = \
u("""\
2\n\
⎛⎧x for x > 0⎞ \n\
⎜⎨ ⎟ \n\
⎝⎩y otherwise⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_ITE():
expr = ITE(x, y, z)
assert pretty(expr) == (
'/y for x \n'
'< \n'
'\\z otherwise'
)
assert upretty(expr) == u("""\
⎧y for x \n\
⎨ \n\
⎩z otherwise\
""")
def test_pretty_seq():
expr = ()
ascii_str = \
"""\
()\
"""
ucode_str = \
u("""\
()\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = []
ascii_str = \
"""\
[]\
"""
ucode_str = \
u("""\
[]\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = {}
expr_2 = {}
ascii_str = \
"""\
{}\
"""
ucode_str = \
u("""\
{}\
""")
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
expr = (1/x,)
ascii_str = \
"""\
1 \n\
(-,)\n\
x \
"""
ucode_str = \
u("""\
⎛1 ⎞\n\
⎜─,⎟\n\
⎝x ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = [x**2, 1/x, x, y, sin(th)**2/cos(ph)**2]
ascii_str = \
"""\
2 \n\
2 1 sin (theta) \n\
[x , -, x, y, -----------]\n\
x 2 \n\
cos (phi) \
"""
ucode_str = \
u("""\
⎡ 2 ⎤\n\
⎢ 2 1 sin (θ)⎥\n\
⎢x , ─, x, y, ───────⎥\n\
⎢ x 2 ⎥\n\
⎣ cos (φ)⎦\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2, 1/x, x, y, sin(th)**2/cos(ph)**2)
ascii_str = \
"""\
2 \n\
2 1 sin (theta) \n\
(x , -, x, y, -----------)\n\
x 2 \n\
cos (phi) \
"""
ucode_str = \
u("""\
⎛ 2 ⎞\n\
⎜ 2 1 sin (θ)⎟\n\
⎜x , ─, x, y, ───────⎟\n\
⎜ x 2 ⎟\n\
⎝ cos (φ)⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Tuple(x**2, 1/x, x, y, sin(th)**2/cos(ph)**2)
ascii_str = \
"""\
2 \n\
2 1 sin (theta) \n\
(x , -, x, y, -----------)\n\
x 2 \n\
cos (phi) \
"""
ucode_str = \
u("""\
⎛ 2 ⎞\n\
⎜ 2 1 sin (θ)⎟\n\
⎜x , ─, x, y, ───────⎟\n\
⎜ x 2 ⎟\n\
⎝ cos (φ)⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = {x: sin(x)}
expr_2 = Dict({x: sin(x)})
ascii_str = \
"""\
{x: sin(x)}\
"""
ucode_str = \
u("""\
{x: sin(x)}\
""")
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
expr = {1/x: 1/y, x: sin(x)**2}
expr_2 = Dict({1/x: 1/y, x: sin(x)**2})
ascii_str = \
"""\
1 1 2 \n\
{-: -, x: sin (x)}\n\
x y \
"""
ucode_str = \
u("""\
⎧1 1 2 ⎫\n\
⎨─: ─, x: sin (x)⎬\n\
⎩x y ⎭\
""")
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
# There used to be a bug with pretty-printing sequences of even height.
expr = [x**2]
ascii_str = \
"""\
2 \n\
[x ]\
"""
ucode_str = \
u("""\
⎡ 2⎤\n\
⎣x ⎦\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2,)
ascii_str = \
"""\
2 \n\
(x ,)\
"""
ucode_str = \
u("""\
⎛ 2 ⎞\n\
⎝x ,⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Tuple(x**2)
ascii_str = \
"""\
2 \n\
(x ,)\
"""
ucode_str = \
u("""\
⎛ 2 ⎞\n\
⎝x ,⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = {x**2: 1}
expr_2 = Dict({x**2: 1})
ascii_str = \
"""\
2 \n\
{x : 1}\
"""
ucode_str = \
u("""\
⎧ 2 ⎫\n\
⎨x : 1⎬\n\
⎩ ⎭\
""")
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
def test_any_object_in_sequence():
# Cf. issue 5306
b1 = Basic()
b2 = Basic(Basic())
expr = [b2, b1]
assert pretty(expr) == "[Basic(Basic()), Basic()]"
assert upretty(expr) == u"[Basic(Basic()), Basic()]"
expr = {b2, b1}
assert pretty(expr) == "{Basic(), Basic(Basic())}"
assert upretty(expr) == u"{Basic(), Basic(Basic())}"
expr = {b2: b1, b1: b2}
expr2 = Dict({b2: b1, b1: b2})
assert pretty(expr) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
assert pretty(
expr2) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
assert upretty(
expr) == u"{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
assert upretty(
expr2) == u"{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
def test_print_builtin_set():
assert pretty(set()) == 'set()'
assert upretty(set()) == u'set()'
assert pretty(frozenset()) == 'frozenset()'
assert upretty(frozenset()) == u'frozenset()'
s1 = {1/x, x}
s2 = frozenset(s1)
assert pretty(s1) == \
"""\
1 \n\
{-, x}
x \
"""
assert upretty(s1) == \
u"""\
⎧1 ⎫
⎨─, x⎬
⎩x ⎭\
"""
assert pretty(s2) == \
"""\
1 \n\
frozenset({-, x})
x \
"""
assert upretty(s2) == \
u"""\
⎛⎧1 ⎫⎞
frozenset⎜⎨─, x⎬⎟
⎝⎩x ⎭⎠\
"""
def test_pretty_sets():
s = FiniteSet
assert pretty(s(*[x*y, x**2])) == \
"""\
2 \n\
{x , x*y}\
"""
assert pretty(s(*range(1, 6))) == "{1, 2, 3, 4, 5}"
assert pretty(s(*range(1, 13))) == "{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}"
assert pretty(set([x*y, x**2])) == \
"""\
2 \n\
{x , x*y}\
"""
assert pretty(set(range(1, 6))) == "{1, 2, 3, 4, 5}"
assert pretty(set(range(1, 13))) == \
"{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}"
assert pretty(frozenset([x*y, x**2])) == \
"""\
2 \n\
frozenset({x , x*y})\
"""
assert pretty(frozenset(range(1, 6))) == "frozenset({1, 2, 3, 4, 5})"
assert pretty(frozenset(range(1, 13))) == \
"frozenset({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12})"
assert pretty(Range(0, 3, 1)) == '{0, 1, 2}'
ascii_str = '{0, 1, ..., 29}'
ucode_str = u'{0, 1, …, 29}'
assert pretty(Range(0, 30, 1)) == ascii_str
assert upretty(Range(0, 30, 1)) == ucode_str
ascii_str = '{30, 29, ..., 2}'
ucode_str = u('{30, 29, …, 2}')
assert pretty(Range(30, 1, -1)) == ascii_str
assert upretty(Range(30, 1, -1)) == ucode_str
ascii_str = '{0, 2, ...}'
ucode_str = u'{0, 2, …}'
assert pretty(Range(0, oo, 2)) == ascii_str
assert upretty(Range(0, oo, 2)) == ucode_str
ascii_str = '{..., 2, 0}'
ucode_str = u('{…, 2, 0}')
assert pretty(Range(oo, -2, -2)) == ascii_str
assert upretty(Range(oo, -2, -2)) == ucode_str
ascii_str = '{-2, -3, ...}'
ucode_str = u('{-2, -3, …}')
assert pretty(Range(-2, -oo, -1)) == ascii_str
assert upretty(Range(-2, -oo, -1)) == ucode_str
def test_pretty_SetExpr():
iv = Interval(1, 3)
se = SetExpr(iv)
ascii_str = "SetExpr([1, 3])"
ucode_str = u("SetExpr([1, 3])")
assert pretty(se) == ascii_str
assert upretty(se) == ucode_str
def test_pretty_ImageSet():
imgset = ImageSet(Lambda((x, y), x + y), {1, 2, 3}, {3, 4})
ascii_str = '{x + y | x in {1, 2, 3} , y in {3, 4}}'
ucode_str = u('{x + y | x ∊ {1, 2, 3} , y ∊ {3, 4}}')
assert pretty(imgset) == ascii_str
assert upretty(imgset) == ucode_str
imgset = ImageSet(Lambda(x, x**2), S.Naturals)
ascii_str = \
' 2 \n'\
'{x | x in Naturals}'
ucode_str = u('''\
⎧ 2 ⎫\n\
⎨x | x ∊ ℕ⎬\n\
⎩ ⎭''')
assert pretty(imgset) == ascii_str
assert upretty(imgset) == ucode_str
def test_pretty_ConditionSet():
from sympy import ConditionSet
ascii_str = '{x | x in (-oo, oo) and sin(x) = 0}'
ucode_str = u'{x | x ∊ ℝ ∧ sin(x) = 0}'
assert pretty(ConditionSet(x, Eq(sin(x), 0), S.Reals)) == ascii_str
assert upretty(ConditionSet(x, Eq(sin(x), 0), S.Reals)) == ucode_str
assert pretty(ConditionSet(x, Contains(x, S.Reals, evaluate=False), FiniteSet(1))) == '{1}'
assert upretty(ConditionSet(x, Contains(x, S.Reals, evaluate=False), FiniteSet(1))) == u'{1}'
assert pretty(ConditionSet(x, And(x > 1, x < -1), FiniteSet(1, 2, 3))) == "EmptySet()"
assert upretty(ConditionSet(x, And(x > 1, x < -1), FiniteSet(1, 2, 3))) == u"∅"
assert pretty(ConditionSet(x, Or(x > 1, x < -1), FiniteSet(1, 2))) == '{2}'
assert upretty(ConditionSet(x, Or(x > 1, x < -1), FiniteSet(1, 2))) == u'{2}'
def test_pretty_ComplexRegion():
from sympy import ComplexRegion
ucode_str = u'{x + y⋅ⅈ | x, y ∊ [3, 5] × [4, 6]}'
assert upretty(ComplexRegion(Interval(3, 5)*Interval(4, 6))) == ucode_str
ucode_str = u'{r⋅(ⅈ⋅sin(θ) + cos(θ)) | r, θ ∊ [0, 1] × [0, 2⋅π)}'
assert upretty(ComplexRegion(Interval(0, 1)*Interval(0, 2*pi), polar=True)) == ucode_str
def test_pretty_Union_issue_10414():
a, b = Interval(2, 3), Interval(4, 7)
ucode_str = u'[2, 3] ∪ [4, 7]'
ascii_str = '[2, 3] U [4, 7]'
assert upretty(Union(a, b)) == ucode_str
assert pretty(Union(a, b)) == ascii_str
def test_pretty_Intersection_issue_10414():
x, y, z, w = symbols('x, y, z, w')
a, b = Interval(x, y), Interval(z, w)
ucode_str = u'[x, y] ∩ [z, w]'
ascii_str = '[x, y] n [z, w]'
assert upretty(Intersection(a, b)) == ucode_str
assert pretty(Intersection(a, b)) == ascii_str
def test_ProductSet_paranthesis():
ucode_str = u'([4, 7] × {1, 2}) ∪ ([2, 3] × [4, 7])'
a, b, c = Interval(2, 3), Interval(4, 7), Interval(1, 9)
assert upretty(Union(a*b, b*FiniteSet(1, 2))) == ucode_str
def test_ProductSet_prod_char_issue_10413():
ascii_str = '[2, 3] x [4, 7]'
ucode_str = u'[2, 3] × [4, 7]'
a, b = Interval(2, 3), Interval(4, 7)
assert pretty(a*b) == ascii_str
assert upretty(a*b) == ucode_str
def test_pretty_sequences():
s1 = SeqFormula(a**2, (0, oo))
s2 = SeqPer((1, 2))
ascii_str = '[0, 1, 4, 9, ...]'
ucode_str = u'[0, 1, 4, 9, …]'
assert pretty(s1) == ascii_str
assert upretty(s1) == ucode_str
ascii_str = '[1, 2, 1, 2, ...]'
ucode_str = u'[1, 2, 1, 2, …]'
assert pretty(s2) == ascii_str
assert upretty(s2) == ucode_str
s3 = SeqFormula(a**2, (0, 2))
s4 = SeqPer((1, 2), (0, 2))
ascii_str = '[0, 1, 4]'
ucode_str = u'[0, 1, 4]'
assert pretty(s3) == ascii_str
assert upretty(s3) == ucode_str
ascii_str = '[1, 2, 1]'
ucode_str = u'[1, 2, 1]'
assert pretty(s4) == ascii_str
assert upretty(s4) == ucode_str
s5 = SeqFormula(a**2, (-oo, 0))
s6 = SeqPer((1, 2), (-oo, 0))
ascii_str = '[..., 9, 4, 1, 0]'
ucode_str = u'[…, 9, 4, 1, 0]'
assert pretty(s5) == ascii_str
assert upretty(s5) == ucode_str
ascii_str = '[..., 2, 1, 2, 1]'
ucode_str = u'[…, 2, 1, 2, 1]'
assert pretty(s6) == ascii_str
assert upretty(s6) == ucode_str
ascii_str = '[1, 3, 5, 11, ...]'
ucode_str = u'[1, 3, 5, 11, …]'
assert pretty(SeqAdd(s1, s2)) == ascii_str
assert upretty(SeqAdd(s1, s2)) == ucode_str
ascii_str = '[1, 3, 5]'
ucode_str = u'[1, 3, 5]'
assert pretty(SeqAdd(s3, s4)) == ascii_str
assert upretty(SeqAdd(s3, s4)) == ucode_str
ascii_str = '[..., 11, 5, 3, 1]'
ucode_str = u'[…, 11, 5, 3, 1]'
assert pretty(SeqAdd(s5, s6)) == ascii_str
assert upretty(SeqAdd(s5, s6)) == ucode_str
ascii_str = '[0, 2, 4, 18, ...]'
ucode_str = u'[0, 2, 4, 18, …]'
assert pretty(SeqMul(s1, s2)) == ascii_str
assert upretty(SeqMul(s1, s2)) == ucode_str
ascii_str = '[0, 2, 4]'
ucode_str = u'[0, 2, 4]'
assert pretty(SeqMul(s3, s4)) == ascii_str
assert upretty(SeqMul(s3, s4)) == ucode_str
ascii_str = '[..., 18, 4, 2, 0]'
ucode_str = u'[…, 18, 4, 2, 0]'
assert pretty(SeqMul(s5, s6)) == ascii_str
assert upretty(SeqMul(s5, s6)) == ucode_str
# Sequences with symbolic limits, issue 12629
s7 = SeqFormula(a**2, (a, 0, x))
raises(NotImplementedError, lambda: pretty(s7))
raises(NotImplementedError, lambda: upretty(s7))
b = Symbol('b')
s8 = SeqFormula(b*a**2, (a, 0, 2))
ascii_str = u'[0, b, 4*b]'
ucode_str = u'[0, b, 4⋅b]'
assert pretty(s8) == ascii_str
assert upretty(s8) == ucode_str
def test_pretty_FourierSeries():
f = fourier_series(x, (x, -pi, pi))
ascii_str = \
"""\
2*sin(3*x) \n\
2*sin(x) - sin(2*x) + ---------- + ...\n\
3 \
"""
ucode_str = \
u("""\
2⋅sin(3⋅x) \n\
2⋅sin(x) - sin(2⋅x) + ────────── + …\n\
3 \
""")
assert pretty(f) == ascii_str
assert upretty(f) == ucode_str
def test_pretty_FormalPowerSeries():
f = fps(log(1 + x))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ -k k \n\
\\ -(-1) *x \n\
/ -----------\n\
/ k \n\
/___, \n\
k = 1 \
"""
ucode_str = \
u("""\
∞ \n\
____ \n\
╲ \n\
╲ -k k \n\
╲ -(-1) ⋅x \n\
╱ ───────────\n\
╱ k \n\
╱ \n\
‾‾‾‾ \n\
k = 1 \
""")
assert pretty(f) == ascii_str
assert upretty(f) == ucode_str
def test_pretty_limits():
expr = Limit(x, x, oo)
ascii_str = \
"""\
lim x\n\
x->oo \
"""
ucode_str = \
u("""\
lim x\n\
x─→∞ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x**2, x, 0)
ascii_str = \
"""\
2\n\
lim x \n\
x->0+ \
"""
ucode_str = \
u("""\
2\n\
lim x \n\
x─→0⁺ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(1/x, x, 0)
ascii_str = \
"""\
1\n\
lim -\n\
x->0+x\
"""
ucode_str = \
u("""\
1\n\
lim ─\n\
x─→0⁺x\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(sin(x)/x, x, 0)
ascii_str = \
"""\
/sin(x)\\\n\
lim |------|\n\
x->0+\\ x /\
"""
ucode_str = \
u("""\
⎛sin(x)⎞\n\
lim ⎜──────⎟\n\
x─→0⁺⎝ x ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(sin(x)/x, x, 0, "-")
ascii_str = \
"""\
/sin(x)\\\n\
lim |------|\n\
x->0-\\ x /\
"""
ucode_str = \
u("""\
⎛sin(x)⎞\n\
lim ⎜──────⎟\n\
x─→0⁻⎝ x ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x + sin(x), x, 0)
ascii_str = \
"""\
lim (x + sin(x))\n\
x->0+ \
"""
ucode_str = \
u("""\
lim (x + sin(x))\n\
x─→0⁺ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x, x, 0)**2
ascii_str = \
"""\
2\n\
/ lim x\\ \n\
\\x->0+ / \
"""
ucode_str = \
u("""\
2\n\
⎛ lim x⎞ \n\
⎝x─→0⁺ ⎠ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x*Limit(y/2,y,0), x, 0)
ascii_str = \
"""\
/ /y\\\\\n\
lim |x* lim |-||\n\
x->0+\\ y->0+\\2//\
"""
ucode_str = \
u("""\
⎛ ⎛y⎞⎞\n\
lim ⎜x⋅ lim ⎜─⎟⎟\n\
x─→0⁺⎝ y─→0⁺⎝2⎠⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2*Limit(x*Limit(y/2,y,0), x, 0)
ascii_str = \
"""\
/ /y\\\\\n\
2* lim |x* lim |-||\n\
x->0+\\ y->0+\\2//\
"""
ucode_str = \
u("""\
⎛ ⎛y⎞⎞\n\
2⋅ lim ⎜x⋅ lim ⎜─⎟⎟\n\
x─→0⁺⎝ y─→0⁺⎝2⎠⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(sin(x), x, 0, dir='+-')
ascii_str = \
"""\
lim sin(x)\n\
x->0 \
"""
ucode_str = \
u("""\
lim sin(x)\n\
x─→0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_ComplexRootOf():
expr = rootof(x**5 + 11*x - 2, 0)
ascii_str = \
"""\
/ 5 \\\n\
CRootOf\\x + 11*x - 2, 0/\
"""
ucode_str = \
u("""\
⎛ 5 ⎞\n\
CRootOf⎝x + 11⋅x - 2, 0⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_RootSum():
expr = RootSum(x**5 + 11*x - 2, auto=False)
ascii_str = \
"""\
/ 5 \\\n\
RootSum\\x + 11*x - 2/\
"""
ucode_str = \
u("""\
⎛ 5 ⎞\n\
RootSum⎝x + 11⋅x - 2⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = RootSum(x**5 + 11*x - 2, Lambda(z, exp(z)))
ascii_str = \
"""\
/ 5 z\\\n\
RootSum\\x + 11*x - 2, z -> e /\
"""
ucode_str = \
u("""\
⎛ 5 z⎞\n\
RootSum⎝x + 11⋅x - 2, z ↦ ℯ ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_GroebnerBasis():
expr = groebner([], x, y)
ascii_str = \
"""\
GroebnerBasis([], x, y, domain=ZZ, order=lex)\
"""
ucode_str = \
u("""\
GroebnerBasis([], x, y, domain=ℤ, order=lex)\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
F = [x**2 - 3*y - x + 1, y**2 - 2*x + y - 1]
expr = groebner(F, x, y, order='grlex')
ascii_str = \
"""\
/[ 2 2 ] \\\n\
GroebnerBasis\\[x - x - 3*y + 1, y - 2*x + y - 1], x, y, domain=ZZ, order=grlex/\
"""
ucode_str = \
u("""\
⎛⎡ 2 2 ⎤ ⎞\n\
GroebnerBasis⎝⎣x - x - 3⋅y + 1, y - 2⋅x + y - 1⎦, x, y, domain=ℤ, order=grlex⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = expr.fglm('lex')
ascii_str = \
"""\
/[ 2 4 3 2 ] \\\n\
GroebnerBasis\\[2*x - y - y + 1, y + 2*y - 3*y - 16*y + 7], x, y, domain=ZZ, order=lex/\
"""
ucode_str = \
u("""\
⎛⎡ 2 4 3 2 ⎤ ⎞\n\
GroebnerBasis⎝⎣2⋅x - y - y + 1, y + 2⋅y - 3⋅y - 16⋅y + 7⎦, x, y, domain=ℤ, order=lex⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_Boolean():
expr = Not(x, evaluate=False)
assert pretty(expr) == "Not(x)"
assert upretty(expr) == u"¬x"
expr = And(x, y)
assert pretty(expr) == "And(x, y)"
assert upretty(expr) == u"x ∧ y"
expr = Or(x, y)
assert pretty(expr) == "Or(x, y)"
assert upretty(expr) == u"x ∨ y"
syms = symbols('a:f')
expr = And(*syms)
assert pretty(expr) == "And(a, b, c, d, e, f)"
assert upretty(expr) == u"a ∧ b ∧ c ∧ d ∧ e ∧ f"
expr = Or(*syms)
assert pretty(expr) == "Or(a, b, c, d, e, f)"
assert upretty(expr) == u"a ∨ b ∨ c ∨ d ∨ e ∨ f"
expr = Xor(x, y, evaluate=False)
assert pretty(expr) == "Xor(x, y)"
assert upretty(expr) == u"x ⊻ y"
expr = Nand(x, y, evaluate=False)
assert pretty(expr) == "Nand(x, y)"
assert upretty(expr) == u"x ⊼ y"
expr = Nor(x, y, evaluate=False)
assert pretty(expr) == "Nor(x, y)"
assert upretty(expr) == u"x ⊽ y"
expr = Implies(x, y, evaluate=False)
assert pretty(expr) == "Implies(x, y)"
assert upretty(expr) == u"x → y"
# don't sort args
expr = Implies(y, x, evaluate=False)
assert pretty(expr) == "Implies(y, x)"
assert upretty(expr) == u"y → x"
expr = Equivalent(x, y, evaluate=False)
assert pretty(expr) == "Equivalent(x, y)"
assert upretty(expr) == u"x ⇔ y"
expr = Equivalent(y, x, evaluate=False)
assert pretty(expr) == "Equivalent(x, y)"
assert upretty(expr) == u"x ⇔ y"
def test_pretty_Domain():
expr = FF(23)
assert pretty(expr) == "GF(23)"
assert upretty(expr) == u"ℤ₂₃"
expr = ZZ
assert pretty(expr) == "ZZ"
assert upretty(expr) == u"ℤ"
expr = QQ
assert pretty(expr) == "QQ"
assert upretty(expr) == u"ℚ"
expr = RR
assert pretty(expr) == "RR"
assert upretty(expr) == u"ℝ"
expr = QQ[x]
assert pretty(expr) == "QQ[x]"
assert upretty(expr) == u"ℚ[x]"
expr = QQ[x, y]
assert pretty(expr) == "QQ[x, y]"
assert upretty(expr) == u"ℚ[x, y]"
expr = ZZ.frac_field(x)
assert pretty(expr) == "ZZ(x)"
assert upretty(expr) == u"ℤ(x)"
expr = ZZ.frac_field(x, y)
assert pretty(expr) == "ZZ(x, y)"
assert upretty(expr) == u"ℤ(x, y)"
expr = QQ.poly_ring(x, y, order=grlex)
assert pretty(expr) == "QQ[x, y, order=grlex]"
assert upretty(expr) == u"ℚ[x, y, order=grlex]"
expr = QQ.poly_ring(x, y, order=ilex)
assert pretty(expr) == "QQ[x, y, order=ilex]"
assert upretty(expr) == u"ℚ[x, y, order=ilex]"
def test_pretty_prec():
assert xpretty(S("0.3"), full_prec=True, wrap_line=False) == "0.300000000000000"
assert xpretty(S("0.3"), full_prec="auto", wrap_line=False) == "0.300000000000000"
assert xpretty(S("0.3"), full_prec=False, wrap_line=False) == "0.3"
assert xpretty(S("0.3")*x, full_prec=True, use_unicode=False, wrap_line=False) in [
"0.300000000000000*x",
"x*0.300000000000000"
]
assert xpretty(S("0.3")*x, full_prec="auto", use_unicode=False, wrap_line=False) in [
"0.3*x",
"x*0.3"
]
assert xpretty(S("0.3")*x, full_prec=False, use_unicode=False, wrap_line=False) in [
"0.3*x",
"x*0.3"
]
def test_pprint():
import sys
from sympy.core.compatibility import StringIO
fd = StringIO()
sso = sys.stdout
sys.stdout = fd
try:
pprint(pi, use_unicode=False, wrap_line=False)
finally:
sys.stdout = sso
assert fd.getvalue() == 'pi\n'
def test_pretty_class():
"""Test that the printer dispatcher correctly handles classes."""
class C:
pass # C has no .__class__ and this was causing problems
class D(object):
pass
assert pretty( C ) == str( C )
assert pretty( D ) == str( D )
def test_pretty_no_wrap_line():
huge_expr = 0
for i in range(20):
huge_expr += i*sin(i + x)
assert xpretty(huge_expr ).find('\n') != -1
assert xpretty(huge_expr, wrap_line=False).find('\n') == -1
def test_settings():
raises(TypeError, lambda: pretty(S(4), method="garbage"))
def test_pretty_sum():
from sympy.abc import x, a, b, k, m, n
expr = Sum(k**k, (k, 0, n))
ascii_str = \
"""\
n \n\
___ \n\
\\ ` \n\
\\ k\n\
/ k \n\
/__, \n\
k = 0 \
"""
ucode_str = \
u("""\
n \n\
___ \n\
╲ \n\
╲ k\n\
╱ k \n\
╱ \n\
‾‾‾ \n\
k = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**k, (k, oo, n))
ascii_str = \
"""\
n \n\
___ \n\
\\ ` \n\
\\ k\n\
/ k \n\
/__, \n\
k = oo \
"""
ucode_str = \
u("""\
n \n\
___ \n\
╲ \n\
╲ k\n\
╱ k \n\
╱ \n\
‾‾‾ \n\
k = ∞ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(Integral(x**n, (x, -oo, oo))), (k, 0, n**n))
ascii_str = \
"""\
n \n\
n \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
k = 0 \
"""
ucode_str = \
u("""\
n \n\
n \n\
______ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╲ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
╱ \n\
‾‾‾‾‾‾ \n\
k = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(
Integral(x**n, (x, -oo, oo))), (k, 0, Integral(x**x, (x, -oo, oo))))
ascii_str = \
"""\
oo \n\
/ \n\
| \n\
| x \n\
| x dx \n\
| \n\
/ \n\
-oo \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
k = 0 \
"""
ucode_str = \
u("""\
∞ \n\
⌠ \n\
⎮ x \n\
⎮ x dx \n\
⌡ \n\
-∞ \n\
______ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╲ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
╱ \n\
‾‾‾‾‾‾ \n\
k = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(Integral(x**n, (x, -oo, oo))), (
k, x + n + x**2 + n**2 + (x/n) + (1/x), Integral(x**x, (x, -oo, oo))))
ascii_str = \
"""\
oo \n\
/ \n\
| \n\
| x \n\
| x dx \n\
| \n\
/ \n\
-oo \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
2 2 1 x \n\
k = n + n + x + x + - + - \n\
x n \
"""
ucode_str = \
u("""\
∞ \n\
⌠ \n\
⎮ x \n\
⎮ x dx \n\
⌡ \n\
-∞ \n\
______ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╲ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
╱ \n\
‾‾‾‾‾‾ \n\
2 2 1 x \n\
k = n + n + x + x + ─ + ─ \n\
x n \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(
Integral(x**n, (x, -oo, oo))), (k, 0, x + n + x**2 + n**2 + (x/n) + (1/x)))
ascii_str = \
"""\
2 2 1 x \n\
n + n + x + x + - + - \n\
x n \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
k = 0 \
"""
ucode_str = \
u("""\
2 2 1 x \n\
n + n + x + x + ─ + ─ \n\
x n \n\
______ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╲ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
╱ \n\
‾‾‾‾‾‾ \n\
k = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x, (x, 0, oo))
ascii_str = \
"""\
oo \n\
__ \n\
\\ ` \n\
) x\n\
/_, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
___ \n\
╲ \n\
╲ x\n\
╱ \n\
╱ \n\
‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x**2, (x, 0, oo))
ascii_str = \
u("""\
oo \n\
___ \n\
\\ ` \n\
\\ 2\n\
/ x \n\
/__, \n\
x = 0 \
""")
ucode_str = \
u("""\
∞ \n\
___ \n\
╲ \n\
╲ 2\n\
╱ x \n\
╱ \n\
‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x/2, (x, 0, oo))
ascii_str = \
"""\
oo \n\
___ \n\
\\ ` \n\
\\ x\n\
) -\n\
/ 2\n\
/__, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
____ \n\
╲ \n\
╲ x\n\
╲ ─\n\
╱ 2\n\
╱ \n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x**3/2, (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ 3\n\
\\ x \n\
/ --\n\
/ 2 \n\
/___, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
____ \n\
╲ \n\
╲ 3\n\
╲ x \n\
╱ ──\n\
╱ 2 \n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum((x**3*y**(x/2))**n, (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ n\n\
\\ / x\\ \n\
) | -| \n\
/ | 3 2| \n\
/ \\x *y / \n\
/___, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
_____ \n\
╲ \n\
╲ n\n\
╲ ⎛ x⎞ \n\
╲ ⎜ ─⎟ \n\
╱ ⎜ 3 2⎟ \n\
╱ ⎝x ⋅y ⎠ \n\
╱ \n\
╱ \n\
‾‾‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(1/x**2, (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ 1 \n\
\\ --\n\
/ 2\n\
/ x \n\
/___, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
____ \n\
╲ \n\
╲ 1 \n\
╲ ──\n\
╱ 2\n\
╱ x \n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(1/y**(a/b), (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ -a \n\
\\ ---\n\
/ b \n\
/ y \n\
/___, \n\
x = 0 \
"""
ucode_str = \
u("""\
∞ \n\
____ \n\
╲ \n\
╲ -a \n\
╲ ───\n\
╱ b \n\
╱ y \n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(1/y**(a/b), (x, 0, oo), (y, 1, 2))
ascii_str = \
"""\
2 oo \n\
____ ____ \n\
\\ ` \\ ` \n\
\\ \\ -a\n\
\\ \\ --\n\
/ / b \n\
/ / y \n\
/___, /___, \n\
y = 1 x = 0 \
"""
ucode_str = \
u("""\
2 ∞ \n\
____ ____ \n\
╲ ╲ \n\
╲ ╲ -a\n\
╲ ╲ ──\n\
╱ ╱ b \n\
╱ ╱ y \n\
╱ ╱ \n\
‾‾‾‾ ‾‾‾‾ \n\
y = 1 x = 0 \
""")
expr = Sum(1/(1 + 1/(
1 + 1/k)) + 1, (k, 111, 1 + 1/n), (k, 1/(1 + m), oo)) + 1/(1 + 1/k)
ascii_str = \
"""\
1 \n\
1 + - \n\
oo n \n\
_____ _____ \n\
\\ ` \\ ` \n\
\\ \\ / 1 \\ \n\
\\ \\ |1 + ---------| \n\
\\ \\ | 1 | 1 \n\
) ) | 1 + -----| + -----\n\
/ / | 1| 1\n\
/ / | 1 + -| 1 + -\n\
/ / \\ k/ k\n\
/____, /____, \n\
1 k = 111 \n\
k = ----- \n\
m + 1 \
"""
ucode_str = \
u("""\
1 \n\
1 + ─ \n\
∞ n \n\
______ ______ \n\
╲ ╲ \n\
╲ ╲ ⎛ 1 ⎞ \n\
╲ ╲ ⎜1 + ─────────⎟ \n\
╲ ╲ ⎜ 1 ⎟ \n\
╲ ╲ ⎜ 1 + ─────⎟ 1 \n\
╱ ╱ ⎜ 1⎟ + ─────\n\
╱ ╱ ⎜ 1 + ─⎟ 1\n\
╱ ╱ ⎝ k⎠ 1 + ─\n\
╱ ╱ k\n\
╱ ╱ \n\
‾‾‾‾‾‾ ‾‾‾‾‾‾ \n\
1 k = 111 \n\
k = ───── \n\
m + 1 \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_units():
expr = joule
ascii_str1 = \
"""\
2\n\
kilogram*meter \n\
---------------\n\
2 \n\
second \
"""
unicode_str1 = \
u("""\
2\n\
kilogram⋅meter \n\
───────────────\n\
2 \n\
second \
""")
ascii_str2 = \
"""\
2\n\
3*x*y*kilogram*meter \n\
---------------------\n\
2 \n\
second \
"""
unicode_str2 = \
u("""\
2\n\
3⋅x⋅y⋅kilogram⋅meter \n\
─────────────────────\n\
2 \n\
second \
""")
from sympy.physics.units import kg, m, s
assert upretty(expr) == u("joule")
assert pretty(expr) == "joule"
assert upretty(expr.convert_to(kg*m**2/s**2)) == unicode_str1
assert pretty(expr.convert_to(kg*m**2/s**2)) == ascii_str1
assert upretty(3*kg*x*m**2*y/s**2) == unicode_str2
assert pretty(3*kg*x*m**2*y/s**2) == ascii_str2
def test_pretty_Subs():
f = Function('f')
expr = Subs(f(x), x, ph**2)
ascii_str = \
"""\
(f(x))| 2\n\
|x=phi \
"""
unicode_str = \
u("""\
(f(x))│ 2\n\
│x=φ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Subs(f(x).diff(x), x, 0)
ascii_str = \
"""\
/d \\| \n\
|--(f(x))|| \n\
\\dx /|x=0\
"""
unicode_str = \
u("""\
⎛d ⎞│ \n\
⎜──(f(x))⎟│ \n\
⎝dx ⎠│x=0\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Subs(f(x).diff(x)/y, (x, y), (0, Rational(1, 2)))
ascii_str = \
"""\
/d \\| \n\
|--(f(x))|| \n\
|dx || \n\
|--------|| \n\
\\ y /|x=0, y=1/2\
"""
unicode_str = \
u("""\
⎛d ⎞│ \n\
⎜──(f(x))⎟│ \n\
⎜dx ⎟│ \n\
⎜────────⎟│ \n\
⎝ y ⎠│x=0, y=1/2\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
def test_gammas():
assert upretty(lowergamma(x, y)) == u"γ(x, y)"
assert upretty(uppergamma(x, y)) == u"Γ(x, y)"
assert xpretty(gamma(x), use_unicode=True) == u'Γ(x)'
assert xpretty(gamma, use_unicode=True) == u'Γ'
assert xpretty(symbols('gamma', cls=Function)(x), use_unicode=True) == u'γ(x)'
assert xpretty(symbols('gamma', cls=Function), use_unicode=True) == u'γ'
def test_beta():
assert xpretty(beta(x,y), use_unicode=True) == u'Β(x, y)'
assert xpretty(beta(x,y), use_unicode=False) == u'B(x, y)'
assert xpretty(beta, use_unicode=True) == u'Β'
assert xpretty(beta, use_unicode=False) == u'B'
mybeta = Function('beta')
assert xpretty(mybeta(x), use_unicode=True) == u'β(x)'
assert xpretty(mybeta(x, y, z), use_unicode=False) == u'beta(x, y, z)'
assert xpretty(mybeta, use_unicode=True) == u'β'
# test that notation passes to subclasses of the same name only
def test_function_subclass_different_name():
class mygamma(gamma):
pass
assert xpretty(mygamma, use_unicode=True) == r"mygamma"
assert xpretty(mygamma(x), use_unicode=True) == r"mygamma(x)"
def test_SingularityFunction():
assert xpretty(SingularityFunction(x, 0, n), use_unicode=True) == (
"""\
n\n\
<x> \
""")
assert xpretty(SingularityFunction(x, 1, n), use_unicode=True) == (
"""\
n\n\
<x - 1> \
""")
assert xpretty(SingularityFunction(x, -1, n), use_unicode=True) == (
"""\
n\n\
<x + 1> \
""")
assert xpretty(SingularityFunction(x, a, n), use_unicode=True) == (
"""\
n\n\
<-a + x> \
""")
assert xpretty(SingularityFunction(x, y, n), use_unicode=True) == (
"""\
n\n\
<x - y> \
""")
assert xpretty(SingularityFunction(x, 0, n), use_unicode=False) == (
"""\
n\n\
<x> \
""")
assert xpretty(SingularityFunction(x, 1, n), use_unicode=False) == (
"""\
n\n\
<x - 1> \
""")
assert xpretty(SingularityFunction(x, -1, n), use_unicode=False) == (
"""\
n\n\
<x + 1> \
""")
assert xpretty(SingularityFunction(x, a, n), use_unicode=False) == (
"""\
n\n\
<-a + x> \
""")
assert xpretty(SingularityFunction(x, y, n), use_unicode=False) == (
"""\
n\n\
<x - y> \
""")
def test_deltas():
assert xpretty(DiracDelta(x), use_unicode=True) == u'δ(x)'
assert xpretty(DiracDelta(x, 1), use_unicode=True) == \
u("""\
(1) \n\
δ (x)\
""")
assert xpretty(x*DiracDelta(x, 1), use_unicode=True) == \
u("""\
(1) \n\
x⋅δ (x)\
""")
def test_hyper():
expr = hyper((), (), z)
ucode_str = \
u("""\
┌─ ⎛ │ ⎞\n\
├─ ⎜ │ z⎟\n\
0╵ 0 ⎝ │ ⎠\
""")
ascii_str = \
"""\
_ \n\
|_ / | \\\n\
| | | z|\n\
0 0 \\ | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper((), (1,), x)
ucode_str = \
u("""\
┌─ ⎛ │ ⎞\n\
├─ ⎜ │ x⎟\n\
0╵ 1 ⎝1 │ ⎠\
""")
ascii_str = \
"""\
_ \n\
|_ / | \\\n\
| | | x|\n\
0 1 \\1 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper([2], [1], x)
ucode_str = \
u("""\
┌─ ⎛2 │ ⎞\n\
├─ ⎜ │ x⎟\n\
1╵ 1 ⎝1 │ ⎠\
""")
ascii_str = \
"""\
_ \n\
|_ /2 | \\\n\
| | | x|\n\
1 1 \\1 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper((pi/3, -2*k), (3, 4, 5, -3), x)
ucode_str = \
u("""\
⎛ π │ ⎞\n\
┌─ ⎜ ─, -2⋅k │ ⎟\n\
├─ ⎜ 3 │ x⎟\n\
2╵ 4 ⎜ │ ⎟\n\
⎝3, 4, 5, -3 │ ⎠\
""")
ascii_str = \
"""\
\n\
_ / pi | \\\n\
|_ | --, -2*k | |\n\
| | 3 | x|\n\
2 4 | | |\n\
\\3, 4, 5, -3 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper((pi, S('2/3'), -2*k), (3, 4, 5, -3), x**2)
ucode_str = \
u("""\
┌─ ⎛π, 2/3, -2⋅k │ 2⎞\n\
├─ ⎜ │ x ⎟\n\
3╵ 4 ⎝3, 4, 5, -3 │ ⎠\
""")
ascii_str = \
"""\
_ \n\
|_ /pi, 2/3, -2*k | 2\\\n\
| | | x |\n\
3 4 \\ 3, 4, 5, -3 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper([1, 2], [3, 4], 1/(1/(1/(1/x + 1) + 1) + 1))
ucode_str = \
u("""\
⎛ │ 1 ⎞\n\
⎜ │ ─────────────⎟\n\
⎜ │ 1 ⎟\n\
┌─ ⎜1, 2 │ 1 + ─────────⎟\n\
├─ ⎜ │ 1 ⎟\n\
2╵ 2 ⎜3, 4 │ 1 + ─────⎟\n\
⎜ │ 1⎟\n\
⎜ │ 1 + ─⎟\n\
⎝ │ x⎠\
""")
ascii_str = \
"""\
\n\
/ | 1 \\\n\
| | -------------|\n\
_ | | 1 |\n\
|_ |1, 2 | 1 + ---------|\n\
| | | 1 |\n\
2 2 |3, 4 | 1 + -----|\n\
| | 1|\n\
| | 1 + -|\n\
\\ | x/\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_meijerg():
expr = meijerg([pi, pi, x], [1], [0, 1], [1, 2, 3], z)
ucode_str = \
u("""\
╭─╮2, 3 ⎛π, π, x 1 │ ⎞\n\
│╶┐ ⎜ │ z⎟\n\
╰─╯4, 5 ⎝ 0, 1 1, 2, 3 │ ⎠\
""")
ascii_str = \
"""\
__2, 3 /pi, pi, x 1 | \\\n\
/__ | | z|\n\
\\_|4, 5 \\ 0, 1 1, 2, 3 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = meijerg([1, pi/7], [2, pi, 5], [], [], z**2)
ucode_str = \
u("""\
⎛ π │ ⎞\n\
╭─╮0, 2 ⎜1, ─ 2, π, 5 │ 2⎟\n\
│╶┐ ⎜ 7 │ z ⎟\n\
╰─╯5, 0 ⎜ │ ⎟\n\
⎝ │ ⎠\
""")
ascii_str = \
"""\
/ pi | \\\n\
__0, 2 |1, -- 2, pi, 5 | 2|\n\
/__ | 7 | z |\n\
\\_|5, 0 | | |\n\
\\ | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ucode_str = \
u("""\
╭─╮ 1, 10 ⎛1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1 │ ⎞\n\
│╶┐ ⎜ │ z⎟\n\
╰─╯11, 2 ⎝ 1 1 │ ⎠\
""")
ascii_str = \
"""\
__ 1, 10 /1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1 | \\\n\
/__ | | z|\n\
\\_|11, 2 \\ 1 1 | /\
"""
expr = meijerg([1]*10, [1], [1], [1], z)
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = meijerg([1, 2, ], [4, 3], [3], [4, 5], 1/(1/(1/(1/x + 1) + 1) + 1))
ucode_str = \
u("""\
⎛ │ 1 ⎞\n\
⎜ │ ─────────────⎟\n\
⎜ │ 1 ⎟\n\
╭─╮1, 2 ⎜1, 2 4, 3 │ 1 + ─────────⎟\n\
│╶┐ ⎜ │ 1 ⎟\n\
╰─╯4, 3 ⎜ 3 4, 5 │ 1 + ─────⎟\n\
⎜ │ 1⎟\n\
⎜ │ 1 + ─⎟\n\
⎝ │ x⎠\
""")
ascii_str = \
"""\
/ | 1 \\\n\
| | -------------|\n\
| | 1 |\n\
__1, 2 |1, 2 4, 3 | 1 + ---------|\n\
/__ | | 1 |\n\
\\_|4, 3 | 3 4, 5 | 1 + -----|\n\
| | 1|\n\
| | 1 + -|\n\
\\ | x/\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(expr, x)
ucode_str = \
u("""\
⌠ \n\
⎮ ⎛ │ 1 ⎞ \n\
⎮ ⎜ │ ─────────────⎟ \n\
⎮ ⎜ │ 1 ⎟ \n\
⎮ ╭─╮1, 2 ⎜1, 2 4, 3 │ 1 + ─────────⎟ \n\
⎮ │╶┐ ⎜ │ 1 ⎟ dx\n\
⎮ ╰─╯4, 3 ⎜ 3 4, 5 │ 1 + ─────⎟ \n\
⎮ ⎜ │ 1⎟ \n\
⎮ ⎜ │ 1 + ─⎟ \n\
⎮ ⎝ │ x⎠ \n\
⌡ \
""")
ascii_str = \
"""\
/ \n\
| \n\
| / | 1 \\ \n\
| | | -------------| \n\
| | | 1 | \n\
| __1, 2 |1, 2 4, 3 | 1 + ---------| \n\
| /__ | | 1 | dx\n\
| \\_|4, 3 | 3 4, 5 | 1 + -----| \n\
| | | 1| \n\
| | | 1 + -| \n\
| \\ | x/ \n\
| \n\
/ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_noncommutative():
A, B, C = symbols('A,B,C', commutative=False)
expr = A*B*C**-1
ascii_str = \
"""\
-1\n\
A*B*C \
"""
ucode_str = \
u("""\
-1\n\
A⋅B⋅C \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = C**-1*A*B
ascii_str = \
"""\
-1 \n\
C *A*B\
"""
ucode_str = \
u("""\
-1 \n\
C ⋅A⋅B\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A*C**-1*B
ascii_str = \
"""\
-1 \n\
A*C *B\
"""
ucode_str = \
u("""\
-1 \n\
A⋅C ⋅B\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A*C**-1*B/x
ascii_str = \
"""\
-1 \n\
A*C *B\n\
-------\n\
x \
"""
ucode_str = \
u("""\
-1 \n\
A⋅C ⋅B\n\
───────\n\
x \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_special_functions():
x, y = symbols("x y")
# atan2
expr = atan2(y/sqrt(200), sqrt(x))
ascii_str = \
"""\
/ ___ \\\n\
|\\/ 2 *y ___|\n\
atan2|-------, \\/ x |\n\
\\ 20 /\
"""
ucode_str = \
u("""\
⎛√2⋅y ⎞\n\
atan2⎜────, √x⎟\n\
⎝ 20 ⎠\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_geometry():
e = Segment((0, 1), (0, 2))
assert pretty(e) == 'Segment2D(Point2D(0, 1), Point2D(0, 2))'
e = Ray((1, 1), angle=4.02*pi)
assert pretty(e) == 'Ray2D(Point2D(1, 1), Point2D(2, tan(pi/50) + 1))'
def test_expint():
expr = Ei(x)
string = 'Ei(x)'
assert pretty(expr) == string
assert upretty(expr) == string
expr = expint(1, z)
ucode_str = u"E₁(z)"
ascii_str = "expint(1, z)"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
assert pretty(Shi(x)) == 'Shi(x)'
assert pretty(Si(x)) == 'Si(x)'
assert pretty(Ci(x)) == 'Ci(x)'
assert pretty(Chi(x)) == 'Chi(x)'
assert upretty(Shi(x)) == 'Shi(x)'
assert upretty(Si(x)) == 'Si(x)'
assert upretty(Ci(x)) == 'Ci(x)'
assert upretty(Chi(x)) == 'Chi(x)'
def test_elliptic_functions():
ascii_str = \
"""\
/ 1 \\\n\
K|-----|\n\
\\z + 1/\
"""
ucode_str = \
u("""\
⎛ 1 ⎞\n\
K⎜─────⎟\n\
⎝z + 1⎠\
""")
expr = elliptic_k(1/(z + 1))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ | 1 \\\n\
F|1|-----|\n\
\\ |z + 1/\
"""
ucode_str = \
u("""\
⎛ │ 1 ⎞\n\
F⎜1│─────⎟\n\
⎝ │z + 1⎠\
""")
expr = elliptic_f(1, 1/(1 + z))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ 1 \\\n\
E|-----|\n\
\\z + 1/\
"""
ucode_str = \
u("""\
⎛ 1 ⎞\n\
E⎜─────⎟\n\
⎝z + 1⎠\
""")
expr = elliptic_e(1/(z + 1))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ | 1 \\\n\
E|1|-----|\n\
\\ |z + 1/\
"""
ucode_str = \
u("""\
⎛ │ 1 ⎞\n\
E⎜1│─────⎟\n\
⎝ │z + 1⎠\
""")
expr = elliptic_e(1, 1/(1 + z))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ |4\\\n\
Pi|3|-|\n\
\\ |x/\
"""
ucode_str = \
u("""\
⎛ │4⎞\n\
Π⎜3│─⎟\n\
⎝ │x⎠\
""")
expr = elliptic_pi(3, 4/x)
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ 4| \\\n\
Pi|3; -|6|\n\
\\ x| /\
"""
ucode_str = \
u("""\
⎛ 4│ ⎞\n\
Π⎜3; ─│6⎟\n\
⎝ x│ ⎠\
""")
expr = elliptic_pi(3, 4/x, 6)
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
X = Normal('x1', 0, 1)
assert upretty(where(X > 0)) == u"Domain: 0 < x₁ ∧ x₁ < ∞"
D = Die('d1', 6)
assert upretty(where(D > 4)) == u'Domain: d₁ = 5 ∨ d₁ = 6'
A = Exponential('a', 1)
B = Exponential('b', 1)
assert upretty(pspace(Tuple(A, B)).domain) == \
u'Domain: 0 ≤ a ∧ 0 ≤ b ∧ a < ∞ ∧ b < ∞'
def test_PrettyPoly():
F = QQ.frac_field(x, y)
R = QQ.poly_ring(x, y)
expr = F.convert(x/(x + y))
assert pretty(expr) == "x/(x + y)"
assert upretty(expr) == u"x/(x + y)"
expr = R.convert(x + y)
assert pretty(expr) == "x + y"
assert upretty(expr) == u"x + y"
def test_issue_6285():
assert pretty(Pow(2, -5, evaluate=False)) == '1 \n--\n 5\n2 '
assert pretty(Pow(x, (1/pi))) == 'pi___\n\\/ x '
def test_issue_6359():
assert pretty(Integral(x**2, x)**2) == \
"""\
2
/ / \\ \n\
| | | \n\
| | 2 | \n\
| | x dx| \n\
| | | \n\
\\/ / \
"""
assert upretty(Integral(x**2, x)**2) == \
u("""\
2
⎛⌠ ⎞ \n\
⎜⎮ 2 ⎟ \n\
⎜⎮ x dx⎟ \n\
⎝⌡ ⎠ \
""")
assert pretty(Sum(x**2, (x, 0, 1))**2) == \
"""\
2
/ 1 \\ \n\
| ___ | \n\
| \\ ` | \n\
| \\ 2| \n\
| / x | \n\
| /__, | \n\
\\x = 0 / \
"""
assert upretty(Sum(x**2, (x, 0, 1))**2) == \
u("""\
2
⎛ 1 ⎞ \n\
⎜ ___ ⎟ \n\
⎜ ╲ ⎟ \n\
⎜ ╲ 2⎟ \n\
⎜ ╱ x ⎟ \n\
⎜ ╱ ⎟ \n\
⎜ ‾‾‾ ⎟ \n\
⎝x = 0 ⎠ \
""")
assert pretty(Product(x**2, (x, 1, 2))**2) == \
"""\
2
/ 2 \\ \n\
|______ | \n\
| | | 2| \n\
| | | x | \n\
| | | | \n\
\\x = 1 / \
"""
assert upretty(Product(x**2, (x, 1, 2))**2) == \
u("""\
2
⎛ 2 ⎞ \n\
⎜─┬──┬─ ⎟ \n\
⎜ │ │ 2⎟ \n\
⎜ │ │ x ⎟ \n\
⎜ │ │ ⎟ \n\
⎝x = 1 ⎠ \
""")
f = Function('f')
assert pretty(Derivative(f(x), x)**2) == \
"""\
2
/d \\ \n\
|--(f(x))| \n\
\\dx / \
"""
assert upretty(Derivative(f(x), x)**2) == \
u("""\
2
⎛d ⎞ \n\
⎜──(f(x))⎟ \n\
⎝dx ⎠ \
""")
def test_issue_6739():
ascii_str = \
"""\
1 \n\
-----\n\
___\n\
\\/ x \
"""
ucode_str = \
u("""\
1 \n\
──\n\
√x\
""")
assert pretty(1/sqrt(x)) == ascii_str
assert upretty(1/sqrt(x)) == ucode_str
def test_complicated_symbol_unchanged():
for symb_name in ["dexpr2_d1tau", "dexpr2^d1tau"]:
assert pretty(Symbol(symb_name)) == symb_name
def test_categories():
from sympy.categories import (Object, IdentityMorphism,
NamedMorphism, Category, Diagram, DiagramGrid)
A1 = Object("A1")
A2 = Object("A2")
A3 = Object("A3")
f1 = NamedMorphism(A1, A2, "f1")
f2 = NamedMorphism(A2, A3, "f2")
id_A1 = IdentityMorphism(A1)
K1 = Category("K1")
assert pretty(A1) == "A1"
assert upretty(A1) == u"A₁"
assert pretty(f1) == "f1:A1-->A2"
assert upretty(f1) == u"f₁:A₁——▶A₂"
assert pretty(id_A1) == "id:A1-->A1"
assert upretty(id_A1) == u"id:A₁——▶A₁"
assert pretty(f2*f1) == "f2*f1:A1-->A3"
assert upretty(f2*f1) == u"f₂∘f₁:A₁——▶A₃"
assert pretty(K1) == "K1"
assert upretty(K1) == u"K₁"
# Test how diagrams are printed.
d = Diagram()
assert pretty(d) == "EmptySet()"
assert upretty(d) == u"∅"
d = Diagram({f1: "unique", f2: S.EmptySet})
assert pretty(d) == "{f2*f1:A1-->A3: EmptySet(), id:A1-->A1: " \
"EmptySet(), id:A2-->A2: EmptySet(), id:A3-->A3: " \
"EmptySet(), f1:A1-->A2: {unique}, f2:A2-->A3: EmptySet()}"
assert upretty(d) == u("{f₂∘f₁:A₁——▶A₃: ∅, id:A₁——▶A₁: ∅, " \
"id:A₂——▶A₂: ∅, id:A₃——▶A₃: ∅, f₁:A₁——▶A₂: {unique}, f₂:A₂——▶A₃: ∅}")
d = Diagram({f1: "unique", f2: S.EmptySet}, {f2 * f1: "unique"})
assert pretty(d) == "{f2*f1:A1-->A3: EmptySet(), id:A1-->A1: " \
"EmptySet(), id:A2-->A2: EmptySet(), id:A3-->A3: " \
"EmptySet(), f1:A1-->A2: {unique}, f2:A2-->A3: EmptySet()}" \
" ==> {f2*f1:A1-->A3: {unique}}"
assert upretty(d) == u("{f₂∘f₁:A₁——▶A₃: ∅, id:A₁——▶A₁: ∅, id:A₂——▶A₂: " \
"∅, id:A₃——▶A₃: ∅, f₁:A₁——▶A₂: {unique}, f₂:A₂——▶A₃: ∅}" \
" ══▶ {f₂∘f₁:A₁——▶A₃: {unique}}")
grid = DiagramGrid(d)
assert pretty(grid) == "A1 A2\n \nA3 "
assert upretty(grid) == u"A₁ A₂\n \nA₃ "
def test_PrettyModules():
R = QQ.old_poly_ring(x, y)
F = R.free_module(2)
M = F.submodule([x, y], [1, x**2])
ucode_str = \
u("""\
2\n\
ℚ[x, y] \
""")
ascii_str = \
"""\
2\n\
QQ[x, y] \
"""
assert upretty(F) == ucode_str
assert pretty(F) == ascii_str
ucode_str = \
u("""\
╱ ⎡ 2⎤╲\n\
╲[x, y], ⎣1, x ⎦╱\
""")
ascii_str = \
"""\
2 \n\
<[x, y], [1, x ]>\
"""
assert upretty(M) == ucode_str
assert pretty(M) == ascii_str
I = R.ideal(x**2, y)
ucode_str = \
u("""\
╱ 2 ╲\n\
╲x , y╱\
""")
ascii_str = \
"""\
2 \n\
<x , y>\
"""
assert upretty(I) == ucode_str
assert pretty(I) == ascii_str
Q = F / M
ucode_str = \
u("""\
2 \n\
ℚ[x, y] \n\
─────────────────\n\
╱ ⎡ 2⎤╲\n\
╲[x, y], ⎣1, x ⎦╱\
""")
ascii_str = \
"""\
2 \n\
QQ[x, y] \n\
-----------------\n\
2 \n\
<[x, y], [1, x ]>\
"""
assert upretty(Q) == ucode_str
assert pretty(Q) == ascii_str
ucode_str = \
u("""\
╱⎡ 3⎤ ╲\n\
│⎢ x ⎥ ╱ ⎡ 2⎤╲ ╱ ⎡ 2⎤╲│\n\
│⎢1, ──⎥ + ╲[x, y], ⎣1, x ⎦╱, [2, y] + ╲[x, y], ⎣1, x ⎦╱│\n\
╲⎣ 2 ⎦ ╱\
""")
ascii_str = \
"""\
3 \n\
x 2 2 \n\
<[1, --] + <[x, y], [1, x ]>, [2, y] + <[x, y], [1, x ]>>\n\
2 \
"""
def test_QuotientRing():
R = QQ.old_poly_ring(x)/[x**2 + 1]
ucode_str = \
u("""\
ℚ[x] \n\
────────\n\
╱ 2 ╲\n\
╲x + 1╱\
""")
ascii_str = \
"""\
QQ[x] \n\
--------\n\
2 \n\
<x + 1>\
"""
assert upretty(R) == ucode_str
assert pretty(R) == ascii_str
ucode_str = \
u("""\
╱ 2 ╲\n\
1 + ╲x + 1╱\
""")
ascii_str = \
"""\
2 \n\
1 + <x + 1>\
"""
assert upretty(R.one) == ucode_str
assert pretty(R.one) == ascii_str
def test_Homomorphism():
from sympy.polys.agca import homomorphism
R = QQ.old_poly_ring(x)
expr = homomorphism(R.free_module(1), R.free_module(1), [0])
ucode_str = \
u("""\
1 1\n\
[0] : ℚ[x] ──> ℚ[x] \
""")
ascii_str = \
"""\
1 1\n\
[0] : QQ[x] --> QQ[x] \
"""
assert upretty(expr) == ucode_str
assert pretty(expr) == ascii_str
expr = homomorphism(R.free_module(2), R.free_module(2), [0, 0])
ucode_str = \
u("""\
⎡0 0⎤ 2 2\n\
⎢ ⎥ : ℚ[x] ──> ℚ[x] \n\
⎣0 0⎦ \
""")
ascii_str = \
"""\
[0 0] 2 2\n\
[ ] : QQ[x] --> QQ[x] \n\
[0 0] \
"""
assert upretty(expr) == ucode_str
assert pretty(expr) == ascii_str
expr = homomorphism(R.free_module(1), R.free_module(1) / [[x]], [0])
ucode_str = \
u("""\
1\n\
1 ℚ[x] \n\
[0] : ℚ[x] ──> ─────\n\
<[x]>\
""")
ascii_str = \
"""\
1\n\
1 QQ[x] \n\
[0] : QQ[x] --> ------\n\
<[x]> \
"""
assert upretty(expr) == ucode_str
assert pretty(expr) == ascii_str
def test_Tr():
A, B = symbols('A B', commutative=False)
t = Tr(A*B)
assert pretty(t) == r'Tr(A*B)'
assert upretty(t) == u'Tr(A⋅B)'
def test_pretty_Add():
eq = Mul(-2, x - 2, evaluate=False) + 5
assert pretty(eq) == '5 - 2*(x - 2)'
def test_issue_7179():
assert upretty(Not(Equivalent(x, y))) == u'x ⇎ y'
assert upretty(Not(Implies(x, y))) == u'x ↛ y'
def test_issue_7180():
assert upretty(Equivalent(x, y)) == u'x ⇔ y'
def test_pretty_Complement():
assert pretty(S.Reals - S.Naturals) == '(-oo, oo) \\ Naturals'
assert upretty(S.Reals - S.Naturals) == u'ℝ \\ ℕ'
assert pretty(S.Reals - S.Naturals0) == '(-oo, oo) \\ Naturals0'
assert upretty(S.Reals - S.Naturals0) == u'ℝ \\ ℕ₀'
def test_pretty_SymmetricDifference():
from sympy import SymmetricDifference, Interval
from sympy.utilities.pytest import raises
assert upretty(SymmetricDifference(Interval(2,3), Interval(3,5), \
evaluate = False)) == u'[2, 3] ∆ [3, 5]'
with raises(NotImplementedError):
pretty(SymmetricDifference(Interval(2,3), Interval(3,5), evaluate = False))
def test_pretty_Contains():
assert pretty(Contains(x, S.Integers)) == 'Contains(x, Integers)'
assert upretty(Contains(x, S.Integers)) == u'x ∈ ℤ'
def test_issue_8292():
from sympy.core import sympify
e = sympify('((x+x**4)/(x-1))-(2*(x-1)**4/(x-1)**4)', evaluate=False)
ucode_str = \
u("""\
4 4 \n\
2⋅(x - 1) x + x\n\
- ────────── + ──────\n\
4 x - 1 \n\
(x - 1) \
""")
ascii_str = \
"""\
4 4 \n\
2*(x - 1) x + x\n\
- ---------- + ------\n\
4 x - 1 \n\
(x - 1) \
"""
assert pretty(e) == ascii_str
assert upretty(e) == ucode_str
def test_issue_4335():
y = Function('y')
expr = -y(x).diff(x)
ucode_str = \
u("""\
d \n\
-──(y(x))\n\
dx \
""")
ascii_str = \
"""\
d \n\
- --(y(x))\n\
dx \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_issue_8344():
from sympy.core import sympify
e = sympify('2*x*y**2/1**2 + 1', evaluate=False)
ucode_str = \
u("""\
2 \n\
2⋅x⋅y \n\
────── + 1\n\
2 \n\
1 \
""")
assert upretty(e) == ucode_str
def test_issue_6324():
x = Pow(2, 3, evaluate=False)
y = Pow(10, -2, evaluate=False)
e = Mul(x, y, evaluate=False)
ucode_str = \
u("""\
3\n\
2 \n\
───\n\
2\n\
10 \
""")
assert upretty(e) == ucode_str
def test_issue_7927():
e = sin(x/2)**cos(x/2)
ucode_str = \
u("""\
⎛x⎞\n\
cos⎜─⎟\n\
⎝2⎠\n\
⎛ ⎛x⎞⎞ \n\
⎜sin⎜─⎟⎟ \n\
⎝ ⎝2⎠⎠ \
""")
assert upretty(e) == ucode_str
e = sin(x)**(S(11)/13)
ucode_str = \
u("""\
11\n\
──\n\
13\n\
(sin(x)) \
""")
assert upretty(e) == ucode_str
def test_issue_6134():
from sympy.abc import lamda, t
phi = Function('phi')
e = lamda*x*Integral(phi(t)*pi*sin(pi*t), (t, 0, 1)) + lamda*x**2*Integral(phi(t)*2*pi*sin(2*pi*t), (t, 0, 1))
ucode_str = \
u("""\
1 1 \n\
2 ⌠ ⌠ \n\
λ⋅x ⋅⎮ 2⋅π⋅φ(t)⋅sin(2⋅π⋅t) dt + λ⋅x⋅⎮ π⋅φ(t)⋅sin(π⋅t) dt\n\
⌡ ⌡ \n\
0 0 \
""")
assert upretty(e) == ucode_str
def test_issue_9877():
ucode_str1 = u'(2, 3) ∪ ([1, 2] \\ {x})'
a, b, c = Interval(2, 3, True, True), Interval(1, 2), FiniteSet(x)
assert upretty(Union(a, Complement(b, c))) == ucode_str1
ucode_str2 = u'{x} ∩ {y} ∩ ({z} \\ [1, 2])'
d, e, f, g = FiniteSet(x), FiniteSet(y), FiniteSet(z), Interval(1, 2)
assert upretty(Intersection(d, e, Complement(f, g))) == ucode_str2
def test_issue_13651():
expr1 = c + Mul(-1, a + b, evaluate=False)
assert pretty(expr1) == 'c - (a + b)'
expr2 = c + Mul(-1, a - b + d, evaluate=False)
assert pretty(expr2) == 'c - (a - b + d)'
def test_pretty_primenu():
from sympy.ntheory.factor_ import primenu
ascii_str1 = "nu(n)"
ucode_str1 = u("ν(n)")
n = symbols('n', integer=True)
assert pretty(primenu(n)) == ascii_str1
assert upretty(primenu(n)) == ucode_str1
def test_pretty_primeomega():
from sympy.ntheory.factor_ import primeomega
ascii_str1 = "Omega(n)"
ucode_str1 = u("Ω(n)")
n = symbols('n', integer=True)
assert pretty(primeomega(n)) == ascii_str1
assert upretty(primeomega(n)) == ucode_str1
def test_pretty_Mod():
from sympy.core import Mod
ascii_str1 = "x mod 7"
ucode_str1 = u("x mod 7")
ascii_str2 = "(x + 1) mod 7"
ucode_str2 = u("(x + 1) mod 7")
ascii_str3 = "2*x mod 7"
ucode_str3 = u("2⋅x mod 7")
ascii_str4 = "(x mod 7) + 1"
ucode_str4 = u("(x mod 7) + 1")
ascii_str5 = "2*(x mod 7)"
ucode_str5 = u("2⋅(x mod 7)")
x = symbols('x', integer=True)
assert pretty(Mod(x, 7)) == ascii_str1
assert upretty(Mod(x, 7)) == ucode_str1
assert pretty(Mod(x + 1, 7)) == ascii_str2
assert upretty(Mod(x + 1, 7)) == ucode_str2
assert pretty(Mod(2 * x, 7)) == ascii_str3
assert upretty(Mod(2 * x, 7)) == ucode_str3
assert pretty(Mod(x, 7) + 1) == ascii_str4
assert upretty(Mod(x, 7) + 1) == ucode_str4
assert pretty(2 * Mod(x, 7)) == ascii_str5
assert upretty(2 * Mod(x, 7)) == ucode_str5
def test_issue_11801():
assert pretty(Symbol("")) == ""
assert upretty(Symbol("")) == ""
def test_pretty_UnevaluatedExpr():
x = symbols('x')
he = UnevaluatedExpr(1/x)
ucode_str = \
u("""\
1\n\
─\n\
x\
""")
assert upretty(he) == ucode_str
ucode_str = \
u("""\
2\n\
⎛1⎞ \n\
⎜─⎟ \n\
⎝x⎠ \
""")
assert upretty(he**2) == ucode_str
ucode_str = \
u("""\
1\n\
1 + ─\n\
x\
""")
assert upretty(he + 1) == ucode_str
ucode_str = \
u('''\
1\n\
x⋅─\n\
x\
''')
assert upretty(x*he) == ucode_str
def test_issue_10472():
M = (Matrix([[0, 0], [0, 0]]), Matrix([0, 0]))
ucode_str = \
u("""\
⎛⎡0 0⎤ ⎡0⎤⎞
⎜⎢ ⎥, ⎢ ⎥⎟
⎝⎣0 0⎦ ⎣0⎦⎠\
""")
assert upretty(M) == ucode_str
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
ascii_str1 = "A_00"
ucode_str1 = u("A₀₀")
assert pretty(A[0, 0]) == ascii_str1
assert upretty(A[0, 0]) == ucode_str1
ascii_str1 = "3*A_00"
ucode_str1 = u("3⋅A₀₀")
assert pretty(3*A[0, 0]) == ascii_str1
assert upretty(3*A[0, 0]) == ucode_str1
ascii_str1 = "(-B + A)[0, 0]"
ucode_str1 = u("(-B + A)[0, 0]")
F = C[0, 0].subs(C, A - B)
assert pretty(F) == ascii_str1
assert upretty(F) == ucode_str1
def test_issue_12675():
from sympy.vector import CoordSys3D
x, y, t, j = symbols('x y t j')
e = CoordSys3D('e')
ucode_str = \
u("""\
⎛ t⎞ \n\
⎜⎛x⎞ ⎟ j_e\n\
⎜⎜─⎟ ⎟ \n\
⎝⎝y⎠ ⎠ \
""")
assert upretty((x/y)**t*e.j) == ucode_str
ucode_str = \
u("""\
⎛1⎞ \n\
⎜─⎟ j_e\n\
⎝y⎠ \
""")
assert upretty((1/y)*e.j) == ucode_str
def test_MatrixSymbol_printing():
# test cases for issue #14237
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert pretty(-A*B*C) == "-A*B*C"
assert pretty(A - B) == "-B + A"
assert pretty(A*B*C - A*B - B*C) == "-A*B -B*C + A*B*C"
# issue #14814
x = MatrixSymbol('x', n, n)
y = MatrixSymbol('y*', n, n)
assert pretty(x + y) == "x + y*"
ascii_str = \
"""\
2 \n\
-2*y* -a*x\
"""
assert pretty(-a*x + -2*y*y) == ascii_str
def test_degree_printing():
expr1 = 90*degree
assert pretty(expr1) == u'90°'
expr2 = x*degree
assert pretty(expr2) == u'x°'
expr3 = cos(x*degree + 90*degree)
assert pretty(expr3) == u'cos(x° + 90°)'
def test_vector_expr_pretty_printing():
A = CoordSys3D('A')
assert upretty(Cross(A.i, A.x*A.i+3*A.y*A.j)) == u("(i_A)×((x_A) i_A + (3⋅y_A) j_A)")
assert upretty(x*Cross(A.i, A.j)) == u('x⋅(i_A)×(j_A)')
assert upretty(Curl(A.x*A.i + 3*A.y*A.j)) == u("∇×((x_A) i_A + (3⋅y_A) j_A)")
assert upretty(Divergence(A.x*A.i + 3*A.y*A.j)) == u("∇⋅((x_A) i_A + (3⋅y_A) j_A)")
assert upretty(Dot(A.i, A.x*A.i+3*A.y*A.j)) == u("(i_A)⋅((x_A) i_A + (3⋅y_A) j_A)")
assert upretty(Gradient(A.x+3*A.y)) == u("∇(x_A + 3⋅y_A)")
assert upretty(Laplacian(A.x+3*A.y)) == u("∆(x_A + 3⋅y_A)")
# TODO: add support for ASCII pretty.
def test_pretty_print_tensor_expr():
L = TensorIndexType("L")
i, j, k = tensor_indices("i j k", L)
i0 = tensor_indices("i_0", L)
A, B, C, D = tensorhead("A B C D", [L], [[1]])
H = tensorhead("H", [L, L], [[1], [1]])
expr = -i
ascii_str = \
"""\
-i\
"""
ucode_str = \
u("""\
-i\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i)
ascii_str = \
"""\
i\n\
A \n\
\
"""
ucode_str = \
u("""\
i\n\
A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i0)
ascii_str = \
"""\
i_0\n\
A \n\
\
"""
ucode_str = \
u("""\
i₀\n\
A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(-i)
ascii_str = \
"""\
\n\
A \n\
i\
"""
ucode_str = \
u("""\
\n\
A \n\
i\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -3*A(-i)
ascii_str = \
"""\
\n\
-3*A \n\
i\
"""
ucode_str = \
u("""\
\n\
-3⋅A \n\
i\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = H(i, -j)
ascii_str = \
"""\
i \n\
H \n\
j\
"""
ucode_str = \
u("""\
i \n\
H \n\
j\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = H(i, -i)
ascii_str = \
"""\
L_0 \n\
H \n\
L_0\
"""
ucode_str = \
u("""\
L₀ \n\
H \n\
L₀\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = H(i, -j)*A(j)*B(k)
ascii_str = \
"""\
i L_0 k\n\
H *A *B \n\
L_0 \
"""
ucode_str = \
u("""\
i L₀ k\n\
H ⋅A ⋅B \n\
L₀ \
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (1+x)*A(i)
ascii_str = \
"""\
i\n\
(x + 1)*A \n\
\
"""
ucode_str = \
u("""\
i\n\
(x + 1)⋅A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i) + 3*B(i)
ascii_str = \
"""\
i i\n\
A + 3*B \n\
\
"""
ucode_str = \
u("""\
i i\n\
A + 3⋅B \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_print_tensor_partial_deriv():
from sympy.tensor.toperators import PartialDerivative
from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensorhead
L = TensorIndexType("L")
i, j, k = tensor_indices("i j k", L)
i0 = tensor_indices("i0", L)
A, B, C, D = tensorhead("A B C D", [L], [[1]])
H = tensorhead("H", [L, L], [[1], [1]])
expr = PartialDerivative(A(i), A(j))
ascii_str = \
"""\
d / i\\\n\
---|A |\n\
j\\ /\n\
dA \n\
\
"""
ucode_str = \
u("""\
∂ ⎛ i⎞\n\
───⎜A ⎟\n\
j⎝ ⎠\n\
∂A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i)*PartialDerivative(H(k, -i), A(j))
ascii_str = \
"""\
L_0 d / k \\\n\
A *---|H |\n\
j\\ L_0/\n\
dA \n\
\
"""
ucode_str = \
u("""\
L₀ ∂ ⎛ k ⎞\n\
A ⋅───⎜H ⎟\n\
j⎝ L₀⎠\n\
∂A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i)*PartialDerivative(B(k)*C(-i) + 3*H(k, -i), A(j))
ascii_str = \
"""\
L_0 d / k k \\\n\
A *---|B *C + 3*H |\n\
j\\ L_0 L_0/\n\
dA \n\
\
"""
ucode_str = \
u("""\
L₀ ∂ ⎛ k k ⎞\n\
A ⋅───⎜B ⋅C + 3⋅H ⎟\n\
j⎝ L₀ L₀⎠\n\
∂A \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (A(i) + B(i))*PartialDerivative(C(-j), D(j))
ascii_str = \
"""\
/ i i\\ d / \\\n\
|A + B |*-----|C |\n\
\\ / L_0\\ L_0/\n\
dD \n\
\
"""
ucode_str = \
u("""\
⎛ i i⎞ ∂ ⎛ ⎞\n\
⎜A + B ⎟⋅────⎜C ⎟\n\
⎝ ⎠ L₀⎝ L₀⎠\n\
∂D \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (A(i) + B(i))*PartialDerivative(C(-i), D(j))
ascii_str = \
"""\
/ L_0 L_0\\ d / \\\n\
|A + B |*---|C |\n\
\\ / j\\ L_0/\n\
dD \n\
\
"""
ucode_str = \
u("""\
⎛ L₀ L₀⎞ ∂ ⎛ ⎞\n\
⎜A + B ⎟⋅───⎜C ⎟\n\
⎝ ⎠ j⎝ L₀⎠\n\
∂D \n\
\
""")
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = TensorElement(H(i, j), {i:1})
ascii_str = \
"""\
i=1,j\n\
H \n\
\
"""
ucode_str = ascii_str
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = TensorElement(H(i, j), {i:1, j:1})
ascii_str = \
"""\
i=1,j=1\n\
H \n\
\
"""
ucode_str = ascii_str
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = TensorElement(H(i, j), {j:1})
ascii_str = \
"""\
i,j=1\n\
H \n\
\
"""
ucode_str = ascii_str
expr = TensorElement(H(-i, j), {-i:1})
ascii_str = \
"""\
j\n\
H \n\
i=1 \
"""
ucode_str = ascii_str
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_issue_15560():
a = MatrixSymbol('a', 1, 1)
e = pretty(a*(KroneckerProduct(a, a)))
result = 'a*(a x a)'
assert e == result
def test_print_lerchphi():
# Part of issue 6013
a = Symbol('a')
pretty(lerchphi(a, 1, 2))
uresult = u'Φ(a, 1, 2)'
aresult = 'lerchphi(a, 1, 2)'
assert pretty(lerchphi(a, 1, 2)) == aresult
assert upretty(lerchphi(a, 1, 2)) == uresult
def test_issue_15583():
N = mechanics.ReferenceFrame('N')
result = '(n_x, n_y, n_z)'
e = pretty((N.x, N.y, N.z))
assert e == result
def test_matrixSymbolBold():
# Issue 15871
def boldpretty(expr):
return xpretty(expr, use_unicode=True, wrap_line=False, mat_symbol_style="bold")
from sympy import trace
A = MatrixSymbol("A", 2, 2)
assert boldpretty(trace(A)) == u'tr(𝐀)'
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert boldpretty(-A) == u'-𝐀'
assert boldpretty(A - A*B - B) == u'-𝐁 -𝐀⋅𝐁 + 𝐀'
assert boldpretty(-A*B - A*B*C - B) == u'-𝐁 -𝐀⋅𝐁 -𝐀⋅𝐁⋅𝐂'
A = MatrixSymbol("Addot", 3, 3)
assert boldpretty(A) == u'𝐀̈'
omega = MatrixSymbol("omega", 3, 3)
assert boldpretty(omega) == u'ω'
omega = MatrixSymbol("omeganorm", 3, 3)
assert boldpretty(omega) == u'‖ω‖'
a = Symbol('alpha')
b = Symbol('b')
c = MatrixSymbol("c", 3, 1)
d = MatrixSymbol("d", 3, 1)
assert boldpretty(a*B*c+b*d) == u'b⋅𝐝 + α⋅𝐁⋅𝐜'
d = MatrixSymbol("delta", 3, 1)
B = MatrixSymbol("Beta", 3, 3)
assert boldpretty(a*B*c+b*d) == u'b⋅δ + α⋅Β⋅𝐜'
A = MatrixSymbol("A_2", 3, 3)
assert boldpretty(A) == u'𝐀₂'
def test_center_accent():
assert center_accent('a', u'\N{COMBINING TILDE}') == u'ã'
assert center_accent('aa', u'\N{COMBINING TILDE}') == u'aã'
assert center_accent('aaa', u'\N{COMBINING TILDE}') == u'aãa'
assert center_accent('aaaa', u'\N{COMBINING TILDE}') == u'aaãa'
assert center_accent('aaaaa', u'\N{COMBINING TILDE}') == u'aaãaa'
assert center_accent('abcdefg', u'\N{COMBINING FOUR DOTS ABOVE}') == u'abcd⃜efg'
def test_imaginary_unit():
from sympy import pretty # As it is redefined above
assert pretty(1 + I, use_unicode=False) == '1 + I'
assert pretty(1 + I, use_unicode=True) == u'1 + ⅈ'
assert pretty(1 + I, use_unicode=False, imaginary_unit='j') == '1 + I'
assert pretty(1 + I, use_unicode=True, imaginary_unit='j') == u'1 + ⅉ'
raises(TypeError, lambda: pretty(I, imaginary_unit=I))
raises(ValueError, lambda: pretty(I, imaginary_unit="kkk"))
|
cd5e720adf90e78a73d320c04a7f0b07d7f233fbd51288d2f348417d67f7a419
|
'''
Utility functions for Rubi integration.
See: http://www.apmaths.uwo.ca/~arich/IntegrationRules/PortableDocumentFiles/Integration%20utility%20functions.pdf
'''
from sympy.external import import_module
matchpy = import_module("matchpy")
from sympy.utilities.decorator import doctest_depends_on
from sympy.functions.elementary.integers import floor, frac
from sympy.functions import (log as sym_log , sin, cos, tan, cot, csc, sec, sqrt, erf, gamma, uppergamma, polygamma, digamma,
loggamma, factorial, zeta, LambertW)
from sympy.functions.elementary.hyperbolic import acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch
from sympy.functions.elementary.trigonometric import atan, acsc, asin, acot, acos, asec, atan2
from sympy.polys.polytools import Poly, quo, rem, total_degree, degree
from sympy.simplify.simplify import fraction, simplify, cancel, powsimp
from sympy.core.sympify import sympify
from sympy.utilities.iterables import postorder_traversal
from sympy.functions.special.error_functions import fresnelc, fresnels, erfc, erfi, Ei, expint, li, Si, Ci, Shi, Chi
from sympy.functions.elementary.complexes import im, re, Abs
from sympy.core.exprtools import factor_terms
from sympy import (Basic, E, polylog, N, Wild, WildFunction, factor, gcd, Sum, S, I, Mul, Integer, Float, Dict, Symbol, Rational,
Add, hyper, symbols, sqf_list, sqf, Max, factorint, factorrat, Min, sign, E, Function, collect, FiniteSet, nsimplify,
expand_trig, expand, poly, apart, lcm, And, Pow, pi, zoo, oo, Integral, UnevaluatedExpr, PolynomialError, Dummy, exp as sym_exp,
powdenest, PolynomialDivisionFailed, discriminant, UnificationFailed, appellf1)
from sympy.functions.special.hyper import TupleArg
from sympy.functions.special.elliptic_integrals import elliptic_f, elliptic_e, elliptic_pi
from sympy.utilities.iterables import flatten
from random import randint
from sympy.logic.boolalg import Or
class rubi_unevaluated_expr(UnevaluatedExpr):
'''
This is needed to convert `exp` as `Pow`.
sympy's UnevaluatedExpr has an issue with `is_commutative`.
'''
@property
def is_commutative(self):
from sympy.core.logic import fuzzy_and
return fuzzy_and(a.is_commutative for a in self.args)
_E = rubi_unevaluated_expr(E)
class exp(Function):
'''
sympy's exp is not identified as `Pow`. So it is not matched with `Pow`.
Like `a = exp(2)` is not identified as `Pow(E, 2)`. Rubi rules need it.
So, another exp has been created only for rubi module.
Examples
========
>>> from sympy import Pow, exp as sym_exp
>>> isinstance(sym_exp(2), Pow)
False
>>> from sympy.integrals.rubi.utility_function import exp
>>> isinstance(exp(2), Pow)
True
'''
@classmethod
def eval(cls, *args):
return Pow(_E, args[0])
class log(Function):
'''
For rule matching different `exp` has been used. So for proper results,
`log` is modified little only for case when it encounters rubi's `exp`.
For other cases it is same.
Examples
========
>>> from sympy.integrals.rubi.utility_function import exp, log
>>> a = exp(2)
>>> log(a)
2
'''
@classmethod
def eval(cls, *args):
if args[0].has(_E):
return sym_log(args[0]).doit()
else:
return sym_log(args[0])
if matchpy:
from matchpy import Arity, Operation, CommutativeOperation, AssociativeOperation, OneIdentityOperation, CustomConstraint, Pattern, ReplacementRule, ManyToOneReplacer
from matchpy.expressions.functions import register_operation_iterator, register_operation_factory
from sympy.integrals.rubi.symbol import WC
from matchpy import is_match, replace_all
class UtilityOperator(Operation):
name = 'UtilityOperator'
arity = Arity.variadic
commutative=False
associative=True
Operation.register(Integral)
register_operation_iterator(Integral, lambda a: (a._args[0],) + a._args[1], lambda a: len((a._args[0],) + a._args[1]))
Operation.register(Pow)
OneIdentityOperation.register(Pow)
register_operation_iterator(Pow, lambda a: a._args, lambda a: len(a._args))
Operation.register(Add)
OneIdentityOperation.register(Add)
CommutativeOperation.register(Add)
AssociativeOperation.register(Add)
register_operation_iterator(Add, lambda a: a._args, lambda a: len(a._args))
Operation.register(Mul)
OneIdentityOperation.register(Mul)
CommutativeOperation.register(Mul)
AssociativeOperation.register(Mul)
register_operation_iterator(Mul, lambda a: a._args, lambda a: len(a._args))
Operation.register(exp)
register_operation_iterator(exp, lambda a: a._args, lambda a: len(a._args))
Operation.register(log)
register_operation_iterator(log, lambda a: a._args, lambda a: len(a._args))
Operation.register(sym_log)
register_operation_iterator(sym_log, lambda a: a._args, lambda a: len(a._args))
Operation.register(gamma)
register_operation_iterator(gamma, lambda a: a._args, lambda a: len(a._args))
Operation.register(uppergamma)
register_operation_iterator(uppergamma, lambda a: a._args, lambda a: len(a._args))
Operation.register(fresnels)
register_operation_iterator(fresnels, lambda a: a._args, lambda a: len(a._args))
Operation.register(fresnelc)
register_operation_iterator(fresnelc, lambda a: a._args, lambda a: len(a._args))
Operation.register(erf)
register_operation_iterator(erf, lambda a: a._args, lambda a: len(a._args))
Operation.register(Ei)
register_operation_iterator(Ei, lambda a: a._args, lambda a: len(a._args))
Operation.register(erfc)
register_operation_iterator(erfc, lambda a: a._args, lambda a: len(a._args))
Operation.register(erfi)
register_operation_iterator(erfi, lambda a: a._args, lambda a: len(a._args))
Operation.register(sin)
register_operation_iterator(sin, lambda a: a._args, lambda a: len(a._args))
Operation.register(cos)
register_operation_iterator(cos, lambda a: a._args, lambda a: len(a._args))
Operation.register(tan)
register_operation_iterator(tan, lambda a: a._args, lambda a: len(a._args))
Operation.register(cot)
register_operation_iterator(cot, lambda a: a._args, lambda a: len(a._args))
Operation.register(csc)
register_operation_iterator(csc, lambda a: a._args, lambda a: len(a._args))
Operation.register(sec)
register_operation_iterator(sec, lambda a: a._args, lambda a: len(a._args))
Operation.register(sinh)
register_operation_iterator(sinh, lambda a: a._args, lambda a: len(a._args))
Operation.register(cosh)
register_operation_iterator(cosh, lambda a: a._args, lambda a: len(a._args))
Operation.register(tanh)
register_operation_iterator(tanh, lambda a: a._args, lambda a: len(a._args))
Operation.register(coth)
register_operation_iterator(coth, lambda a: a._args, lambda a: len(a._args))
Operation.register(csch)
register_operation_iterator(csch, lambda a: a._args, lambda a: len(a._args))
Operation.register(sech)
register_operation_iterator(sech, lambda a: a._args, lambda a: len(a._args))
Operation.register(asin)
register_operation_iterator(asin, lambda a: a._args, lambda a: len(a._args))
Operation.register(acos)
register_operation_iterator(acos, lambda a: a._args, lambda a: len(a._args))
Operation.register(atan)
register_operation_iterator(atan, lambda a: a._args, lambda a: len(a._args))
Operation.register(acot)
register_operation_iterator(acot, lambda a: a._args, lambda a: len(a._args))
Operation.register(acsc)
register_operation_iterator(acsc, lambda a: a._args, lambda a: len(a._args))
Operation.register(asec)
register_operation_iterator(asec, lambda a: a._args, lambda a: len(a._args))
Operation.register(asinh)
register_operation_iterator(asinh, lambda a: a._args, lambda a: len(a._args))
Operation.register(acosh)
register_operation_iterator(acosh, lambda a: a._args, lambda a: len(a._args))
Operation.register(atanh)
register_operation_iterator(atanh, lambda a: a._args, lambda a: len(a._args))
Operation.register(acoth)
register_operation_iterator(acoth, lambda a: a._args, lambda a: len(a._args))
Operation.register(acsch)
register_operation_iterator(acsch, lambda a: a._args, lambda a: len(a._args))
Operation.register(asech)
register_operation_iterator(asech, lambda a: a._args, lambda a: len(a._args))
def sympy_op_factory(old_operation, new_operands, variable_name):
return type(old_operation)(*new_operands)
register_operation_factory(Basic, sympy_op_factory)
A_, B_, C_, F_, G_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, z_ = [WC(i) for i in 'ABCFGabcdefghijklmnpqrtuvswxz']
a, b, c, d, e = symbols('a b c d e')
class Int(Function):
'''
Integrates given `expr` by matching rubi rules.
'''
@classmethod
def eval(cls, expr, var):
if isinstance(expr, (int, Integer, float, Float)):
return S(expr)*var
from sympy.integrals.rubi.rubi import util_rubi_integrate
return util_rubi_integrate(expr, var)
def replace_pow_exp(z):
'''
This function converts back rubi's `exp` to general sympy's `exp`.
Examples
========
>>> from sympy.integrals.rubi.utility_function import exp as rubi_exp, replace_pow_exp
>>> expr = rubi_exp(5)
>>> expr
E**5
>>> replace_pow_exp(expr)
exp(5)
'''
z = S(z)
if z.has(_E):
z = z.replace(_E, E)
return z
def Simplify(expr):
expr = simplify(expr)
return expr
def Set(expr, value):
return {expr: value}
def With(subs, expr):
if isinstance(subs, dict):
k = list(subs.keys())[0]
expr = expr.xreplace({k: subs[k]})
else:
for i in subs:
k = list(i.keys())[0]
expr = expr.xreplace({k: i[k]})
return expr
def Module(subs, expr):
return With(subs, expr)
def Scan(f, expr):
# evaluates f applied to each element of expr in turn.
for i in expr:
yield f(i)
def MapAnd(f, l, x=None):
# MapAnd[f,l] applies f to the elements of list l until False is returned; else returns True
if x:
for i in l:
if f(i, x) == False:
return False
return True
else:
for i in l:
if f(i) == False:
return False
return True
def FalseQ(u):
if isinstance(u, (Dict, dict)):
return FalseQ(*list(u.values()))
return u == False
def ZeroQ(*expr):
if len(expr) == 1:
if isinstance(expr[0], list):
return list(ZeroQ(i) for i in expr[0])
else:
return Simplify(expr[0]) == 0
else:
return all(ZeroQ(i) for i in expr)
def OneQ(a):
if a == S(1):
return True
return False
def NegativeQ(u):
u = Simplify(u)
if u in (zoo, oo):
return False
if u.is_comparable:
res = u < 0
if not res.is_Relational:
return res
return False
def NonzeroQ(expr):
return Simplify(expr) != 0
def FreeQ(nodes, var):
if var == Int:
return FreeQ(nodes, Integral)
if isinstance(nodes, list):
return not any(S(expr).has(var) for expr in nodes)
else:
nodes = S(nodes)
return not nodes.has(var)
def NFreeQ(nodes, var):
''' Note that in rubi 4.10.8 this function was not defined in `Integration Utility Functions.m`,
but was used in rules. So explicitly its returning `False`
'''
return False
# return not FreeQ(nodes, var)
def List(*var):
return list(var)
def PositiveQ(var):
var = Simplify(var)
if var in (zoo, oo):
return False
if var.is_comparable:
res = var > 0
if not res.is_Relational:
return res
return False
def PositiveIntegerQ(*args):
return all(var.is_Integer and PositiveQ(var) for var in args)
def NegativeIntegerQ(*args):
return all(var.is_Integer and NegativeQ(var) for var in args)
def IntegerQ(var):
var = Simplify(var)
if isinstance(var, (int, Integer)):
return True
else:
return var.is_Integer
def IntegersQ(*var):
return all(IntegerQ(i) for i in var)
def _ComplexNumberQ(var):
i = S(im(var))
if isinstance(i, (Integer, Float)):
return i != 0
else:
return False
def ComplexNumberQ(*var):
"""
ComplexNumberQ(m, n,...) returns True if m, n, ... are all explicit complex numbers, else it returns False.
Examples
========
>>> from sympy.integrals.rubi.utility_function import ComplexNumberQ
>>> from sympy import I
>>> ComplexNumberQ(1 + I*2, I)
True
>>> ComplexNumberQ(2, I)
False
"""
return all(_ComplexNumberQ(i) for i in var)
def PureComplexNumberQ(*var):
return all((_ComplexNumberQ(i) and re(i)==0) for i in var)
def RealNumericQ(u):
return u.is_real
def PositiveOrZeroQ(u):
return u.is_real and u >= 0
def NegativeOrZeroQ(u):
return u.is_real and u <= 0
def FractionOrNegativeQ(u):
return FractionQ(u) or NegativeQ(u)
def NegQ(var):
return Not(PosQ(var)) and NonzeroQ(var)
def Equal(a, b):
return a == b
def Unequal(a, b):
return a != b
def IntPart(u):
# IntPart[u] returns the sum of the integer terms of u.
if ProductQ(u):
if IntegerQ(First(u)):
return First(u)*IntPart(Rest(u))
elif IntegerQ(u):
return u
elif FractionQ(u):
return IntegerPart(u)
elif SumQ(u):
res = 0
for i in u.args:
res += IntPart(i)
return res
return 0
def FracPart(u):
# FracPart[u] returns the sum of the non-integer terms of u.
if ProductQ(u):
if IntegerQ(First(u)):
return First(u)*FracPart(Rest(u))
if IntegerQ(u):
return 0
elif FractionQ(u):
return FractionalPart(u)
elif SumQ(u):
res = 0
for i in u.args:
res += FracPart(i)
return res
else:
return u
def RationalQ(*nodes):
return all(var.is_Rational for var in nodes)
def ProductQ(expr):
return S(expr).is_Mul
def SumQ(expr):
return expr.is_Add
def NonsumQ(expr):
return not SumQ(expr)
def Subst(a, x, y):
if None in [a, x, y]:
return None
if a.has(Function('Integrate')):
# substituting in `Function(Integrate)` won't take care of properties of Integral
a = a.replace(Function('Integrate'), Integral)
return a.subs(x, y)
# return a.xreplace({x: y})
def First(expr, d=None):
"""
Gives the first element if it exists, or d otherwise.
Examples
========
>>> from sympy.integrals.rubi.utility_function import First
>>> from sympy.abc import a, b, c
>>> First(a + b + c)
a
>>> First(a*b*c)
a
"""
if isinstance(expr, list):
return expr[0]
if isinstance(expr, Symbol):
return expr
else:
if SumQ(expr) or ProductQ(expr):
l = Sort(expr.args)
return l[0]
else:
return expr.args[0]
def Rest(expr):
"""
Gives rest of the elements if it exists
Examples
========
>>> from sympy.integrals.rubi.utility_function import Rest
>>> from sympy.abc import a, b, c
>>> Rest(a + b + c)
b + c
>>> Rest(a*b*c)
b*c
"""
if isinstance(expr, list):
return expr[1:]
else:
if SumQ(expr) or ProductQ(expr):
l = Sort(expr.args)
return expr.func(*l[1:])
else:
return expr.args[1]
def SqrtNumberQ(expr):
# SqrtNumberQ[u] returns True if u^2 is a rational number; else it returns False.
if PowerQ(expr):
m = expr.base
n = expr.exp
return (IntegerQ(n) and SqrtNumberQ(m)) or (IntegerQ(n-S(1)/2) and RationalQ(m))
elif expr.is_Mul:
return all(SqrtNumberQ(i) for i in expr.args)
else:
return RationalQ(expr) or expr == I
def SqrtNumberSumQ(u):
return SumQ(u) and SqrtNumberQ(First(u)) and SqrtNumberQ(Rest(u)) or ProductQ(u) and SqrtNumberQ(First(u)) and SqrtNumberSumQ(Rest(u))
def LinearQ(expr, x):
"""
LinearQ(expr, x) returns True iff u is a polynomial of degree 1.
Examples
========
>>> from sympy.integrals.rubi.utility_function import LinearQ
>>> from sympy.abc import x, y, a
>>> LinearQ(a, x)
False
>>> LinearQ(3*x + y**2, x)
True
>>> LinearQ(3*x + y**2, y)
False
"""
if isinstance(expr, list):
return all(LinearQ(i, x) for i in expr)
elif expr.is_polynomial(x):
if degree(Poly(expr, x), gen=x) == 1:
return True
return False
def Sqrt(a):
return sqrt(a)
def ArcCosh(a):
return acosh(a)
class Util_Coefficient(Function):
def doit(self):
if len(self.args) == 2:
n = 1
else:
n = Simplify(self.args[2])
if NumericQ(n):
expr = expand(self.args[0])
if isinstance(n, (int, Integer)):
return expr.coeff(self.args[1], n)
else:
return expr.coeff(self.args[1]**n)
else:
return self
def Coefficient(expr, var, n=1):
"""
Coefficient(expr, var) gives the coefficient of form in the polynomial expr.
Coefficient(expr, var, n) gives the coefficient of var**n in expr.
Examples
========
>>> from sympy.integrals.rubi.utility_function import Coefficient
>>> from sympy.abc import x, a, b, c
>>> Coefficient(7 + 2*x + 4*x**3, x, 1)
2
>>> Coefficient(a + b*x + c*x**3, x, 0)
a
>>> Coefficient(a + b*x + c*x**3, x, 4)
0
>>> Coefficient(b*x + c*x**3, x, 3)
c
"""
if NumericQ(n):
if expr == 0 or n in (zoo, oo):
return 0
expr = expand(expr)
if isinstance(n, (int, Integer)):
return expr.coeff(var, n)
else:
return expr.coeff(var**n)
return Util_Coefficient(expr, var, n)
def Denominator(var):
var = Simplify(var)
if isinstance(var, Pow):
if isinstance(var.exp, Integer):
if var.exp > 0:
return Pow(Denominator(var.base), var.exp)
elif var.exp < 0:
return Pow(Numerator(var.base), -1*var.exp)
elif isinstance(var, Add):
var = factor(var)
return fraction(var)[1]
def Hypergeometric2F1(a, b, c, z):
return hyper([a, b], [c], z)
def Not(var):
if isinstance(var, bool):
return not var
elif var.is_Relational:
var = False
return not var
def FractionalPart(a):
return frac(a)
def IntegerPart(a):
return floor(a)
def AppellF1(a, b1, b2, c, x, y):
return appellf1(a, b1, b2, c, x, y)
def EllipticPi(*args):
return elliptic_pi(*args)
def EllipticE(*args):
return elliptic_e(*args)
def EllipticF(Phi, m):
return elliptic_f(Phi, m)
def ArcTan(a, b = None):
if b is None:
return atan(a)
else:
return atan2(a, b)
def ArcCot(a):
return acot(a)
def ArcCoth(a):
return acoth(a)
def ArcTanh(a):
return atanh(a)
def ArcSin(a):
return asin(a)
def ArcSinh(a):
return asinh(a)
def ArcCos(a):
return acos(a)
def ArcCsc(a):
return acsc(a)
def ArcSec(a):
return asec(a)
def ArcCsch(a):
return acsch(a)
def ArcSech(a):
return asech(a)
def Sinh(u):
return sinh(u)
def Tanh(u):
return tanh(u)
def Cosh(u):
return cosh(u)
def Sech(u):
return sech(u)
def Csch(u):
return csch(u)
def Coth(u):
return coth(u)
def LessEqual(*args):
for i in range(0, len(args) - 1):
try:
if args[i] > args[i + 1]:
return False
except:
return False
return True
def Less(*args):
for i in range(0, len(args) - 1):
try:
if args[i] >= args[i + 1]:
return False
except:
return False
return True
def Greater(*args):
for i in range(0, len(args) - 1):
try:
if args[i] <= args[i + 1]:
return False
except:
return False
return True
def GreaterEqual(*args):
for i in range(0, len(args) - 1):
try:
if args[i] < args[i + 1]:
return False
except:
return False
return True
def FractionQ(*args):
"""
FractionQ(m, n,...) returns True if m, n, ... are all explicit fractions, else it returns False.
Examples
========
>>> from sympy import S
>>> from sympy.integrals.rubi.utility_function import FractionQ
>>> FractionQ(S('3'))
False
>>> FractionQ(S('3')/S('2'))
True
"""
return all(i.is_Rational for i in args) and all(Denominator(i)!= S(1) for i in args)
def IntLinearcQ(a, b, c, d, m, n, x):
# returns True iff (a+b*x)^m*(c+d*x)^n is integrable wrt x in terms of non-hypergeometric functions.
return IntegerQ(m) or IntegerQ(n) or IntegersQ(S(3)*m, S(3)*n) or IntegersQ(S(4)*m, S(4)*n) or IntegersQ(S(2)*m, S(6)*n) or IntegersQ(S(6)*m, S(2)*n) or IntegerQ(m + n)
Defer = UnevaluatedExpr
def Expand(expr):
return expr.expand()
def IndependentQ(u, x):
"""
If u is free from x IndependentQ(u, x) returns True else False.
Examples
========
>>> from sympy.integrals.rubi.utility_function import IndependentQ
>>> from sympy.abc import x, a, b
>>> IndependentQ(a + b*x, x)
False
>>> IndependentQ(a + b, x)
True
"""
return FreeQ(u, x)
def PowerQ(expr):
return expr.is_Pow or ExpQ(expr)
def IntegerPowerQ(u):
if isinstance(u, sym_exp): #special case for exp
return IntegerQ(u.args[0])
return PowerQ(u) and IntegerQ(u.args[1])
def PositiveIntegerPowerQ(u):
if isinstance(u, sym_exp):
return IntegerQ(u.args[0]) and PositiveQ(u.args[0])
return PowerQ(u) and IntegerQ(u.args[1]) and PositiveQ(u.args[1])
def FractionalPowerQ(u):
if isinstance(u, sym_exp):
return FractionQ(u.args[0])
return PowerQ(u) and FractionQ(u.args[1])
def AtomQ(expr):
expr = sympify(expr)
if isinstance(expr, list):
return False
if expr in [None, True, False, _E]: # [None, True, False] are atoms in mathematica and _E is also an atom
return True
# elif isinstance(expr, list):
# return all(AtomQ(i) for i in expr)
else:
return expr.is_Atom
def ExpQ(u):
u = replace_pow_exp(u)
return Head(u) in (sym_exp, exp)
def LogQ(u):
return u.func in (sym_log, log)
def Head(u):
return u.func
def MemberQ(l, u):
if isinstance(l, list):
return u in l
else:
return u in l.args
def TrigQ(u):
if AtomQ(u):
x = u
else:
x = Head(u)
return MemberQ([sin, cos, tan, cot, sec, csc], x)
def SinQ(u):
return Head(u) == sin
def CosQ(u):
return Head(u) == cos
def TanQ(u):
return Head(u) == tan
def CotQ(u):
return Head(u) == cot
def SecQ(u):
return Head(u) == sec
def CscQ(u):
return Head(u) == csc
def Sin(u):
return sin(u)
def Cos(u):
return cos(u)
def Tan(u):
return tan(u)
def Cot(u):
return cot(u)
def Sec(u):
return sec(u)
def Csc(u):
return csc(u)
def HyperbolicQ(u):
if AtomQ(u):
x = u
else:
x = Head(u)
return MemberQ([sinh, cosh, tanh, coth, sech, csch], x)
def SinhQ(u):
return Head(u) == sinh
def CoshQ(u):
return Head(u) == cosh
def TanhQ(u):
return Head(u) == tanh
def CothQ(u):
return Head(u) == coth
def SechQ(u):
return Head(u) == sech
def CschQ(u):
return Head(u) == csch
def InverseTrigQ(u):
if AtomQ(u):
x = u
else:
x = Head(u)
return MemberQ([asin, acos, atan, acot, asec, acsc], x)
def SinCosQ(f):
return MemberQ([sin, cos, sec, csc], Head(f))
def SinhCoshQ(f):
return MemberQ([sinh, cosh, sech, csch], Head(f))
def LeafCount(expr):
return len(list(postorder_traversal(expr)))
def Numerator(u):
u = Simplify(u)
if isinstance(u, Pow):
if isinstance(u.exp, Integer):
if u.exp > 0:
return Pow(Numerator(u.base), u.exp)
elif u.exp < 0:
return Pow(Denominator(u.base), -1*u.exp)
elif isinstance(u, Add):
u = factor(u)
return fraction(u)[0]
def NumberQ(u):
if isinstance(u, (int, float)):
return True
return u.is_number
def NumericQ(u):
return N(u).is_number
def Length(expr):
"""
Returns number of elements in the experssion just as sympy's len.
Examples
========
>>> from sympy.integrals.rubi.utility_function import Length
>>> from sympy.abc import x, a, b
>>> from sympy import cos, sin
>>> Length(a + b)
2
>>> Length(sin(a)*cos(a))
2
"""
if isinstance(expr, list):
return len(expr)
return len(expr.args)
def ListQ(u):
return isinstance(u, list)
def Im(u):
u = S(u)
return im(u.doit())
def Re(u):
u = S(u)
return re(u.doit())
def InverseHyperbolicQ(u):
if not u.is_Atom:
u = Head(u)
return u in [acosh, asinh, atanh, acoth, acsch, acsch]
def InverseFunctionQ(u):
# returns True if u is a call on an inverse function; else returns False.
return LogQ(u) or InverseTrigQ(u) and Length(u) <= 1 or InverseHyperbolicQ(u) or u.func == polylog
def TrigHyperbolicFreeQ(u, x):
# If u is free of trig, hyperbolic and calculus functions involving x, TrigHyperbolicFreeQ[u,x] returns true; else it returns False.
if AtomQ(u):
return True
else:
if TrigQ(u) | HyperbolicQ(u) | CalculusQ(u):
return FreeQ(u, x)
else:
for i in u.args:
if not TrigHyperbolicFreeQ(i, x):
return False
return True
def InverseFunctionFreeQ(u, x):
# If u is free of inverse, calculus and hypergeometric functions involving x, InverseFunctionFreeQ[u,x] returns true; else it returns False.
if AtomQ(u):
return True
else:
if InverseFunctionQ(u) or CalculusQ(u) or u.func == hyper or u.func == appellf1:
return FreeQ(u, x)
else:
for i in u.args:
if not ElementaryFunctionQ(i):
return False
return True
def RealQ(u):
if ListQ(u):
return MapAnd(RealQ, u)
elif NumericQ(u):
return ZeroQ(Im(N(u)))
elif PowerQ(u):
u = u.base
v = u.exp
return RealQ(u) & RealQ(v) & (IntegerQ(v) | PositiveOrZeroQ(u))
elif u.is_Mul:
return all(RealQ(i) for i in u.args)
elif u.is_Add:
return all(RealQ(i) for i in u.args)
elif u.is_Function:
f = u.func
u = u.args[0]
if f in [sin, cos, tan, cot, sec, csc, atan, acot, erf]:
return RealQ(u)
else:
if f in [asin, acos]:
return LE(-1, u, 1)
else:
if f == sym_log:
return PositiveOrZeroQ(u)
else:
return False
else:
return False
def EqQ(u, v):
return ZeroQ(u - v)
def FractionalPowerFreeQ(u):
if AtomQ(u):
return True
elif FractionalPowerQ(u):
return False
def ComplexFreeQ(u):
if AtomQ(u) and Not(ComplexNumberQ(u)):
return True
else:
return False
def PolynomialQ(u, x = None):
if x is None :
return u.is_polynomial()
if isinstance(x, Pow):
if isinstance(x.exp, Integer):
deg = degree(u, x.base)
if u.is_polynomial(x):
if deg % x.exp !=0 :
return False
try:
p = Poly(u, x.base)
except PolynomialError:
return False
c_list = p.all_coeffs()
coeff_list = c_list[:-1:x.exp]
coeff_list += [c_list[-1]]
for i in coeff_list:
if not i == 0:
index = c_list.index(i)
c_list[index] = 0
if all(i == 0 for i in c_list):
return True
else:
return False
return u.is_polynomial(x)
else:
return False
elif isinstance(x.exp, (Float, Rational)): #not full - proof
if FreeQ(simplify(u), x.base) and Exponent(u, x.base) == 0:
if not all(FreeQ(u, i) for i in x.base.free_symbols):
return False
if isinstance(x, Mul):
return all(PolynomialQ(u, i) for i in x.args)
return u.is_polynomial(x)
def FactorSquareFree(u):
return sqf(u)
def PowerOfLinearQ(expr, x):
u = Wild('u')
w = Wild('w')
m = Wild('m')
n = Wild('n')
Match = expr.match(u**m)
if PolynomialQ(Match[u], x) and FreeQ(Match[m], x):
if IntegerQ(Match[m]):
e = FactorSquareFree(Match[u]).match(w**n)
if FreeQ(e[n], x) and LinearQ(e[w], x):
return True
else:
return False
else:
return LinearQ(Match[u], x)
else:
return False
def Exponent(expr, x, h = None):
expr = Expand(S(expr))
if h is None:
if S(expr).is_number or (not expr.has(x)):
return 0
if PolynomialQ(expr, x):
if isinstance(x, Rational):
return degree(Poly(expr, x), x)
return degree(expr, gen = x)
else:
return 0
else:
if S(expr).is_number or (not expr.has(x)):
res = [0]
if expr.is_Add:
expr = collect(expr, x)
lst = []
k = 1
for t in expr.args:
if t.has(x):
if isinstance(x, Rational):
lst += [degree(Poly(t, x), x)]
else:
lst += [degree(t, gen = x)]
else:
if k == 1:
lst += [0]
k += 1
lst.sort()
res = lst
else:
if isinstance(x, Rational):
res = [degree(Poly(expr, x), x)]
else:
res = [degree(expr, gen = x)]
return h(*res)
def QuadraticQ(u, x):
# QuadraticQ(u, x) returns True iff u is a polynomial of degree 2 and not a monomial of the form a x^2
if ListQ(u):
for expr in u:
if Not(PolyQ(expr, x, 2) and Not(Coefficient(expr, x, 0) == 0 and Coefficient(expr, x, 1) == 0)):
return False
return True
else:
return PolyQ(u, x, 2) and Not(Coefficient(u, x, 0) == 0 and Coefficient(u, x, 1) == 0)
def LinearPairQ(u, v, x):
# LinearPairQ(u, v, x) returns True iff u and v are linear not equal x but u/v is a constant wrt x
return LinearQ(u, x) and LinearQ(v, x) and NonzeroQ(u-x) and ZeroQ(Coefficient(u, x, 0)*Coefficient(v, x, 1)-Coefficient(u, x, 1)*Coefficient(v, x, 0))
def BinomialParts(u, x):
if PolynomialQ(u, x):
if Exponent(u, x) > 0:
lst = Exponent(u, x, List)
if len(lst)==1:
return [0, Coefficient(u, x, Exponent(u, x)), Exponent(u,x)]
elif len(lst) == 2 and lst[0] == 0:
return [Coefficient(u, x, 0), Coefficient(u, x, Exponent(u, x)), Exponent(u, x)]
else:
return False
else:
return False
elif PowerQ(u):
if u.base == x and FreeQ(u.exp, x):
return [0, 1, u.exp]
else:
return False
elif ProductQ(u):
if FreeQ(First(u), x):
lst2 = BinomialParts(Rest(u), x)
if AtomQ(lst2):
return False
else:
return [First(u)*lst2[0], First(u)*lst2[1], lst2[2]]
elif FreeQ(Rest(u), x):
lst1 = BinomialParts(First(u), x)
if AtomQ(lst1):
return False
else:
return [Rest(u)*lst1[0], Rest(u)*lst1[1], lst1[2]]
lst1 = BinomialParts(First(u), x)
if AtomQ(lst1):
return False
lst2 = BinomialParts(Rest(u), x)
if AtomQ(lst2):
return False
a = lst1[0]
b = lst1[1]
m = lst1[2]
c = lst2[0]
d = lst2[1]
n = lst2[2]
if ZeroQ(a):
if ZeroQ(c):
return [0, b*d, m + n]
elif ZeroQ(m + n):
return [b*d, b*c, m]
else:
return False
if ZeroQ(c):
if ZeroQ(m + n):
return [b*d, a*d, n]
else:
return False
if EqQ(m, n) and ZeroQ(a*d + b*c):
return [a*c, b*d, 2*m]
else:
return False
elif SumQ(u):
if FreeQ(First(u),x):
lst2 = BinomialParts(Rest(u), x)
if AtomQ(lst2):
return False
else:
return [First(u) + lst2[0], lst2[1], lst2[2]]
elif FreeQ(Rest(u), x):
lst1 = BinomialParts(First(u), x)
if AtomQ(lst1):
return False
else:
return[Rest(u) + lst1[0], lst1[1], lst1[2]]
lst1 = BinomialParts(First(u), x)
if AtomQ(lst1):
return False
lst2 = BinomialParts(Rest(u),x)
if AtomQ(lst2):
return False
if EqQ(lst1[2], lst2[2]):
return [lst1[0] + lst2[0], lst1[1] + lst2[1], lst1[2]]
else:
return False
else:
return False
def TrinomialParts(u, x):
# If u is equivalent to a trinomial of the form a + b*x^n + c*x^(2*n) where n!=0, b!=0 and c!=0, TrinomialParts[u,x] returns the list {a,b,c,n}; else it returns False.
u = sympify(u)
if PolynomialQ(u, x):
lst = CoefficientList(u, x)
if len(lst)<3 or EvenQ(sympify(len(lst))) or ZeroQ((len(lst)+1)/2):
return False
#Catch(
# Scan(Function(if ZeroQ(lst), Null, Throw(False), Drop(Drop(Drop(lst, [(len(lst)+1)/2]), 1), -1];
# [First(lst), lst[(len(lst)+1)/2], Last(lst), (len(lst)-1)/2]):
if PowerQ(u):
if EqQ(u.exp, 2):
lst = BinomialParts(u.base, x)
if not lst or ZeroQ(lst[0]):
return False
else:
return [lst[0]**2, 2*lst[0]*lst[1], lst[1]**2, lst[2]]
else:
return False
if ProductQ(u):
if FreeQ(First(u), x):
lst2 = TrinomialParts(Rest(u), x)
if not lst2:
return False
else:
return [First(u)*lst2[0], First(u)*lst2[1], First(u)*lst2[2], lst2[3]]
if FreeQ(Rest(u), x):
lst1 = TrinomialParts(First(u), x)
if not lst1:
return False
else:
return [Rest(u)*lst1[0], Rest(u)*lst1[1], Rest(u)*lst1[2], lst1[3]]
lst1 = BinomialParts(First(u), x)
if not lst1:
return False
lst2 = BinomialParts(Rest(u), x)
if not lst2:
return False
a = lst1[0]
b = lst1[1]
m = lst1[2]
c = lst2[0]
d = lst2[1]
n = lst2[2]
if EqQ(m, n) and NonzeroQ(a*d+b*c):
return [a*c, a*d + b*c, b*d, m]
else:
return False
if SumQ(u):
if FreeQ(First(u), x):
lst2 = TrinomialParts(Rest(u), x)
if not lst2:
return False
else:
return [First(u)+lst2[0], lst2[1], lst2[2], lst2[3]]
if FreeQ(Rest(u), x):
lst1 = TrinomialParts(First(u), x)
if not lst1:
return False
else:
return [Rest(u)+lst1[0], lst1[1], lst1[2], lst1[3]]
lst1 = TrinomialParts(First(u), x)
if not lst1:
lst3 = BinomialParts(First(u), x)
if not lst3:
return False
lst2 = TrinomialParts(Rest(u), x)
if not lst2:
lst4 = BinomialParts(Rest(u), x)
if not lst4:
return False
if EqQ(lst3[2], 2*lst4[2]):
return [lst3[0]+lst4[0], lst4[1], lst3[1], lst4[2]]
if EqQ(lst4[2], 2*lst3[2]):
return [lst3[0]+lst4[0], lst3[1], lst4[1], lst3[2]]
else:
return False
if EqQ(lst3[2], lst2[3]) and NonzeroQ(lst3[1]+lst2[1]):
return [lst3[0]+lst2[0], lst3[1]+lst2[1], lst2[2], lst2[3]]
if EqQ(lst3[2], 2*lst2[3]) and NonzeroQ(lst3[1]+lst2[2]):
return [lst3[0]+lst2[0], lst2[1], lst3[1]+lst2[2], lst2[3]]
else:
return False
lst2 = TrinomialParts(Rest(u), x)
if AtomQ(lst2):
lst4 = BinomialParts(Rest(u), x)
if not lst4:
return False
if EqQ(lst4[2], lst1[3]) and NonzeroQ(lst1[1]+lst4[0]):
return [lst1[0]+lst4[0], lst1[1]+lst4[1], lst1[2], lst1[3]]
if EqQ(lst4[2], 2*lst1[3]) and NonzeroQ(lst1[2]+lst4[1]):
return [lst1[0]+lst4[0], lst1[1], lst1[2]+lst4[1], lst1[3]]
else:
return False
if EqQ(lst1[3], lst2[3]) and NonzeroQ(lst1[1]+lst2[1]) and NonzeroQ(lst1[2]+lst2[2]):
return [lst1[0]+lst2[0], lst1[1]+lst2[1], lst1[2]+lst2[2], lst1[3]]
else:
return False
else:
return False
def PolyQ(u, x, n=None):
# returns True iff u is a polynomial of degree n.
if ListQ(u):
return all(PolyQ(i, x) for i in u)
if n==None:
if u == x:
return False
elif isinstance(x, Pow):
n = x.exp
x_base = x.base
if FreeQ(n, x_base):
if PositiveIntegerQ(n):
return PolyQ(u, x_base) and (PolynomialQ(u, x) or PolynomialQ(Together(u), x))
elif AtomQ(n):
return PolynomialQ(u, x) and FreeQ(CoefficientList(u, x), x_base)
else:
return False
return PolynomialQ(u, x) or PolynomialQ(u, Together(x))
else:
return PolynomialQ(u, x) and Coefficient(u, x, n) != 0 and Exponent(u, x) == n
def EvenQ(u):
# gives True if expr is an even integer, and False otherwise.
return isinstance(u, (Integer, int)) and u%2 == 0
def OddQ(u):
# gives True if expr is an odd integer, and False otherwise.
return isinstance(u, (Integer, int)) and u%2 == 1
def PerfectSquareQ(u):
# (* If u is a rational number whose squareroot is rational or if u is of the form u1^n1 u2^n2 ...
# and n1, n2, ... are even, PerfectSquareQ[u] returns True; else it returns False. *)
if RationalQ(u):
return Greater(u, 0) and RationalQ(Sqrt(u))
elif PowerQ(u):
return EvenQ(u.exp)
elif ProductQ(u):
return PerfectSquareQ(First(u)) and PerfectSquareQ(Rest(u))
elif SumQ(u):
s = Simplify(u)
if NonsumQ(s):
return PerfectSquareQ(s)
return False
else:
return False
def NiceSqrtAuxQ(u):
if RationalQ(u):
return u > 0
elif PowerQ(u):
return EvenQ(u.exp)
elif ProductQ(u):
return NiceSqrtAuxQ(First(u)) and NiceSqrtAuxQ(Rest(u))
elif SumQ(u):
s = Simplify(u)
return NonsumQ(s) and NiceSqrtAuxQ(s)
else:
return False
def NiceSqrtQ(u):
return Not(NegativeQ(u)) and NiceSqrtAuxQ(u)
def Together(u):
return factor(u)
def PosAux(u):
if RationalQ(u):
return u>0
elif NumberQ(u):
if ZeroQ(Re(u)):
return Im(u) > 0
else:
return Re(u) > 0
elif NumericQ(u):
v = N(u)
if ZeroQ(Re(v)):
return Im(v) > 0
else:
return Re(v) > 0
elif PowerQ(u):
if OddQ(u.exp):
return PosAux(u.base)
else:
return True
elif ProductQ(u):
if PosAux(First(u)):
return PosAux(Rest(u))
else:
return not PosAux(Rest(u))
elif SumQ(u):
return PosAux(First(u))
else:
res = u > 0
if res in(True, False):
return res
return True
def PosQ(u):
# If u is not 0 and has a positive form, PosQ[u] returns True, else it returns False.
return PosAux(TogetherSimplify(u))
def CoefficientList(u, x):
if PolynomialQ(u, x):
return list(reversed(Poly(u, x).all_coeffs()))
else:
return []
def ReplaceAll(expr, args):
if isinstance(args, list):
n_args = {}
for i in args:
n_args.update(i)
return expr.subs(n_args)
return expr.subs(args)
def ExpandLinearProduct(v, u, a, b, x):
# If u is a polynomial in x, ExpandLinearProduct[v,u,a,b,x] expands v*u into a sum of terms of the form c*v*(a+b*x)^n.
if FreeQ([a, b], x) and PolynomialQ(u, x):
lst = CoefficientList(ReplaceAll(u, {x: (x - a)/b}), x)
lst = [SimplifyTerm(i, x) for i in lst]
res = 0
for k in range(1, len(lst)+1):
res = res + Simplify(v*lst[k-1]*(a + b*x)**(k - 1))
return res
return u*v
def GCD(*args):
args = S(args)
if len(args) == 1:
if isinstance(args[0], (int, Integer)):
return args[0]
else:
return S(1)
return gcd(*args)
def ContentFactor(expn):
return factor_terms(expn)
def NumericFactor(u):
# returns the real numeric factor of u.
if NumberQ(u):
if ZeroQ(Im(u)):
return u
elif ZeroQ(Re(u)):
return Im(u)
else:
return S(1)
elif PowerQ(u):
if RationalQ(u.base) and RationalQ(u.exp):
if u.exp > 0:
return 1/Denominator(u.base)
else:
return 1/(1/Denominator(u.base))
else:
return S(1)
elif ProductQ(u):
return Mul(*[NumericFactor(i) for i in u.args])
elif SumQ(u):
if LeafCount(u) < 50:
c = ContentFactor(u)
if SumQ(c):
return S(1)
else:
return NumericFactor(c)
else:
m = NumericFactor(First(u))
n = NumericFactor(Rest(u))
if m < 0 and n < 0:
return -GCD(-m, -n)
else:
return GCD(m, n)
return S(1)
def NonnumericFactors(u):
if NumberQ(u):
if ZeroQ(Im(u)):
return S(1)
elif ZeroQ(Re(u)):
return I
return u
elif PowerQ(u):
if RationalQ(u.base) and FractionQ(u.exp):
return u/NumericFactor(u)
return u
elif ProductQ(u):
result = 1
for i in u.args:
result *= NonnumericFactors(i)
return result
elif SumQ(u):
if LeafCount(u) < 50:
i = ContentFactor(u)
if SumQ(i):
return u
else:
return NonnumericFactors(i)
n = NumericFactor(u)
result = 0
for i in u.args:
result += i/n
return result
return u
def MakeAssocList(u, x, alst=None):
# (* MakeAssocList[u,x,alst] returns an association list of gensymed symbols with the nonatomic
# parameters of a u that are not integer powers, products or sums. *)
if alst is None:
alst = []
u = replace_pow_exp(u)
x = replace_pow_exp(x)
if AtomQ(u):
return alst
elif IntegerPowerQ(u):
return MakeAssocList(u.base, x, alst)
elif ProductQ(u) or SumQ(u):
return MakeAssocList(Rest(u), x, MakeAssocList(First(u), x, alst))
elif FreeQ(u, x):
tmp = []
for i in alst:
if PowerQ(i):
if i.exp == u:
tmp.append(i)
break
elif len(i.args) > 1: # make sure args has length > 1, else causes index error some times
if i.args[1] == u:
tmp.append(i)
break
if tmp == []:
alst.append(u)
return alst
return alst
def GensymSubst(u, x, alst=None):
# (* GensymSubst[u,x,alst] returns u with the kernels in alst free of x replaced by gensymed names. *)
if alst is None:
alst =[]
u = replace_pow_exp(u)
x = replace_pow_exp(x)
if AtomQ(u):
return u
elif IntegerPowerQ(u):
return GensymSubst(u.base, x, alst)**u.exp
elif ProductQ(u) or SumQ(u):
return u.func(*[GensymSubst(i, x, alst) for i in u.args])
elif FreeQ(u, x):
tmp = []
for i in alst:
if PowerQ(i):
if i.exp == u:
tmp.append(i)
break
elif len(i.args) > 1: # make sure args has length > 1, else causes index error some times
if i.args[1] == u:
tmp.append(i)
break
if tmp == []:
return u
return tmp[0][0]
return u
def KernelSubst(u, x, alst):
# (* KernelSubst[u,x,alst] returns u with the gensymed names in alst replaced by kernels free of x. *)
if AtomQ(u):
tmp = []
for i in alst:
if i.args[0] == u:
tmp.append(i)
break
if tmp == []:
return u
elif len(tmp[0].args) > 1: # make sure args has length > 1, else causes index error some times
return tmp[0].args[1]
elif IntegerPowerQ(u):
tmp = KernelSubst(u.base, x, alst)
if u.exp < 0 and ZeroQ(tmp):
return 'Indeterminate'
return tmp**u.exp
elif ProductQ(u) or SumQ(u):
return u.func(*[KernelSubst(i, x, alst) for i in u.args])
return u
def ExpandExpression(u, x):
if AlgebraicFunctionQ(u, x) and Not(RationalFunctionQ(u, x)):
v = ExpandAlgebraicFunction(u, x)
else:
v = S(0)
if SumQ(v):
return ExpandCleanup(v, x)
v = SmartApart(u, x)
if SumQ(v):
return ExpandCleanup(v, x)
v = SmartApart(RationalFunctionFactors(u, x), x, x)
if SumQ(v):
w = NonrationalFunctionFactors(u, x)
return ExpandCleanup(v.func(*[i*w for i in v.args]), x)
v = Expand(u)
if SumQ(v):
return ExpandCleanup(v, x)
v = Expand(u)
if SumQ(v):
return ExpandCleanup(v, x)
return SimplifyTerm(u, x)
def Apart(u, x):
if RationalFunctionQ(u, x):
return apart(u, x)
return u
def SmartApart(*args):
if len(args) == 2:
u, x = args
alst = MakeAssocList(u, x)
tmp = KernelSubst(Apart(GensymSubst(u, x, alst), x), x, alst)
if tmp == 'Indeterminate':
return u
return tmp
u, v, x = args
alst = MakeAssocList(u, x)
tmp = KernelSubst(Apart(GensymSubst(u, x, alst), x), x, alst)
if tmp == 'Indeterminate':
return u
return tmp
def MatchQ(expr, pattern, *var):
# returns the matched arguments after matching pattern with expression
match = expr.match(pattern)
if match:
return tuple(match[i] for i in var)
else:
return None
def PolynomialQuotientRemainder(p, q, x):
return [PolynomialQuotient(p, q, x), PolynomialRemainder(p, q, x)]
def FreeFactors(u, x):
# returns the product of the factors of u free of x.
if ProductQ(u):
result = 1
for i in u.args:
if FreeQ(i, x):
result *= i
return result
elif FreeQ(u, x):
return u
else:
return S(1)
def NonfreeFactors(u, x):
"""
Returns the product of the factors of u not free of x.
Examples
========
>>> from sympy.integrals.rubi.utility_function import NonfreeFactors
>>> from sympy.abc import x, a, b
>>> NonfreeFactors(a, x)
1
>>> NonfreeFactors(x + a, x)
a + x
>>> NonfreeFactors(a*b*x, x)
x
"""
if ProductQ(u):
result = 1
for i in u.args:
if not FreeQ(i, x):
result *= i
return result
elif FreeQ(u, x):
return 1
else:
return u
def RemoveContentAux(expr, x):
return RemoveContentAux_replacer.replace(UtilityOperator(expr, x))
def RemoveContent(u, x):
v = NonfreeFactors(u, x)
w = Together(v)
if EqQ(FreeFactors(w, x), 1):
return RemoveContentAux(v, x)
else:
return RemoveContentAux(NonfreeFactors(w, x), x)
def FreeTerms(u, x):
"""
Returns the sum of the terms of u free of x.
Examples
========
>>> from sympy.integrals.rubi.utility_function import FreeTerms
>>> from sympy.abc import x, a, b
>>> FreeTerms(a, x)
a
>>> FreeTerms(x*a, x)
0
>>> FreeTerms(a*x + b, x)
b
"""
if SumQ(u):
result = 0
for i in u.args:
if FreeQ(i, x):
result += i
return result
elif FreeQ(u, x):
return u
else:
return 0
def NonfreeTerms(u, x):
# returns the sum of the terms of u free of x.
if SumQ(u):
result = S(0)
for i in u.args:
if not FreeQ(i, x):
result += i
return result
elif not FreeQ(u, x):
return u
else:
return S(0)
def ExpandAlgebraicFunction(expr, x):
if ProductQ(expr):
u_ = Wild('u', exclude=[x])
n_ = Wild('n', exclude=[x])
v_ = Wild('v')
pattern = u_*v_
match = expr.match(pattern)
if match:
keys = [u_, v_]
if len(keys) == len(match):
u, v = tuple([match[i] for i in keys])
if SumQ(v):
u, v = v, u
if not FreeQ(u, x) and SumQ(u):
result = 0
for i in u.args:
result += i*v
return result
pattern = u_**n_*v_
match = expr.match(pattern)
if match:
keys = [u_, n_, v_]
if len(keys) == len(match):
u, n, v = tuple([match[i] for i in keys])
if PositiveIntegerQ(n) and SumQ(u):
w = Expand(u**n)
result = 0
for i in w.args:
result += i*v
return result
return expr
def CollectReciprocals(expr, x):
# Basis: e/(a+b x)+f/(c+d x)==(c e+a f+(d e+b f) x)/(a c+(b c+a d) x+b d x^2)
if SumQ(expr):
u_ = Wild('u')
a_ = Wild('a', exclude=[x])
b_ = Wild('b', exclude=[x])
c_ = Wild('c', exclude=[x])
d_ = Wild('d', exclude=[x])
e_ = Wild('e', exclude=[x])
f_ = Wild('f', exclude=[x])
pattern = u_ + e_/(a_ + b_*x) + f_/(c_+d_*x)
match = expr.match(pattern)
if match:
try: # .match() does not work peoperly always
keys = [u_, a_, b_, c_, d_, e_, f_]
u, a, b, c, d, e, f = tuple([match[i] for i in keys])
if ZeroQ(b*c + a*d) & ZeroQ(d*e + b*f):
return CollectReciprocals(u + (c*e + a*f)/(a*c + b*d*x**2),x)
elif ZeroQ(b*c + a*d) & ZeroQ(c*e + a*f):
return CollectReciprocals(u + (d*e + b*f)*x/(a*c + b*d*x**2),x)
except:
pass
return expr
def ExpandCleanup(u, x):
v = CollectReciprocals(u, x)
if SumQ(v):
res = 0
for i in v.args:
res += SimplifyTerm(i, x)
v = res
if SumQ(v):
return UnifySum(v, x)
else:
return v
else:
return v
def AlgebraicFunctionQ(u, x, flag=False):
if ListQ(u):
if u == []:
return True
elif AlgebraicFunctionQ(First(u), x, flag):
return AlgebraicFunctionQ(Rest(u), x, flag)
else:
return False
elif AtomQ(u) or FreeQ(u, x):
return True
elif PowerQ(u):
if RationalQ(u.exp) | flag & FreeQ(u.exp, x):
return AlgebraicFunctionQ(u.base, x, flag)
elif ProductQ(u) | SumQ(u):
for i in u.args:
if not AlgebraicFunctionQ(i, x, flag):
return False
return True
return False
def Coeff(expr, form, n=1):
if n == 1:
return Coefficient(Together(expr), form, n)
else:
coef1 = Coefficient(expr, form, n)
coef2 = Coefficient(Together(expr), form, n)
if Simplify(coef1 - coef2) == 0:
return coef1
else:
return coef2
def LeadTerm(u):
if SumQ(u):
return First(u)
return u
def RemainingTerms(u):
if SumQ(u):
return Rest(u)
return u
def LeadFactor(u):
# returns the leading factor of u.
if ComplexNumberQ(u) and Re(u) == 0:
if Im(u) == S(1):
return u
else:
return LeadFactor(Im(u))
elif ProductQ(u):
return LeadFactor(First(u))
return u
def RemainingFactors(u):
# returns the remaining factors of u.
if ComplexNumberQ(u) and Re(u) == 0:
if Im(u) == 1:
return S(1)
else:
return I*RemainingFactors(Im(u))
elif ProductQ(u):
return RemainingFactors(First(u))*Rest(u)
return S(1)
def LeadBase(u):
"""
returns the base of the leading factor of u.
Examples
========
>>> from sympy.integrals.rubi.utility_function import LeadBase
>>> from sympy.abc import a, b, c
>>> LeadBase(a**b)
a
>>> LeadBase(a**b*c)
a
"""
v = LeadFactor(u)
if PowerQ(v):
return v.base
return v
def LeadDegree(u):
# returns the degree of the leading factor of u.
v = LeadFactor(u)
if PowerQ(v):
return v.exp
return v
def Numer(expr):
# returns the numerator of u.
if PowerQ(expr):
if expr.exp < 0:
return 1
if ProductQ(expr):
return Mul(*[Numer(i) for i in expr.args])
return Numerator(expr)
def Denom(u):
# returns the denominator of u
if PowerQ(u):
if u.exp < 0:
return u.args[0]**(-u.args[1])
elif ProductQ(u):
return Mul(*[Denom(i) for i in u.args])
return Denominator(u)
def hypergeom(n, d, z):
return hyper(n, d, z)
def Expon(expr, form, h=None):
if h:
return Exponent(Together(expr), form, h)
else:
return Exponent(Together(expr), form)
def MergeMonomials(expr, x):
u_ = Wild('u')
p_ = Wild('p', exclude=[x, 1, 0])
a_ = Wild('a', exclude=[x])
b_ = Wild('b', exclude=[x, 0])
c_ = Wild('c', exclude=[x])
d_ = Wild('d', exclude=[x, 0])
n_ = Wild('n', exclude=[x])
m_ = Wild('m', exclude=[x])
# Basis: If m/n\[Element]\[DoubleStruckCapitalZ], then z^m (c z^n)^p==(c z^n)^(m/n+p)/c^(m/n)
pattern = u_*(a_ + b_*x)**m_*(c_*(a_ + b_*x)**n_)**p_
match = expr.match(pattern)
if match:
keys = [u_, a_, b_, m_, c_, n_, p_]
if len(keys) == len(match):
u, a, b, m, c, n, p = tuple([match[i] for i in keys])
if IntegerQ(m/n):
if u*(c*(a + b*x)**n)**(m/n + p)/c**(m/n) == S.NaN:
return expr
else:
return u*(c*(a + b*x)**n)**(m/n + p)/c**(m/n)
# Basis: If m\[Element]\[DoubleStruckCapitalZ] \[And] b c-a d==0, then (a+b z)^m==b^m/d^m (c+d z)^m
pattern = u_*(a_ + b_*x)**m_*(c_ + d_*x)**n_
match = expr.match(pattern)
if match:
keys = [u_, a_, b_, m_, c_, d_, n_]
if len(keys) == len(match):
u, a, b, m, c, d, n = tuple([match[i] for i in keys])
if IntegerQ(m) and ZeroQ(b*c - a*d):
if u*b**m/d**m*(c + d*x)**(m + n) == S.NaN:
return expr
else:
return u*b**m/d**m*(c + d*x)**(m + n)
return expr
def PolynomialDivide(u, v, x):
quo = PolynomialQuotient(u, v, x)
rem = PolynomialRemainder(u, v, x)
s = 0
for i in Exponent(quo, x, List):
s += Simp(Together(Coefficient(quo, x, i)*x**i), x)
quo = s
rem = Together(rem)
free = FreeFactors(rem, x)
rem = NonfreeFactors(rem, x)
monomial = x**Exponent(rem, x, Min)
if NegQ(Coefficient(rem, x, 0)):
monomial = -monomial
s = 0
for i in Exponent(rem, x, List):
s += Simp(Together(Coefficient(rem, x, i)*x**i/monomial), x)
rem = s
if BinomialQ(v, x):
return quo + free*monomial*rem/ExpandToSum(v, x)
else:
return quo + free*monomial*rem/v
def BinomialQ(u, x, n=None):
"""
If u is equivalent to an expression of the form a + b*x**n, BinomialQ(u, x, n) returns True, else it returns False.
Examples
========
>>> from sympy.integrals.rubi.utility_function import BinomialQ
>>> from sympy.abc import x
>>> BinomialQ(x**9, x)
True
>>> BinomialQ((1 + x)**3, x)
False
"""
if ListQ(u):
for i in u:
if Not(BinomialQ(i, x, n)):
return False
return True
elif NumberQ(x):
return False
return ListQ(BinomialParts(u, x))
def TrinomialQ(u, x):
"""
If u is equivalent to an expression of the form a + b*x**n + c*x**(2*n) where n, b and c are not 0,
TrinomialQ(u, x) returns True, else it returns False.
Examples
========
>>> from sympy.integrals.rubi.utility_function import TrinomialQ
>>> from sympy.abc import x
>>> TrinomialQ((7 + 2*x**6 + 3*x**12), x)
True
>>> TrinomialQ(x**2, x)
False
"""
if ListQ(u):
for i in u.args:
if Not(TrinomialQ(i, x)):
return False
return True
check = False
u = replace_pow_exp(u)
if PowerQ(u):
if u.exp == 2 and BinomialQ(u.base, x):
check = True
return ListQ(TrinomialParts(u,x)) and Not(QuadraticQ(u, x)) and Not(check)
def GeneralizedBinomialQ(u, x):
"""
If u is equivalent to an expression of the form a*x**q+b*x**n where n, q and b are not 0,
GeneralizedBinomialQ(u, x) returns True, else it returns False.
Examples
========
>>> from sympy.integrals.rubi.utility_function import GeneralizedBinomialQ
>>> from sympy.abc import a, x, q, b, n
>>> GeneralizedBinomialQ(a*x**q, x)
False
"""
if ListQ(u):
return all(GeneralizedBinomialQ(i, x) for i in u)
return ListQ(GeneralizedBinomialParts(u, x))
def GeneralizedTrinomialQ(u, x):
"""
If u is equivalent to an expression of the form a*x**q+b*x**n+c*x**(2*n-q) where n, q, b and c are not 0,
GeneralizedTrinomialQ(u, x) returns True, else it returns False.
Examples
========
>>> from sympy.integrals.rubi.utility_function import GeneralizedTrinomialQ
>>> from sympy.abc import x
>>> GeneralizedTrinomialQ(7 + 2*x**6 + 3*x**12, x)
False
"""
if ListQ(u):
return all(GeneralizedTrinomialQ(i, x) for i in u)
return ListQ(GeneralizedTrinomialParts(u, x))
def FactorSquareFreeList(poly):
r = sqf_list(poly)
result = [[1, 1]]
for i in r[1]:
result.append(list(i))
return result
def PerfectPowerTest(u, x):
# If u (x) is equivalent to a polynomial raised to an integer power greater than 1,
# PerfectPowerTest[u,x] returns u (x) as an expanded polynomial raised to the power;
# else it returns False.
if PolynomialQ(u, x):
lst = FactorSquareFreeList(u)
gcd = 0
v = 1
if lst[0] == [1, 1]:
lst = Rest(lst)
for i in lst:
gcd = GCD(gcd, i[1])
if gcd > 1:
for i in lst:
v = v*i[0]**(i[1]/gcd)
return Expand(v)**gcd
else:
return False
return False
def SquareFreeFactorTest(u, x):
# If u (x) can be square free factored, SquareFreeFactorTest[u,x] returns u (x) in
# factored form; else it returns False.
if PolynomialQ(u, x):
v = FactorSquareFree(u)
if PowerQ(v) or ProductQ(v):
return v
return False
return False
def RationalFunctionQ(u, x):
# If u is a rational function of x, RationalFunctionQ[u,x] returns True; else it returns False.
if AtomQ(u) or FreeQ(u, x):
return True
elif IntegerPowerQ(u):
return RationalFunctionQ(u.base, x)
elif ProductQ(u) or SumQ(u):
for i in u.args:
if Not(RationalFunctionQ(i, x)):
return False
return True
return False
def RationalFunctionFactors(u, x):
# RationalFunctionFactors[u,x] returns the product of the factors of u that are rational functions of x.
if ProductQ(u):
res = 1
for i in u.args:
if RationalFunctionQ(i, x):
res *= i
return res
elif RationalFunctionQ(u, x):
return u
return S(1)
def NonrationalFunctionFactors(u, x):
if ProductQ(u):
res = 1
for i in u.args:
if not RationalFunctionQ(i, x):
res *= i
return res
elif RationalFunctionQ(u, x):
return S(1)
return u
def Reverse(u):
if isinstance(u, list):
return list(reversed(u))
else:
l = list(u.args)
return u.func(*list(reversed(l)))
def RationalFunctionExponents(u, x):
"""
u is a polynomial or rational function of x.
RationalFunctionExponents(u, x) returns a list of the exponent of the
numerator of u and the exponent of the denominator of u.
Examples
========
>>> from sympy.integrals.rubi.utility_function import RationalFunctionExponents
>>> from sympy.abc import x, a
>>> RationalFunctionExponents(x, x)
[1, 0]
>>> RationalFunctionExponents(x**(-1), x)
[0, 1]
>>> RationalFunctionExponents(x**(-1)*a, x)
[0, 1]
"""
if PolynomialQ(u, x):
return [Exponent(u, x), 0]
elif IntegerPowerQ(u):
if PositiveQ(u.exp):
return u.exp*RationalFunctionExponents(u.base, x)
return (-u.exp)*Reverse(RationalFunctionExponents(u.base, x))
elif ProductQ(u):
lst1 = RationalFunctionExponents(First(u), x)
lst2 = RationalFunctionExponents(Rest(u), x)
return [lst1[0] + lst2[0], lst1[1] + lst2[1]]
elif SumQ(u):
v = Together(u)
if SumQ(v):
lst1 = RationalFunctionExponents(First(u), x)
lst2 = RationalFunctionExponents(Rest(u), x)
return [Max(lst1[0] + lst2[1], lst2[0] + lst1[1]), lst1[1] + lst2[1]]
else:
return RationalFunctionExponents(v, x)
return [0, 0]
def RationalFunctionExpand(expr, x):
# expr is a polynomial or rational function of x.
# RationalFunctionExpand[u,x] returns the expansion of the factors of u that are rational functions times the other factors.
def cons_f1(n):
return FractionQ(n)
cons1 = CustomConstraint(cons_f1)
def cons_f2(x, v):
if not isinstance(x, Symbol):
return False
return UnsameQ(v, x)
cons2 = CustomConstraint(cons_f2)
def With1(n, u, x, v):
w = RationalFunctionExpand(u, x)
return If(SumQ(w), Add(*[i*v**n for i in w.args]), v**n*w)
pattern1 = Pattern(UtilityOperator(u_*v_**n_, x_), cons1, cons2)
rule1 = ReplacementRule(pattern1, With1)
def With2(u, x):
v = ExpandIntegrand(u, x)
def _consf_u(a, b, c, d, p, m, n, x):
return And(FreeQ(List(a, b, c, d, p), x), IntegersQ(m, n), Equal(m, Add(n, S(-1))))
cons_u = CustomConstraint(_consf_u)
pat = Pattern(UtilityOperator(x_**WC('m', S(1))*(x_*WC('d', S(1)) + c_)**p_/(x_**n_*WC('b', S(1)) + a_), x_), cons_u)
result_matchq = is_match(UtilityOperator(u, x), pat)
if UnsameQ(v, u) and not result_matchq:
return v
else:
v = ExpandIntegrand(RationalFunctionFactors(u, x), x)
w = NonrationalFunctionFactors(u, x)
if SumQ(v):
return Add(*[i*w for i in v.args])
else:
return v*w
pattern2 = Pattern(UtilityOperator(u_, x_))
rule2 = ReplacementRule(pattern2, With2)
expr = expr.replace(sym_exp, exp)
res = replace_all(UtilityOperator(expr, x), [rule1, rule2])
return replace_pow_exp(res)
def ExpandIntegrand(expr, x, extra=None):
expr = replace_pow_exp(expr)
if not extra is None:
extra, x = x, extra
w = ExpandIntegrand(extra, x)
r = NonfreeTerms(w, x)
if SumQ(r):
result = [expr*FreeTerms(w, x)]
for i in r.args:
result.append(MergeMonomials(expr*i, x))
return r.func(*result)
else:
return expr*FreeTerms(w, x) + MergeMonomials(expr*r, x)
else:
u_ = Wild('u', exclude=[0, 1])
a_ = Wild('a', exclude=[x])
b_ = Wild('b', exclude=[x, 0])
F_ = Wild('F', exclude=[0])
c_ = Wild('c', exclude=[x])
d_ = Wild('d', exclude=[x, 0])
n_ = Wild('n', exclude=[0, 1])
pattern = u_*(a_ + b_*F_)**n_
match = expr.match(pattern)
if match:
if MemberQ([asin, acos, asinh, acosh], match[F_].func):
keys = [u_, a_, b_, F_, n_]
if len(match) == len(keys):
u, a, b, F, n = tuple([match[i] for i in keys])
match = F.args[0].match(c_ + d_*x)
if match:
keys = c_, d_
if len(keys) == len(match):
c, d = tuple([match[i] for i in keys])
if PolynomialQ(u, x):
F = F.func
return ExpandLinearProduct((a + b*F(c + d*x))**n, u, c, d, x)
expr = expr.replace(sym_exp, exp)
res = replace_all(UtilityOperator(expr, x), ExpandIntegrand_rules, max_count = 1)
return replace_pow_exp(res)
def SimplerQ(u, v):
# If u is simpler than v, SimplerQ(u, v) returns True, else it returns False. SimplerQ(u, u) returns False
if IntegerQ(u):
if IntegerQ(v):
if Abs(u)==Abs(v):
return v<0
else:
return Abs(u)<Abs(v)
else:
return True
elif IntegerQ(v):
return False
elif FractionQ(u):
if FractionQ(v):
if Denominator(u) == Denominator(v):
return SimplerQ(Numerator(u), Numerator(v))
else:
return Denominator(u)<Denominator(v)
else:
return True
elif FractionQ(v):
return False
elif (Re(u)==0 or Re(u) == 0) and (Re(v)==0 or Re(v) == 0):
return SimplerQ(Im(u), Im(v))
elif ComplexNumberQ(u):
if ComplexNumberQ(v):
if Re(u) == Re(v):
return SimplerQ(Im(u), Im(v))
else:
return SimplerQ(Re(u),Re(v))
else:
return False
elif NumberQ(u):
if NumberQ(v):
return OrderedQ([u,v])
else:
return True
elif NumberQ(v):
return False
elif AtomQ(u) or (Head(u) == re) or (Head(u) == im):
if AtomQ(v) or (Head(u) == re) or (Head(u) == im):
return OrderedQ([u,v])
else:
return True
elif AtomQ(v) or (Head(u) == re) or (Head(u) == im):
return False
elif Head(u) == Head(v):
if Length(u) == Length(v):
for i in range(len(u.args)):
if not u.args[i] == v.args[i]:
return SimplerQ(u.args[i], v.args[i])
return False
return Length(u) < Length(v)
elif LeafCount(u) < LeafCount(v):
return True
elif LeafCount(v) < LeafCount(u):
return False
return Not(OrderedQ([v,u]))
def SimplerSqrtQ(u, v):
# If Rt(u, 2) is simpler than Rt(v, 2), SimplerSqrtQ(u, v) returns True, else it returns False. SimplerSqrtQ(u, u) returns False
if NegativeQ(v) and Not(NegativeQ(u)):
return True
if NegativeQ(u) and Not(NegativeQ(v)):
return False
sqrtu = Rt(u, S(2))
sqrtv = Rt(v, S(2))
if IntegerQ(sqrtu):
if IntegerQ(sqrtv):
return sqrtu<sqrtv
else:
return True
if IntegerQ(sqrtv):
return False
if RationalQ(sqrtu):
if RationalQ(sqrtv):
return sqrtu<sqrtv
else:
return True
if RationalQ(sqrtv):
return False
if PosQ(u):
if PosQ(v):
return LeafCount(sqrtu)<LeafCount(sqrtv)
else:
return True
if PosQ(v):
return False
if LeafCount(sqrtu)<LeafCount(sqrtv):
return True
if LeafCount(sqrtv)<LeafCount(sqrtu):
return False
else:
return Not(OrderedQ([v, u]))
def SumSimplerQ(u, v):
"""
If u + v is simpler than u, SumSimplerQ(u, v) returns True, else it returns False.
If for every term w of v there is a term of u equal to n*w where n<-1/2, u + v will be simpler than u.
Examples
========
>>> from sympy.integrals.rubi.utility_function import SumSimplerQ
>>> from sympy.abc import x
>>> from sympy import S
>>> SumSimplerQ(S(4 + x),S(3 + x**3))
False
"""
if RationalQ(u, v):
if v == S(0):
return False
elif v > S(0):
return u < -S(1)
else:
return u >= -v
else:
return SumSimplerAuxQ(Expand(u), Expand(v))
def BinomialDegree(u, x):
# if u is a binomial. BinomialDegree[u,x] returns the degree of x in u.
bp = BinomialParts(u, x)
if bp == False:
return bp
return bp[2]
def TrinomialDegree(u, x):
# If u is equivalent to a trinomial of the form a + b*x^n + c*x^(2*n) where n!=0, b!=0 and c!=0, TrinomialDegree[u,x] returns n
t = TrinomialParts(u, x)
if t:
return t[3]
return t
def CancelCommonFactors(u, v):
def _delete_cases(a, b):
# only for CancelCommonFactors
lst = []
deleted = False
for i in a.args:
if i == b and not deleted:
deleted = True
continue
lst.append(i)
return a.func(*lst)
# CancelCommonFactors[u,v] returns {u',v'} are the noncommon factors of u and v respectively.
if ProductQ(u):
if ProductQ(v):
if MemberQ(v, First(u)):
return CancelCommonFactors(Rest(u), _delete_cases(v, First(u)))
else:
lst = CancelCommonFactors(Rest(u), v)
return [First(u)*lst[0], lst[1]]
else:
if MemberQ(u, v):
return [_delete_cases(u, v), 1]
else:
return[u, v]
elif ProductQ(v):
if MemberQ(v, u):
return [1, _delete_cases(v, u)]
else:
return [u, v]
return[u, v]
def SimplerIntegrandQ(u, v, x):
lst = CancelCommonFactors(u, v)
u1 = lst[0]
v1 = lst[1]
if Head(u1) == Head(v1) and Length(u1) == 1 and Length(v1) == 1:
return SimplerIntegrandQ(u1.args[0], v1.args[0], x)
if LeafCount(u1)<3/4*LeafCount(v1):
return True
if RationalFunctionQ(u1, x):
if RationalFunctionQ(v1, x):
t1 = 0
t2 = 0
for i in RationalFunctionExponents(u1, x):
t1 += i
for i in RationalFunctionExponents(v1, x):
t2 += i
return t1 < t2
else:
return True
else:
return False
def GeneralizedBinomialDegree(u, x):
b = GeneralizedBinomialParts(u, x)
if b:
return b[2] - b[3]
def GeneralizedBinomialParts(expr, x):
expr = Expand(expr)
if GeneralizedBinomialMatchQ(expr, x):
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
n = Wild('n', exclude=[x])
q = Wild('q', exclude=[x])
Match = expr.match(a*x**q + b*x**n)
if Match and PosQ(Match[q] - Match[n]):
return [Match[b], Match[a], Match[q], Match[n]]
else:
return False
def GeneralizedTrinomialDegree(u, x):
t = GeneralizedTrinomialParts(u, x)
if t:
return t[3] - t[4]
def GeneralizedTrinomialParts(expr, x):
expr = Expand(expr)
if GeneralizedTrinomialMatchQ(expr, x):
a = Wild('a', exclude=[x, 0])
b = Wild('b', exclude=[x, 0])
c = Wild('c', exclude=[x])
n = Wild('n', exclude=[x, 0])
q = Wild('q', exclude=[x])
Match = expr.match(a*x**q + b*x**n+c*x**(2*n-q))
if Match and expr.is_Add:
return [Match[c], Match[b], Match[a], Match[n], 2*Match[n]-Match[q]]
else:
return False
def MonomialQ(u, x):
# If u is of the form a*x^n where n!=0 and a!=0, MonomialQ[u,x] returns True; else False
if isinstance(u, list):
return all(MonomialQ(i, x) for i in u)
else:
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
re = u.match(a*x**b)
if re:
return True
return False
def MonomialSumQ(u, x):
# if u(x) is a sum and each term is free of x or an expression of the form a*x^n, MonomialSumQ(u, x) returns True; else it returns False
if SumQ(u):
for i in u.args:
if Not(FreeQ(i, x) or MonomialQ(i, x)):
return False
return True
@doctest_depends_on(modules=('matchpy',))
def MinimumMonomialExponent(u, x):
"""
u is sum whose terms are monomials. MinimumMonomialExponent(u, x) returns the exponent of the term having the smallest exponent
Examples
========
>>> from sympy.integrals.rubi.utility_function import MinimumMonomialExponent
>>> from sympy.abc import x
>>> MinimumMonomialExponent(x**2 + 5*x**2 + 3*x**5, x)
2
>>> MinimumMonomialExponent(x**2 + 5*x**2 + 1, x)
0
"""
n =MonomialExponent(First(u), x)
for i in u.args:
if PosQ(n - MonomialExponent(i, x)):
n = MonomialExponent(i, x)
return n
def MonomialExponent(u, x):
# u is a monomial. MonomialExponent(u, x) returns the exponent of x in u
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
re = u.match(a*x**b)
if re:
return re[b]
def LinearMatchQ(u, x):
# LinearMatchQ(u, x) returns True iff u matches patterns of the form a+b*x where a and b are free of x
if isinstance(u, list):
return all(LinearMatchQ(i, x) for i in u)
else:
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
re = u.match(a + b*x)
if re:
return True
return False
def PowerOfLinearMatchQ(u, x):
if isinstance(u, list):
for i in u:
if not PowerOfLinearMatchQ(i, x):
return False
return True
else:
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x, 0])
m = Wild('m', exclude=[x, 0])
Match = u.match((a + b*x)**m)
if Match:
return True
else:
return False
def QuadraticMatchQ(u, x):
if ListQ(u):
return all(QuadraticMatchQ(i, x) for i in u)
pattern1 = Pattern(UtilityOperator(x_**2*WC('c', 1) + x_*WC('b', 1) + WC('a', 0), x_), CustomConstraint(lambda a, b, c, x: FreeQ([a, b, c], x)))
pattern2 = Pattern(UtilityOperator(x_**2*WC('c', 1) + WC('a', 0), x_), CustomConstraint(lambda a, c, x: FreeQ([a, c], x)))
u1 = UtilityOperator(u, x)
return is_match(u1, pattern1) or is_match(u1, pattern2)
def CubicMatchQ(u, x):
if isinstance(u, list):
return all(CubicMatchQ(i, x) for i in u)
else:
pattern1 = Pattern(UtilityOperator(x_**3*WC('d', 1) + x_**2*WC('c', 1) + x_*WC('b', 1) + WC('a', 0), x_), CustomConstraint(lambda a, b, c, d, x: FreeQ([a, b, c, d], x)))
pattern2 = Pattern(UtilityOperator(x_**3*WC('d', 1) + x_*WC('b', 1) + WC('a', 0), x_), CustomConstraint(lambda a, b, d, x: FreeQ([a, b, d], x)))
pattern3 = Pattern(UtilityOperator(x_**3*WC('d', 1) + x_**2*WC('c', 1) + WC('a', 0), x_), CustomConstraint(lambda a, c, d, x: FreeQ([a, c, d], x)))
pattern4 = Pattern(UtilityOperator(x_**3*WC('d', 1) + WC('a', 0), x_), CustomConstraint(lambda a, d, x: FreeQ([a, d], x)))
u1 = UtilityOperator(u, x)
if is_match(u1, pattern1) or is_match(u1, pattern2) or is_match(u1, pattern3) or is_match(u1, pattern4):
return True
else:
return False
def BinomialMatchQ(u, x):
if isinstance(u, list):
return all(BinomialMatchQ(i, x) for i in u)
else:
pattern = Pattern(UtilityOperator(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)), x_) , CustomConstraint(lambda a, b, n, x: FreeQ([a,b,n],x)))
u = UtilityOperator(u, x)
return is_match(u, pattern)
def TrinomialMatchQ(u, x):
if isinstance(u, list):
return all(TrinomialMatchQ(i, x) for i in u)
else:
pattern = Pattern(UtilityOperator(x_**WC('j', S(1))*WC('c', S(1)) + x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)), x_) , CustomConstraint(lambda a, b, c, n, x: FreeQ([a, b, c, n], x)), CustomConstraint(lambda j, n: ZeroQ(j-2*n) ))
u = UtilityOperator(u, x)
return is_match(u, pattern)
def GeneralizedBinomialMatchQ(u, x):
if isinstance(u, list):
return all(GeneralizedBinomialMatchQ(i, x) for i in u)
else:
a = Wild('a', exclude=[x, 0])
b = Wild('b', exclude=[x, 0])
n = Wild('n', exclude=[x, 0])
q = Wild('q', exclude=[x, 0])
Match = u.match(a*x**q + b*x**n)
if Match and len(Match) == 4 and Match[q] != 0 and Match[n] != 0:
return True
else:
return False
def GeneralizedTrinomialMatchQ(u, x):
if isinstance(u, list):
return all(GeneralizedTrinomialMatchQ(i, x) for i in u)
else:
a = Wild('a', exclude=[x, 0])
b = Wild('b', exclude=[x, 0])
n = Wild('n', exclude=[x, 0])
c = Wild('c', exclude=[x, 0])
q = Wild('q', exclude=[x, 0])
Match = u.match(a*x**q + b*x**n + c*x**(2*n - q))
if Match and len(Match) == 5 and 2*Match[n] - Match[q] != 0 and Match[n] != 0:
return True
else:
return False
def QuotientOfLinearsMatchQ(u, x):
if isinstance(u, list):
return all(QuotientOfLinearsMatchQ(i, x) for i in u)
else:
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
d = Wild('d', exclude=[x])
c = Wild('c', exclude=[x])
e = Wild('e')
Match = u.match(e*(a + b*x)/(c + d*x))
if Match and len(Match) == 5:
return True
else:
return False
def PolynomialTermQ(u, x):
a = Wild('a', exclude=[x])
n = Wild('n', exclude=[x])
Match = u.match(a*x**n)
if Match and IntegerQ(Match[n]) and Greater(Match[n], S(0)):
return True
else:
return False
def PolynomialTerms(u, x):
s = 0
for i in u.args:
if PolynomialTermQ(i, x):
s = s + i
return s
def NonpolynomialTerms(u, x):
s = 0
for i in u.args:
if not PolynomialTermQ(i, x):
s = s + i
return s
def PseudoBinomialParts(u, x):
if PolynomialQ(u, x) and Greater(Expon(u, x), S(2)):
n = Expon(u, x)
d = Rt(Coefficient(u, x, n), n)
c = d**(-n + S(1))*Coefficient(u, x, n + S(-1))/n
a = Simplify(u - (c + d*x)**n)
if NonzeroQ(a) and FreeQ(a, x):
return [a, S(1), c, d, n]
else:
return False
else:
return False
def NormalizePseudoBinomial(u, x):
lst = PseudoBinomialParts(u, x)
if lst:
return (lst[0] + lst[1]*(lst[2] + lst[3]*x)**lst[4])
def PseudoBinomialPairQ(u, v, x):
lst1 = PseudoBinomialParts(u, x)
if AtomQ(lst1):
return False
else:
lst2 = PseudoBinomialParts(v, x)
if AtomQ(lst2):
return False
else:
return Drop(lst1, 2) == Drop(lst2, 2)
def PseudoBinomialQ(u, x):
lst = PseudoBinomialParts(u, x)
if lst:
return True
else:
return False
def PolynomialGCD(f, g):
return gcd(f, g)
def PolyGCD(u, v, x):
# (* u and v are polynomials in x. *)
# (* PolyGCD[u,v,x] returns the factors of the gcd of u and v dependent on x. *)
return NonfreeFactors(PolynomialGCD(u, v), x)
def AlgebraicFunctionFactors(u, x, flag=False):
# (* AlgebraicFunctionFactors[u,x] returns the product of the factors of u that are algebraic functions of x. *)
if ProductQ(u):
result = 1
for i in u.args:
if AlgebraicFunctionQ(i, x, flag):
result *= i
return result
if AlgebraicFunctionQ(u, x, flag):
return u
return 1
def NonalgebraicFunctionFactors(u, x):
"""
NonalgebraicFunctionFactors[u,x] returns the product of the factors of u that are not algebraic functions of x.
Examples
========
>>> from sympy.integrals.rubi.utility_function import NonalgebraicFunctionFactors
>>> from sympy.abc import x
>>> from sympy import sin
>>> NonalgebraicFunctionFactors(sin(x), x)
sin(x)
>>> NonalgebraicFunctionFactors(x, x)
1
"""
if ProductQ(u):
result = 1
for i in u.args:
if not AlgebraicFunctionQ(i, x):
result *= i
return result
if AlgebraicFunctionQ(u, x):
return 1
return u
def QuotientOfLinearsP(u, x):
if LinearQ(u, x):
return True
elif SumQ(u):
if FreeQ(u.args[0], x):
return QuotientOfLinearsP(Rest(u), x)
elif LinearQ(Numerator(u), x) and LinearQ(Denominator(u), x):
return True
elif ProductQ(u):
if FreeQ(First(u), x):
return QuotientOfLinearsP(Rest(u), x)
elif Numerator(u) == 1 and PowerQ(u):
return QuotientOfLinearsP(Denominator(u), x)
return u == x or FreeQ(u, x)
def QuotientOfLinearsParts(u, x):
# If u is equivalent to an expression of the form (a+b*x)/(c+d*x), QuotientOfLinearsParts[u,x]
# returns the list {a, b, c, d}.
if LinearQ(u, x):
return [Coefficient(u, x, 0), Coefficient(u, x, 1), 1, 0]
elif PowerQ(u):
if Numerator(u) == 1:
u = Denominator(u)
r = QuotientOfLinearsParts(u, x)
return [r[2], r[3], r[0], r[1]]
elif SumQ(u):
a = First(u)
if FreeQ(a, x):
u = Rest(u)
r = QuotientOfLinearsParts(u, x)
return [r[0] + a*r[2], r[1] + a*r[3], r[2], r[3]]
elif ProductQ(u):
a = First(u)
if FreeQ(a, x):
r = QuotientOfLinearsParts(Rest(u), x)
return [a*r[0], a*r[1], r[2], r[3]]
a = Numerator(u)
d = Denominator(u)
if LinearQ(a, x) and LinearQ(d, x):
return [Coefficient(a, x, 0), Coefficient(a, x, 1), Coefficient(d, x, 0), Coefficient(d, x, 1)]
elif u == x:
return [0, 1, 1, 0]
elif FreeQ(u, x):
return [u, 0, 1, 0]
return [u, 0, 1, 0]
def QuotientOfLinearsQ(u, x):
# (*QuotientOfLinearsQ[u,x] returns True iff u is equivalent to an expression of the form (a+b x)/(c+d x) where b!=0 and d!=0.*)
if ListQ(u):
for i in u:
if not QuotientOfLinearsQ(i, x):
return False
return True
q = QuotientOfLinearsParts(u, x)
return QuotientOfLinearsP(u, x) and NonzeroQ(q[1]) and NonzeroQ(q[3])
def Flatten(l):
return flatten(l)
def Sort(u, r=False):
return sorted(u, key=lambda x: x.sort_key(), reverse=r)
# (*Definition: A number is absurd if it is a rational number, a positive rational number raised to a fractional power, or a product of absurd numbers.*)
def AbsurdNumberQ(u):
# (* AbsurdNumberQ[u] returns True if u is an absurd number, else it returns False. *)
if PowerQ(u):
v = u.exp
u = u.base
return RationalQ(u) and u > 0 and FractionQ(v)
elif ProductQ(u):
return all(AbsurdNumberQ(i) for i in u.args)
return RationalQ(u)
def AbsurdNumberFactors(u):
# (* AbsurdNumberFactors[u] returns the product of the factors of u that are absurd numbers. *)
if AbsurdNumberQ(u):
return u
elif ProductQ(u):
result = S(1)
for i in u.args:
if AbsurdNumberQ(i):
result *= i
return result
return NumericFactor(u)
def NonabsurdNumberFactors(u):
# (* NonabsurdNumberFactors[u] returns the product of the factors of u that are not absurd numbers. *)
if AbsurdNumberQ(u):
return S(1)
elif ProductQ(u):
result = 1
for i in u.args:
result *= NonabsurdNumberFactors(i)
return result
return NonnumericFactors(u)
def SumSimplerAuxQ(u, v):
if SumQ(v):
return (RationalQ(First(v)) or SumSimplerAuxQ(u,First(v))) and (RationalQ(Rest(v)) or SumSimplerAuxQ(u,Rest(v)))
elif SumQ(u):
return SumSimplerAuxQ(First(u), v) or SumSimplerAuxQ(Rest(u), v)
else:
return v!=0 and NonnumericFactors(u)==NonnumericFactors(v) and (NumericFactor(u)/NumericFactor(v)<-1/2 or NumericFactor(u)/NumericFactor(v)==-1/2 and NumericFactor(u)<0)
def Prepend(l1, l2):
if not isinstance(l2, list):
return [l2] + l1
return l2 + l1
def Drop(lst, n):
if isinstance(lst, list):
if isinstance(n, list):
lst = lst[:(n[0]-1)] + lst[n[1]:]
elif n > 0:
lst = lst[n:]
elif n < 0:
lst = lst[:-n]
else:
return lst
return lst
return lst.func(*[i for i in Drop(list(lst.args), n)])
def CombineExponents(lst):
if Length(lst) < 2:
return lst
elif lst[0][0] == lst[1][0]:
return CombineExponents(Prepend(Drop(lst,2),[lst[0][0], lst[0][1] + lst[1][1]]))
return Prepend(CombineExponents(Rest(lst)), First(lst))
def FactorInteger(n, l=None):
if isinstance(n, (int, Integer)):
return sorted(factorint(n, limit=l).items())
else:
return sorted(factorrat(n, limit=l).items())
def FactorAbsurdNumber(m):
# (* m must be an absurd number. FactorAbsurdNumber[m] returns the prime factorization of m *)
# (* as list of base-degree pairs where the bases are prime numbers and the degrees are rational. *)
if RationalQ(m):
return FactorInteger(m)
elif PowerQ(m):
r = FactorInteger(m.base)
return [r[0], r[1]*m.exp]
# CombineExponents[Sort[Flatten[Map[FactorAbsurdNumber,Apply[List,m]],1], Function[i1[[1]]<i2[[1]]]]]
return list((m.as_base_exp(),))
def SubstForInverseFunction(*args):
"""
SubstForInverseFunction(u, v, w, x) returns u with subexpressions equal to v replaced by x and x replaced by w.
Examples
========
>>> from sympy.integrals.rubi.utility_function import SubstForInverseFunction
>>> from sympy.abc import x, a, b
>>> SubstForInverseFunction(a, a, b, x)
a
>>> SubstForInverseFunction(x**a, x**a, b, x)
x
>>> SubstForInverseFunction(a*x**a, a, b, x)
a*b**a
"""
if len(args) == 3:
u, v, x = args[0], args[1], args[2]
return SubstForInverseFunction(u, v, (-Coefficient(v.args[0], x, 0) + InverseFunction(Head(v))(x))/Coefficient(v.args[0], x, 1), x)
elif len(args) == 4:
u, v, w, x = args[0], args[1], args[2], args[3]
if AtomQ(u):
if u == x:
return w
return u
elif Head(u) == Head(v) and ZeroQ(u.args[0] - v.args[0]):
return x
res = [SubstForInverseFunction(i, v, w, x) for i in u.args]
return u.func(*res)
def SubstForFractionalPower(u, v, n, w, x):
# (* SubstForFractionalPower[u,v,n,w,x] returns u with subexpressions equal to v^(m/n) replaced
# by x^m and x replaced by w. *)
if AtomQ(u):
if u == x:
return w
return u
elif FractionalPowerQ(u):
if ZeroQ(u.base - v):
return x**(n*u.exp)
res = [SubstForFractionalPower(i, v, n, w, x) for i in u.args]
return u.func(*res)
def SubstForFractionalPowerOfQuotientOfLinears(u, x):
# (* If u has a subexpression of the form ((a+b*x)/(c+d*x))^(m/n) where m and n>1 are integers,
# SubstForFractionalPowerOfQuotientOfLinears[u,x] returns the list {v,n,(a+b*x)/(c+d*x),b*c-a*d} where v is u
# with subexpressions of the form ((a+b*x)/(c+d*x))^(m/n) replaced by x^m and x replaced
lst = FractionalPowerOfQuotientOfLinears(u, 1, False, x)
if AtomQ(lst) or AtomQ(lst[1]):
return False
n = lst[0]
tmp = lst[1]
lst = QuotientOfLinearsParts(tmp, x)
a, b, c, d = lst[0], lst[1], lst[2], lst[3]
if ZeroQ(d):
return False
lst = Simplify(x**(n - 1)*SubstForFractionalPower(u, tmp, n, (-a + c*x**n)/(b - d*x**n), x)/(b - d*x**n)**2)
return [NonfreeFactors(lst, x), n, tmp, FreeFactors(lst, x)*(b*c - a*d)]
def FractionalPowerOfQuotientOfLinears(u, n, v, x):
# (* If u has a subexpression of the form ((a+b*x)/(c+d*x))^(m/n),
# FractionalPowerOfQuotientOfLinears[u,1,False,x] returns {n,(a+b*x)/(c+d*x)}; else it returns False. *)
if AtomQ(u) or FreeQ(u, x):
return [n, v]
elif CalculusQ(u):
return False
elif FractionalPowerQ(u):
if QuotientOfLinearsQ(u.base, x) and Not(LinearQ(u.base, x)) and (FalseQ(v) or ZeroQ(u.base - v)):
return [LCM(Denominator(u.exp), n), u.base]
lst = [n, v]
for i in u.args:
lst = FractionalPowerOfQuotientOfLinears(i, lst[0], lst[1],x)
if AtomQ(lst):
return False
return lst
def SubstForFractionalPowerQ(u, v, x):
# (* If the substitution x=v^(1/n) will not complicate algebraic subexpressions of u,
# SubstForFractionalPowerQ[u,v,x] returns True; else it returns False. *)
if AtomQ(u) or FreeQ(u, x):
return True
elif FractionalPowerQ(u):
return SubstForFractionalPowerAuxQ(u, v, x)
return all(SubstForFractionalPowerQ(i, v, x) for i in u.args)
def SubstForFractionalPowerAuxQ(u, v, x):
if AtomQ(u):
return False
elif FractionalPowerQ(u):
if ZeroQ(u.base - v):
return True
return any(SubstForFractionalPowerAuxQ(i, v, x) for i in u.args)
def FractionalPowerOfSquareQ(u):
# (* If a subexpression of u is of the form ((v+w)^2)^n where n is a fraction, *)
# (* FractionalPowerOfSquareQ[u] returns (v+w)^2; else it returns False. *)
if AtomQ(u):
return False
elif FractionalPowerQ(u):
a_ = Wild('a', exclude=[0])
b_ = Wild('b', exclude=[0])
c_ = Wild('c', exclude=[0])
match = u.base.match(a_*(b_ + c_)**(S(2)))
if match:
keys = [a_, b_, c_]
if len(keys) == len(match):
a, b, c = tuple(match[i] for i in keys)
if NonsumQ(a):
return (b + c)**S(2)
for i in u.args:
tmp = FractionalPowerOfSquareQ(i)
if Not(FalseQ(tmp)):
return tmp
return False
def FractionalPowerSubexpressionQ(u, v, w):
# (* If a subexpression of u is of the form w^n where n is a fraction but not equal to v, *)
# (* FractionalPowerSubexpressionQ[u,v,w] returns True; else it returns False. *)
if AtomQ(u):
return False
elif FractionalPowerQ(u):
if PositiveQ(u.base/w):
return Not(u.base == v) and LeafCount(w) < 3*LeafCount(v)
for i in u.args:
if FractionalPowerSubexpressionQ(i, v, w):
return True
return False
def Apply(f, lst):
return f(*lst)
def FactorNumericGcd(u):
# (* FactorNumericGcd[u] returns u with the gcd of the numeric coefficients of terms of sums factored out. *)
if PowerQ(u):
if RationalQ(u.exp):
return FactorNumericGcd(u.base)**u.exp
elif ProductQ(u):
res = [FactorNumericGcd(i) for i in u.args]
return Mul(*res)
elif SumQ(u):
g = GCD([NumericFactor(i) for i in u.args])
r = Add(*[i/g for i in u.args])
return g*r
return u
def MergeableFactorQ(bas, deg, v):
# (* MergeableFactorQ[bas,deg,v] returns True iff bas equals the base of a factor of v or bas is a factor of every term of v. *)
if bas == v:
return RationalQ(deg + S(1)) and (deg + 1>=0 or RationalQ(deg) and deg>0)
elif PowerQ(v):
if bas == v.base:
return RationalQ(deg+v.exp) and (deg+v.exp>=0 or RationalQ(deg) and deg>0)
return SumQ(v.base) and IntegerQ(v.exp) and (Not(IntegerQ(deg) or IntegerQ(deg/v.exp))) and MergeableFactorQ(bas, deg/v.exp, v.base)
elif ProductQ(v):
return MergeableFactorQ(bas, deg, First(v)) or MergeableFactorQ(bas, deg, Rest(v))
return SumQ(v) and MergeableFactorQ(bas, deg, First(v)) and MergeableFactorQ(bas, deg, Rest(v))
def MergeFactor(bas, deg, v):
# (* If MergeableFactorQ[bas,deg,v], MergeFactor[bas,deg,v] return the product of bas^deg and v,
# but with bas^deg merged into the factor of v whose base equals bas. *)
if bas == v:
return bas**(deg + 1)
elif PowerQ(v):
if bas == v.base:
return bas**(deg + v.exp)
return MergeFactor(bas, deg/v.exp, v.base**v.exp)
elif ProductQ(v):
if MergeableFactorQ(bas, deg, First(v)):
return MergeFactor(bas, deg, First(v))*Rest(v)
return First(v)*MergeFactor(bas, deg, Rest(v))
return MergeFactor(bas, deg, First(v)) + MergeFactor(bas, deg, Rest(v))
def MergeFactors(u, v):
# (* MergeFactors[u,v] returns the product of u and v, but with the mergeable factors of u merged into v. *)
if ProductQ(u):
return MergeFactors(Rest(u), MergeFactors(First(u), v))
elif PowerQ(u):
if MergeableFactorQ(u.base, u.exp, v):
return MergeFactor(u.base, u.exp, v)
elif RationalQ(u.exp) and u.exp < -1 and MergeableFactorQ(u.base, -S(1), v):
return MergeFactors(u.base**(u.exp + 1), MergeFactor(u.base, -S(1), v))
return u*v
elif MergeableFactorQ(u, S(1), v):
return MergeFactor(u, S(1), v)
return u*v
def TrigSimplifyQ(u):
# (* TrigSimplifyQ[u] returns True if TrigSimplify[u] actually simplifies u; else False. *)
return ActivateTrig(u) != TrigSimplify(u)
def TrigSimplify(u):
# (* TrigSimplify[u] returns a bottom-up trig simplification of u. *)
return ActivateTrig(TrigSimplifyRecur(u))
def TrigSimplifyRecur(u):
if AtomQ(u):
return u
return TrigSimplifyAux(u.func(*[TrigSimplifyRecur(i) for i in u.args]))
def Order(expr1, expr2):
if expr1 == expr2:
return 0
elif expr1.sort_key() > expr2.sort_key():
return -1
return 1
def FactorOrder(u, v):
if u == 1:
if v == 1:
return 0
return -1
elif v == 1:
return 1
return Order(u, v)
def Smallest(num1, num2=None):
if num2 is None:
lst = num1
num = lst[0]
for i in Rest(lst):
num = Smallest(num, i)
return num
return Min(num1, num2)
def OrderedQ(l):
return l == Sort(l)
def MinimumDegree(deg1, deg2):
if RationalQ(deg1):
if RationalQ(deg2):
return Min(deg1, deg2)
return deg1
elif RationalQ(deg2):
return deg2
deg = Simplify(deg1- deg2)
if RationalQ(deg):
if deg > 0:
return deg2
return deg1
elif OrderedQ([deg1, deg2]):
return deg1
return deg2
def PositiveFactors(u):
# (* PositiveFactors[u] returns the positive factors of u *)
if ZeroQ(u):
return S(1)
elif RationalQ(u):
return Abs(u)
elif PositiveQ(u):
return u
elif ProductQ(u):
res = 1
for i in u.args:
res *= PositiveFactors(i)
return res
return 1
def Sign(u):
return sign(u)
def NonpositiveFactors(u):
# (* NonpositiveFactors[u] returns the nonpositive factors of u *)
if ZeroQ(u):
return u
elif RationalQ(u):
return Sign(u)
elif PositiveQ(u):
return S(1)
elif ProductQ(u):
res = S(1)
for i in u.args:
res *= NonpositiveFactors(i)
return res
return u
def PolynomialInAuxQ(u, v, x):
if u == v:
return True
elif AtomQ(u):
return u != x
elif PowerQ(u):
if PowerQ(v):
if u.base == v.base:
return PositiveIntegerQ(u.exp/v.exp)
return PositiveIntegerQ(u.exp) and PolynomialInAuxQ(u.base, v, x)
elif SumQ(u) or ProductQ(u):
for i in u.args:
if Not(PolynomialInAuxQ(i, v, x)):
return False
return True
return False
def PolynomialInQ(u, v, x):
"""
If u is a polynomial in v(x), PolynomialInQ(u, v, x) returns True, else it returns False.
Examples
========
>>> from sympy.integrals.rubi.utility_function import PolynomialInQ
>>> from sympy.abc import x
>>> from sympy import log, S
>>> PolynomialInQ(S(1), log(x), x)
True
>>> PolynomialInQ(log(x), log(x), x)
True
>>> PolynomialInQ(1 + log(x)**2, log(x), x)
True
"""
return PolynomialInAuxQ(u, NonfreeFactors(NonfreeTerms(v, x), x), x)
def ExponentInAux(u, v, x):
if u == v:
return S(1)
elif AtomQ(u):
return S(0)
elif PowerQ(u):
if PowerQ(v):
if u.base == v.base:
return u.exp/v.exp
return u.exp*ExponentInAux(u.base, v, x)
elif ProductQ(u):
return Add(*[ExponentInAux(i, v, x) for i in u.args])
return Max(*[ExponentInAux(i, v, x) for i in u.args])
def ExponentIn(u, v, x):
return ExponentInAux(u, NonfreeFactors(NonfreeTerms(v, x), x), x)
def PolynomialInSubstAux(u, v, x):
if u == v:
return x
elif AtomQ(u):
return u
elif PowerQ(u):
if PowerQ(v):
if u.base == v.base:
return x**(u.exp/v.exp)
return PolynomialInSubstAux(u.base, v, x)**u.exp
return u.func(*[PolynomialInSubstAux(i, v, x) for i in u.args])
def PolynomialInSubst(u, v, x):
# If u is a polynomial in v[x], PolynomialInSubst[u,v,x] returns the polynomial u in x.
w = NonfreeTerms(v, x)
return ReplaceAll(PolynomialInSubstAux(u, NonfreeFactors(w, x), x), {x: x - FreeTerms(v, x)/FreeFactors(w, x)})
def Distrib(u, v):
# Distrib[u,v] returns the sum of u times each term of v.
if SumQ(v):
return Add(*[u*i for i in v.args])
return u*v
def DistributeDegree(u, m):
# DistributeDegree[u,m] returns the product of the factors of u each raised to the mth degree.
if AtomQ(u):
return u**m
elif PowerQ(u):
return u.base**(u.exp*m)
elif ProductQ(u):
return Mul(*[DistributeDegree(i, m) for i in u.args])
return u**m
def FunctionOfPower(*args):
"""
FunctionOfPower[u,x] returns the gcd of the integer degrees of x in u.
Examples
========
>>> from sympy.integrals.rubi.utility_function import FunctionOfPower
>>> from sympy.abc import x
>>> FunctionOfPower(x, x)
1
>>> FunctionOfPower(x**3, x)
3
"""
if len(args) == 2:
return FunctionOfPower(args[0], None, args[1])
u, n, x = args
if FreeQ(u, x):
return n
elif u == x:
return S(1)
elif PowerQ(u):
if u.base == x and IntegerQ(u.exp):
if n is None:
return u.exp
return GCD(n, u.exp)
tmp = n
for i in u.args:
tmp = FunctionOfPower(i, tmp, x)
return tmp
def DivideDegreesOfFactors(u, n):
"""
DivideDegreesOfFactors[u,n] returns the product of the base of the factors of u raised to the degree of the factors divided by n.
Examples
========
>>> from sympy import S
>>> from sympy.integrals.rubi.utility_function import DivideDegreesOfFactors
>>> from sympy.abc import a, b
>>> DivideDegreesOfFactors(a**b, S(3))
a**(b/3)
"""
if ProductQ(u):
return Mul(*[LeadBase(i)**(LeadDegree(i)/n) for i in u.args])
return LeadBase(u)**(LeadDegree(u)/n)
def MonomialFactor(u, x):
# MonomialFactor[u,x] returns the list {n,v} where x^n*v==u and n is free of x.
if AtomQ(u):
if u == x:
return [S(1), S(1)]
return [S(0), u]
elif PowerQ(u):
if IntegerQ(u.exp):
lst = MonomialFactor(u.base, x)
return [lst[0]*u.exp, lst[1]**u.exp]
elif u.base == x and FreeQ(u.exp, x):
return [u.exp, S(1)]
return [S(0), u]
elif ProductQ(u):
lst1 = MonomialFactor(First(u), x)
lst2 = MonomialFactor(Rest(u), x)
return [lst1[0] + lst2[0], lst1[1]*lst2[1]]
elif SumQ(u):
lst = [MonomialFactor(i, x) for i in u.args]
deg = lst[0][0]
for i in Rest(lst):
deg = MinimumDegree(deg, i[0])
if ZeroQ(deg) or RationalQ(deg) and deg < 0:
return [S(0), u]
return [deg, Add(*[x**(i[0] - deg)*i[1] for i in lst])]
return [S(0), u]
def FullSimplify(expr):
return Simplify(expr)
def FunctionOfLinearSubst(u, a, b, x):
if FreeQ(u, x):
return u
elif LinearQ(u, x):
tmp = Coefficient(u, x, 1)
if tmp == b:
tmp = S(1)
else:
tmp = tmp/b
return Coefficient(u, x, S(0)) - a*tmp + tmp*x
elif PowerQ(u):
if FreeQ(u.base, x):
return E**(FullSimplify(FunctionOfLinearSubst(Log(u.base)*u.exp, a, b, x)))
lst = MonomialFactor(u, x)
if ProductQ(u) and NonzeroQ(lst[0]):
if RationalQ(LeadFactor(lst[1])) and LeadFactor(lst[1]) < 0:
return -FunctionOfLinearSubst(DivideDegreesOfFactors(-lst[1], lst[0])*x, a, b, x)**lst[0]
return FunctionOfLinearSubst(DivideDegreesOfFactors(lst[1], lst[0])*x, a, b, x)**lst[0]
return u.func(*[FunctionOfLinearSubst(i, a, b, x) for i in u.args])
def FunctionOfLinear(*args):
# (* If u (x) is equivalent to an expression of the form f (a+b*x) and not the case that a==0 and
# b==1, FunctionOfLinear[u,x] returns the list {f (x),a,b}; else it returns False. *)
if len(args) == 2:
u, x = args
lst = FunctionOfLinear(u, False, False, x, False)
if AtomQ(lst) or FalseQ(lst[0]) or (lst[0] == 0 and lst[1] == 1):
return False
return [FunctionOfLinearSubst(u, lst[0], lst[1], x), lst[0], lst[1]]
u, a, b, x, flag = args
if FreeQ(u, x):
return [a, b]
elif CalculusQ(u):
return False
elif LinearQ(u, x):
if FalseQ(a):
return [Coefficient(u, x, 0), Coefficient(u, x, 1)]
lst = CommonFactors([b, Coefficient(u, x, 1)])
if ZeroQ(Coefficient(u, x, 0)) and Not(flag):
return [0, lst[0]]
elif ZeroQ(b*Coefficient(u, x, 0) - a*Coefficient(u, x, 1)):
return [a/lst[1], lst[0]]
return [0, 1]
elif PowerQ(u):
if FreeQ(u.base, x):
return FunctionOfLinear(Log(u.base)*u.exp, a, b, x, False)
lst = MonomialFactor(u, x)
if ProductQ(u) and NonzeroQ(lst[0]):
if False and IntegerQ(lst[0]) and lst[0] != -1 and FreeQ(lst[1], x):
if RationalQ(LeadFactor(lst[1])) and LeadFactor(lst[1]) < 0:
return FunctionOfLinear(DivideDegreesOfFactors(-lst[1], lst[0])*x, a, b, x, False)
return FunctionOfLinear(DivideDegreesOfFactors(lst[1], lst[0])*x, a, b, x, False)
return False
lst = [a, b]
for i in u.args:
lst = FunctionOfLinear(i, lst[0], lst[1], x, SumQ(u))
if AtomQ(lst):
return False
return lst
def NormalizeIntegrand(u, x):
v = NormalizeLeadTermSigns(NormalizeIntegrandAux(u, x))
if v == NormalizeLeadTermSigns(u):
return u
else:
return v
def NormalizeIntegrandAux(u, x):
if SumQ(u):
l = 0
for i in u.args:
l += NormalizeIntegrandAux(i, x)
return l
if ProductQ(MergeMonomials(u, x)):
l = 1
for i in MergeMonomials(u, x).args:
l *= NormalizeIntegrandFactor(i, x)
return l
else:
return NormalizeIntegrandFactor(MergeMonomials(u, x), x)
def NormalizeIntegrandFactor(u, x):
if PowerQ(u):
if FreeQ(u.exp, x):
bas = NormalizeIntegrandFactorBase(u.base, x)
deg = u.exp
if IntegerQ(deg) and SumQ(bas):
if all(MonomialQ(i, x) for i in bas.args):
mi = MinimumMonomialExponent(bas, x)
q = 0
for i in bas.args:
q += Simplify(i/x**mi)
return x**(mi*deg)*q**deg
else:
return bas**deg
else:
return bas**deg
if PowerQ(u):
if FreeQ(u.base, x):
return u.base**NormalizeIntegrandFactorBase(u.exp, x)
bas = NormalizeIntegrandFactorBase(u, x)
if SumQ(bas):
if all(MonomialQ(i, x) for i in bas.args):
mi = MinimumMonomialExponent(bas, x)
z = 0
for j in bas.args:
z += j/x**mi
return x**mi*z
else:
return bas
else:
return bas
def NormalizeIntegrandFactorBase(expr, x):
m = Wild('m', exclude=[x])
u = Wild('u')
match = expr.match(x**m*u)
if match and SumQ(u):
l = 0
for i in u.args:
l += NormalizeIntegrandFactorBase((x**m*i), x)
return l
if BinomialQ(expr, x):
if BinomialMatchQ(expr, x):
return expr
else:
return ExpandToSum(expr, x)
elif TrinomialQ(expr, x):
if TrinomialMatchQ(expr, x):
return expr
else:
return ExpandToSum(expr, x)
elif ProductQ(expr):
l = 1
for i in expr.args:
l *= NormalizeIntegrandFactor(i, x)
return l
elif PolynomialQ(expr, x) and Exponent(expr, x)<=4:
return ExpandToSum(expr, x)
elif SumQ(expr):
w = Wild('w')
m = Wild('m', exclude=[x])
v = TogetherSimplify(expr)
if SumQ(v) or v.match(x**m*w) and SumQ(w) or LeafCount(v)>LeafCount(expr)+2:
return UnifySum(expr, x)
else:
return NormalizeIntegrandFactorBase(v, x)
else:
return expr
def NormalizeTogether(u):
return NormalizeLeadTermSigns(Together(u))
def NormalizeLeadTermSigns(u):
if ProductQ(u):
t = 1
for i in u.args:
lst = SignOfFactor(i)
if lst[0] == 1:
t *= lst[1]
else:
t *= AbsorbMinusSign(lst[1])
return t
else:
lst = SignOfFactor(u)
if lst[0] == 1:
return lst[1]
else:
return AbsorbMinusSign(lst[1])
def AbsorbMinusSign(expr, *x):
m = Wild('m', exclude=[x])
u = Wild('u')
v = Wild('v')
match = expr.match(u*v**m)
if match:
if len(match) == 3:
if SumQ(match[v]) and OddQ(match[m]):
return match[u]*(-match[v])**match[m]
return -expr
def NormalizeSumFactors(u):
if AtomQ(u):
return u
elif ProductQ(u):
k = 1
for i in u.args:
k *= NormalizeSumFactors(i)
return SignOfFactor(k)[0]*SignOfFactor(k)[1]
elif SumQ(u):
k = 0
for i in u.args:
k += NormalizeSumFactors(i)
return k
else:
return u
def SignOfFactor(u):
if RationalQ(u) and u < 0 or SumQ(u) and NumericFactor(First(u)) < 0:
return [-1, -u]
elif IntegerPowerQ(u):
if SumQ(u.base) and NumericFactor(First(u.base)) < 0:
return [(-1)**u.exp, (-u.base)**u.exp]
elif ProductQ(u):
k = 1
h = 1
for i in u.args:
k *= SignOfFactor(i)[0]
h *= SignOfFactor(i)[1]
return [k, h]
return [1, u]
def NormalizePowerOfLinear(u, x):
v = FactorSquareFree(u)
if PowerQ(v):
if LinearQ(v.base, x) and FreeQ(v.exp, x):
return ExpandToSum(v.base, x)**v.exp
return ExpandToSum(v, x)
def SimplifyIntegrand(u, x):
v = NormalizeLeadTermSigns(NormalizeIntegrandAux(Simplify(u), x))
if LeafCount(v) < 4/5*LeafCount(u):
return v
if v != NormalizeLeadTermSigns(u):
return v
else:
return u
def SimplifyTerm(u, x):
v = Simplify(u)
w = Together(v)
if LeafCount(v) < LeafCount(w):
return NormalizeIntegrand(v, x)
else:
return NormalizeIntegrand(w, x)
def TogetherSimplify(u):
v = Together(Simplify(Together(u)))
return FixSimplify(v)
def SmartSimplify(u):
v = Simplify(u)
w = factor(v)
if LeafCount(w) < LeafCount(v):
v = w
if Not(FalseQ(w == FractionalPowerOfSquareQ(v))) and FractionalPowerSubexpressionQ(u, w, Expand(w)):
v = SubstForExpn(v, w, Expand(w))
else:
v = FactorNumericGcd(v)
return FixSimplify(v)
def SubstForExpn(u, v, w):
if u == v:
return w
if AtomQ(u):
return u
else:
k = 0
for i in u.args:
k += SubstForExpn(i, v, w)
return k
def ExpandToSum(u, *x):
if len(x) == 1:
x = x[0]
expr = 0
if PolyQ(S(u), x):
for t in Exponent(u, x, List):
expr += Coeff(u, x, t)*x**t
return expr
if BinomialQ(u, x):
i = BinomialParts(u, x)
expr += i[0] + i[1]*x**i[2]
return expr
if TrinomialQ(u, x):
i = TrinomialParts(u, x)
expr += i[0] + i[1]*x**i[3] + i[2]*x**(2*i[3])
return expr
if GeneralizedBinomialMatchQ(u, x):
i = GeneralizedBinomialParts(u, x)
expr += i[0]*x**i[3] + i[1]*x**i[2]
return expr
if GeneralizedTrinomialMatchQ(u, x):
i = GeneralizedTrinomialParts(u, x)
expr += i[0]*x**i[4] + i[1]*x**i[3] + i[2]*x**(2*i[3]-i[4])
return expr
else:
return Expand(u)
else:
v = x[0]
x = x[1]
w = ExpandToSum(v, x)
r = NonfreeTerms(w, x)
if SumQ(r):
k = u*FreeTerms(w, x)
for i in r.args:
k += MergeMonomials(u*i, x)
return k
else:
return u*FreeTerms(w, x) + MergeMonomials(u*r, x)
def UnifySum(u, x):
if SumQ(u):
t = 0
lst = []
for i in u.args:
lst += [i]
for j in UnifyTerms(lst, x):
t += j
return t
else:
return SimplifyTerm(u, x)
def UnifyTerms(lst, x):
if lst==[]:
return lst
else:
return UnifyTerm(First(lst), UnifyTerms(Rest(lst), x), x)
def UnifyTerm(term, lst, x):
if lst==[]:
return [term]
tmp = Simplify(First(lst)/term)
if FreeQ(tmp, x):
return Prepend(Rest(lst), [(1+tmp)*term])
else:
return Prepend(UnifyTerm(term, Rest(lst), x), [First(lst)])
def CalculusQ(u):
return False
def FunctionOfInverseLinear(*args):
# (* If u is a function of an inverse linear binomial of the form 1/(a+b*x),
# FunctionOfInverseLinear[u,x] returns the list {a,b}; else it returns False. *)
if len(args) == 2:
u, x = args
return FunctionOfInverseLinear(u, None, x)
u, lst, x = args
if FreeQ(u, x):
return lst
elif u == x:
return False
elif QuotientOfLinearsQ(u, x):
tmp = Drop(QuotientOfLinearsParts(u, x), 2)
if tmp[1] == 0:
return False
elif lst is None:
return tmp
elif ZeroQ(lst[0]*tmp[1] - lst[1]*tmp[0]):
return lst
return False
elif CalculusQ(u):
return False
tmp = lst
for i in u.args:
tmp = FunctionOfInverseLinear(i, tmp, x)
if AtomQ(tmp):
return False
return tmp
def PureFunctionOfSinhQ(u, v, x):
# (* If u is a pure function of Sinh[v] and/or Csch[v], PureFunctionOfSinhQ[u,v,x] returns True;
# else it returns False. *)
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif HyperbolicQ(u) and ZeroQ(u.args[0] - v):
return SinhQ(u) or CschQ(u)
for i in u.args:
if Not(PureFunctionOfSinhQ(i, v, x)):
return False
return True
def PureFunctionOfTanhQ(u, v , x):
# (* If u is a pure function of Tanh[v] and/or Coth[v], PureFunctionOfTanhQ[u,v,x] returns True;
# else it returns False. *)
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif HyperbolicQ(u) and ZeroQ(u.args[0] - v):
return TanhQ(u) or CothQ(u)
for i in u.args:
if Not(PureFunctionOfTanhQ(i, v, x)):
return False
return True
def PureFunctionOfCoshQ(u, v, x):
# (* If u is a pure function of Cosh[v] and/or Sech[v], PureFunctionOfCoshQ[u,v,x] returns True;
# else it returns False. *)
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif HyperbolicQ(u) and ZeroQ(u.args[0] - v):
return CoshQ(u) or SechQ(u)
for i in u.args:
if Not(PureFunctionOfCoshQ(i, v, x)):
return False
return True
def IntegerQuotientQ(u, v):
# (* If u/v is an integer, IntegerQuotientQ[u,v] returns True; else it returns False. *)
return IntegerQ(Simplify(u/v))
def OddQuotientQ(u, v):
# (* If u/v is odd, OddQuotientQ[u,v] returns True; else it returns False. *)
return OddQ(Simplify(u/v))
def EvenQuotientQ(u, v):
# (* If u/v is even, EvenQuotientQ[u,v] returns True; else it returns False. *)
return EvenQ(Simplify(u/v))
def FindTrigFactor(func1, func2, u, v, flag):
# (* If func[w]^m is a factor of u where m is odd and w is an integer multiple of v,
# FindTrigFactor[func1,func2,u,v,True] returns the list {w,u/func[w]^n}; else it returns False. *)
# (* If func[w]^m is a factor of u where m is odd and w is an integer multiple of v not equal to v,
# FindTrigFactor[func1,func2,u,v,False] returns the list {w,u/func[w]^n}; else it returns False. *)
if u == 1:
return False
elif (Head(LeadBase(u)) == func1 or Head(LeadBase(u)) == func2) and OddQ(LeadDegree(u)) and IntegerQuotientQ(LeadBase(u).args[0], v) and (flag or NonzeroQ(LeadBase(u).args[0] - v)):
return [LeadBase[u].args[0], RemainingFactors(u)]
lst = FindTrigFactor(func1, func2, RemainingFactors(u), v, flag)
if AtomQ(lst):
return False
return [lst[0], LeadFactor(u)*lst[1]]
def FunctionOfSinhQ(u, v, x):
# (* If u is a function of Sinh[v], FunctionOfSinhQ[u,v,x] returns True; else it returns False. *)
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif HyperbolicQ(u) and IntegerQuotientQ(u.args[0], v):
if OddQuotientQ(u.args[0], v):
# (* Basis: If m odd, Sinh[m*v]^n is a function of Sinh[v]. *)
return SinhQ(u) or CschQ(u)
# (* Basis: If m even, Cos[m*v]^n is a function of Sinh[v]. *)
return CoshQ(u) or SechQ(u)
elif IntegerPowerQ(u):
if HyperbolicQ(u.base) and IntegerQuotientQ(u.base.args[0], v):
if EvenQ(u.exp):
# (* Basis: If m integer and n even, Hyper[m*v]^n is a function of Sinh[v]. *)
return True
return FunctionOfSinhQ(u.base, v, x)
elif ProductQ(u):
if CoshQ(u.args[0]) and SinhQ(u.args[1]) and ZeroQ(u.args[0].args[0] - v/2) and ZeroQ(u.args[1].args[0] - v/2):
return FunctionOfSinhQ(Drop(u, 2), v, x)
lst = FindTrigFactor(Sinh, Csch, u, v, False)
if ListQ(lst) and EvenQuotientQ(lst[0], v):
# (* Basis: If m even and n odd, Sinh[m*v]^n == Cosh[v]*u where u is a function of Sinh[v]. *)
return FunctionOfSinhQ(Cosh(v)*lst[1], v, x)
lst = FindTrigFactor(Cosh, Sech, u, v, False)
if ListQ(lst) and OddQuotientQ(lst[0], v):
# (* Basis: If m odd and n odd, Cosh[m*v]^n == Cosh[v]*u where u is a function of Sinh[v]. *)
return FunctionOfSinhQ(Cosh(v)*lst[1], v, x)
lst = FindTrigFactor(Tanh, Coth, u, v, True)
if ListQ(lst):
# (* Basis: If m integer and n odd, Tanh[m*v]^n == Cosh[v]*u where u is a function of Sinh[v]. *)
return FunctionOfSinhQ(Cosh(v)*lst[1], v, x)
return all(FunctionOfSinhQ(i, v, x) for i in u.args)
return all(FunctionOfSinhQ(i, v, x) for i in u.args)
def FunctionOfCoshQ(u, v, x):
#(* If u is a function of Cosh[v], FunctionOfCoshQ[u,v,x] returns True; else it returns False. *)
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif HyperbolicQ(u) and IntegerQuotientQ(u.args[0], v):
# (* Basis: If m integer, Cosh[m*v]^n is a function of Cosh[v]. *)
return CoshQ(u) or SechQ(u)
elif IntegerPowerQ(u):
if HyperbolicQ(u.base) and IntegerQuotientQ(u.base.args[0], v):
if EvenQ(u.exp):
# (* Basis: If m integer and n even, Hyper[m*v]^n is a function of Cosh[v]. *)
return True
return FunctionOfCoshQ(u.base, v, x)
elif ProductQ(u):
lst = FindTrigFactor(Sinh, Csch, u, v, False)
if ListQ(lst):
# (* Basis: If m integer and n odd, Sinh[m*v]^n == Sinh[v]*u where u is a function of Cosh[v]. *)
return FunctionOfCoshQ(Sinh(v)*lst[1], v, x)
lst = FindTrigFactor(Tanh, Coth, u, v, True)
if ListQ(lst):
# (* Basis: If m integer and n odd, Tanh[m*v]^n == Sinh[v]*u where u is a function of Cosh[v]. *)
return FunctionOfCoshQ(Sinh(v)*lst[1], v, x)
return all(FunctionOfCoshQ(i, v, x) for i in u.args)
return all(FunctionOfCoshQ(i, v, x) for i in u.args)
def OddHyperbolicPowerQ(u, v, x):
if SinhQ(u) or CoshQ(u) or SechQ(u) or CschQ(u):
return OddQuotientQ(u.args[0], v)
if PowerQ(u):
return OddQ(u.exp) and OddHyperbolicPowerQ(u.base, v, x)
if ProductQ(u):
if Not(EqQ(FreeFactors(u, x), 1)):
return OddHyperbolicPowerQ(NonfreeFactors(u, x), v, x)
lst = []
for i in u.args:
if Not(FunctionOfTanhQ(i, v, x)):
lst.append(i)
if lst == []:
return True
return Length(lst)==1 and OddHyperbolicPowerQ(lst[0], v, x)
if SumQ(u):
return all(OddHyperbolicPowerQ(i, v, x) for i in u.args)
return False
def FunctionOfTanhQ(u, v, x):
#(* If u is a function of the form f[Tanh[v],Coth[v]] where f is independent of x,
# FunctionOfTanhQ[u,v,x] returns True; else it returns False. *)
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif HyperbolicQ(u) and IntegerQuotientQ(u.args[0], v):
return TanhQ(u) or CothQ(u) or EvenQuotientQ(u.args[0], v)
elif PowerQ(u):
if EvenQ(u.exp) and HyperbolicQ(u.base) and IntegerQuotientQ(u.base.args[0], v):
return True
elif EvenQ(u.args[1]) and SumQ(u.args[0]):
return FunctionOfTanhQ(Expand(u.args[0]**2, v, x))
if ProductQ(u):
lst = []
for i in u.args:
if Not(FunctionOfTanhQ(i, v, x)):
lst.append(i)
if lst == []:
return True
return Length(lst)==2 and OddHyperbolicPowerQ(lst[0], v, x) and OddHyperbolicPowerQ(lst[1], v, x)
return all(FunctionOfTanhQ(i, v, x) for i in u.args)
def FunctionOfTanhWeight(u, v, x):
"""
u is a function of the form f(tanh(v), coth(v)) where f is independent of x.
FunctionOfTanhWeight(u, v, x) returns a nonnegative number if u is best considered a function of tanh(v), else it returns a negative number.
Examples
========
>>> from sympy import sinh, log, tanh
>>> from sympy.abc import x
>>> from sympy.integrals.rubi.utility_function import FunctionOfTanhWeight
>>> FunctionOfTanhWeight(x, log(x), x)
0
>>> FunctionOfTanhWeight(sinh(log(x)), log(x), x)
0
>>> FunctionOfTanhWeight(tanh(log(x)), log(x), x)
1
"""
if AtomQ(u):
return S(0)
elif CalculusQ(u):
return S(0)
elif HyperbolicQ(u) and IntegerQuotientQ(u.args[0], v):
if TanhQ(u) and ZeroQ(u.args[0] - v):
return S(1)
elif CothQ(u) and ZeroQ(u.args[0] - v):
return S(-1)
return S(0)
elif PowerQ(u):
if EvenQ(u.exp) and HyperbolicQ(u.base) and IntegerQuotientQ(u.base.args[0], v):
if TanhQ(u.base) or CoshQ(u.base) or SechQ(u.base):
return S(1)
return S(-1)
if ProductQ(u):
if all(FunctionOfTanhQ(i, v, x) for i in u.args):
return Add(*[FunctionOfTanhWeight(i, v, x) for i in u.args])
return S(0)
return Add(*[FunctionOfTanhWeight(i, v, x) for i in u.args])
def FunctionOfHyperbolicQ(u, v, x):
# (* If u (x) is equivalent to a function of the form f (Sinh[v],Cosh[v],Tanh[v],Coth[v],Sech[v],Csch[v])
# where f is independent of x, FunctionOfHyperbolicQ[u,v,x] returns True; else it returns False. *)
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif HyperbolicQ(u) and IntegerQuotientQ(u.args[0], v):
return True
return all(FunctionOfHyperbolicQ(i, v, x) for i in u.args)
def SmartNumerator(expr):
if PowerQ(expr):
n = expr.exp
u = expr.base
if RationalQ(n) and n < 0:
return SmartDenominator(u**(-n))
elif ProductQ(expr):
return Mul(*[SmartNumerator(i) for i in expr.args])
return Numerator(expr)
def SmartDenominator(expr):
if PowerQ(expr):
u = expr.base
n = expr.exp
if RationalQ(n) and n < 0:
return SmartNumerator(u**(-n))
elif ProductQ(expr):
return Mul(*[SmartDenominator(i) for i in expr.args])
return Denominator(expr)
def ActivateTrig(u):
return u
def ExpandTrig(*args):
if len(args) == 2:
u, x = args
return ActivateTrig(ExpandIntegrand(u, x))
u, v, x = args
w = ExpandTrig(v, x)
z = ActivateTrig(u)
if SumQ(w):
return w.func(*[z*i for i in w.args])
return z*w
def TrigExpand(u):
return expand_trig(u)
# SubstForTrig[u_,sin_,cos_,v_,x_] :=
# If[AtomQ[u],
# u,
# If[TrigQ[u] && IntegerQuotientQ[u[[1]],v],
# If[u[[1]]===v || ZeroQ[u[[1]]-v],
# If[SinQ[u],
# sin,
# If[CosQ[u],
# cos,
# If[TanQ[u],
# sin/cos,
# If[CotQ[u],
# cos/sin,
# If[SecQ[u],
# 1/cos,
# 1/sin]]]]],
# Map[Function[SubstForTrig[#,sin,cos,v,x]],
# ReplaceAll[TrigExpand[Head[u][Simplify[u[[1]]/v]*x]],x->v]]],
# If[ProductQ[u] && CosQ[u[[1]]] && SinQ[u[[2]]] && ZeroQ[u[[1,1]]-v/2] && ZeroQ[u[[2,1]]-v/2],
# sin/2*SubstForTrig[Drop[u,2],sin,cos,v,x],
# Map[Function[SubstForTrig[#,sin,cos,v,x]],u]]]]
def SubstForTrig(u, sin_ , cos_, v, x):
# (* u (v) is an expression of the form f (Sin[v],Cos[v],Tan[v],Cot[v],Sec[v],Csc[v]). *)
# (* SubstForTrig[u,sin,cos,v,x] returns the expression f (sin,cos,sin/cos,cos/sin,1/cos,1/sin). *)
if AtomQ(u):
return u
elif TrigQ(u) and IntegerQuotientQ(u.args[0], v):
if u.args[0] == v or ZeroQ(u.args[0] - v):
if SinQ(u):
return sin_
elif CosQ(u):
return cos_
elif TanQ(u):
return sin_/cos_
elif CotQ(u):
return cos_/sin_
elif SecQ(u):
return 1/cos_
return 1/sin_
r = ReplaceAll(TrigExpand(Head(u)(Simplify(u.args[0]/v*x))), {x: v})
return r.func(*[SubstForTrig(i, sin_, cos_, v, x) for i in r.args])
if ProductQ(u) and CosQ(u.args[0]) and SinQ(u.args[1]) and ZeroQ(u.args[0].args[0] - v/2) and ZeroQ(u.args[1].args[0] - v/2):
return sin(x)/2*SubstForTrig(Drop(u, 2), sin_, cos_, v, x)
return u.func(*[SubstForTrig(i, sin_, cos_, v, x) for i in u.args])
def SubstForHyperbolic(u, sinh_, cosh_, v, x):
# (* u (v) is an expression of the form f (Sinh[v],Cosh[v],Tanh[v],Coth[v],Sech[v],Csch[v]). *)
# (* SubstForHyperbolic[u,sinh,cosh,v,x] returns the expression
# f (sinh,cosh,sinh/cosh,cosh/sinh,1/cosh,1/sinh). *)
if AtomQ(u):
return u
elif HyperbolicQ(u) and IntegerQuotientQ(u.args[0], v):
if u.args[0] == v or ZeroQ(u.args[0] - v):
if SinhQ(u):
return sinh_
elif CoshQ(u):
return cosh_
elif TanhQ(u):
return sinh_/cosh_
elif CothQ(u):
return cosh_/sinh_
if SechQ(u):
return 1/cosh_
return 1/sinh_
r = ReplaceAll(TrigExpand(Head(u)(Simplify(u.args[0]/v)*x)), {x: v})
return r.func(*[SubstForHyperbolic(i, sinh_, cosh_, v, x) for i in r.args])
elif ProductQ(u) and CoshQ(u.args[0]) and SinhQ(u.args[1]) and ZeroQ(u.args[0].args[0] - v/2) and ZeroQ(u.args[1].args[0] - v/2):
return sinh(x)/2*SubstForHyperbolic(Drop(u, 2), sinh_, cosh_, v, x)
return u.func(*[SubstForHyperbolic(i, sinh_, cosh_, v, x) for i in u.args])
def InertTrigFreeQ(u):
return FreeQ(u, sin) and FreeQ(u, cos) and FreeQ(u, tan) and FreeQ(u, cot) and FreeQ(u, sec) and FreeQ(u, csc)
def LCM(a, b):
return lcm(a, b)
def SubstForFractionalPowerOfLinear(u, x):
# (* If u has a subexpression of the form (a+b*x)^(m/n) where m and n>1 are integers,
# SubstForFractionalPowerOfLinear[u,x] returns the list {v,n,a+b*x,1/b} where v is u
# with subexpressions of the form (a+b*x)^(m/n) replaced by x^m and x replaced
# by -a/b+x^n/b, and all times x^(n-1); else it returns False. *)
lst = FractionalPowerOfLinear(u, S(1), False, x)
if AtomQ(lst) or FalseQ(lst[1]):
return False
n = lst[0]
a = Coefficient(lst[1], x, 0)
b = Coefficient(lst[1], x, 1)
tmp = Simplify(x**(n-1)*SubstForFractionalPower(u, lst[1], n, -a/b + x**n/b, x))
return [NonfreeFactors(tmp, x), n, lst[1], FreeFactors(tmp, x)/b]
def FractionalPowerOfLinear(u, n, v, x):
# If u has a subexpression of the form (a + b*x)**(m/n), FractionalPowerOfLinear(u, 1, False, x) returns [n, a + b*x], else it returns False.
if AtomQ(u) or FreeQ(u, x):
return [n, v]
elif CalculusQ(u):
return False
elif FractionalPowerQ(u):
if LinearQ(u.base, x) and (FalseQ(v) or ZeroQ(u.base - v)):
return [LCM(Denominator(u.exp), n), u.base]
lst = [n, v]
for i in u.args:
lst = FractionalPowerOfLinear(i, lst[0], lst[1], x)
if AtomQ(lst):
return False
return lst
def InverseFunctionOfLinear(u, x):
# (* If u has a subexpression of the form g[a+b*x] where g is an inverse function,
# InverseFunctionOfLinear[u,x] returns g[a+b*x]; else it returns False. *)
if AtomQ(u) or CalculusQ(u) or FreeQ(u, x):
return False
elif InverseFunctionQ(u) and LinearQ(u.args[0], x):
return u
for i in u.args:
tmp = InverseFunctionOfLinear(i, x)
if Not(AtomQ(tmp)):
return tmp
return False
def InertTrigQ(*args):
if len(args) == 1:
f = args[0]
l = [sin,cos,tan,cot,sec,csc]
return any(Head(f) == i for i in l)
elif len(args) == 2:
f, g = args
if f == g:
return InertTrigQ(f)
return InertReciprocalQ(f, g) or InertReciprocalQ(g, f)
else:
f, g, h = args
return InertTrigQ(g, f) and InertTrigQ(g, h)
def InertReciprocalQ(f, g):
return (f.func == sin and g.func == csc) or (f.func == cos and g.func == sec) or (f.func == tan and g.func == cot)
def DeactivateTrig(u, x):
# (* u is a function of trig functions of a linear function of x. *)
# (* DeactivateTrig[u,x] returns u with the trig functions replaced with inert trig functions. *)
return FixInertTrigFunction(DeactivateTrigAux(u, x), x)
def FixInertTrigFunction(u, x):
return u
def DeactivateTrigAux(u, x):
if AtomQ(u):
return u
elif TrigQ(u) and LinearQ(u.args[0], x):
v = ExpandToSum(u.args[0], x)
if SinQ(u):
return sin(v)
elif CosQ(u):
return cos(v)
elif TanQ(u):
return tan(u)
elif CotQ(u):
return cot(v)
elif SecQ(u):
return sec(v)
return csc(v)
elif HyperbolicQ(u) and LinearQ(u.args[0], x):
v = ExpandToSum(I*u.args[0], x)
if SinhQ(u):
return -I*sin(v)
elif CoshQ(u):
return cos(v)
elif TanhQ(u):
return -I*tan(v)
elif CothQ(u):
I*cot(v)
elif SechQ(u):
return sec(v)
return I*csc(v)
return u.func(*[DeactivateTrigAux(i, x) for i in u.args])
def PowerOfInertTrigSumQ(u, func, x):
p_ = Wild('p', exclude=[x])
q_ = Wild('q', exclude=[x])
a_ = Wild('a', exclude=[x])
b_ = Wild('b', exclude=[x])
c_ = Wild('c', exclude=[x])
d_ = Wild('d', exclude=[x])
n_ = Wild('n', exclude=[x])
w_ = Wild('w')
pattern = (a_ + b_*(c_*func(w_))**p_)**n_
match = u.match(pattern)
if match:
keys = [a_, b_, c_, n_, p_, w_]
if len(keys) == len(match):
return True
pattern = (a_ + b_*(d_*func(w_))**p_ + c_*(d_*func(w_))**q_)**n_
match = u.match(pattern)
if match:
keys = [a_, b_, c_, d_, n_, p_, q_, w_]
if len(keys) == len(match):
return True
return False
def PiecewiseLinearQ(*args):
# (* If the derivative of u wrt x is a constant wrt x, PiecewiseLinearQ[u,x] returns True;
# else it returns False. *)
if len(args) == 3:
u, v, x = args
return PiecewiseLinearQ(u, x) and PiecewiseLinearQ(v, x)
u, x = args
if LinearQ(u, x):
return True
c_ = Wild('c', exclude=[x])
F_ = Wild('F', exclude=[x])
v_ = Wild('v')
match = u.match(log(c_*F_**v_))
if match:
if len(match) == 3:
if LinearQ(match[v_], x):
return True
try:
F = type(u)
G = type(u.args[0])
v = u.args[0].args[0]
if LinearQ(v, x):
if MemberQ([[atanh, tanh], [atanh, coth], [acoth, coth], [acoth, tanh], [atan, tan], [atan, cot], [acot, cot], [acot, tan]], [F, G]):
return True
except:
pass
return False
def KnownTrigIntegrandQ(lst, u, x):
if u == 1:
return True
a_ = Wild('a', exclude=[x])
b_ = Wild('b', exclude=[x, 0])
func_ = WildFunction('func')
m_ = Wild('m', exclude=[x])
A_ = Wild('A', exclude=[x])
B_ = Wild('B', exclude=[x, 0])
C_ = Wild('C', exclude=[x, 0])
match = u.match((a_ + b_*func_)**m_)
if match:
func = match[func_]
if LinearQ(func.args[0], x) and MemberQ(lst, func.func):
return True
match = u.match((a_ + b_*func_)**m_*(A_ + B_*func_))
if match:
func = match[func_]
if LinearQ(func.args[0], x) and MemberQ(lst, func.func):
return True
match = u.match(A_ + C_*func_**2)
if match:
func = match[func_]
if LinearQ(func.args[0], x) and MemberQ(lst, func.func):
return True
match = u.match(A_ + B_*func_ + C_*func_**2)
if match:
func = match[func_]
if LinearQ(func.args[0], x) and MemberQ(lst, func.func):
return True
match = u.match((a_ + b_*func_)**m_*(A_ + C_*func_**2))
if match:
func = match[func_]
if LinearQ(func.args[0], x) and MemberQ(lst, func.func):
return True
match = u.match((a_ + b_*func_)**m_*(A_ + B_*func_ + C_*func_**2))
if match:
func = match[func_]
if LinearQ(func.args[0], x) and MemberQ(lst, func.func):
return True
return False
def KnownSineIntegrandQ(u, x):
return KnownTrigIntegrandQ([sin, cos], u, x)
def KnownTangentIntegrandQ(u, x):
return KnownTrigIntegrandQ([tan], u, x)
def KnownCotangentIntegrandQ(u, x):
return KnownTrigIntegrandQ([cot], u, x)
def KnownSecantIntegrandQ(u, x):
return KnownTrigIntegrandQ([sec, csc], u, x)
def TryPureTanSubst(u, x):
a_ = Wild('a', exclude=[x])
b_ = Wild('b', exclude=[x])
c_ = Wild('c', exclude=[x])
G_ = Wild('G')
F = u.func
try:
if MemberQ([atan, acot, atanh, acoth], F):
match = u.args[0].match(c_*(a_ + b_*G_))
if match:
if len(match) == 4:
G = match[G_]
if MemberQ([tan, cot, tanh, coth], G.func):
if LinearQ(G.args[0], x):
return True
except:
pass
return False
def TryTanhSubst(u, x):
if LogQ(u):
return False
elif not FalseQ(FunctionOfLinear(u, x)):
return False
a_ = Wild('a', exclude=[x])
m_ = Wild('m', exclude=[x])
p_ = Wild('p', exclude=[x])
r_, s_, t_, n_, b_, f_, g_ = map(Wild, 'rstnbfg')
match = u.match(r_*(s_ + t_)**n_)
if match:
if len(match) == 4:
r, s, t, n = [match[i] for i in [r_, s_, t_, n_]]
if IntegerQ(n) and PositiveQ(n):
return False
match = u.match(1/(a_ + b_*f_**n_))
if match:
if len(match) == 4:
a, b, f, n = [match[i] for i in [a_, b_, f_, n_]]
if SinhCoshQ(f) and IntegerQ(n) and n > 2:
return False
match = u.match(f_*g_)
if match:
if len(match) == 2:
f, g = match[f_], match[g_]
if SinhCoshQ(f) and SinhCoshQ(g):
if IntegersQ(f.args[0]/x, g.args[0]/x):
return False
match = u.match(r_*(a_*s_**m_)**p_)
if match:
if len(match) == 5:
r, a, s, m, p = [match[i] for i in [r_, a_, s_, m_, p_]]
if Not(m==2 and (s == Sech(x) or s == Csch(x))):
return False
if u != ExpandIntegrand(u, x):
return False
return True
def TryPureTanhSubst(u, x):
F = u.func
a_ = Wild('a', exclude=[x])
G_ = Wild('G')
if F == sym_log:
return False
match = u.args[0].match(a_*G_)
if match and len(match) == 2:
G = match[G_].func
if MemberQ([atanh, acoth], F) and MemberQ([tanh, coth], G):
return False
if u != ExpandIntegrand(u, x):
return False
return True
def AbsurdNumberGCD(*seq):
# (* m, n, ... must be absurd numbers. AbsurdNumberGCD[m,n,...] returns the gcd of m, n, ... *)
lst = list(seq)
if Length(lst) == 1:
return First(lst)
return AbsurdNumberGCDList(FactorAbsurdNumber(First(lst)), FactorAbsurdNumber(AbsurdNumberGCD(*Rest(lst))))
def AbsurdNumberGCDList(lst1, lst2):
# (* lst1 and lst2 must be absurd number prime factorization lists. *)
# (* AbsurdNumberGCDList[lst1,lst2] returns the gcd of the absurd numbers represented by lst1 and lst2. *)
if lst1 == []:
return Mul(*[i[0]**Min(i[1],0) for i in lst2])
elif lst2 == []:
return Mul(*[i[0]**Min(i[1],0) for i in lst1])
elif lst1[0][0] == lst2[0][0]:
if lst1[0][1] <= lst2[0][1]:
return lst1[0][0]**lst1[0][1]*AbsurdNumberGCDList(Rest(lst1), Rest(lst2))
return lst1[0][0]**lst2[0][1]*AbsurdNumberGCDList(Rest(lst1), Rest(lst2))
elif lst1[0][0] < lst2[0][0]:
if lst1[0][1] < 0:
return lst1[0][0]**lst1[0][1]*AbsurdNumberGCDList(Rest(lst1), lst2)
return AbsurdNumberGCDList(Rest(lst1), lst2)
elif lst2[0][1] < 0:
return lst2[0][0]**lst2[0][1]*AbsurdNumberGCDList(lst1, Rest(lst2))
return AbsurdNumberGCDList(lst1, Rest(lst2))
def ExpandTrigExpand(u, F, v, m, n, x):
w = Expand(TrigExpand(F.xreplace({x: n*x}))**m).xreplace({x: v})
if SumQ(w):
t = 0
for i in w.args:
t += u*i
return t
else:
return u*w
def ExpandTrigReduce(*args):
if len(args) == 3:
u = args[0]
v = args[1]
x = args[2]
w = ExpandTrigReduce(v, x)
if SumQ(w):
t = 0
for i in w.args:
t += u*i
return t
else:
return u*w
else:
u = args[0]
x = args[1]
return ExpandTrigReduceAux(u, x)
def ExpandTrigReduceAux(u, x):
v = TrigReduce(u).expand()
if SumQ(v):
t = 0
for i in v.args:
t += NormalizeTrig(i, x)
return t
return NormalizeTrig(v, x)
def NormalizeTrig(v, x):
a = Wild('a', exclude=[x])
n = Wild('n', exclude=[x, 0])
F = Wild('F')
expr = a*F**n
M = v.match(expr)
if M and len(M[F].args) == 1 and PolynomialQ(M[F].args[0], x) and Exponent(M[F].args[0], x)>0:
u = M[F].args[0]
return M[a]*M[F].xreplace({u: ExpandToSum(u, x)})**M[n]
else:
return v
#=================================
def TrigToExp(expr):
ex = expr.rewrite(sin, exp).rewrite(cos, exp).rewrite(tan, exp).rewrite(sec, exp).rewrite(csc, exp).rewrite(cot, exp)
return ex.replace(sym_exp, exp)
def ExpandTrigToExp(u, *args):
if len(args) == 1:
x = args[0]
return ExpandTrigToExp(1, u, x)
else:
v = args[0]
x = args[1]
w = TrigToExp(v)
k = 0
if SumQ(w):
for i in w.args:
k += SimplifyIntegrand(u*i, x)
w = k
else:
w = SimplifyIntegrand(u*w, x)
return ExpandIntegrand(FreeFactors(w, x), NonfreeFactors(w, x),x)
#======================================
def TrigReduce(i):
"""
TrigReduce(expr) rewrites products and powers of trigonometric functions in expr in terms of trigonometric functions with combined arguments.
Examples
========
>>> from sympy import sin, cos
>>> from sympy.integrals.rubi.utility_function import TrigReduce
>>> from sympy.abc import x
>>> TrigReduce(cos(x)**2)
cos(2*x)/2 + 1/2
>>> TrigReduce(cos(x)**2*sin(x))
sin(x)/4 + sin(3*x)/4
>>> TrigReduce(cos(x)**2+sin(x))
sin(x) + cos(2*x)/2 + 1/2
"""
if SumQ(i):
t = 0
for k in i.args:
t += TrigReduce(k)
return t
if ProductQ(i):
if any(PowerQ(k) for k in i.args):
if (i.rewrite((sin, sinh), sym_exp).rewrite((cos, cosh), sym_exp).expand().rewrite(sym_exp, sin)).has(I, cosh, sinh):
return i.rewrite((sin, sinh), sym_exp).rewrite((cos, cosh), sym_exp).expand().rewrite(sym_exp, sin).simplify()
else:
return i.rewrite((sin, sinh), sym_exp).rewrite((cos, cosh), sym_exp).expand().rewrite(sym_exp, sin)
else:
a = Wild('a')
b = Wild('b')
v = Wild('v')
Match = i.match(v*sin(a)*cos(b))
if Match:
a = Match[a]
b = Match[b]
v = Match[v]
return i.subs(v*sin(a)*cos(b), v*S(1)/2*(sin(a + b) + sin(a - b)))
Match = i.match(v*sin(a)*sin(b))
if Match:
a = Match[a]
b = Match[b]
v = Match[v]
return i.subs(v*sin(a)*sin(b), v*S(1)/2*cos(a - b) - cos(a + b))
Match = i.match(v*cos(a)*cos(b))
if Match:
a = Match[a]
b = Match[b]
v = Match[v]
return i.subs(v*cos(a)*cos(b), v*S(1)/2*cos(a + b) + cos(a - b))
Match = i.match(v*sinh(a)*cosh(b))
if Match:
a = Match[a]
b = Match[b]
v = Match[v]
return i.subs(v*sinh(a)*cosh(b), v*S(1)/2*(sinh(a + b) + sinh(a - b)))
Match = i.match(v*sinh(a)*sinh(b))
if Match:
a = Match[a]
b = Match[b]
v = Match[v]
return i.subs(v*sinh(a)*sinh(b), v*S(1)/2*cosh(a - b) - cosh(a + b))
Match = i.match(v*cosh(a)*cosh(b))
if Match:
a = Match[a]
b = Match[b]
v = Match[v]
return i.subs(v*cosh(a)*cosh(b), v*S(1)/2*cosh(a + b) + cosh(a - b))
if PowerQ(i):
if i.has(sin, sinh):
if (i.rewrite((sin, sinh), sym_exp).expand().rewrite(sym_exp, sin)).has(I, cosh, sinh):
return i.rewrite((sin, sinh), sym_exp).expand().rewrite(sym_exp, sin).simplify()
else:
return i.rewrite((sin, sinh), sym_exp).expand().rewrite(sym_exp, sin)
if i.has(cos, cosh):
if (i.rewrite((cos, cosh), sym_exp).expand().rewrite(sym_exp, cos)).has(I, cosh, sinh):
return i.rewrite((cos, cosh), sym_exp).expand().rewrite(sym_exp, cos).simplify()
else:
return i.rewrite((cos, cosh), sym_exp).expand().rewrite(sym_exp, cos)
return i
def FunctionOfTrig(u, *args):
# If u is a function of trig functions of v where v is a linear function of x,
# FunctionOfTrig[u,x] returns v; else it returns False.
if len(args) == 1:
x = args[0]
v = FunctionOfTrig(u, None, x)
if v:
return v
else:
return False
else:
v, x = args
if AtomQ(u):
if u == x:
return False
else:
return v
if TrigQ(u) and LinearQ(u.args[0], x):
if v is None:
return u.args[0]
else:
a = Coefficient(v, x, 0)
b = Coefficient(v, x, 1)
c = Coefficient(u.args[0], x, 0)
d = Coefficient(u.args[0], x, 1)
if ZeroQ(a*d - b*c) and RationalQ(b/d):
return a/Numerator(b/d) + b*x/Numerator(b/d)
else:
return False
if HyperbolicQ(u) and LinearQ(u.args[0], x):
if v is None:
return I*u.args[0]
a = Coefficient(v, x, 0)
b = Coefficient(v, x, 1)
c = I*Coefficient(u.args[0], x, 0)
d = I*Coefficient(u.args[0], x, 1)
if ZeroQ(a*d - b*c) and RationalQ(b/d):
return a/Numerator(b/d) + b*x/Numerator(b/d)
else:
return False
if CalculusQ(u):
return False
else:
w = v
for i in u.args:
w = FunctionOfTrig(i, w, x)
if FalseQ(w):
return False
return w
def AlgebraicTrigFunctionQ(u, x):
# If u is algebraic function of trig functions, AlgebraicTrigFunctionQ(u,x) returns True; else it returns False.
if AtomQ(u):
return True
elif TrigQ(u) and LinearQ(u.args[0], x):
return True
elif HyperbolicQ(u) and LinearQ(u.args[0], x):
return True
elif PowerQ(u):
if FreeQ(u.exp, x):
return AlgebraicTrigFunctionQ(u.base, x)
elif ProductQ(u) or SumQ(u):
for i in u.args:
if not AlgebraicTrigFunctionQ(i, x):
return False
return True
return False
def FunctionOfHyperbolic(u, *x):
# If u is a function of hyperbolic trig functions of v where v is linear in x,
# FunctionOfHyperbolic(u,x) returns v; else it returns False.
if len(x) == 1:
x = x[0]
v = FunctionOfHyperbolic(u, None, x)
if v==None:
return False
else:
return v
else:
v = x[0]
x = x[1]
if AtomQ(u):
if u == x:
return False
return v
if HyperbolicQ(u) and LinearQ(u.args[0], x):
if v is None:
return u.args[0]
a = Coefficient(v, x, 0)
b = Coefficient(v, x, 1)
c = Coefficient(u.args[0], x, 0)
d = Coefficient(u.args[0], x, 1)
if ZeroQ(a*d - b*c) and RationalQ(b/d):
return a/Numerator(b/d) + b*x/Numerator(b/d)
else:
return False
if CalculusQ(u):
return False
w = v
for i in u.args:
if w == FunctionOfHyperbolic(i, w, x):
return False
return w
def FunctionOfQ(v, u, x, PureFlag=False):
# v is a function of x. If u is a function of v, FunctionOfQ(v, u, x) returns True; else it returns False. *)
if FreeQ(u, x):
return False
elif AtomQ(v):
return True
elif ProductQ(v) and Not(EqQ(FreeFactors(v, x), 1)):
return FunctionOfQ(NonfreeFactors(v, x), u, x, PureFlag)
elif PureFlag:
if SinQ(v) or CscQ(v):
return PureFunctionOfSinQ(u, v.args[0], x)
elif CosQ(v) or SecQ(v):
return PureFunctionOfCosQ(u, v.args[0], x)
elif TanQ(v):
return PureFunctionOfTanQ(u, v.args[0], x)
elif CotQ(v):
return PureFunctionOfCotQ(u, v.args[0], x)
elif SinhQ(v) or CschQ(v):
return PureFunctionOfSinhQ(u, v.args[0], x)
elif CoshQ(v) or SechQ(v):
return PureFunctionOfCoshQ(u, v.args[0], x)
elif TanhQ(v):
return PureFunctionOfTanhQ(u, v.args[0], x)
elif CothQ(v):
return PureFunctionOfCothQ(u, v.args[0], x)
else:
return FunctionOfExpnQ(u, v, x) != False
elif SinQ(v) or CscQ(v):
return FunctionOfSinQ(u, v.args[0], x)
elif CosQ(v) or SecQ(v):
return FunctionOfCosQ(u, v.args[0], x)
elif TanQ(v) or CotQ(v):
FunctionOfTanQ(u, v.args[0], x)
elif SinhQ(v) or CschQ(v):
return FunctionOfSinhQ(u, v.args[0], x)
elif CoshQ(v) or SechQ(v):
return FunctionOfCoshQ(u, v.args[0], x)
elif TanhQ(v) or CothQ(v):
return FunctionOfTanhQ(u, v.args[0], x)
return FunctionOfExpnQ(u, v, x) != False
def FunctionOfExpnQ(u, v, x):
if u == v:
return 1
if AtomQ(u):
if u == x:
return False
else:
return 0
if CalculusQ(u):
return False
if PowerQ(u):
if FreeQ(u.exp, x):
if ZeroQ(u.base - v):
if IntegerQ(u.exp):
return u.exp
else:
return 1
if PowerQ(v):
if FreeQ(v.exp, x) and ZeroQ(u.base-v.base):
if RationalQ(v.exp):
if RationalQ(u.exp) and IntegerQ(u.exp/v.exp) and (v.exp>0 or u.exp<0):
return u.exp/v.exp
else:
return False
if IntegerQ(Simplify(u.exp/v.exp)):
return Simplify(u.exp/v.exp)
else:
return False
return FunctionOfExpnQ(u.base, v, x)
if ProductQ(u) and Not(EqQ(FreeFactors(u, x), 1)):
return FunctionOfExpnQ(NonfreeFactors(u, x), v, x)
if ProductQ(u) and ProductQ(v):
deg1 = FunctionOfExpnQ(First(u), First(v), x)
if deg1==False:
return False
deg2 = FunctionOfExpnQ(Rest(u), Rest(v), x);
if deg1==deg2 and FreeQ(Simplify(u/v^deg1), x):
return deg1
else:
return False
lst = []
for i in u.args:
if FunctionOfExpnQ(i, v, x) is False:
return False
lst.append(FunctionOfExpnQ(i, v, x))
return Apply(GCD, lst)
def PureFunctionOfSinQ(u, v, x):
# If u is a pure function of Sin(v) and/or Csc(v), PureFunctionOfSinQ(u, v, x) returns True; else it returns False.
if AtomQ(u):
return u!=x
if CalculusQ(u):
return False
if TrigQ(u) and ZeroQ(u.args[0]-v):
return SinQ(u) or CscQ(u)
for i in u.args:
if Not(PureFunctionOfSinQ(i, v, x)):
return False
return True
def PureFunctionOfCosQ(u, v, x):
# If u is a pure function of Cos(v) and/or Sec(v), PureFunctionOfCosQ(u, v, x) returns True; else it returns False.
if AtomQ(u):
return u!=x
if CalculusQ(u):
return False
if TrigQ(u) and ZeroQ(u.args[0]-v):
return CosQ(u) or SecQ(u)
for i in u.args:
if Not(PureFunctionOfCosQ(i, v, x)):
return False
return True
def PureFunctionOfTanQ(u, v, x):
# If u is a pure function of Tan(v) and/or Cot(v), PureFunctionOfTanQ(u, v, x) returns True; else it returns False.
if AtomQ(u):
return u!=x
if CalculusQ(u):
return False
if TrigQ(u) and ZeroQ(u.args[0]-v):
return TanQ(u) or CotQ(u)
for i in u.args:
if Not(PureFunctionOfTanQ(i, v, x)):
return False
return True
def PureFunctionOfCotQ(u, v, x):
# If u is a pure function of Cot(v), PureFunctionOfCotQ(u, v, x) returns True; else it returns False.
if AtomQ(u):
return u!=x
if CalculusQ(u):
return False
if TrigQ(u) and ZeroQ(u.args[0]-v):
return CotQ(u)
for i in u.args:
if Not(PureFunctionOfCotQ(i, v, x)):
return False
return True
def FunctionOfCosQ(u, v, x):
# If u is a function of Cos[v], FunctionOfCosQ[u,v,x] returns True; else it returns False.
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif TrigQ(u) and IntegerQuotientQ(u.args[0], v):
# Basis: If m integer, Cos[m*v]^n is a function of Cos[v]. *)
return CosQ(u) or SecQ(u)
elif IntegerPowerQ(u):
if TrigQ(u.base) and IntegerQuotientQ(u.base.args[0], v):
if EvenQ(u.exp):
# Basis: If m integer and n even, Trig[m*v]^n is a function of Cos[v]. *)
return True
return FunctionOfCosQ(u.base, v, x)
elif ProductQ(u):
lst = FindTrigFactor(sin, csc, u, v, False)
if ListQ(lst):
# (* Basis: If m integer and n odd, Sin[m*v]^n == Sin[v]*u where u is a function of Cos[v]. *)
return FunctionOfCosQ(Sin(v)*lst[1], v, x)
lst = FindTrigFactor(tan, cot, u, v, True)
if ListQ(lst):
# (* Basis: If m integer and n odd, Tan[m*v]^n == Sin[v]*u where u is a function of Cos[v]. *)
return FunctionOfCosQ(Sin(v)*lst[1], v, x)
return all(FunctionOfCosQ(i, v, x) for i in u.args)
return all(FunctionOfCosQ(i, v, x) for i in u.args)
def FunctionOfSinQ(u, v, x):
# If u is a function of Sin[v], FunctionOfSinQ[u,v,x] returns True; else it returns False.
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif TrigQ(u) and IntegerQuotientQ(u.args[0], v):
if OddQuotientQ(u.args[0], v):
# Basis: If m odd, Sin[m*v]^n is a function of Sin[v].
return SinQ(u) or CscQ(u)
# Basis: If m even, Cos[m*v]^n is a function of Sin[v].
return CosQ(u) or SecQ(u)
elif IntegerPowerQ(u):
if TrigQ(u.base) and IntegerQuotientQ(u.base.args[0], v):
if EvenQ(u.exp):
# Basis: If m integer and n even, Hyper[m*v]^n is a function of Sin[v].
return True
return FunctionOfSinQ(u.base, v, x)
elif ProductQ(u):
if CosQ(u.args[0]) and SinQ(u.args[1]) and ZeroQ(u.args[0].args[0] - v/2) and ZeroQ(u.args[1].args[0] - v/2):
return FunctionOfSinQ(Drop(u, 2), v, x)
lst = FindTrigFactor(sin, csch, u, v, False)
if ListQ(lst) and EvenQuotientQ(lst[0], v):
# Basis: If m even and n odd, Sin[m*v]^n == Cos[v]*u where u is a function of Sin[v].
return FunctionOfSinQ(Cos(v)*lst[1], v, x)
lst = FindTrigFactor(cos, sec, u, v, False)
if ListQ(lst) and OddQuotientQ(lst[0], v):
# Basis: If m odd and n odd, Cos[m*v]^n == Cos[v]*u where u is a function of Sin[v].
return FunctionOfSinQ(Cos(v)*lst[1], v, x)
lst = FindTrigFactor(tan, cot, u, v, True)
if ListQ(lst):
# Basis: If m integer and n odd, Tan[m*v]^n == Cos[v]*u where u is a function of Sin[v].
return FunctionOfSinQ(Cos(v)*lst[1], v, x)
return all(FunctionOfSinQ(i, v, x) for i in u.args)
return all(FunctionOfSinQ(i, v, x) for i in u.args)
def OddTrigPowerQ(u, v, x):
if SinQ(u) or CosQ(u) or SecQ(u) or CscQ(u):
return OddQuotientQ(u.args[0], v)
if PowerQ(u):
return OddQ(u.exp) and OddTrigPowerQ(u.base, v, x)
if ProductQ(u):
if not FreeFactors(u, x) == 1:
return OddTrigPowerQ(NonfreeFactors(u, x), v, x)
lst = []
for i in u.args:
if Not(FunctionOfTanQ(i, v, x)):
lst.append(i)
if lst == []:
return True
return Length(lst)==1 and OddTrigPowerQ(lst[0], v, x)
if SumQ(u):
return all(OddTrigPowerQ(i, v, x) for i in u.args)
return False
def FunctionOfTanQ(u, v, x):
# If u is a function of the form f[Tan[v],Cot[v]] where f is independent of x,
# FunctionOfTanQ[u,v,x] returns True; else it returns False.
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif TrigQ(u) and IntegerQuotientQ(u.args[0], v):
return TanQ(u) or CotQ(u) or EvenQuotientQ(u.args[0], v)
elif PowerQ(u):
if EvenQ(u.exp) and TrigQ(u.base) and IntegerQuotientQ(u.base.args[0], v):
return True
elif EvenQ(u.exp) and SumQ(u.base):
return FunctionOfTanQ(Expand(u.base**2, v, x))
if ProductQ(u):
lst = []
for i in u.args:
if Not(FunctionOfTanQ(i, v, x)):
lst.append(i)
if lst == []:
return True
return Length(lst)==2 and OddTrigPowerQ(lst[0], v, x) and OddTrigPowerQ(lst[1], v, x)
return all(FunctionOfTanQ(i, v, x) for i in u.args)
def FunctionOfTanWeight(u, v, x):
# (* u is a function of the form f[Tan[v],Cot[v]] where f is independent of x.
# FunctionOfTanWeight[u,v,x] returns a nonnegative number if u is best considered a function
# of Tan[v]; else it returns a negative number. *)
if AtomQ(u):
return S(0)
elif CalculusQ(u):
return S(0)
elif TrigQ(u) and IntegerQuotientQ(u.args[0], v):
if TanQ(u) and ZeroQ(u.args[0] - v):
return S(1)
elif CotQ(u) and ZeroQ(u.args[0] - v):
return S(-1)
return S(0)
elif PowerQ(u):
if EvenQ(u.exp) and TrigQ(u.base) and IntegerQuotientQ(u.base.args[0], v):
if TanQ(u.base) or CosQ(u.base) or SecQ(u.base):
return S(1)
return S(-1)
if ProductQ(u):
if all(FunctionOfTanQ(i, v, x) for i in u.args):
return Add(*[FunctionOfTanWeight(i, v, x) for i in u.args])
return S(0)
return Add(*[FunctionOfTanWeight(i, v, x) for i in u.args])
def FunctionOfTrigQ(u, v, x):
# If u (x) is equivalent to a function of the form f (Sin[v],Cos[v],Tan[v],Cot[v],Sec[v],Csc[v]) where f is independent of x, FunctionOfTrigQ[u,v,x] returns True; else it returns False.
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif TrigQ(u) and IntegerQuotientQ(u.args[0], v):
return True
return all(FunctionOfTrigQ(i, v, x) for i in u.args)
def FunctionOfDensePolynomialsQ(u, x):
# If all occurrences of x in u (x) are in dense polynomials, FunctionOfDensePolynomialsQ[u,x] returns True; else it returns False.
if FreeQ(u, x):
return True
if PolynomialQ(u, x):
return Length(Exponent(u,x,List))>1
return all(FunctionOfDensePolynomialsQ(i, x) for i in u.args)
def FunctionOfLog(u, *args):
# If u (x) is equivalent to an expression of the form f (Log[a*x^n]), FunctionOfLog[u,x] returns
# the list {f (x),a*x^n,n}; else it returns False.
if len(args) == 1:
x = args[0]
lst = FunctionOfLog(u, False, False, x)
if AtomQ(lst) or FalseQ(lst[1]) or not isinstance(x, Symbol):
return False
else:
return lst
else:
v = args[0]
n = args[1]
x = args[2]
if AtomQ(u):
if u==x:
return False
else:
return [u, v, n]
if CalculusQ(u):
return False
lst = BinomialParts(u.args[0], x)
if LogQ(u) and ListQ(lst) and ZeroQ(lst[0]):
if FalseQ(v) or u.args[0] == v:
return [x, u.args[0], lst[2]]
else:
return False
lst = [0, v, n]
l = []
for i in u.args:
lst = FunctionOfLog(i, lst[1], lst[2], x)
if AtomQ(lst):
return False
else:
l.append(lst[0])
return [u.func(*l), lst[1], lst[2]]
def PowerVariableExpn(u, m, x):
# If m is an integer, u is an expression of the form f((c*x)**n) and g=GCD(m,n)>1,
# PowerVariableExpn(u,m,x) returns the list {x**(m/g)*f((c*x)**(n/g)),g,c}; else it returns False.
if IntegerQ(m):
lst = PowerVariableDegree(u, m, 1, x)
if not lst:
return False
else:
return [x**(m/lst[0])*PowerVariableSubst(u, lst[0], x), lst[0], lst[1]]
else:
return False
def PowerVariableDegree(u, m, c, x):
if FreeQ(u, x):
return [m, c]
if AtomQ(u) or CalculusQ(u):
return False
if PowerQ(u):
if FreeQ(u.base/x, x):
if ZeroQ(m) or m == u.exp and c == u.base/x:
return [u.exp, u.base/x]
if IntegerQ(u.exp) and IntegerQ(m) and GCD(m, u.exp)>1 and c==u.base/x:
return [GCD(m, u.exp), c]
else:
return False
lst = [m, c]
for i in u.args:
if PowerVariableDegree(i, lst[0], lst[1], x) == False:
return False
lst1 = PowerVariableDegree(i, lst[0], lst[1], x)
if not lst1:
return False
else:
return lst1
def PowerVariableSubst(u, m, x):
if FreeQ(u, x) or AtomQ(u) or CalculusQ(u):
return u
if PowerQ(u):
if FreeQ(u.base/x, x):
return x**(u.exp/m)
if ProductQ(u):
l = 1
for i in u.args:
l *= (PowerVariableSubst(i, m, x))
return l
if SumQ(u):
l = 0
for i in u.args:
l += (PowerVariableSubst(i, m, x))
return l
return u
def EulerIntegrandQ(expr, x):
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
n = Wild('n', exclude=[x, 0])
m = Wild('m', exclude=[x, 0])
p = Wild('p', exclude=[x, 0])
u = Wild('u')
v = Wild('v')
# Pattern 1
M = expr.match((a*x + b*u**n)**p)
if M:
if len(M) == 5 and FreeQ([M[a], M[b]], x) and IntegerQ(M[n] + 1/2) and QuadraticQ(M[u], x) and Not(RationalQ(M[p])) or NegativeIntegerQ(M[p]) and Not(BinomialQ(M[u], x)):
return True
# Pattern 2
M = expr.match(v**m*(a*x + b*u**n)**p)
if M:
if len(M) == 6 and FreeQ([M[a], M[b]], x) and ZeroQ(M[u] - M[v]) and IntegersQ(2*M[m], M[n] + 1/2) and QuadraticQ(M[u], x) and Not(RationalQ(M[p])) or NegativeIntegerQ(M[p]) and Not(BinomialQ(M[u], x)):
return True
# Pattern 3
M = expr.match(u**n*v**p)
if M:
if len(M) == 3 and NegativeIntegerQ(M[p]) and IntegerQ(M[n] + 1/2) and QuadraticQ(M[u], x) and QuadraticQ(M[v], x) and Not(BinomialQ(M[v], x)):
return True
else:
return False
def FunctionOfSquareRootOfQuadratic(u, *args):
if len(args) == 1:
x = args[0]
pattern = Pattern(UtilityOperator(x_**WC('m', 1)*(a_ + x**WC('n', 1)*WC('b', 1))**p_, x), CustomConstraint(lambda a, b, m, n, p, x: FreeQ([a, b, m, n, p], x)))
M = is_match(UtilityOperator(u, args[0]), pattern)
if M:
return False
tmp = FunctionOfSquareRootOfQuadratic(u, False, x)
if AtomQ(tmp) or FalseQ(tmp[0]):
return False
tmp = tmp[0]
a = Coefficient(tmp, x, 0)
b = Coefficient(tmp, x, 1)
c = Coefficient(tmp, x, 2)
if ZeroQ(a) and ZeroQ(b) or ZeroQ(b**2-4*a*c):
return False
if PosQ(c):
sqrt = Rt(c, S(2));
q = a*sqrt + b*x + sqrt*x**2
r = b + 2*sqrt*x
return [Simplify(SquareRootOfQuadraticSubst(u, q/r, (-a+x**2)/r, x)*q/r**2), Simplify(sqrt*x + Sqrt(tmp)), 2]
if PosQ(a):
sqrt = Rt(a, S(2))
q = c*sqrt - b*x + sqrt*x**2
r = c - x**2
return [Simplify(SquareRootOfQuadraticSubst(u, q/r, (-b+2*sqrt*x)/r, x)*q/r**2), Simplify((-sqrt+Sqrt(tmp))/x), 1]
sqrt = Rt(b**2 - 4*a*c, S(2))
r = c - x**2
return[Simplify(-sqrt*SquareRootOfQuadraticSubst(u, -sqrt*x/r, -(b*c+c*sqrt+(-b+sqrt)*x**2)/(2*c*r), x)*x/r**2), FullSimplify(2*c*Sqrt(tmp)/(b-sqrt+2*c*x)), 3]
else:
v = args[0]
x = args[1]
if AtomQ(u) or FreeQ(u, x):
return [v]
if PowerQ(u):
if FreeQ(u.exp, x):
if FractionQ(u.exp) and Denominator(u.exp)==2 and PolynomialQ(u.base, x) and Exponent(u.base, x)==2:
if FalseQ(v) or u.base == v:
return [u.base]
else:
return False
return FunctionOfSquareRootOfQuadratic(u.base, v, x)
if ProductQ(u) or SumQ(u):
lst = [v]
lst1 = []
for i in u.args:
if FunctionOfSquareRootOfQuadratic(i, lst[0], x) == False:
return False
lst1 = FunctionOfSquareRootOfQuadratic(i, lst[0], x)
return lst1
else:
return False
def SquareRootOfQuadraticSubst(u, vv, xx, x):
# SquareRootOfQuadraticSubst(u, vv, xx, x) returns u with fractional powers replaced by vv raised to the power and x replaced by xx.
if AtomQ(u) or FreeQ(u, x):
if u==x:
return xx
return u
if PowerQ(u):
if FreeQ(u.exp, x):
if FractionQ(u.exp) and Denominator(u.exp)==2 and PolynomialQ(u.base, x) and Exponent(u.base, x)==2:
return vv**Numerator(u.exp)
return SquareRootOfQuadraticSubst(u.base, vv, xx, x)**u.exp
elif SumQ(u):
t = 0
for i in u.args:
t += SquareRootOfQuadraticSubst(i, vv, xx, x)
return t
elif ProductQ(u):
t = 1
for i in u.args:
t *= SquareRootOfQuadraticSubst(i, vv, xx, x)
return t
def Divides(y, u, x):
# If u divided by y is free of x, Divides[y,u,x] returns the quotient; else it returns False.
v = Simplify(u/y)
if FreeQ(v, x):
return v
else:
return False
def DerivativeDivides(y, u, x):
'''
If y not equal to x, y is easy to differentiate wrt x, and u divided by the derivative of y
is free of x, DerivativeDivides[y,u,x] returns the quotient; else it returns False.
'''
from matchpy import is_match
pattern0 = Pattern(Mul(a , b_), CustomConstraint(lambda a, b : FreeQ(a, b)))
def f1(y, u, x):
if PolynomialQ(y, x):
return PolynomialQ(u, x) and Exponent(u, x)==Exponent(y, x)-1
else:
return EasyDQ(y, x)
if is_match(y, pattern0):
return False
elif f1(y, u, x):
v = D(y ,x)
if EqQ(v, 0):
return False
else:
v = Simplify(u/v)
if FreeQ(v, x):
return v
else:
return False
else:
return False
def EasyDQ(expr, x):
# If u is easy to differentiate wrt x, EasyDQ(u, x) returns True; else it returns False *)
u = Wild('u',exclude=[1])
m = Wild('m',exclude=[x, 0])
M = expr.match(u*x**m)
if M:
return EasyDQ(M[u], x)
if AtomQ(expr) or FreeQ(expr, x) or Length(expr)==0:
return True
elif CalculusQ(expr):
return False
elif Length(expr)==1:
return EasyDQ(expr.args[0], x)
elif BinomialQ(expr, x) or ProductOfLinearPowersQ(expr, x):
return True
elif RationalFunctionQ(expr, x) and RationalFunctionExponents(expr, x)==[1, 1]:
return True
elif ProductQ(expr):
if FreeQ(First(expr), x):
return EasyDQ(Rest(expr), x)
elif FreeQ(Rest(expr), x):
return EasyDQ(First(expr), x)
else:
return False
elif SumQ(expr):
return EasyDQ(First(expr), x) and EasyDQ(Rest(expr), x)
elif Length(expr)==2:
if FreeQ(expr.args[0], x):
EasyDQ(expr.args[1], x)
elif FreeQ(expr.args[1], x):
return EasyDQ(expr.args[0], x)
else:
return False
return False
def ProductOfLinearPowersQ(u, x):
# ProductOfLinearPowersQ(u, x) returns True iff u is a product of factors of the form v^n where v is linear in x
v = Wild('v')
n = Wild('n', exclude=[x])
M = u.match(v**n)
return FreeQ(u, x) or M and LinearQ(M[v], x) or ProductQ(u) and ProductOfLinearPowersQ(First(u), x) and ProductOfLinearPowersQ(Rest(u), x)
def Rt(u, n):
return RtAux(TogetherSimplify(u), n)
def NthRoot(u, n):
return nsimplify(u**(1/n))
def AtomBaseQ(u):
# If u is an atom or an atom raised to an odd degree, AtomBaseQ(u) returns True; else it returns False
return AtomQ(u) or PowerQ(u) and OddQ(u.args[1]) and AtomBaseQ(u.args[0])
def SumBaseQ(u):
# If u is a sum or a sum raised to an odd degree, SumBaseQ(u) returns True; else it returns False
return SumQ(u) or PowerQ(u) and OddQ(u.args[1]) and SumBaseQ(u.args[0])
def NegSumBaseQ(u):
# If u is a sum or a sum raised to an odd degree whose lead term has a negative form, NegSumBaseQ(u) returns True; else it returns False
return SumQ(u) and NegQ(First(u)) or PowerQ(u) and OddQ(u.args[1]) and NegSumBaseQ(u.args[0])
def AllNegTermQ(u):
# If all terms of u have a negative form, AllNegTermQ(u) returns True; else it returns False
if PowerQ(u):
if OddQ(u.exp):
return AllNegTermQ(u.base)
if SumQ(u):
return NegQ(First(u)) and AllNegTermQ(Rest(u))
return NegQ(u)
def SomeNegTermQ(u):
# If some term of u has a negative form, SomeNegTermQ(u) returns True; else it returns False
if PowerQ(u):
if OddQ(u.exp):
return SomeNegTermQ(u.base)
if SumQ(u):
return NegQ(First(u)) or SomeNegTermQ(Rest(u))
return NegQ(u)
def TrigSquareQ(u):
# If u is an expression of the form Sin(z)^2 or Cos(z)^2, TrigSquareQ(u) returns True, else it returns False
return PowerQ(u) and EqQ(u.args[1], 2) and MemberQ([sin, cos], Head(u.args[0]))
def RtAux(u, n):
if PowerQ(u):
return u.base**(u.exp/n)
if ComplexNumberQ(u):
a = Re(u)
b = Im(u)
if Not(IntegerQ(a) and IntegerQ(b)) and IntegerQ(a/(a**2 + b**2)) and IntegerQ(b/(a**2 + b**2)):
# Basis: a+b*I==1/(a/(a^2+b^2)-b/(a^2+b^2)*I)
return S(1)/RtAux(a/(a**2 + b**2) - b/(a**2 + b**2)*I, n)
else:
return NthRoot(u, n)
if ProductQ(u):
lst = SplitProduct(PositiveQ, u)
if ListQ(lst):
return RtAux(lst[0], n)*RtAux(lst[1], n)
lst = SplitProduct(NegativeQ, u)
if ListQ(lst):
if EqQ(lst[0], -1):
v = lst[1]
if PowerQ(v):
if NegativeQ(v.exp):
return 1/RtAux(-v.base**(-v.exp), n)
if ProductQ(v):
if ListQ(SplitProduct(SumBaseQ, v)):
lst = SplitProduct(AllNegTermQ, v)
if ListQ(lst):
return RtAux(-lst[0], n)*RtAux(lst[1], n)
lst = SplitProduct(NegSumBaseQ, v)
if ListQ(lst):
return RtAux(-lst[0], n)*RtAux(lst[1], n)
lst = SplitProduct(SomeNegTermQ, v)
if ListQ(lst):
return RtAux(-lst[0], n)*RtAux(lst[1], n)
lst = SplitProduct(SumBaseQ, v)
return RtAux(-lst[0], n)*RtAux(lst[1], n)
lst = SplitProduct(AtomBaseQ, v)
if ListQ(lst):
return RtAux(-lst[0], n)*RtAux(lst[1], n)
else:
return RtAux(-First(v), n)*RtAux(Rest(v), n)
if OddQ(n):
return -RtAux(v, n)
else:
return NthRoot(u, n)
else:
return RtAux(-lst[0], n)*RtAux(-lst[1], n)
lst = SplitProduct(AllNegTermQ, u)
if ListQ(lst) and ListQ(SplitProduct(SumBaseQ, lst[1])):
return RtAux(-lst[0], n)*RtAux(-lst[1], n)
lst = SplitProduct(NegSumBaseQ, u)
if ListQ(lst) and ListQ(SplitProduct(NegSumBaseQ, lst[1])):
return RtAux(-lst[0], n)*RtAux(-lst[1], n)
return u.func(*[RtAux(i, n) for i in u.args])
v = TrigSquare(u)
if Not(AtomQ(v)):
return RtAux(v, n)
if OddQ(n) and NegativeQ(u):
return -RtAux(-u, n)
if OddQ(n) and NegQ(u) and PosQ(-u):
return -RtAux(-u, n)
else:
return NthRoot(u, n)
def TrigSquare(u):
# If u is an expression of the form a-a*Sin(z)^2 or a-a*Cos(z)^2, TrigSquare(u) returns Cos(z)^2 or Sin(z)^2 respectively,
# else it returns False.
if SumQ(u):
for i in u.args:
v = SplitProduct(TrigSquareQ, i)
if v == False or SplitSum(v, u) == False:
return False
lst = SplitSum(SplitProduct(TrigSquareQ, i))
if lst and ZeroQ(lst[1][2] + lst[1]):
if Head(lst[0][0].args[0]) == sin:
return lst[1]*cos(lst[1][1][1][1])**2
return lst[1]*sin(lst[1][1][1][1])**2
else:
return False
else:
return False
def IntSum(u, x):
# If u is free of x or of the form c*(a+b*x)^m, IntSum[u,x] returns the antiderivative of u wrt x;
# else it returns d*Int[v,x] where d*v=u and d is free of x.
return Add(*[Integral(i, x) for i in u.args])
return Simp(FreeTerms(u, x)*x, x) + IntTerm(NonfreeTerms(u, x), x)
def IntTerm(expr, x):
# If u is of the form c*(a+b*x)**m, IntTerm(u,x) returns the antiderivative of u wrt x;
# else it returns d*Int(v,x) where d*v=u and d is free of x.
c = Wild('c', exclude=[x])
m = Wild('m', exclude=[x, 0])
v = Wild('v')
M = expr.match(c/v)
if M and len(M) == 2 and FreeQ(M[c], x) and LinearQ(M[v], x):
return Simp(M[c]*Log(RemoveContent(M[v], x))/Coefficient(M[v], x, 1), x)
M = expr.match(c*v**m)
if M and len(M) == 3 and NonzeroQ(M[m] + 1) and LinearQ(M[v], x):
return Simp(M[c]*M[v]**(M[m] + 1)/(Coefficient(M[v], x, 1)*(M[m] + 1)), x)
if SumQ(expr):
t = 0
for i in expr.args:
t += IntTerm(i, x)
return t
else:
u = expr
return Dist(FreeFactors(u,x), Integral(NonfreeFactors(u, x), x), x)
def Map2(f, lst1, lst2):
result = []
for i in range(0, len(lst1)):
result.append(f(lst1[i], lst2[i]))
return result
def ConstantFactor(u, x):
# (* ConstantFactor[u,x] returns a 2-element list of the factors of u[x] free of x and the
# factors not free of u[x]. Common constant factors of the terms of sums are also collected. *)
if FreeQ(u, x):
return [u, S(1)]
elif AtomQ(u):
return [S(1), u]
elif PowerQ(u):
if FreeQ(u.exp, x):
lst = ConstantFactor(u.base, x)
if IntegerQ(u.exp):
return [lst[0]**u.exp, lst[1]**u.exp]
tmp = PositiveFactors(lst[0])
if tmp == 1:
return [S(1), u]
return [tmp**u.exp, (NonpositiveFactors(lst[0])*lst[1])**u.exp]
elif ProductQ(u):
lst = [ConstantFactor(i, x) for i in u.args]
return [Mul(*[First(i) for i in lst]), Mul(*[i[1] for i in lst])]
elif SumQ(u):
lst1 = [ConstantFactor(i, x) for i in u.args]
if SameQ(*[i[1] for i in lst1]):
return [Add(*[i[0] for i in lst]), lst1[0][1]]
lst2 = CommonFactors([First(i) for i in lst1])
return [First(lst2), Add(*Map2(Mul, Rest(lst2), [i[1] for i in lst1]))]
return [S(1), u]
def SameQ(*args):
for i in range(0, len(args) - 1):
if args[i] != args[i+1]:
return False
return True
def ReplacePart(lst, a, b):
lst[b] = a
return lst
def CommonFactors(lst):
# (* If lst is a list of n terms, CommonFactors[lst] returns a n+1-element list whose first
# element is the product of the factors common to all terms of lst, and whose remaining
# elements are quotients of each term divided by the common factor. *)
lst1 = [NonabsurdNumberFactors(i) for i in lst]
lst2 = [AbsurdNumberFactors(i) for i in lst]
num = AbsurdNumberGCD(*lst2)
common = num
lst2 = [i/num for i in lst2]
while (True):
lst3 = [LeadFactor(i) for i in lst1]
if SameQ(*lst3):
common = common*lst3[0]
lst1 = [RemainingFactors(i) for i in lst1]
elif (all((LogQ(i) and IntegerQ(First(i)) and First(i) > 0) for i in lst3) and
all(RationalQ(i) for i in [FullSimplify(j/First(lst3)) for j in lst3])):
lst4 = [FullSimplify(j/First(lst3)) for j in lst3]
num = GCD(*lst4)
common = common*Log((First(lst3)[0])**num)
lst2 = [lst2[i]*lst4[i]/num for i in range(0, len(lst2))]
lst1 = [RemainingFactors(i) for i in lst1]
lst4 = [LeadDegree(i) for i in lst1]
if SameQ(*[LeadBase(i) for i in lst1]) and RationalQ(*lst4):
num = Smallest(lst4)
base = LeadBase(lst1[0])
if num != 0:
common = common*base**num
lst2 = [lst2[i]*base**(lst4[i] - num) for i in range(0, len(lst2))]
lst1 = [RemainingFactors(i) for i in lst1]
elif (Length(lst1) == 2 and ZeroQ(LeadBase(lst1[0]) + LeadBase(lst1[1])) and
NonzeroQ(lst1[0] - 1) and IntegerQ(lst4[0]) and FractionQ(lst4[1])):
num = Min(lst4)
base = LeadBase(lst1[1])
if num != 0:
common = common*base**num
lst2 = [lst2[0]*(-1)**lst4[0], lst2[1]]
lst2 = [lst2[i]*base**(lst4[i] - num) for i in range(0, len(lst2))]
lst1 = [RemainingFactors(i) for i in lst1]
elif (Length(lst1) == 2 and ZeroQ(lst1[0] + LeadBase(lst1[1])) and
NonzeroQ(lst1[1] - 1) and IntegerQ(lst1[1]) and FractionQ(lst4[0])):
num = Min(lst4)
base = LeadBase(lst1[0])
if num != 0:
common = common*base**num
lst2 = [lst2[0], lst2[1]*(-1)**lst4[1]]
lst2 = [lst2[i]*base**(lst4[i] - num) for i in range(0, len(lst2))]
lst1 = [RemainingFactors(i) for i in lst1]
else:
num = MostMainFactorPosition(lst3)
lst2 = ReplacePart(lst2, lst3[num]*lst2[num], num)
lst1 = ReplacePart(lst1, RemainingFactors(lst1[num]), num)
if all(i==1 for i in lst1):
return Prepend(lst2, common)
def MostMainFactorPosition(lst):
factor = S(1)
num = 0
for i in range(0, Length(lst)):
if FactorOrder(lst[i], factor) > 0:
factor = lst[i]
num = i
return num
SbaseS, SexponS = None, None
SexponFlagS = False
def FunctionOfExponentialQ(u, x):
# (* FunctionOfExponentialQ[u,x] returns True iff u is a function of F^v where F is a constant and v is linear in x, *)
# (* and such an exponential explicitly occurs in u (i.e. not just implicitly in hyperbolic functions). *)
global SbaseS, SexponS, SexponFlagS
SbaseS, SexponS = None, None
SexponFlagS = False
res = FunctionOfExponentialTest(u, x)
return res and SexponFlagS
def FunctionOfExponential(u, x):
global SbaseS, SexponS, SexponFlagS
# (* u is a function of F^v where v is linear in x. FunctionOfExponential[u,x] returns F^v. *)
SbaseS, SexponS = None, None
SexponFlagS = False
FunctionOfExponentialTest(u, x)
return SbaseS**SexponS
def FunctionOfExponentialFunction(u, x):
global SbaseS, SexponS, SexponFlagS
# (* u is a function of F^v where v is linear in x. FunctionOfExponentialFunction[u,x] returns u with F^v replaced by x. *)
SbaseS, SexponS = None, None
SexponFlagS = False
FunctionOfExponentialTest(u, x)
return SimplifyIntegrand(FunctionOfExponentialFunctionAux(u, x), x)
def FunctionOfExponentialFunctionAux(u, x):
# (* u is a function of F^v where v is linear in x, and the fluid variables $base$=F and $expon$=v. *)
# (* FunctionOfExponentialFunctionAux[u,x] returns u with F^v replaced by x. *)
global SbaseS, SexponS, SexponFlagS
if AtomQ(u):
return u
elif PowerQ(u):
if FreeQ(u.base, x) and LinearQ(u.exp, x):
if ZeroQ(Coefficient(SexponS, x, 0)):
return u.base**Coefficient(u.exp, x, 0)*x**FullSimplify(Log(u.base)*Coefficient(u.exp, x, 1)/(Log(SbaseS)*Coefficient(SexponS, x, 1)))
return x**FullSimplify(Log(u.base)*Coefficient(u.exp, x, 1)/(Log(SbaseS)*Coefficient(SexponS, x, 1)))
elif HyperbolicQ(u) and LinearQ(u.args[0], x):
tmp = x**FullSimplify(Coefficient(u.args[0], x, 1)/(Log(SbaseS)*Coefficient(SexponS, x, 1)))
if SinhQ(u):
return tmp/2 - 1/(2*tmp)
elif CoshQ(u):
return tmp/2 + 1/(2*tmp)
elif TanhQ(u):
return (tmp - 1/tmp)/(tmp + 1/tmp)
elif CothQ(u):
return (tmp + 1/tmp)/(tmp - 1/tmp)
elif SechQ(u):
return 2/(tmp + 1/tmp)
return 2/(tmp - 1/tmp)
if PowerQ(u):
if FreeQ(u.base, x) and SumQ(u.exp):
return FunctionOfExponentialFunctionAux(u.base**First(u.exp), x)*FunctionOfExponentialFunctionAux(u.base**Rest(u.exp), x)
return u.func(*[FunctionOfExponentialFunctionAux(i, x) for i in u.args])
def FunctionOfExponentialTest(u, x):
# (* FunctionOfExponentialTest[u,x] returns True iff u is a function of F^v where F is a constant and v is linear in x. *)
# (* Before it is called, the fluid variables $base$ and $expon$ should be set to Null and $exponFlag$ to False. *)
# (* If u is a function of F^v, $base$ and $expon$ are set to F and v, respectively. *)
# (* If an explicit exponential occurs in u, $exponFlag$ is set to True. *)
global SbaseS, SexponS, SexponFlagS
if FreeQ(u, x):
return True
elif u == x or CalculusQ(u):
return False
elif PowerQ(u):
if FreeQ(u.base, x) and LinearQ(u.exp, x):
SexponFlagS = True
return FunctionOfExponentialTestAux(u.base, u.exp, x)
elif HyperbolicQ(u) and LinearQ(u.args[0], x):
return FunctionOfExponentialTestAux(E, u.args[0], x)
if PowerQ(u):
if FreeQ(u.base, x) and SumQ(u.exp):
return FunctionOfExponentialTest(u.base**First(u.exp), x) and FunctionOfExponentialTest(u.base**Rest(u.exp), x)
return all(FunctionOfExponentialTest(i, x) for i in u.args)
def FunctionOfExponentialTestAux(base, expon, x):
global SbaseS, SexponS, SexponFlagS
if SbaseS is None:
SbaseS = base
SexponS = expon
return True
tmp = FullSimplify(Log(base)*Coefficient(expon, x, 1)/(Log(SbaseS)*Coefficient(SexponS, x, 1)))
if Not(RationalQ(tmp)):
return False
elif ZeroQ(Coefficient(SexponS, x, 0)) or NonzeroQ(tmp - FullSimplify(Log(base)*Coefficient(expon, x, 0)/(Log(SbaseS)*Coefficient(SexponS, x, 0)))):
if PositiveIntegerQ(base, SbaseS) and base<SbaseS:
SbaseS = base
SexponS = expon
tmp = 1/tmp
SexponS = Coefficient(SexponS, x, 1)*x/Denominator(tmp)
if tmp < 0 and NegQ(Coefficient(SexponS, x, 1)):
SexponS = -SexponS
return True
else:
return True
if PositiveIntegerQ(base, SbaseS) and base < SbaseS:
SbaseS = base
SexponS = expon
tmp = 1/tmp
SexponS = SexponS/Denominator(tmp)
if tmp < 0 and NegQ(Coefficient(SexponS, x, 1)):
SexponS = -SexponS
return True
return True
def stdev(lst):
"""Calculates the standard deviation for a list of numbers."""
num_items = len(lst)
mean = sum(lst) / num_items
differences = [x - mean for x in lst]
sq_differences = [d ** 2 for d in differences]
ssd = sum(sq_differences)
variance = ssd / num_items
sd = sqrt(variance)
return sd
def rubi_test(expr, x, optimal_output, expand=False, _hyper_check=False, _diff=False, _numerical=False):
#Returns True if (expr - optimal_output) is equal to 0 or a constant
#expr: integrated expression
#x: integration variable
#expand=True equates `expr` with `optimal_output` in expanded form
#_hyper_check=True evaluates numerically
#_diff=True differentiates the expressions before equating
#_numerical=True equates the expressions at random `x`. Normally used for large expressions.
from sympy import nsimplify
if not expr.has(csc, sec, cot, csch, sech, coth):
optimal_output = process_trig(optimal_output)
if expr == optimal_output:
return True
if simplify(expr) == simplify(optimal_output):
return True
if nsimplify(expr) == nsimplify(optimal_output):
return True
if expr.has(sym_exp):
expr = powsimp(powdenest(expr), force=True)
if simplify(expr) == simplify(powsimp(optimal_output, force=True)):
return True
res = expr - optimal_output
if _numerical:
args = res.free_symbols
rand_val = []
try:
for i in range(0, 5): # check at 5 random points
rand_x = randint(1, 40)
substitutions = dict((s, rand_x) for s in args)
rand_val.append(float(abs(res.subs(substitutions).n())))
if stdev(rand_val) < Pow(10, -3):
return True
except:
pass
# return False
dres = res.diff(x)
if _numerical:
args = dres.free_symbols
rand_val = []
try:
for i in range(0, 5): # check at 5 random points
rand_x = randint(1, 40)
substitutions = dict((s, rand_x) for s in args)
rand_val.append(float(abs(dres.subs(substitutions).n())))
if stdev(rand_val) < Pow(10, -3):
return True
# return False
except:
pass
# return False
r = Simplify(nsimplify(res))
if r == 0 or (not r.has(x)):
return True
if _diff:
if dres == 0:
return True
elif Simplify(dres) == 0:
return True
if expand: # expands the expression and equates
e = res.expand()
if Simplify(e) == 0 or (not e.has(x)):
return True
return False
def If(cond, t, f):
# returns t if condition is true else f
if cond:
return t
return f
def IntQuadraticQ(a, b, c, d, e, m, p, x):
# (* IntQuadraticQ[a,b,c,d,e,m,p,x] returns True iff (d+e*x)^m*(a+b*x+c*x^2)^p is integrable wrt x in terms of non-Appell functions. *)
return IntegerQ(p) or PositiveIntegerQ(m) or IntegersQ(2*m, 2*p) or IntegersQ(m, 4*p) or IntegersQ(m, p + S(1)/3) and (ZeroQ(c**2*d**2 - b*c*d*e + b**2*e**2 - 3*a*c*e**2) or ZeroQ(c**2*d**2 - b*c*d*e - 2*b**2*e**2 + 9*a*c*e**2))
def IntBinomialQ(*args):
#(* IntBinomialQ(a,b,c,n,m,p,x) returns True iff (c*x)^m*(a+b*x^n)^p is integrable wrt x in terms of non-hypergeometric functions. *)
if len(args) == 8:
a, b, c, d, n, p, q, x = args
return IntegersQ(p,q) or PositiveIntegerQ(p) or PositiveIntegerQ(q) or (ZeroQ(n-2) or ZeroQ(n-4)) and (IntegersQ(p,4*q) or IntegersQ(4*p,q)) or ZeroQ(n-2) and (IntegersQ(2*p,2*q) or IntegersQ(3*p,q) and ZeroQ(b*c+3*a*d) or IntegersQ(p,3*q) and ZeroQ(3*b*c+a*d))
elif len(args) == 7:
a, b, c, n, m, p, x = args
return IntegerQ(2*p) or IntegerQ((m+1)/n + p) or (ZeroQ(n - 2) or ZeroQ(n - 4)) and IntegersQ(2*m, 4*p) or ZeroQ(n - 2) and IntegerQ(6*p) and (IntegerQ(m) or IntegerQ(m - p))
elif len(args) == 10:
a, b, c, d, e, m, n, p, q, x = args
return IntegersQ(p,q) or PositiveIntegerQ(p) or PositiveIntegerQ(q) or ZeroQ(n-2) and IntegerQ(m) and IntegersQ(2*p,2*q) or ZeroQ(n-4) and (IntegersQ(m,p,2*q) or IntegersQ(m,2*p,q))
def RectifyTangent(*args):
# (* RectifyTangent(u,a,b,r,x) returns an expression whose derivative equals the derivative of r*ArcTan(a+b*Tan(u)) wrt x. *)
if len(args) == 5:
u, a, b, r, x = args
t1 = Together(a)
t2 = Together(b)
if (PureComplexNumberQ(t1) or (ProductQ(t1) and any(PureComplexNumberQ(i) for i in t1.args))) and (PureComplexNumberQ(t2) or ProductQ(t2) and any(PureComplexNumberQ(i) for i in t2.args)):
c = a/I
d = b/I
if NegativeQ(d):
return RectifyTangent(u, -a, -b, -r, x)
e = SmartDenominator(Together(c + d*x))
c = c*e
d = d*e
if EvenQ(Denominator(NumericFactor(Together(u)))):
return I*r*Log(RemoveContent(Simplify((c+e)**2+d**2)+Simplify((c+e)**2-d**2)*Cos(2*u)+Simplify(2*(c+e)*d)*Sin(2*u),x))/4 - I*r*Log(RemoveContent(Simplify((c-e)**2+d**2)+Simplify((c-e)**2-d**2)*Cos(2*u)+Simplify(2*(c-e)*d)*Sin(2*u),x))/4
return I*r*Log(RemoveContent(Simplify((c+e)**2)+Simplify(2*(c+e)*d)*Cos(u)*Sin(u)-Simplify((c+e)**2-d**2)*Sin(u)**2,x))/4 - I*r*Log(RemoveContent(Simplify((c-e)**2)+Simplify(2*(c-e)*d)*Cos(u)*Sin(u)-Simplify((c-e)**2-d**2)*Sin(u)**2,x))/4
elif NegativeQ(b):
return RectifyTangent(u, -a, -b, -r, x)
elif EvenQ(Denominator(NumericFactor(Together(u)))):
return r*SimplifyAntiderivative(u,x) + r*ArcTan(Simplify((2*a*b*Cos(2*u)-(1+a**2-b**2)*Sin(2*u))/(a**2+(1+b)**2+(1+a**2-b**2)*Cos(2*u)+2*a*b*Sin(2*u))))
return r*SimplifyAntiderivative(u,x) - r*ArcTan(ActivateTrig(Simplify((a*b-2*a*b*cos(u)**2+(1+a**2-b**2)*cos(u)*sin(u))/(b*(1+b)+(1+a**2-b**2)*cos(u)**2+2*a*b*cos(u)*sin(u)))))
u, a, b, x = args
t = Together(a)
if PureComplexNumberQ(t) or (ProductQ(t) and any(PureComplexNumberQ(i) for i in t.args)):
c = a/I
if NegativeQ(c):
return RectifyTangent(u, -a, -b, x)
if ZeroQ(c - 1):
if EvenQ(Denominator(NumericFactor(Together(u)))):
return I*b*ArcTanh(Sin(2*u))/2
return I*b*ArcTanh(2*cos(u)*sin(u))/2
e = SmartDenominator(c)
c = c*e
return I*b*Log(RemoveContent(e*Cos(u)+c*Sin(u),x))/2 - I*b*Log(RemoveContent(e*Cos(u)-c*Sin(u),x))/2
elif NegativeQ(a):
return RectifyTangent(u, -a, -b, x)
elif ZeroQ(a - 1):
return b*SimplifyAntiderivative(u, x)
elif EvenQ(Denominator(NumericFactor(Together(u)))):
c = Simplify((1 + a)/(1 - a))
numr = SmartNumerator(c)
denr = SmartDenominator(c)
return b*SimplifyAntiderivative(u,x) - b*ArcTan(NormalizeLeadTermSigns(denr*Sin(2*u)/(numr+denr*Cos(2*u)))),
elif PositiveQ(a - 1):
c = Simplify(1/(a - 1))
numr = SmartNumerator(c)
denr = SmartDenominator(c)
return b*SimplifyAntiderivative(u,x) + b*ArcTan(NormalizeLeadTermSigns(denr*Cos(u)*Sin(u)/(numr+denr*Sin(u)**2))),
c = Simplify(a/(1 - a))
numr = SmartNumerator(c)
denr = SmartDenominator(c)
return b*SimplifyAntiderivative(u,x) - b*ArcTan(NormalizeLeadTermSigns(denr*Cos(u)*Sin(u)/(numr+denr*Cos(u)**2)))
def RectifyCotangent(*args):
#(* RectifyCotangent[u,a,b,r,x] returns an expression whose derivative equals the derivative of r*ArcTan[a+b*Cot[u]] wrt x. *)
if len(args) == 5:
u, a, b, r, x = args
t1 = Together(a)
t2 = Together(b)
if (PureComplexNumberQ(t1) or (ProductQ(t1) and any(PureComplexNumberQ(i) for i in t1.args))) and (PureComplexNumberQ(t2) or ProductQ(t2) and any(PureComplexNumberQ(i) for i in t2.args)):
c = a/I
d = b/I
if NegativeQ(d):
return RectifyTangent(u,-a,-b,-r,x)
e = SmartDenominator(Together(c + d*x))
c = c*e
d = d*e
if EvenQ(Denominator(NumericFactor(Together(u)))):
return I*r*Log(RemoveContent(Simplify((c+e)**2+d**2)-Simplify((c+e)**2-d**2)*Cos(2*u)+Simplify(2*(c+e)*d)*Sin(2*u),x))/4 - I*r*Log(RemoveContent(Simplify((c-e)**2+d**2)-Simplify((c-e)**2-d**2)*Cos(2*u)+Simplify(2*(c-e)*d)*Sin(2*u),x))/4
return I*r*Log(RemoveContent(Simplify((c+e)**2)-Simplify((c+e)**2-d**2)*Cos(u)**2+Simplify(2*(c+e)*d)*Cos(u)*Sin(u),x))/4 - I*r*Log(RemoveContent(Simplify((c-e)**2)-Simplify((c-e)**2-d**2)*Cos(u)**2+Simplify(2*(c-e)*d)*Cos(u)*Sin(u),x))/4
elif NegativeQ(b):
return RectifyCotangent(u,-a,-b,-r,x)
elif EvenQ(Denominator(NumericFactor(Together(u)))):
return -r*SimplifyAntiderivative(u,x) - r*ArcTan(Simplify((2*a*b*Cos(2*u)+(1+a**2-b**2)*Sin(2*u))/(a**2+(1+b)**2-(1+a**2-b**2)*Cos(2*u)+2*a*b*Sin(2*u))))
return -r*SimplifyAntiderivative(u,x) - r*ArcTan(ActivateTrig(Simplify((a*b-2*a*b*sin(u)**2+(1+a**2-b**2)*cos(u)*sin(u))/(b*(1+b)+(1+a**2-b**2)*sin(u)**2+2*a*b*cos(u)*sin(u)))))
u, a, b, x = args
t = Together(a)
if PureComplexNumberQ(t) or (ProductQ(t) and any(PureComplexNumberQ(i) for i in t.args)):
c = a/I
if NegativeQ(c):
return RectifyCotangent(u,-a,-b,x)
elif ZeroQ(c - 1):
if EvenQ(Denominator(NumericFactor(Together(u)))):
return -I*b*ArcTanh(Sin(2*u))/2
return -I*b*ArcTanh(2*Cos(u)*Sin(u))/2
e = SmartDenominator(c)
c = c*e
return -I*b*Log(RemoveContent(c*Cos(u)+e*Sin(u),x))/2 + I*b*Log(RemoveContent(c*Cos(u)-e*Sin(u),x))/2
elif NegativeQ(a):
return RectifyCotangent(u,-a,-b,x)
elif ZeroQ(a-1):
return b*SimplifyAntiderivative(u,x)
elif EvenQ(Denominator(NumericFactor(Together(u)))):
c = Simplify(a - 1)
numr = SmartNumerator(c)
denr = SmartDenominator(c)
return b*SimplifyAntiderivative(u,x) - b*ArcTan(NormalizeLeadTermSigns(denr*Cos(u)*Sin(u)/(numr+denr*Cos(u)**2)))
c = Simplify(a/(1-a))
numr = SmartNumerator(c)
denr = SmartDenominator(c)
return b*SimplifyAntiderivative(u,x) + b*ArcTan(NormalizeLeadTermSigns(denr*Cos(u)*Sin(u)/(numr+denr*Sin(u)**2)))
def Inequality(*args):
f = args[1::2]
e = args[0::2]
r = []
for i in range(0, len(f)):
r.append(f[i](e[i], e[i + 1]))
return all(r)
def Condition(r, c):
# returns r if c is True
if c:
return r
else:
raise NotImplementedError('In Condition()')
def Simp(u, x):
u = replace_pow_exp(u)
return NormalizeSumFactors(SimpHelp(u, x))
def SimpHelp(u, x):
if AtomQ(u):
return u
elif FreeQ(u, x):
v = SmartSimplify(u)
if LeafCount(v) <= LeafCount(u):
return v
return u
elif ProductQ(u):
#m = MatchQ[Rest[u],a_.+n_*Pi+b_.*v_ /; FreeQ[{a,b},x] && Not[FreeQ[v,x]] && EqQ[n^2,1/4]]
#if EqQ(First(u), S(1)/2) and m:
# if
#If[EqQ[First[u],1/2] && MatchQ[Rest[u],a_.+n_*Pi+b_.*v_ /; FreeQ[{a,b},x] && Not[FreeQ[v,x]] && EqQ[n^2,1/4]],
# If[MatchQ[Rest[u],n_*Pi+b_.*v_ /; FreeQ[b,x] && Not[FreeQ[v,x]] && EqQ[n^2,1/4]],
# Map[Function[1/2*#],Rest[u]],
# If[MatchQ[Rest[u],m_*a_.+n_*Pi+p_*b_.*v_ /; FreeQ[{a,b},x] && Not[FreeQ[v,x]] && IntegersQ[m/2,p/2]],
# Map[Function[1/2*#],Rest[u]],
# u]],
v = FreeFactors(u, x)
w = NonfreeFactors(u, x)
v = NumericFactor(v)*SmartSimplify(NonnumericFactors(v)*x**2)/x**2
if ProductQ(w):
w = Mul(*[SimpHelp(i,x) for i in w.args])
else:
w = SimpHelp(w, x)
w = FactorNumericGcd(w)
v = MergeFactors(v, w)
if ProductQ(v):
return Mul(*[SimpFixFactor(i, x) for i in v.args])
return v
elif SumQ(u):
Pi = pi
a_ = Wild('a', exclude=[x])
b_ = Wild('b', exclude=[x, 0])
n_ = Wild('n', exclude=[x, 0, 0])
pattern = a_ + n_*Pi + b_*x
match = u.match(pattern)
m = False
if match:
if EqQ(match[n_]**3, S(1)/16):
m = True
if m:
return u
elif PolynomialQ(u, x) and Exponent(u, x)<=0:
return SimpHelp(Coefficient(u, x, 0), x)
elif PolynomialQ(u, x) and Exponent(u, x) == 1 and Coefficient(u, x, 0) == 0:
return SimpHelp(Coefficient(u, x, 1), x)*x
v = 0
w = 0
for i in u.args:
if FreeQ(i, x):
v = i + v
else:
w = i + w
v = SmartSimplify(v)
if SumQ(w):
w = Add(*[SimpHelp(i, x) for i in w.args])
else:
w = SimpHelp(w, x)
return v + w
return u.func(*[SimpHelp(i, x) for i in u.args])
def SplitProduct(func, u):
#(* If func[v] is True for a factor v of u, SplitProduct[func,u] returns {v, u/v} where v is the first such factor; else it returns False. *)
if ProductQ(u):
if func(First(u)):
return [First(u), Rest(u)]
lst = SplitProduct(func, Rest(u))
if AtomQ(lst):
return False
return [lst[0], First(u)*lst[1]]
if func(u):
return [u, 1]
return False
def SplitSum(func, u):
# (* If func[v] is nonatomic for a term v of u, SplitSum[func,u] returns {func[v], u-v} where v is the first such term; else it returns False. *)
if SumQ(u):
if Not(AtomQ(func(First(u)))):
return [func(First(u)), Rest(u)]
lst = SplitSum(func, Rest(u))
if AtomQ(lst):
return False
return [lst[0], First(u) + lst[1]]
elif Not(AtomQ(func(u))):
return [func(u), 0]
return False
def SubstFor(*args):
if len(args) == 4:
w, v, u, x = args
# u is a function of v. SubstFor(w,v,u,x) returns w times u with v replaced by x.
return SimplifyIntegrand(w*SubstFor(v, u, x), x)
v, u, x = args
# u is a function of v. SubstFor(v, u, x) returns u with v replaced by x.
if AtomQ(v):
return Subst(u, v, x)
elif Not(EqQ(FreeFactors(v, x), 1)):
return SubstFor(NonfreeFactors(v, x), u, x/FreeFactors(v, x))
elif SinQ(v):
return SubstForTrig(u, x, Sqrt(1 - x**2), v.args[0], x)
elif CosQ(v):
return SubstForTrig(u, Sqrt(1 - x**2), x, v.args[0], x)
elif TanQ(v):
return SubstForTrig(u, x/Sqrt(1 + x**2), 1/Sqrt(1 + x**2), v.args[0], x)
elif CotQ(v):
return SubstForTrig(u, 1/Sqrt(1 + x**2), x/Sqrt(1 + x**2), v.args[0], x)
elif SecQ(v):
return SubstForTrig(u, 1/Sqrt(1 - x**2), 1/x, v.args[0], x)
elif CscQ(v):
return SubstForTrig(u, 1/x, 1/Sqrt(1 - x**2), v.args[0], x)
elif SinhQ(v):
return SubstForHyperbolic(u, x, Sqrt(1 + x**2), v.args[0], x)
elif CoshQ(v):
return SubstForHyperbolic(u, Sqrt( - 1 + x**2), x, v.args[0], x)
elif TanhQ(v):
return SubstForHyperbolic(u, x/Sqrt(1 - x**2), 1/Sqrt(1 - x**2), v.args[0], x)
elif CothQ(v):
return SubstForHyperbolic(u, 1/Sqrt( - 1 + x**2), x/Sqrt( - 1 + x**2), v.args[0], x)
elif SechQ(v):
return SubstForHyperbolic(u, 1/Sqrt( - 1 + x**2), 1/x, v.args[0], x)
elif CschQ(v):
return SubstForHyperbolic(u, 1/x, 1/Sqrt(1 + x**2), v.args[0], x)
else:
return SubstForAux(u, v, x)
def SubstForAux(u, v, x):
# u is a function of v. SubstForAux(u, v, x) returns u with v replaced by x.
if u==v:
return x
elif AtomQ(u):
if PowerQ(v):
if FreeQ(v.exp, x) and ZeroQ(u - v.base):
return x**Simplify(1/v.exp)
return u
elif PowerQ(u):
if FreeQ(u.exp, x):
if ZeroQ(u.base - v):
return x**u.exp
if PowerQ(v):
if FreeQ(v.exp, x) and ZeroQ(u.base - v.base):
return x**Simplify(u.exp/v.exp)
return SubstForAux(u.base, v, x)**u.exp
elif ProductQ(u) and Not(EqQ(FreeFactors(u, x), 1)):
return FreeFactors(u, x)*SubstForAux(NonfreeFactors(u, x), v, x)
elif ProductQ(u) and ProductQ(v):
return SubstForAux(First(u), First(v), x)
return u.func(*[SubstForAux(i, v, x) for i in u.args])
def FresnelS(x):
return fresnels(x)
def FresnelC(x):
return fresnelc(x)
def Erf(x):
return erf(x)
def Erfc(x):
return erfc(x)
def Erfi(x):
return erfi(x)
class Gamma(Function):
@classmethod
def eval(cls,*args):
a = args[0]
if len(args) == 1:
return gamma(a)
else:
b = args[1]
if (NumericQ(a) and NumericQ(b)) or a == 1:
return uppergamma(a, b)
def FunctionOfTrigOfLinearQ(u, x):
# If u is an algebraic function of trig functions of a linear function of x,
# FunctionOfTrigOfLinearQ[u,x] returns True; else it returns False.
if FunctionOfTrig(u, None, x) and AlgebraicTrigFunctionQ(u, x) and FunctionOfLinear(FunctionOfTrig(u, None, x), x):
return True
else:
return False
def ElementaryFunctionQ(u):
# ElementaryExpressionQ[u] returns True if u is a sum, product, or power and all the operands
# are elementary expressions; or if u is a call on a trig, hyperbolic, or inverse function
# and all the arguments are elementary expressions; else it returns False.
if AtomQ(u):
return True
elif SumQ(u) or ProductQ(u) or PowerQ(u) or TrigQ(u) or HyperbolicQ(u) or InverseFunctionQ(u):
for i in u.args:
if not ElementaryFunctionQ(i):
return False
return True
return False
def Complex(a, b):
return a + I*b
def UnsameQ(a, b):
return a != b
@doctest_depends_on(modules=('matchpy',))
def _SimpFixFactor():
replacer = ManyToOneReplacer()
pattern1 = Pattern(UtilityOperator(Pow(Add(Mul(Complex(S(0), c_), WC('a', S(1))), Mul(Complex(S(0), d_), WC('b', S(1)))), WC('p', S(1))), x_), CustomConstraint(lambda p: IntegerQ(p)))
rule1 = ReplacementRule(pattern1, lambda b, c, x, a, p, d : Mul(Pow(I, p), SimpFixFactor(Pow(Add(Mul(a, c), Mul(b, d)), p), x)))
replacer.add(rule1)
pattern2 = Pattern(UtilityOperator(Pow(Add(Mul(Complex(S(0), d_), WC('a', S(1))), Mul(Complex(S(0), e_), WC('b', S(1))), Mul(Complex(S(0), f_), WC('c', S(1)))), WC('p', S(1))), x_), CustomConstraint(lambda p: IntegerQ(p)))
rule2 = ReplacementRule(pattern2, lambda b, c, x, f, a, p, e, d : Mul(Pow(I, p), SimpFixFactor(Pow(Add(Mul(a, d), Mul(b, e), Mul(c, f)), p), x)))
replacer.add(rule2)
pattern3 = Pattern(UtilityOperator(Pow(Add(Mul(WC('a', S(1)), Pow(c_, r_)), Mul(WC('b', S(1)), Pow(x_, WC('n', S(1))))), WC('p', S(1))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda n, p: IntegersQ(n, p)), CustomConstraint(lambda c: AtomQ(c)), CustomConstraint(lambda r: RationalQ(r)), CustomConstraint(lambda r: Less(r, S(0))))
rule3 = ReplacementRule(pattern3, lambda b, c, r, n, x, a, p : Mul(Pow(c, Mul(r, p)), SimpFixFactor(Pow(Add(a, Mul(Mul(b, Pow(Pow(c, r), S(-1))), Pow(x, n))), p), x)))
replacer.add(rule3)
pattern4 = Pattern(UtilityOperator(Pow(Add(WC('a', S(0)), Mul(WC('b', S(1)), Pow(c_, r_), Pow(x_, WC('n', S(1))))), WC('p', S(1))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda n, p: IntegersQ(n, p)), CustomConstraint(lambda c: AtomQ(c)), CustomConstraint(lambda r: RationalQ(r)), CustomConstraint(lambda r: Less(r, S(0))))
rule4 = ReplacementRule(pattern4, lambda b, c, r, n, x, a, p : Mul(Pow(c, Mul(r, p)), SimpFixFactor(Pow(Add(Mul(a, Pow(Pow(c, r), S(-1))), Mul(b, Pow(x, n))), p), x)))
replacer.add(rule4)
pattern5 = Pattern(UtilityOperator(Pow(Add(Mul(WC('a', S(1)), Pow(c_, WC('s', S(1)))), Mul(WC('b', S(1)), Pow(c_, WC('r', S(1))), Pow(x_, WC('n', S(1))))), WC('p', S(1))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda n, p: IntegersQ(n, p)), CustomConstraint(lambda r, s: RationalQ(s, r)), CustomConstraint(lambda r, s: Inequality(S(0), Less, s, LessEqual, r)), CustomConstraint(lambda p, c, s: UnsameQ(Pow(c, Mul(s, p)), S(-1))))
rule5 = ReplacementRule(pattern5, lambda b, c, r, n, x, a, p, s : Mul(Pow(c, Mul(s, p)), SimpFixFactor(Pow(Add(a, Mul(b, Pow(c, Add(r, Mul(S(-1), s))), Pow(x, n))), p), x)))
replacer.add(rule5)
pattern6 = Pattern(UtilityOperator(Pow(Add(Mul(WC('a', S(1)), Pow(c_, WC('s', S(1)))), Mul(WC('b', S(1)), Pow(c_, WC('r', S(1))), Pow(x_, WC('n', S(1))))), WC('p', S(1))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda n, p: IntegersQ(n, p)), CustomConstraint(lambda r, s: RationalQ(s, r)), CustomConstraint(lambda s, r: Less(S(0), r, s)), CustomConstraint(lambda p, c, r: UnsameQ(Pow(c, Mul(r, p)), S(-1))))
rule6 = ReplacementRule(pattern6, lambda b, c, r, n, x, a, p, s : Mul(Pow(c, Mul(r, p)), SimpFixFactor(Pow(Add(Mul(a, Pow(c, Add(s, Mul(S(-1), r)))), Mul(b, Pow(x, n))), p), x)))
replacer.add(rule6)
return replacer
@doctest_depends_on(modules=('matchpy',))
def SimpFixFactor(expr, x):
r = SimpFixFactor_replacer.replace(UtilityOperator(expr, x))
if isinstance(r, UtilityOperator):
return expr
return r
@doctest_depends_on(modules=('matchpy',))
def _FixSimplify():
Plus = Add
def cons_f1(n):
return OddQ(n)
cons1 = CustomConstraint(cons_f1)
def cons_f2(m):
return RationalQ(m)
cons2 = CustomConstraint(cons_f2)
def cons_f3(n):
return FractionQ(n)
cons3 = CustomConstraint(cons_f3)
def cons_f4(u):
return SqrtNumberSumQ(u)
cons4 = CustomConstraint(cons_f4)
def cons_f5(v):
return SqrtNumberSumQ(v)
cons5 = CustomConstraint(cons_f5)
def cons_f6(u):
return PositiveQ(u)
cons6 = CustomConstraint(cons_f6)
def cons_f7(v):
return PositiveQ(v)
cons7 = CustomConstraint(cons_f7)
def cons_f8(v):
return SqrtNumberSumQ(S(1)/v)
cons8 = CustomConstraint(cons_f8)
def cons_f9(m):
return IntegerQ(m)
cons9 = CustomConstraint(cons_f9)
def cons_f10(u):
return NegativeQ(u)
cons10 = CustomConstraint(cons_f10)
def cons_f11(n, m, a, b):
return RationalQ(a, b, m, n)
cons11 = CustomConstraint(cons_f11)
def cons_f12(a):
return Greater(a, S(0))
cons12 = CustomConstraint(cons_f12)
def cons_f13(b):
return Greater(b, S(0))
cons13 = CustomConstraint(cons_f13)
def cons_f14(p):
return PositiveIntegerQ(p)
cons14 = CustomConstraint(cons_f14)
def cons_f15(p):
return IntegerQ(p)
cons15 = CustomConstraint(cons_f15)
def cons_f16(p, n):
return Greater(-n + p, S(0))
cons16 = CustomConstraint(cons_f16)
def cons_f17(a, b):
return SameQ(a + b, S(0))
cons17 = CustomConstraint(cons_f17)
def cons_f18(n):
return Not(IntegerQ(n))
cons18 = CustomConstraint(cons_f18)
def cons_f19(c, a, b, d):
return ZeroQ(-a*d + b*c)
cons19 = CustomConstraint(cons_f19)
def cons_f20(a):
return Not(RationalQ(a))
cons20 = CustomConstraint(cons_f20)
def cons_f21(t):
return IntegerQ(t)
cons21 = CustomConstraint(cons_f21)
def cons_f22(n, m):
return RationalQ(m, n)
cons22 = CustomConstraint(cons_f22)
def cons_f23(n, m):
return Inequality(S(0), Less, m, LessEqual, n)
cons23 = CustomConstraint(cons_f23)
def cons_f24(p, n, m):
return RationalQ(m, n, p)
cons24 = CustomConstraint(cons_f24)
def cons_f25(p, n, m):
return Inequality(S(0), Less, m, LessEqual, n, LessEqual, p)
cons25 = CustomConstraint(cons_f25)
def cons_f26(p, n, m, q):
return Inequality(S(0), Less, m, LessEqual, n, LessEqual, p, LessEqual, q)
cons26 = CustomConstraint(cons_f26)
def cons_f27(w):
return Not(RationalQ(w))
cons27 = CustomConstraint(cons_f27)
def cons_f28(n):
return Less(n, S(0))
cons28 = CustomConstraint(cons_f28)
def cons_f29(n, w, v):
return ZeroQ(v + w**(-n))
cons29 = CustomConstraint(cons_f29)
def cons_f30(n):
return IntegerQ(n)
cons30 = CustomConstraint(cons_f30)
def cons_f31(w, v):
return ZeroQ(v + w)
cons31 = CustomConstraint(cons_f31)
def cons_f32(p, n):
return IntegerQ(n/p)
cons32 = CustomConstraint(cons_f32)
def cons_f33(w, v):
return ZeroQ(v - w)
cons33 = CustomConstraint(cons_f33)
def cons_f34(p, n):
return IntegersQ(n, n/p)
cons34 = CustomConstraint(cons_f34)
def cons_f35(a):
return AtomQ(a)
cons35 = CustomConstraint(cons_f35)
def cons_f36(b):
return AtomQ(b)
cons36 = CustomConstraint(cons_f36)
pattern1 = Pattern(UtilityOperator((w_ + Complex(S(0), b_)*WC('v', S(1)))**WC('n', S(1))*Complex(S(0), a_)*WC('u', S(1))), cons1)
def replacement1(n, u, w, v, a, b):
return (S(-1))**(n/S(2) + S(1)/2)*a*u*FixSimplify((b*v - w*Complex(S(0), S(1)))**n)
rule1 = ReplacementRule(pattern1, replacement1)
def With2(m, n, u, w, v):
z = u**(m/GCD(m, n))*v**(n/GCD(m, n))
if Or(AbsurdNumberQ(z), SqrtNumberSumQ(z)):
return True
return False
pattern2 = Pattern(UtilityOperator(u_**WC('m', S(1))*v_**n_*WC('w', S(1))), cons2, cons3, cons4, cons5, cons6, cons7, CustomConstraint(With2))
def replacement2(m, n, u, w, v):
z = u**(m/GCD(m, n))*v**(n/GCD(m, n))
return FixSimplify(w*z**GCD(m, n))
rule2 = ReplacementRule(pattern2, replacement2)
def With3(m, n, u, w, v):
z = u**(m/GCD(m, -n))*v**(n/GCD(m, -n))
if Or(AbsurdNumberQ(z), SqrtNumberSumQ(z)):
return True
return False
pattern3 = Pattern(UtilityOperator(u_**WC('m', S(1))*v_**n_*WC('w', S(1))), cons2, cons3, cons4, cons8, cons6, cons7, CustomConstraint(With3))
def replacement3(m, n, u, w, v):
z = u**(m/GCD(m, -n))*v**(n/GCD(m, -n))
return FixSimplify(w*z**GCD(m, -n))
rule3 = ReplacementRule(pattern3, replacement3)
def With4(m, n, u, w, v):
z = v**(n/GCD(m, n))*(-u)**(m/GCD(m, n))
if Or(AbsurdNumberQ(z), SqrtNumberSumQ(z)):
return True
return False
pattern4 = Pattern(UtilityOperator(u_**WC('m', S(1))*v_**n_*WC('w', S(1))), cons9, cons3, cons4, cons5, cons10, cons7, CustomConstraint(With4))
def replacement4(m, n, u, w, v):
z = v**(n/GCD(m, n))*(-u)**(m/GCD(m, n))
return FixSimplify(-w*z**GCD(m, n))
rule4 = ReplacementRule(pattern4, replacement4)
def With5(m, n, u, w, v):
z = v**(n/GCD(m, -n))*(-u)**(m/GCD(m, -n))
if Or(AbsurdNumberQ(z), SqrtNumberSumQ(z)):
return True
return False
pattern5 = Pattern(UtilityOperator(u_**WC('m', S(1))*v_**n_*WC('w', S(1))), cons9, cons3, cons4, cons8, cons10, cons7, CustomConstraint(With5))
def replacement5(m, n, u, w, v):
z = v**(n/GCD(m, -n))*(-u)**(m/GCD(m, -n))
return FixSimplify(-w*z**GCD(m, -n))
rule5 = ReplacementRule(pattern5, replacement5)
def With6(p, m, n, u, w, v, a, b):
c = a**(m/p)*b**n
if RationalQ(c):
return True
return False
pattern6 = Pattern(UtilityOperator(a_**m_*(b_**n_*WC('v', S(1)) + u_)**WC('p', S(1))*WC('w', S(1))), cons11, cons12, cons13, cons14, CustomConstraint(With6))
def replacement6(p, m, n, u, w, v, a, b):
c = a**(m/p)*b**n
return FixSimplify(w*(a**(m/p)*u + c*v)**p)
rule6 = ReplacementRule(pattern6, replacement6)
pattern7 = Pattern(UtilityOperator(a_**WC('m', S(1))*(a_**n_*WC('u', S(1)) + b_**WC('p', S(1))*WC('v', S(1)))*WC('w', S(1))), cons2, cons3, cons15, cons16, cons17)
def replacement7(p, m, n, u, w, v, a, b):
return FixSimplify(a**(m + n)*w*((S(-1))**p*a**(-n + p)*v + u))
rule7 = ReplacementRule(pattern7, replacement7)
def With8(m, d, n, w, c, a, b):
q = b/d
if FreeQ(q, Plus):
return True
return False
pattern8 = Pattern(UtilityOperator((a_ + b_)**WC('m', S(1))*(c_ + d_)**n_*WC('w', S(1))), cons9, cons18, cons19, CustomConstraint(With8))
def replacement8(m, d, n, w, c, a, b):
q = b/d
return FixSimplify(q**m*w*(c + d)**(m + n))
rule8 = ReplacementRule(pattern8, replacement8)
pattern9 = Pattern(UtilityOperator((a_**WC('m', S(1))*WC('u', S(1)) + a_**WC('n', S(1))*WC('v', S(1)))**WC('t', S(1))*WC('w', S(1))), cons20, cons21, cons22, cons23)
def replacement9(m, n, u, w, v, a, t):
return FixSimplify(a**(m*t)*w*(a**(-m + n)*v + u)**t)
rule9 = ReplacementRule(pattern9, replacement9)
pattern10 = Pattern(UtilityOperator((a_**WC('m', S(1))*WC('u', S(1)) + a_**WC('n', S(1))*WC('v', S(1)) + a_**WC('p', S(1))*WC('z', S(1)))**WC('t', S(1))*WC('w', S(1))), cons20, cons21, cons24, cons25)
def replacement10(p, m, n, u, w, v, a, z, t):
return FixSimplify(a**(m*t)*w*(a**(-m + n)*v + a**(-m + p)*z + u)**t)
rule10 = ReplacementRule(pattern10, replacement10)
pattern11 = Pattern(UtilityOperator((a_**WC('m', S(1))*WC('u', S(1)) + a_**WC('n', S(1))*WC('v', S(1)) + a_**WC('p', S(1))*WC('z', S(1)) + a_**WC('q', S(1))*WC('y', S(1)))**WC('t', S(1))*WC('w', S(1))), cons20, cons21, cons24, cons26)
def replacement11(p, m, n, u, q, w, v, a, z, y, t):
return FixSimplify(a**(m*t)*w*(a**(-m + n)*v + a**(-m + p)*z + a**(-m + q)*y + u)**t)
rule11 = ReplacementRule(pattern11, replacement11)
pattern12 = Pattern(UtilityOperator((sqrt(v_)*WC('b', S(1)) + sqrt(v_)*WC('c', S(1)) + sqrt(v_)*WC('d', S(1)) + sqrt(v_)*WC('a', S(1)) + WC('u', S(0)))*WC('w', S(1))))
def replacement12(d, u, w, v, c, a, b):
return FixSimplify(w*(u + sqrt(v)*FixSimplify(a + b + c + d)))
rule12 = ReplacementRule(pattern12, replacement12)
pattern13 = Pattern(UtilityOperator((sqrt(v_)*WC('b', S(1)) + sqrt(v_)*WC('c', S(1)) + sqrt(v_)*WC('a', S(1)) + WC('u', S(0)))*WC('w', S(1))))
def replacement13(u, w, v, c, a, b):
return FixSimplify(w*(u + sqrt(v)*FixSimplify(a + b + c)))
rule13 = ReplacementRule(pattern13, replacement13)
pattern14 = Pattern(UtilityOperator((sqrt(v_)*WC('b', S(1)) + sqrt(v_)*WC('a', S(1)) + WC('u', S(0)))*WC('w', S(1))))
def replacement14(u, w, v, a, b):
return FixSimplify(w*(u + sqrt(v)*FixSimplify(a + b)))
rule14 = ReplacementRule(pattern14, replacement14)
pattern15 = Pattern(UtilityOperator(v_**m_*w_**n_*WC('u', S(1))), cons2, cons27, cons3, cons28, cons29)
def replacement15(m, n, u, w, v):
return -FixSimplify(u*v**(m + S(-1)))
rule15 = ReplacementRule(pattern15, replacement15)
pattern16 = Pattern(UtilityOperator(v_**m_*w_**WC('n', S(1))*WC('u', S(1))), cons2, cons27, cons30, cons31)
def replacement16(m, n, u, w, v):
return (S(-1))**n*FixSimplify(u*v**(m + n))
rule16 = ReplacementRule(pattern16, replacement16)
pattern17 = Pattern(UtilityOperator(w_**WC('n', S(1))*(-v_**WC('p', S(1)))**m_*WC('u', S(1))), cons2, cons27, cons32, cons33)
def replacement17(p, m, n, u, w, v):
return (S(-1))**(n/p)*FixSimplify(u*(-v**p)**(m + n/p))
rule17 = ReplacementRule(pattern17, replacement17)
pattern18 = Pattern(UtilityOperator(w_**WC('n', S(1))*(-v_**WC('p', S(1)))**m_*WC('u', S(1))), cons2, cons27, cons34, cons31)
def replacement18(p, m, n, u, w, v):
return (S(-1))**(n + n/p)*FixSimplify(u*(-v**p)**(m + n/p))
rule18 = ReplacementRule(pattern18, replacement18)
pattern19 = Pattern(UtilityOperator((a_ - b_)**WC('m', S(1))*(a_ + b_)**WC('m', S(1))*WC('u', S(1))), cons9, cons35, cons36)
def replacement19(m, u, a, b):
return u*(a**S(2) - b**S(2))**m
rule19 = ReplacementRule(pattern19, replacement19)
pattern20 = Pattern(UtilityOperator((S(729)*c - e*(-S(20)*e + S(540)))**WC('m', S(1))*WC('u', S(1))), cons2)
def replacement20(m, u):
return u*(a*e**S(2) - b*d*e + c*d**S(2))**m
rule20 = ReplacementRule(pattern20, replacement20)
pattern21 = Pattern(UtilityOperator((S(729)*c + e*(S(20)*e + S(-540)))**WC('m', S(1))*WC('u', S(1))), cons2)
def replacement21(m, u):
return u*(a*e**S(2) - b*d*e + c*d**S(2))**m
rule21 = ReplacementRule(pattern21, replacement21)
pattern22 = Pattern(UtilityOperator(u_))
def replacement22(u):
return u
rule22 = ReplacementRule(pattern22, replacement22)
return [rule1, rule2, rule3, rule4, rule5, rule6, rule7, rule8, rule9, rule10, rule11, rule12, rule13, rule14, rule15, rule16, rule17, rule18, rule19, rule20, rule21, rule22, ]
@doctest_depends_on(modules=('matchpy',))
def FixSimplify(expr):
if isinstance(expr, (list, tuple, TupleArg)):
return [replace_all(UtilityOperator(i), FixSimplify_rules) for i in expr]
return replace_all(UtilityOperator(expr), FixSimplify_rules)
@doctest_depends_on(modules=('matchpy',))
def _SimplifyAntiderivativeSum():
replacer = ManyToOneReplacer()
pattern1 = Pattern(UtilityOperator(Add(Mul(Log(Add(a_, Mul(WC('b', S(1)), Pow(Tan(u_), WC('n', S(1)))))), WC('A', S(1))), Mul(Log(Cos(u_)), WC('B', S(1))), WC('v', S(0))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda A, x: FreeQ(A, x)), CustomConstraint(lambda B, x: FreeQ(B, x)), CustomConstraint(lambda n: IntegerQ(n)), CustomConstraint(lambda B, A, n: ZeroQ(Add(Mul(n, A), Mul(S(1), B)))))
rule1 = ReplacementRule(pattern1, lambda n, x, v, b, B, A, u, a : Add(SimplifyAntiderivativeSum(v, x), Mul(A, Log(RemoveContent(Add(Mul(a, Pow(Cos(u), n)), Mul(b, Pow(Sin(u), n))), x)))))
replacer.add(rule1)
pattern2 = Pattern(UtilityOperator(Add(Mul(Log(Add(Mul(Pow(Cot(u_), WC('n', S(1))), WC('b', S(1))), a_)), WC('A', S(1))), Mul(Log(Sin(u_)), WC('B', S(1))), WC('v', S(0))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda A, x: FreeQ(A, x)), CustomConstraint(lambda B, x: FreeQ(B, x)), CustomConstraint(lambda n: IntegerQ(n)), CustomConstraint(lambda B, A, n: ZeroQ(Add(Mul(n, A), Mul(S(1), B)))))
rule2 = ReplacementRule(pattern2, lambda n, x, v, b, B, A, a, u : Add(SimplifyAntiderivativeSum(v, x), Mul(A, Log(RemoveContent(Add(Mul(a, Pow(Sin(u), n)), Mul(b, Pow(Cos(u), n))), x)))))
replacer.add(rule2)
pattern3 = Pattern(UtilityOperator(Add(Mul(Log(Add(a_, Mul(WC('b', S(1)), Pow(Tan(u_), WC('n', S(1)))))), WC('A', S(1))), Mul(Log(Add(c_, Mul(WC('d', S(1)), Pow(Tan(u_), WC('n', S(1)))))), WC('B', S(1))), WC('v', S(0))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda d, x: FreeQ(d, x)), CustomConstraint(lambda A, x: FreeQ(A, x)), CustomConstraint(lambda B, x: FreeQ(B, x)), CustomConstraint(lambda n: IntegerQ(n)), CustomConstraint(lambda B, A: ZeroQ(Add(A, B))))
rule3 = ReplacementRule(pattern3, lambda n, x, v, b, A, B, u, c, d, a : Add(SimplifyAntiderivativeSum(v, x), Mul(A, Log(RemoveContent(Add(Mul(a, Pow(Cos(u), n)), Mul(b, Pow(Sin(u), n))), x))), Mul(B, Log(RemoveContent(Add(Mul(c, Pow(Cos(u), n)), Mul(d, Pow(Sin(u), n))), x)))))
replacer.add(rule3)
pattern4 = Pattern(UtilityOperator(Add(Mul(Log(Add(Mul(Pow(Cot(u_), WC('n', S(1))), WC('b', S(1))), a_)), WC('A', S(1))), Mul(Log(Add(Mul(Pow(Cot(u_), WC('n', S(1))), WC('d', S(1))), c_)), WC('B', S(1))), WC('v', S(0))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda d, x: FreeQ(d, x)), CustomConstraint(lambda A, x: FreeQ(A, x)), CustomConstraint(lambda B, x: FreeQ(B, x)), CustomConstraint(lambda n: IntegerQ(n)), CustomConstraint(lambda B, A: ZeroQ(Add(A, B))))
rule4 = ReplacementRule(pattern4, lambda n, x, v, b, A, B, c, a, d, u : Add(SimplifyAntiderivativeSum(v, x), Mul(A, Log(RemoveContent(Add(Mul(b, Pow(Cos(u), n)), Mul(a, Pow(Sin(u), n))), x))), Mul(B, Log(RemoveContent(Add(Mul(d, Pow(Cos(u), n)), Mul(c, Pow(Sin(u), n))), x)))))
replacer.add(rule4)
pattern5 = Pattern(UtilityOperator(Add(Mul(Log(Add(a_, Mul(WC('b', S(1)), Pow(Tan(u_), WC('n', S(1)))))), WC('A', S(1))), Mul(Log(Add(c_, Mul(WC('d', S(1)), Pow(Tan(u_), WC('n', S(1)))))), WC('B', S(1))), Mul(Log(Add(e_, Mul(WC('f', S(1)), Pow(Tan(u_), WC('n', S(1)))))), WC('C', S(1))), WC('v', S(0))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda d, x: FreeQ(d, x)), CustomConstraint(lambda e, x: FreeQ(e, x)), CustomConstraint(lambda f, x: FreeQ(f, x)), CustomConstraint(lambda A, x: FreeQ(A, x)), CustomConstraint(lambda B, x: FreeQ(B, x)), CustomConstraint(lambda C, x: FreeQ(C, x)), CustomConstraint(lambda n: IntegerQ(n)), CustomConstraint(lambda B, A, C: ZeroQ(Add(A, B, C))))
rule5 = ReplacementRule(pattern5, lambda n, e, x, v, b, A, B, u, c, f, d, a, C : Add(SimplifyAntiderivativeSum(v, x), Mul(A, Log(RemoveContent(Add(Mul(a, Pow(Cos(u), n)), Mul(b, Pow(Sin(u), n))), x))), Mul(B, Log(RemoveContent(Add(Mul(c, Pow(Cos(u), n)), Mul(d, Pow(Sin(u), n))), x))), Mul(C, Log(RemoveContent(Add(Mul(e, Pow(Cos(u), n)), Mul(f, Pow(Sin(u), n))), x)))))
replacer.add(rule5)
pattern6 = Pattern(UtilityOperator(Add(Mul(Log(Add(Mul(Pow(Cot(u_), WC('n', S(1))), WC('b', S(1))), a_)), WC('A', S(1))), Mul(Log(Add(Mul(Pow(Cot(u_), WC('n', S(1))), WC('d', S(1))), c_)), WC('B', S(1))), Mul(Log(Add(Mul(Pow(Cot(u_), WC('n', S(1))), WC('f', S(1))), e_)), WC('C', S(1))), WC('v', S(0))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda d, x: FreeQ(d, x)), CustomConstraint(lambda e, x: FreeQ(e, x)), CustomConstraint(lambda f, x: FreeQ(f, x)), CustomConstraint(lambda A, x: FreeQ(A, x)), CustomConstraint(lambda B, x: FreeQ(B, x)), CustomConstraint(lambda C, x: FreeQ(C, x)), CustomConstraint(lambda n: IntegerQ(n)), CustomConstraint(lambda B, A, C: ZeroQ(Add(A, B, C))))
rule6 = ReplacementRule(pattern6, lambda n, e, x, v, b, A, B, c, a, f, d, u, C : Add(SimplifyAntiderivativeSum(v, x), Mul(A, Log(RemoveContent(Add(Mul(b, Pow(Cos(u), n)), Mul(a, Pow(Sin(u), n))), x))), Mul(B, Log(RemoveContent(Add(Mul(d, Pow(Cos(u), n)), Mul(c, Pow(Sin(u), n))), x))), Mul(C, Log(RemoveContent(Add(Mul(f, Pow(Cos(u), n)), Mul(e, Pow(Sin(u), n))), x)))))
replacer.add(rule6)
return replacer
@doctest_depends_on(modules=('matchpy',))
def SimplifyAntiderivativeSum(expr, x):
r = SimplifyAntiderivativeSum_replacer.replace(UtilityOperator(expr, x))
if isinstance(r, UtilityOperator):
return expr
return r
@doctest_depends_on(modules=('matchpy',))
def _SimplifyAntiderivative():
replacer = ManyToOneReplacer()
pattern2 = Pattern(UtilityOperator(Log(Mul(c_, u_)), x_), CustomConstraint(lambda c, x: FreeQ(c, x)))
rule2 = ReplacementRule(pattern2, lambda x, c, u : SimplifyAntiderivative(Log(u), x))
replacer.add(rule2)
pattern3 = Pattern(UtilityOperator(Log(Pow(u_, n_)), x_), CustomConstraint(lambda n, x: FreeQ(n, x)))
rule3 = ReplacementRule(pattern3, lambda x, n, u : Mul(n, SimplifyAntiderivative(Log(u), x)))
replacer.add(rule3)
pattern7 = Pattern(UtilityOperator(Log(Pow(f_, u_)), x_), CustomConstraint(lambda f, x: FreeQ(f, x)))
rule7 = ReplacementRule(pattern7, lambda x, f, u : Mul(Log(f), SimplifyAntiderivative(u, x)))
replacer.add(rule7)
pattern8 = Pattern(UtilityOperator(Log(Add(a_, Mul(WC('b', S(1)), Tan(u_)))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda b, a: ZeroQ(Add(Pow(a, S(2)), Pow(b, S(2))))))
rule8 = ReplacementRule(pattern8, lambda x, b, u, a : Add(Mul(Mul(b, Pow(a, S(1))), SimplifyAntiderivative(u, x)), Mul(S(1), SimplifyAntiderivative(Log(Cos(u)), x))))
replacer.add(rule8)
pattern9 = Pattern(UtilityOperator(Log(Add(Mul(Cot(u_), WC('b', S(1))), a_)), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda b, a: ZeroQ(Add(Pow(a, S(2)), Pow(b, S(2))))))
rule9 = ReplacementRule(pattern9, lambda x, b, u, a : Add(Mul(Mul(Mul(S(1), b), Pow(a, S(1))), SimplifyAntiderivative(u, x)), Mul(S(1), SimplifyAntiderivative(Log(Sin(u)), x))))
replacer.add(rule9)
pattern10 = Pattern(UtilityOperator(ArcTan(Mul(WC('a', S(1)), Tan(u_))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda a: PositiveQ(Pow(a, S(2)))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule10 = ReplacementRule(pattern10, lambda x, u, a : RectifyTangent(u, a, S(1), x))
replacer.add(rule10)
pattern11 = Pattern(UtilityOperator(ArcCot(Mul(WC('a', S(1)), Tan(u_))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda a: PositiveQ(Pow(a, S(2)))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule11 = ReplacementRule(pattern11, lambda x, u, a : RectifyTangent(u, a, S(1), x))
replacer.add(rule11)
pattern12 = Pattern(UtilityOperator(ArcCot(Mul(WC('a', S(1)), Tanh(u_))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule12 = ReplacementRule(pattern12, lambda x, u, a : Mul(S(1), SimplifyAntiderivative(ArcTan(Mul(a, Tanh(u))), x)))
replacer.add(rule12)
pattern13 = Pattern(UtilityOperator(ArcTanh(Mul(WC('a', S(1)), Tan(u_))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda a: PositiveQ(Pow(a, S(2)))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule13 = ReplacementRule(pattern13, lambda x, u, a : RectifyTangent(u, Mul(I, a), Mul(S(1), I), x))
replacer.add(rule13)
pattern14 = Pattern(UtilityOperator(ArcCoth(Mul(WC('a', S(1)), Tan(u_))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda a: PositiveQ(Pow(a, S(2)))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule14 = ReplacementRule(pattern14, lambda x, u, a : RectifyTangent(u, Mul(I, a), Mul(S(1), I), x))
replacer.add(rule14)
pattern15 = Pattern(UtilityOperator(ArcTanh(Tanh(u_)), x_))
rule15 = ReplacementRule(pattern15, lambda x, u : SimplifyAntiderivative(u, x))
replacer.add(rule15)
pattern16 = Pattern(UtilityOperator(ArcCoth(Tanh(u_)), x_))
rule16 = ReplacementRule(pattern16, lambda x, u : SimplifyAntiderivative(u, x))
replacer.add(rule16)
pattern17 = Pattern(UtilityOperator(ArcCot(Mul(Cot(u_), WC('a', S(1)))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda a: PositiveQ(Pow(a, S(2)))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule17 = ReplacementRule(pattern17, lambda x, u, a : RectifyCotangent(u, a, S(1), x))
replacer.add(rule17)
pattern18 = Pattern(UtilityOperator(ArcTan(Mul(Cot(u_), WC('a', S(1)))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda a: PositiveQ(Pow(a, S(2)))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule18 = ReplacementRule(pattern18, lambda x, u, a : RectifyCotangent(u, a, S(1), x))
replacer.add(rule18)
pattern19 = Pattern(UtilityOperator(ArcTan(Mul(Coth(u_), WC('a', S(1)))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule19 = ReplacementRule(pattern19, lambda x, u, a : Mul(S(1), SimplifyAntiderivative(ArcTan(Mul(Tanh(u), Pow(a, S(1)))), x)))
replacer.add(rule19)
pattern20 = Pattern(UtilityOperator(ArcCoth(Mul(Cot(u_), WC('a', S(1)))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda a: PositiveQ(Pow(a, S(2)))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule20 = ReplacementRule(pattern20, lambda x, u, a : RectifyCotangent(u, Mul(I, a), I, x))
replacer.add(rule20)
pattern21 = Pattern(UtilityOperator(ArcTanh(Mul(Cot(u_), WC('a', S(1)))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda a: PositiveQ(Pow(a, S(2)))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule21 = ReplacementRule(pattern21, lambda x, u, a : RectifyCotangent(u, Mul(I, a), I, x))
replacer.add(rule21)
pattern22 = Pattern(UtilityOperator(ArcCoth(Coth(u_)), x_))
rule22 = ReplacementRule(pattern22, lambda x, u : SimplifyAntiderivative(u, x))
replacer.add(rule22)
pattern23 = Pattern(UtilityOperator(ArcTanh(Mul(Coth(u_), WC('a', S(1)))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule23 = ReplacementRule(pattern23, lambda x, u, a : SimplifyAntiderivative(ArcTanh(Mul(Tanh(u), Pow(a, S(1)))), x))
replacer.add(rule23)
pattern24 = Pattern(UtilityOperator(ArcTanh(Coth(u_)), x_))
rule24 = ReplacementRule(pattern24, lambda x, u : SimplifyAntiderivative(u, x))
replacer.add(rule24)
pattern25 = Pattern(UtilityOperator(ArcTan(Mul(WC('c', S(1)), Add(a_, Mul(WC('b', S(1)), Tan(u_))))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda c, a: PositiveQ(Mul(Pow(a, S(2)), Pow(c, S(2))))), CustomConstraint(lambda c, b: PositiveQ(Mul(Pow(b, S(2)), Pow(c, S(2))))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule25 = ReplacementRule(pattern25, lambda x, a, b, u, c : RectifyTangent(u, Mul(a, c), Mul(b, c), S(1), x))
replacer.add(rule25)
pattern26 = Pattern(UtilityOperator(ArcTanh(Mul(WC('c', S(1)), Add(a_, Mul(WC('b', S(1)), Tan(u_))))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda c, a: PositiveQ(Mul(Pow(a, S(2)), Pow(c, S(2))))), CustomConstraint(lambda c, b: PositiveQ(Mul(Pow(b, S(2)), Pow(c, S(2))))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule26 = ReplacementRule(pattern26, lambda x, a, b, u, c : RectifyTangent(u, Mul(I, a, c), Mul(I, b, c), Mul(S(1), I), x))
replacer.add(rule26)
pattern27 = Pattern(UtilityOperator(ArcTan(Mul(WC('c', S(1)), Add(Mul(Cot(u_), WC('b', S(1))), a_))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda c, a: PositiveQ(Mul(Pow(a, S(2)), Pow(c, S(2))))), CustomConstraint(lambda c, b: PositiveQ(Mul(Pow(b, S(2)), Pow(c, S(2))))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule27 = ReplacementRule(pattern27, lambda x, a, b, u, c : RectifyCotangent(u, Mul(a, c), Mul(b, c), S(1), x))
replacer.add(rule27)
pattern28 = Pattern(UtilityOperator(ArcTanh(Mul(WC('c', S(1)), Add(Mul(Cot(u_), WC('b', S(1))), a_))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda c, a: PositiveQ(Mul(Pow(a, S(2)), Pow(c, S(2))))), CustomConstraint(lambda c, b: PositiveQ(Mul(Pow(b, S(2)), Pow(c, S(2))))), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule28 = ReplacementRule(pattern28, lambda x, a, b, u, c : RectifyCotangent(u, Mul(I, a, c), Mul(I, b, c), Mul(S(1), I), x))
replacer.add(rule28)
pattern29 = Pattern(UtilityOperator(ArcTan(Add(WC('a', S(0)), Mul(WC('b', S(1)), Tan(u_)), Mul(WC('c', S(1)), Pow(Tan(u_), S(2))))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule29 = ReplacementRule(pattern29, lambda x, a, b, u, c : If(EvenQ(Denominator(NumericFactor(Together(u)))), ArcTan(NormalizeTogether(Mul(Add(a, c, S(1), Mul(Add(a, Mul(S(1), c), S(1)), Cos(Mul(S(2), u))), Mul(b, Sin(Mul(S(2), u)))), Pow(Add(a, c, S(1), Mul(Add(a, Mul(S(1), c), S(1)), Cos(Mul(S(2), u))), Mul(b, Sin(Mul(S(2), u)))), S(1))))), ArcTan(NormalizeTogether(Mul(Add(c, Mul(Add(a, Mul(S(1), c), S(1)), Pow(Cos(u), S(2))), Mul(b, Cos(u), Sin(u))), Pow(Add(c, Mul(Add(a, Mul(S(1), c), S(1)), Pow(Cos(u), S(2))), Mul(b, Cos(u), Sin(u))), S(1)))))))
replacer.add(rule29)
pattern30 = Pattern(UtilityOperator(ArcTan(Add(WC('a', S(0)), Mul(WC('b', S(1)), Add(WC('d', S(0)), Mul(WC('e', S(1)), Tan(u_)))), Mul(WC('c', S(1)), Pow(Add(WC('f', S(0)), Mul(WC('g', S(1)), Tan(u_))), S(2))))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda b, x: FreeQ(b, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule30 = ReplacementRule(pattern30, lambda x, d, a, e, f, b, u, c, g : SimplifyAntiderivative(ArcTan(Add(a, Mul(b, d), Mul(c, Pow(f, S(2))), Mul(Add(Mul(b, e), Mul(S(2), c, f, g)), Tan(u)), Mul(c, Pow(g, S(2)), Pow(Tan(u), S(2))))), x))
replacer.add(rule30)
pattern31 = Pattern(UtilityOperator(ArcTan(Add(WC('a', S(0)), Mul(WC('c', S(1)), Pow(Tan(u_), S(2))))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule31 = ReplacementRule(pattern31, lambda x, c, u, a : If(EvenQ(Denominator(NumericFactor(Together(u)))), ArcTan(NormalizeTogether(Mul(Add(a, c, S(1), Mul(Add(a, Mul(S(1), c), S(1)), Cos(Mul(S(2), u)))), Pow(Add(a, c, S(1), Mul(Add(a, Mul(S(1), c), S(1)), Cos(Mul(S(2), u)))), S(1))))), ArcTan(NormalizeTogether(Mul(Add(c, Mul(Add(a, Mul(S(1), c), S(1)), Pow(Cos(u), S(2)))), Pow(Add(c, Mul(Add(a, Mul(S(1), c), S(1)), Pow(Cos(u), S(2)))), S(1)))))))
replacer.add(rule31)
pattern32 = Pattern(UtilityOperator(ArcTan(Add(WC('a', S(0)), Mul(WC('c', S(1)), Pow(Add(WC('f', S(0)), Mul(WC('g', S(1)), Tan(u_))), S(2))))), x_), CustomConstraint(lambda a, x: FreeQ(a, x)), CustomConstraint(lambda c, x: FreeQ(c, x)), CustomConstraint(lambda u: ComplexFreeQ(u)))
rule32 = ReplacementRule(pattern32, lambda x, a, f, u, c, g : SimplifyAntiderivative(ArcTan(Add(a, Mul(c, Pow(f, S(2))), Mul(Mul(S(2), c, f, g), Tan(u)), Mul(c, Pow(g, S(2)), Pow(Tan(u), S(2))))), x))
replacer.add(rule32)
return replacer
@doctest_depends_on(modules=('matchpy',))
def SimplifyAntiderivative(expr, x):
r = SimplifyAntiderivative_replacer.replace(UtilityOperator(expr, x))
if isinstance(r, UtilityOperator):
if ProductQ(expr):
u, c = S(1), S(1)
for i in expr.args:
if FreeQ(i, x):
c *= i
else:
u *= i
if FreeQ(c, x) and c != S(1):
v = SimplifyAntiderivative(u, x)
if SumQ(v) and NonsumQ(u):
return Add(*[c*i for i in v.args])
return c*v
elif LogQ(expr):
F = expr.args[0]
if MemberQ([cot, sec, csc, coth, sech, csch], Head(F)):
return -SimplifyAntiderivative(Log(1/F), x)
if MemberQ([log, atan, acot], Head(expr)):
F = Head(expr)
G = expr.args[0]
if MemberQ([cot, sec, csc, coth, sech, csch], Head(G)):
return -SimplifyAntiderivative(F(1/G), x)
if MemberQ([atanh, acoth], Head(expr)):
F = Head(expr)
G = expr.args[0]
if MemberQ([cot, sec, csc, coth, sech, csch], Head(G)):
return SimplifyAntiderivative(F(1/G), x)
u = expr
if FreeQ(u, x):
return S(0)
elif LogQ(u):
return Log(RemoveContent(u.args[0], x))
elif SumQ(u):
return SimplifyAntiderivativeSum(Add(*[SimplifyAntiderivative(i, x) for i in u.args]), x)
return u
else:
return r
@doctest_depends_on(modules=('matchpy',))
def _TrigSimplifyAux():
replacer = ManyToOneReplacer()
pattern1 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(Add(Mul(WC('a', S(1)), Pow(v_, WC('m', S(1)))), Mul(WC('b', S(1)), Pow(v_, WC('n', S(1))))), p_))), CustomConstraint(lambda v: InertTrigQ(v)), CustomConstraint(lambda p: IntegerQ(p)), CustomConstraint(lambda n, m: RationalQ(m, n)), CustomConstraint(lambda n, m: Less(m, n)))
rule1 = ReplacementRule(pattern1, lambda n, a, p, m, u, v, b : Mul(u, Pow(v, Mul(m, p)), Pow(TrigSimplifyAux(Add(a, Mul(b, Pow(v, Add(n, Mul(S(-1), m)))))), p)))
replacer.add(rule1)
pattern2 = Pattern(UtilityOperator(Add(Mul(Pow(cos(u_), S('2')), WC('a', S(1))), WC('v', S(0)), Mul(WC('b', S(1)), Pow(sin(u_), S('2'))))), CustomConstraint(lambda b, a: SameQ(a, b)))
rule2 = ReplacementRule(pattern2, lambda u, v, b, a : Add(a, v))
replacer.add(rule2)
pattern3 = Pattern(UtilityOperator(Add(WC('v', S(0)), Mul(WC('a', S(1)), Pow(sec(u_), S('2'))), Mul(WC('b', S(1)), Pow(tan(u_), S('2'))))), CustomConstraint(lambda b, a: SameQ(a, Mul(S(-1), b))))
rule3 = ReplacementRule(pattern3, lambda u, v, b, a : Add(a, v))
replacer.add(rule3)
pattern4 = Pattern(UtilityOperator(Add(Mul(Pow(csc(u_), S('2')), WC('a', S(1))), Mul(Pow(cot(u_), S('2')), WC('b', S(1))), WC('v', S(0)))), CustomConstraint(lambda b, a: SameQ(a, Mul(S(-1), b))))
rule4 = ReplacementRule(pattern4, lambda u, v, b, a : Add(a, v))
replacer.add(rule4)
pattern5 = Pattern(UtilityOperator(Pow(Add(Mul(Pow(cos(u_), S('2')), WC('a', S(1))), WC('v', S(0)), Mul(WC('b', S(1)), Pow(sin(u_), S('2')))), n_)))
rule5 = ReplacementRule(pattern5, lambda n, a, u, v, b : Pow(Add(Mul(Add(b, Mul(S(-1), a)), Pow(Sin(u), S('2'))), a, v), n))
replacer.add(rule5)
pattern6 = Pattern(UtilityOperator(Add(WC('w', S(0)), u_, Mul(WC('v', S(1)), Pow(sin(z_), S('2'))))), CustomConstraint(lambda u, v: SameQ(u, Mul(S(-1), v))))
rule6 = ReplacementRule(pattern6, lambda u, w, z, v : Add(Mul(u, Pow(Cos(z), S('2'))), w))
replacer.add(rule6)
pattern7 = Pattern(UtilityOperator(Add(Mul(Pow(cos(z_), S('2')), WC('v', S(1))), WC('w', S(0)), u_)), CustomConstraint(lambda u, v: SameQ(u, Mul(S(-1), v))))
rule7 = ReplacementRule(pattern7, lambda z, w, v, u : Add(Mul(u, Pow(Sin(z), S('2'))), w))
replacer.add(rule7)
pattern8 = Pattern(UtilityOperator(Add(WC('w', S(0)), u_, Mul(WC('v', S(1)), Pow(tan(z_), S('2'))))), CustomConstraint(lambda u, v: SameQ(u, v)))
rule8 = ReplacementRule(pattern8, lambda u, w, z, v : Add(Mul(u, Pow(Sec(z), S('2'))), w))
replacer.add(rule8)
pattern9 = Pattern(UtilityOperator(Add(Mul(Pow(cot(z_), S('2')), WC('v', S(1))), WC('w', S(0)), u_)), CustomConstraint(lambda u, v: SameQ(u, v)))
rule9 = ReplacementRule(pattern9, lambda z, w, v, u : Add(Mul(u, Pow(Csc(z), S('2'))), w))
replacer.add(rule9)
pattern10 = Pattern(UtilityOperator(Add(WC('w', S(0)), u_, Mul(WC('v', S(1)), Pow(sec(z_), S('2'))))), CustomConstraint(lambda u, v: SameQ(u, Mul(S(-1), v))))
rule10 = ReplacementRule(pattern10, lambda u, w, z, v : Add(Mul(v, Pow(Tan(z), S('2'))), w))
replacer.add(rule10)
pattern11 = Pattern(UtilityOperator(Add(Mul(Pow(csc(z_), S('2')), WC('v', S(1))), WC('w', S(0)), u_)), CustomConstraint(lambda u, v: SameQ(u, Mul(S(-1), v))))
rule11 = ReplacementRule(pattern11, lambda z, w, v, u : Add(Mul(v, Pow(Cot(z), S('2'))), w))
replacer.add(rule11)
pattern12 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(Add(Mul(cos(v_), WC('b', S(1))), a_), S(-1)), Pow(sin(v_), S('2')))), CustomConstraint(lambda b, a: ZeroQ(Add(Pow(a, S('2')), Mul(S(-1), Pow(b, S('2')))))))
rule12 = ReplacementRule(pattern12, lambda u, v, b, a : Mul(u, Add(Mul(S(1), Pow(a, S(-1))), Mul(S(-1), Mul(Cos(v), Pow(b, S(-1)))))))
replacer.add(rule12)
pattern13 = Pattern(UtilityOperator(Mul(Pow(cos(v_), S('2')), WC('u', S(1)), Pow(Add(a_, Mul(WC('b', S(1)), sin(v_))), S(-1)))), CustomConstraint(lambda b, a: ZeroQ(Add(Pow(a, S('2')), Mul(S(-1), Pow(b, S('2')))))))
rule13 = ReplacementRule(pattern13, lambda u, v, b, a : Mul(u, Add(Mul(S(1), Pow(a, S(-1))), Mul(S(-1), Mul(Sin(v), Pow(b, S(-1)))))))
replacer.add(rule13)
pattern14 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(tan(v_), WC('n', S(1))), Pow(Add(a_, Mul(WC('b', S(1)), Pow(tan(v_), WC('n', S(1))))), S(-1)))), CustomConstraint(lambda n: PositiveIntegerQ(n)), CustomConstraint(lambda a: NonsumQ(a)))
rule14 = ReplacementRule(pattern14, lambda n, a, u, v, b : Mul(u, Pow(Add(b, Mul(a, Pow(Cot(v), n))), S(-1))))
replacer.add(rule14)
pattern15 = Pattern(UtilityOperator(Mul(Pow(cot(v_), WC('n', S(1))), WC('u', S(1)), Pow(Add(Mul(Pow(cot(v_), WC('n', S(1))), WC('b', S(1))), a_), S(-1)))), CustomConstraint(lambda n: PositiveIntegerQ(n)), CustomConstraint(lambda a: NonsumQ(a)))
rule15 = ReplacementRule(pattern15, lambda n, a, u, v, b : Mul(u, Pow(Add(b, Mul(a, Pow(Tan(v), n))), S(-1))))
replacer.add(rule15)
pattern16 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(sec(v_), WC('n', S(1))), Pow(Add(a_, Mul(WC('b', S(1)), Pow(sec(v_), WC('n', S(1))))), S(-1)))), CustomConstraint(lambda n: PositiveIntegerQ(n)), CustomConstraint(lambda a: NonsumQ(a)))
rule16 = ReplacementRule(pattern16, lambda n, a, u, v, b : Mul(u, Pow(Add(b, Mul(a, Pow(Cos(v), n))), S(-1))))
replacer.add(rule16)
pattern17 = Pattern(UtilityOperator(Mul(Pow(csc(v_), WC('n', S(1))), WC('u', S(1)), Pow(Add(Mul(Pow(csc(v_), WC('n', S(1))), WC('b', S(1))), a_), S(-1)))), CustomConstraint(lambda n: PositiveIntegerQ(n)), CustomConstraint(lambda a: NonsumQ(a)))
rule17 = ReplacementRule(pattern17, lambda n, a, u, v, b : Mul(u, Pow(Add(b, Mul(a, Pow(Sin(v), n))), S(-1))))
replacer.add(rule17)
pattern18 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(Add(a_, Mul(WC('b', S(1)), Pow(sec(v_), WC('n', S(1))))), S(-1)), Pow(tan(v_), WC('n', S(1))))), CustomConstraint(lambda n: PositiveIntegerQ(n)), CustomConstraint(lambda a: NonsumQ(a)))
rule18 = ReplacementRule(pattern18, lambda n, a, u, v, b : Mul(u, Mul(Pow(Sin(v), n), Pow(Add(b, Mul(a, Pow(Cos(v), n))), S(-1)))))
replacer.add(rule18)
pattern19 = Pattern(UtilityOperator(Mul(Pow(cot(v_), WC('n', S(1))), WC('u', S(1)), Pow(Add(Mul(Pow(csc(v_), WC('n', S(1))), WC('b', S(1))), a_), S(-1)))), CustomConstraint(lambda n: PositiveIntegerQ(n)), CustomConstraint(lambda a: NonsumQ(a)))
rule19 = ReplacementRule(pattern19, lambda n, a, u, v, b : Mul(u, Mul(Pow(Cos(v), n), Pow(Add(b, Mul(a, Pow(Sin(v), n))), S(-1)))))
replacer.add(rule19)
pattern20 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(Add(Mul(WC('a', S(1)), Pow(sec(v_), WC('n', S(1)))), Mul(WC('b', S(1)), Pow(tan(v_), WC('n', S(1))))), WC('p', S(1))))), CustomConstraint(lambda n, p: IntegersQ(n, p)))
rule20 = ReplacementRule(pattern20, lambda n, a, p, u, v, b : Mul(u, Pow(Sec(v), Mul(n, p)), Pow(Add(a, Mul(b, Pow(Sin(v), n))), p)))
replacer.add(rule20)
pattern21 = Pattern(UtilityOperator(Mul(Pow(Add(Mul(Pow(csc(v_), WC('n', S(1))), WC('a', S(1))), Mul(Pow(cot(v_), WC('n', S(1))), WC('b', S(1)))), WC('p', S(1))), WC('u', S(1)))), CustomConstraint(lambda n, p: IntegersQ(n, p)))
rule21 = ReplacementRule(pattern21, lambda n, a, p, u, v, b : Mul(u, Pow(Csc(v), Mul(n, p)), Pow(Add(a, Mul(b, Pow(Cos(v), n))), p)))
replacer.add(rule21)
pattern22 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(Add(Mul(WC('b', S(1)), Pow(sin(v_), WC('n', S(1)))), Mul(WC('a', S(1)), Pow(tan(v_), WC('n', S(1))))), WC('p', S(1))))), CustomConstraint(lambda n, p: IntegersQ(n, p)))
rule22 = ReplacementRule(pattern22, lambda n, a, p, u, v, b : Mul(u, Pow(Tan(v), Mul(n, p)), Pow(Add(a, Mul(b, Pow(Cos(v), n))), p)))
replacer.add(rule22)
pattern23 = Pattern(UtilityOperator(Mul(Pow(Add(Mul(Pow(cot(v_), WC('n', S(1))), WC('a', S(1))), Mul(Pow(cos(v_), WC('n', S(1))), WC('b', S(1)))), WC('p', S(1))), WC('u', S(1)))), CustomConstraint(lambda n, p: IntegersQ(n, p)))
rule23 = ReplacementRule(pattern23, lambda n, a, p, u, v, b : Mul(u, Pow(Cot(v), Mul(n, p)), Pow(Add(a, Mul(b, Pow(Sin(v), n))), p)))
replacer.add(rule23)
pattern24 = Pattern(UtilityOperator(Mul(Pow(cos(v_), WC('m', S(1))), WC('u', S(1)), Pow(Add(WC('a', S(0)), Mul(WC('c', S(1)), Pow(sec(v_), WC('n', S(1)))), Mul(WC('b', S(1)), Pow(tan(v_), WC('n', S(1))))), WC('p', S(1))))), CustomConstraint(lambda n, p, m: IntegersQ(m, n, p)))
rule24 = ReplacementRule(pattern24, lambda n, a, c, p, m, u, v, b : Mul(u, Pow(Cos(v), Add(m, Mul(S(-1), Mul(n, p)))), Pow(Add(c, Mul(b, Pow(Sin(v), n)), Mul(a, Pow(Cos(v), n))), p)))
replacer.add(rule24)
pattern25 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(sec(v_), WC('m', S(1))), Pow(Add(WC('a', S(0)), Mul(WC('c', S(1)), Pow(sec(v_), WC('n', S(1)))), Mul(WC('b', S(1)), Pow(tan(v_), WC('n', S(1))))), WC('p', S(1))))), CustomConstraint(lambda n, p, m: IntegersQ(m, n, p)))
rule25 = ReplacementRule(pattern25, lambda n, a, c, p, m, u, v, b : Mul(u, Pow(Sec(v), Add(m, Mul(n, p))), Pow(Add(c, Mul(b, Pow(Sin(v), n)), Mul(a, Pow(Cos(v), n))), p)))
replacer.add(rule25)
pattern26 = Pattern(UtilityOperator(Mul(Pow(Add(WC('a', S(0)), Mul(Pow(cot(v_), WC('n', S(1))), WC('b', S(1))), Mul(Pow(csc(v_), WC('n', S(1))), WC('c', S(1)))), WC('p', S(1))), WC('u', S(1)), Pow(sin(v_), WC('m', S(1))))), CustomConstraint(lambda n, p, m: IntegersQ(m, n, p)))
rule26 = ReplacementRule(pattern26, lambda n, a, c, p, m, u, v, b : Mul(u, Pow(Sin(v), Add(m, Mul(S(-1), Mul(n, p)))), Pow(Add(c, Mul(b, Pow(Cos(v), n)), Mul(a, Pow(Sin(v), n))), p)))
replacer.add(rule26)
pattern27 = Pattern(UtilityOperator(Mul(Pow(csc(v_), WC('m', S(1))), Pow(Add(WC('a', S(0)), Mul(Pow(cot(v_), WC('n', S(1))), WC('b', S(1))), Mul(Pow(csc(v_), WC('n', S(1))), WC('c', S(1)))), WC('p', S(1))), WC('u', S(1)))), CustomConstraint(lambda n, p, m: IntegersQ(m, n, p)))
rule27 = ReplacementRule(pattern27, lambda n, a, c, p, m, u, v, b : Mul(u, Pow(Csc(v), Add(m, Mul(n, p))), Pow(Add(c, Mul(b, Pow(Cos(v), n)), Mul(a, Pow(Sin(v), n))), p)))
replacer.add(rule27)
pattern28 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(Add(Mul(Pow(csc(v_), WC('m', S(1))), WC('a', S(1))), Mul(WC('b', S(1)), Pow(sin(v_), WC('n', S(1))))), WC('p', S(1))))), CustomConstraint(lambda n, m: IntegersQ(m, n)))
rule28 = ReplacementRule(pattern28, lambda n, a, p, m, u, v, b : If(And(ZeroQ(Add(m, n, S(-2))), ZeroQ(Add(a, b))), Mul(u, Pow(Mul(a, Mul(Pow(Cos(v), S('2')), Pow(Pow(Sin(v), m), S(-1)))), p)), Mul(u, Pow(Mul(Add(a, Mul(b, Pow(Sin(v), Add(m, n)))), Pow(Pow(Sin(v), m), S(-1))), p))))
replacer.add(rule28)
pattern29 = Pattern(UtilityOperator(Mul(WC('u', S(1)), Pow(Add(Mul(Pow(cos(v_), WC('n', S(1))), WC('b', S(1))), Mul(WC('a', S(1)), Pow(sec(v_), WC('m', S(1))))), WC('p', S(1))))), CustomConstraint(lambda n, m: IntegersQ(m, n)))
rule29 = ReplacementRule(pattern29, lambda n, a, p, m, u, v, b : If(And(ZeroQ(Add(m, n, S(-2))), ZeroQ(Add(a, b))), Mul(u, Pow(Mul(a, Mul(Pow(Sin(v), S('2')), Pow(Pow(Cos(v), m), S(-1)))), p)), Mul(u, Pow(Mul(Add(a, Mul(b, Pow(Cos(v), Add(m, n)))), Pow(Pow(Cos(v), m), S(-1))), p))))
replacer.add(rule29)
pattern30 = Pattern(UtilityOperator(u_))
rule30 = ReplacementRule(pattern30, lambda u : u)
replacer.add(rule30)
return replacer
@doctest_depends_on(modules=('matchpy',))
def TrigSimplifyAux(expr):
return TrigSimplifyAux_replacer.replace(UtilityOperator(expr))
def Cancel(expr):
return cancel(expr)
class Util_Part(Function):
def doit(self):
i = Simplify(self.args[0])
if len(self.args) > 2 :
lst = list(self.args[1:])
else:
lst = self.args[1]
if isinstance(i, (int, Integer)):
if isinstance(lst, list):
return lst[i - 1]
elif AtomQ(lst):
return lst
return lst.args[i - 1]
else:
return self
def Part(lst, i): #see i = -1
if isinstance(lst, list):
return Util_Part(i, *lst).doit()
return Util_Part(i, lst).doit()
def PolyLog(n, p, z=None):
return polylog(n, p)
def D(f, x):
try:
return f.diff(x)
except ValueError:
return Function('D')(f, x)
def IntegralFreeQ(u):
return FreeQ(u, Integral)
def Dist(u, v, x):
#Dist(u,v) returns the sum of u times each term of v, provided v is free of Int
u = replace_pow_exp(u) # to replace back to sympy's exp
v = replace_pow_exp(v)
w = Simp(u*x**2, x)/x**2
if u == 1:
return v
elif u == 0:
return 0
elif NumericFactor(u) < 0 and NumericFactor(-u) > 0:
return -Dist(-u, v, x)
elif SumQ(v):
return Add(*[Dist(u, i, x) for i in v.args])
elif IntegralFreeQ(v):
return Simp(u*v, x)
elif w != u and FreeQ(w, x) and w == Simp(w, x) and w == Simp(w*x**2, x)/x**2:
return Dist(w, v, x)
else:
return Simp(u*v, x)
def PureFunctionOfCothQ(u, v, x):
# If u is a pure function of Coth[v], PureFunctionOfCothQ[u,v,x] returns True;
if AtomQ(u):
return u != x
elif CalculusQ(u):
return False
elif HyperbolicQ(u) and ZeroQ(u.args[0] - v):
return CothQ(u)
return all(PureFunctionOfCothQ(i, v, x) for i in u.args)
def LogIntegral(z):
return li(z)
def ExpIntegralEi(z):
return Ei(z)
def ExpIntegralE(a, b):
return expint(a, b).evalf()
def SinIntegral(z):
return Si(z)
def CosIntegral(z):
return Ci(z)
def SinhIntegral(z):
return Shi(z)
def CoshIntegral(z):
return Chi(z)
class PolyGamma(Function):
@classmethod
def eval(cls, *args):
if len(args) == 2:
return polygamma(args[0], args[1])
return digamma(args[0])
def LogGamma(z):
return loggamma(z)
class ProductLog(Function):
@classmethod
def eval(cls, *args):
if len(args) == 2:
return LambertW(args[1], args[0]).evalf()
return LambertW(args[0]).evalf()
def Factorial(a):
return factorial(a)
def Zeta(*args):
return zeta(*args)
def HypergeometricPFQ(a, b, c):
return hyper(a, b, c)
def Sum_doit(exp, args):
'''
This function perform summation using sympy's `Sum`.
Examples
========
>>> from sympy.integrals.rubi.utility_function import Sum_doit
>>> from sympy.abc import x
>>> Sum_doit(2*x + 2, [x, 0, 1.7])
6
'''
exp = replace_pow_exp(exp)
if not isinstance(args[2], (int, Integer)):
new_args = [args[0], args[1], Floor(args[2])]
return Sum(exp, new_args).doit()
return Sum(exp, args).doit()
def PolynomialQuotient(p, q, x):
try:
p = poly(p, x)
q = poly(q, x)
except:
p = poly(p)
q = poly(q)
try:
return quo(p, q).as_expr()
except (PolynomialDivisionFailed, UnificationFailed):
return p/q
def PolynomialRemainder(p, q, x):
try:
p = poly(p, x)
q = poly(q, x)
except:
p = poly(p)
q = poly(q)
try:
return rem(p, q).as_expr()
except (PolynomialDivisionFailed, UnificationFailed):
return S(0)
def Floor(x, a = None):
if a is None:
return floor(x)
return a*floor(x/a)
def Factor(var):
return factor(var)
def Rule(a, b):
return {a: b}
def Distribute(expr, *args):
if len(args) == 1:
if isinstance(expr, args[0]):
return expr
else:
return expr.expand()
if len(args) == 2:
if isinstance(expr, args[1]):
return expr.expand()
else:
return expr
return expr.expand()
def CoprimeQ(*args):
args = S(args)
g = gcd(*args)
if g == 1:
return True
return False
def Discriminant(a, b):
try:
return discriminant(a, b)
except PolynomialError:
return Function('Discriminant')(a, b)
def Negative(x):
return x < S(0)
def Quotient(m, n):
return Floor(m/n)
def process_trig(expr):
'''
This function processes trigonometric expressions such that all `cot` is
rewritten in terms of `tan`, `sec` in terms of `cos`, `csc` in terms of `sin` and
similarly for `coth`, `sech` and `csch`.
Examples
========
>>> from sympy.integrals.rubi.utility_function import process_trig
>>> from sympy.abc import x
>>> from sympy import coth, cot, csc
>>> process_trig(x*cot(x))
x/tan(x)
>>> process_trig(coth(x)*csc(x))
1/(sin(x)*tanh(x))
'''
expr = expr.replace(lambda x: isinstance(x, cot), lambda x: 1/tan(x.args[0]))
expr = expr.replace(lambda x: isinstance(x, sec), lambda x: 1/cos(x.args[0]))
expr = expr.replace(lambda x: isinstance(x, csc), lambda x: 1/sin(x.args[0]))
expr = expr.replace(lambda x: isinstance(x, coth), lambda x: 1/tanh(x.args[0]))
expr = expr.replace(lambda x: isinstance(x, sech), lambda x: 1/cosh(x.args[0]))
expr = expr.replace(lambda x: isinstance(x, csch), lambda x: 1/sinh(x.args[0]))
return expr
def _ExpandIntegrand():
Plus = Add
Times = Mul
def cons_f1(m):
return PositiveIntegerQ(m)
cons1 = CustomConstraint(cons_f1)
def cons_f2(d, c, b, a):
return ZeroQ(-a*d + b*c)
cons2 = CustomConstraint(cons_f2)
def cons_f3(a, x):
return FreeQ(a, x)
cons3 = CustomConstraint(cons_f3)
def cons_f4(b, x):
return FreeQ(b, x)
cons4 = CustomConstraint(cons_f4)
def cons_f5(c, x):
return FreeQ(c, x)
cons5 = CustomConstraint(cons_f5)
def cons_f6(d, x):
return FreeQ(d, x)
cons6 = CustomConstraint(cons_f6)
def cons_f7(e, x):
return FreeQ(e, x)
cons7 = CustomConstraint(cons_f7)
def cons_f8(f, x):
return FreeQ(f, x)
cons8 = CustomConstraint(cons_f8)
def cons_f9(g, x):
return FreeQ(g, x)
cons9 = CustomConstraint(cons_f9)
def cons_f10(h, x):
return FreeQ(h, x)
cons10 = CustomConstraint(cons_f10)
def cons_f11(e, b, c, f, n, p, F, x, d, m):
if not isinstance(x, Symbol):
return False
return FreeQ(List(F, b, c, d, e, f, m, n, p), x)
cons11 = CustomConstraint(cons_f11)
def cons_f12(F, x):
return FreeQ(F, x)
cons12 = CustomConstraint(cons_f12)
def cons_f13(m, x):
return FreeQ(m, x)
cons13 = CustomConstraint(cons_f13)
def cons_f14(n, x):
return FreeQ(n, x)
cons14 = CustomConstraint(cons_f14)
def cons_f15(p, x):
return FreeQ(p, x)
cons15 = CustomConstraint(cons_f15)
def cons_f16(e, b, c, f, n, a, p, F, x, d, m):
if not isinstance(x, Symbol):
return False
return FreeQ(List(F, a, b, c, d, e, f, m, n, p), x)
cons16 = CustomConstraint(cons_f16)
def cons_f17(n, m):
return IntegersQ(m, n)
cons17 = CustomConstraint(cons_f17)
def cons_f18(n):
return Less(n, S(0))
cons18 = CustomConstraint(cons_f18)
def cons_f19(x, u):
if not isinstance(x, Symbol):
return False
return PolynomialQ(u, x)
cons19 = CustomConstraint(cons_f19)
def cons_f20(G, F, u):
return SameQ(F(u)*G(u), S(1))
cons20 = CustomConstraint(cons_f20)
def cons_f21(q, x):
return FreeQ(q, x)
cons21 = CustomConstraint(cons_f21)
def cons_f22(F):
return MemberQ(List(ArcSin, ArcCos, ArcSinh, ArcCosh), F)
cons22 = CustomConstraint(cons_f22)
def cons_f23(j, n):
return ZeroQ(j - S(2)*n)
cons23 = CustomConstraint(cons_f23)
def cons_f24(A, x):
return FreeQ(A, x)
cons24 = CustomConstraint(cons_f24)
def cons_f25(B, x):
return FreeQ(B, x)
cons25 = CustomConstraint(cons_f25)
def cons_f26(m, u, x):
if not isinstance(x, Symbol):
return False
def _cons_f_u(d, w, c, p, x):
return And(FreeQ(List(c, d), x), IntegerQ(p), Greater(p, m))
cons_u = CustomConstraint(_cons_f_u)
pat = Pattern(UtilityOperator((c_ + x_*WC('d', S(1)))**p_*WC('w', S(1)), x_), cons_u)
result_matchq = is_match(UtilityOperator(u, x), pat)
return Not(And(PositiveIntegerQ(m), result_matchq))
cons26 = CustomConstraint(cons_f26)
def cons_f27(b, v, n, a, x, u, m):
if not isinstance(x, Symbol):
return False
return And(FreeQ(List(a, b, m), x), NegativeIntegerQ(n), Not(IntegerQ(m)), PolynomialQ(u, x), PolynomialQ(v, x),\
RationalQ(m), Less(m, -1), GreaterEqual(Exponent(u, x), (-n - IntegerPart(m))*Exponent(v, x)))
cons27 = CustomConstraint(cons_f27)
def cons_f28(v, n, x, u, m):
if not isinstance(x, Symbol):
return False
return And(FreeQ(List(a, b, m), x), NegativeIntegerQ(n), Not(IntegerQ(m)), PolynomialQ(u, x),\
PolynomialQ(v, x), GreaterEqual(Exponent(u, x), -n*Exponent(v, x)))
cons28 = CustomConstraint(cons_f28)
def cons_f29(n):
return PositiveIntegerQ(n/S(4))
cons29 = CustomConstraint(cons_f29)
def cons_f30(n):
return IntegerQ(n)
cons30 = CustomConstraint(cons_f30)
def cons_f31(n):
return Greater(n, S(1))
cons31 = CustomConstraint(cons_f31)
def cons_f32(n, m):
return Less(S(0), m, n)
cons32 = CustomConstraint(cons_f32)
def cons_f33(n, m):
return OddQ(n/GCD(m, n))
cons33 = CustomConstraint(cons_f33)
def cons_f34(a, b):
return PosQ(a/b)
cons34 = CustomConstraint(cons_f34)
def cons_f35(n, m, p):
return IntegersQ(m, n, p)
cons35 = CustomConstraint(cons_f35)
def cons_f36(n, m, p):
return Less(S(0), m, p, n)
cons36 = CustomConstraint(cons_f36)
def cons_f37(q, n, m, p):
return IntegersQ(m, n, p, q)
cons37 = CustomConstraint(cons_f37)
def cons_f38(n, q, m, p):
return Less(S(0), m, p, q, n)
cons38 = CustomConstraint(cons_f38)
def cons_f39(n):
return IntegerQ(n/S(2))
cons39 = CustomConstraint(cons_f39)
def cons_f40(p):
return NegativeIntegerQ(p)
cons40 = CustomConstraint(cons_f40)
def cons_f41(n, m):
return IntegersQ(m, n/S(2))
cons41 = CustomConstraint(cons_f41)
def cons_f42(n, m):
return Unequal(m, n/S(2))
cons42 = CustomConstraint(cons_f42)
def cons_f43(c, b, a):
return NonzeroQ(-S(4)*a*c + b**S(2))
cons43 = CustomConstraint(cons_f43)
def cons_f44(j, n, m):
return IntegersQ(m, n, j)
cons44 = CustomConstraint(cons_f44)
def cons_f45(n, m):
return Less(S(0), m, S(2)*n)
cons45 = CustomConstraint(cons_f45)
def cons_f46(n, m, p):
return Not(And(Equal(m, n), Equal(p, S(-1))))
cons46 = CustomConstraint(cons_f46)
def cons_f47(v, x):
if not isinstance(x, Symbol):
return False
return PolynomialQ(v, x)
cons47 = CustomConstraint(cons_f47)
def cons_f48(v, x):
if not isinstance(x, Symbol):
return False
return BinomialQ(v, x)
cons48 = CustomConstraint(cons_f48)
def cons_f49(v, x, u):
if not isinstance(x, Symbol):
return False
return Inequality(Exponent(u, x), Equal, Exponent(v, x) + S(-1), GreaterEqual, S(2))
cons49 = CustomConstraint(cons_f49)
def cons_f50(v, x, u):
if not isinstance(x, Symbol):
return False
return GreaterEqual(Exponent(u, x), Exponent(v, x))
cons50 = CustomConstraint(cons_f50)
def cons_f51(p):
return Not(IntegerQ(p))
cons51 = CustomConstraint(cons_f51)
def With2(e, b, c, f, n, a, g, h, x, d, m):
tmp = a*h - b*g
k = Symbol('k')
return f**(e*(c + d*x)**n)*SimplifyTerm(h**(-m)*tmp**m, x)/(g + h*x) + Sum_doit(f**(e*(c + d*x)**n)*(a + b*x)**(-k + m)*SimplifyTerm(b*h**(-k)*tmp**(k - 1), x), List(k, 1, m))
pattern2 = Pattern(UtilityOperator(f_**((x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*WC('e', S(1)))*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))/(x_*WC('h', S(1)) + WC('g', S(0))), x_), cons3, cons4, cons5, cons6, cons7, cons8, cons9, cons10, cons1, cons2)
rule2 = ReplacementRule(pattern2, With2)
pattern3 = Pattern(UtilityOperator(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*WC('b', S(1)))*x_**WC('m', S(1))*(e_ + x_*WC('f', S(1)))**WC('p', S(1)), x_), cons12, cons4, cons5, cons6, cons7, cons8, cons13, cons14, cons15, cons11)
def replacement3(e, b, c, f, n, p, F, x, d, m):
return If(And(PositiveIntegerQ(m, p), LessEqual(m, p), Or(EqQ(n, S(1)), ZeroQ(-c*f + d*e))), ExpandLinearProduct(F**(b*(c + d*x)**n)*(e + f*x)**p, x**m, e, f, x), If(PositiveIntegerQ(p), Distribute(F**(b*(c + d*x)**n)*x**m*(e + f*x)**p, Plus, Times), ExpandIntegrand(F**(b*(c + d*x)**n), x**m*(e + f*x)**p, x)))
rule3 = ReplacementRule(pattern3, replacement3)
pattern4 = Pattern(UtilityOperator(F_**((x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*x_**WC('m', S(1))*(e_ + x_*WC('f', S(1)))**WC('p', S(1)), x_), cons12, cons3, cons4, cons5, cons6, cons7, cons8, cons13, cons14, cons15, cons16)
def replacement4(e, b, c, f, n, a, p, F, x, d, m):
return If(And(PositiveIntegerQ(m, p), LessEqual(m, p), Or(EqQ(n, S(1)), ZeroQ(-c*f + d*e))), ExpandLinearProduct(F**(a + b*(c + d*x)**n)*(e + f*x)**p, x**m, e, f, x), If(PositiveIntegerQ(p), Distribute(F**(a + b*(c + d*x)**n)*x**m*(e + f*x)**p, Plus, Times), ExpandIntegrand(F**(a + b*(c + d*x)**n), x**m*(e + f*x)**p, x)))
rule4 = ReplacementRule(pattern4, replacement4)
def With5(b, v, c, n, a, F, u, x, d, m):
if not isinstance(x, Symbol) or not (FreeQ([F, a, b, c, d], x) and IntegersQ(m, n) and n < 0):
return False
w = ExpandIntegrand((a + b*x)**m*(c + d*x)**n, x)
w = ReplaceAll(w, Rule(x, F**v))
if SumQ(w):
return True
return False
pattern5 = Pattern(UtilityOperator((F_**v_*WC('b', S(1)) + a_)**WC('m', S(1))*(F_**v_*WC('d', S(1)) + c_)**n_*WC('u', S(1)), x_), cons12, cons3, cons4, cons5, cons6, cons17, cons18, CustomConstraint(With5))
def replacement5(b, v, c, n, a, F, u, x, d, m):
w = ReplaceAll(ExpandIntegrand((a + b*x)**m*(c + d*x)**n, x), Rule(x, F**v))
return w.func(*[u*i for i in w.args])
rule5 = ReplacementRule(pattern5, replacement5)
def With6(e, b, c, f, n, a, x, u, d, m):
if not isinstance(x, Symbol) or not (FreeQ([a, b, c, d, e, f, m, n], x) and PolynomialQ(u,x)):
return False
v = ExpandIntegrand(u*(a + b*x)**m, x)
if SumQ(v):
return True
return False
pattern6 = Pattern(UtilityOperator(f_**((x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*WC('e', S(1)))*u_*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1)), x_), cons3, cons4, cons5, cons6, cons7, cons8, cons13, cons14, cons19, CustomConstraint(With6))
def replacement6(e, b, c, f, n, a, x, u, d, m):
v = ExpandIntegrand(u*(a + b*x)**m, x)
return Distribute(f**(e*(c + d*x)**n)*v, Plus, Times)
rule6 = ReplacementRule(pattern6, replacement6)
pattern7 = Pattern(UtilityOperator(u_*(x_*WC('b', S(1)) + WC('a', S(0)))**WC('m', S(1))*log((x_**WC('n', S(1))*WC('e', S(1)) + WC('d', S(0)))**WC('p', S(1))*WC('c', S(1))), x_), cons3, cons4, cons5, cons6, cons7, cons13, cons14, cons15, cons19)
def replacement7(e, b, c, n, a, p, x, u, d, m):
return ExpandIntegrand(log(c*(d + e*x**n)**p), u*(a + b*x)**m, x)
rule7 = ReplacementRule(pattern7, replacement7)
pattern8 = Pattern(UtilityOperator(f_**((x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*WC('e', S(1)))*u_, x_), cons5, cons6, cons7, cons8, cons14, cons19)
def replacement8(e, c, f, n, x, u, d):
return If(EqQ(n, S(1)), ExpandIntegrand(f**(e*(c + d*x)**n), u, x), ExpandLinearProduct(f**(e*(c + d*x)**n), u, c, d, x))
rule8 = ReplacementRule(pattern8, replacement8)
# pattern9 = Pattern(UtilityOperator(F_**u_*(G_*u_*WC('b', S(1)) + a_)**WC('n', S(1)), x_), cons3, cons4, cons17, cons20)
# def replacement9(b, G, n, a, F, u, x, m):
# return ReplaceAll(ExpandIntegrand(x**(-m)*(a + b*x)**n, x), Rule(x, G(u)))
# rule9 = ReplacementRule(pattern9, replacement9)
pattern10 = Pattern(UtilityOperator(u_*(WC('a', S(0)) + WC('b', S(1))*log(((x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*WC('d', S(1)))**WC('q', S(1))*WC('c', S(1))))**n_, x_), cons3, cons4, cons5, cons6, cons7, cons8, cons14, cons15, cons21, cons19)
def replacement10(e, b, c, f, n, a, p, x, u, d, q):
return ExpandLinearProduct((a + b*log(c*(d*(e + f*x)**p)**q))**n, u, e, f, x)
rule10 = ReplacementRule(pattern10, replacement10)
# pattern11 = Pattern(UtilityOperator(u_*(F_*(x_*WC('d', S(1)) + WC('c', S(0)))*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons3, cons4, cons5, cons6, cons14, cons19, cons22)
# def replacement11(b, c, n, a, F, u, x, d):
# return ExpandLinearProduct((a + b*F(c + d*x))**n, u, c, d, x)
# rule11 = ReplacementRule(pattern11, replacement11)
pattern12 = Pattern(UtilityOperator(WC('u', S(1))/(x_**n_*WC('a', S(1)) + sqrt(c_ + x_**j_*WC('d', S(1)))*WC('b', S(1))), x_), cons3, cons4, cons5, cons6, cons14, cons23)
def replacement12(b, c, n, a, x, u, d, j):
return ExpandIntegrand(u*(a*x**n - b*sqrt(c + d*x**(S(2)*n)))/(-b**S(2)*c + x**(S(2)*n)*(a**S(2) - b**S(2)*d)), x)
rule12 = ReplacementRule(pattern12, replacement12)
pattern13 = Pattern(UtilityOperator((a_ + x_*WC('b', S(1)))**m_/(c_ + x_*WC('d', S(1))), x_), cons3, cons4, cons5, cons6, cons1)
def replacement13(b, c, a, x, d, m):
if RationalQ(a, b, c, d):
return ExpandExpression((a + b*x)**m/(c + d*x), x)
else:
tmp = a*d - b*c
k = Symbol("k")
return Sum_doit((a + b*x)**(-k + m)*SimplifyTerm(b*d**(-k)*tmp**(k + S(-1)), x), List(k, S(1), m)) + SimplifyTerm(d**(-m)*tmp**m, x)/(c + d*x)
rule13 = ReplacementRule(pattern13, replacement13)
pattern14 = Pattern(UtilityOperator((A_ + x_*WC('B', S(1)))*(a_ + x_*WC('b', S(1)))**WC('m', S(1))/(c_ + x_*WC('d', S(1))), x_), cons3, cons4, cons5, cons6, cons24, cons25, cons1)
def replacement14(b, B, A, c, a, x, d, m):
if RationalQ(a, b, c, d, A, B):
return ExpandExpression((A + B*x)*(a + b*x)**m/(c + d*x), x)
else:
tmp1 = (A*d - B*c)/d
tmp2 = ExpandIntegrand((a + b*x)**m/(c + d*x), x)
tmp2 = If(SumQ(tmp2), tmp2.func(*[SimplifyTerm(tmp1*i, x) for i in tmp2.args]), SimplifyTerm(tmp1*tmp2, x))
return SimplifyTerm(B/d, x)*(a + b*x)**m + tmp2
rule14 = ReplacementRule(pattern14, replacement14)
def With15(b, a, x, u, m):
tmp1 = Symbol('tmp1')
tmp2 = Symbol('tmp2')
tmp1 = ExpandLinearProduct((a + b*x)**m, u, a, b, x)
if not IntegerQ(m):
return tmp1
else:
tmp2 = ExpandExpression(u*(a + b*x)**m, x)
if SumQ(tmp2) and LessEqual(LeafCount(tmp2), LeafCount(tmp1) + S(2)):
return tmp2
else:
return tmp1
pattern15 = Pattern(UtilityOperator(u_*(a_ + x_*WC('b', S(1)))**m_, x_), cons3, cons4, cons13, cons19, cons26)
rule15 = ReplacementRule(pattern15, With15)
pattern16 = Pattern(UtilityOperator(u_*v_**n_*(a_ + x_*WC('b', S(1)))**m_, x_), cons27)
def replacement16(b, v, n, a, x, u, m):
s = PolynomialQuotientRemainder(u, v**(-n)*(a+b*x)**(-IntegerPart(m)), x)
return ExpandIntegrand((a + b*x)**FractionalPart(m)*s[0], x) + ExpandIntegrand(v**n*(a + b*x)**m*s[1], x)
rule16 = ReplacementRule(pattern16, replacement16)
pattern17 = Pattern(UtilityOperator(u_*v_**n_*(a_ + x_*WC('b', S(1)))**m_, x_), cons28)
def replacement17(b, v, n, a, x, u, m):
s = PolynomialQuotientRemainder(u, v**(-n),x)
return ExpandIntegrand((a + b*x)**(m)*s[0], x) + ExpandIntegrand(v**n*(a + b*x)**m*s[1], x)
rule17 = ReplacementRule(pattern17, replacement17)
def With18(b, n, a, x, u):
r = Numerator(Rt(-a/b, S(2)))
s = Denominator(Rt(-a/b, S(2)))
return r/(S(2)*a*(r + s*u**(n/S(2)))) + r/(S(2)*a*(r - s*u**(n/S(2))))
pattern18 = Pattern(UtilityOperator(S(1)/(a_ + u_**n_*WC('b', S(1))), x_), cons3, cons4, cons29)
rule18 = ReplacementRule(pattern18, With18)
def With19(b, n, a, x, u):
k = Symbol("k")
r = Numerator(Rt(-a/b, n))
s = Denominator(Rt(-a/b, n))
return Sum_doit(r/(a*n*(-(-1)**(2*k/n)*s*u + r)), List(k, 1, n))
pattern19 = Pattern(UtilityOperator(S(1)/(a_ + u_**n_*WC('b', S(1))), x_), cons3, cons4, cons30, cons31)
rule19 = ReplacementRule(pattern19, With19)
def With20(b, n, a, x, u, m):
k = Symbol("k")
g = GCD(m, n)
r = Numerator(Rt(a/b, n/GCD(m, n)))
s = Denominator(Rt(a/b, n/GCD(m, n)))
return If(CoprimeQ(g + m, n), Sum_doit((-1)**(-2*k*m/n)*r*(-r/s)**(m/g)/(a*n*((-1)**(2*g*k/n)*s*u**g + r)), List(k, 1, n/g)), Sum_doit((-1)**(2*k*(g + m)/n)*r*(-r/s)**(m/g)/(a*n*((-1)**(2*g*k/n)*r + s*u**g)), List(k, 1, n/g)))
pattern20 = Pattern(UtilityOperator(u_**WC('m', S(1))/(a_ + u_**n_*WC('b', S(1))), x_), cons3, cons4, cons17, cons32, cons33, cons34)
rule20 = ReplacementRule(pattern20, With20)
def With21(b, n, a, x, u, m):
k = Symbol("k")
g = GCD(m, n)
r = Numerator(Rt(-a/b, n/GCD(m, n)))
s = Denominator(Rt(-a/b, n/GCD(m, n)))
return If(Equal(n/g, S(2)), s/(S(2)*b*(r + s*u**g)) - s/(S(2)*b*(r - s*u**g)), If(CoprimeQ(g + m, n), Sum_doit((S(-1))**(-S(2)*k*m/n)*r*(r/s)**(m/g)/(a*n*(-(S(-1))**(S(2)*g*k/n)*s*u**g + r)), List(k, S(1), n/g)), Sum_doit((S(-1))**(S(2)*k*(g + m)/n)*r*(r/s)**(m/g)/(a*n*((S(-1))**(S(2)*g*k/n)*r - s*u**g)), List(k, S(1), n/g))))
pattern21 = Pattern(UtilityOperator(u_**WC('m', S(1))/(a_ + u_**n_*WC('b', S(1))), x_), cons3, cons4, cons17, cons32)
rule21 = ReplacementRule(pattern21, With21)
def With22(b, c, n, a, x, u, d, m):
k = Symbol("k")
r = Numerator(Rt(-a/b, n))
s = Denominator(Rt(-a/b, n))
return Sum_doit((c*r + (-1)**(-2*k*m/n)*d*r*(r/s)**m)/(a*n*(-(-1)**(2*k/n)*s*u + r)), List(k, 1, n))
pattern22 = Pattern(UtilityOperator((c_ + u_**WC('m', S(1))*WC('d', S(1)))/(a_ + u_**n_*WC('b', S(1))), x_), cons3, cons4, cons5, cons6, cons17, cons32)
rule22 = ReplacementRule(pattern22, With22)
def With23(e, b, c, n, a, p, x, u, d, m):
k = Symbol("k")
r = Numerator(Rt(-a/b, n))
s = Denominator(Rt(-a/b, n))
return Sum_doit((c*r + (-1)**(-2*k*p/n)*e*r*(r/s)**p + (-1)**(-2*k*m/n)*d*r*(r/s)**m)/(a*n*(-(-1)**(2*k/n)*s*u + r)), List(k, 1, n))
pattern23 = Pattern(UtilityOperator((u_**p_*WC('e', S(1)) + u_**WC('m', S(1))*WC('d', S(1)) + WC('c', S(0)))/(a_ + u_**n_*WC('b', S(1))), x_), cons3, cons4, cons5, cons6, cons7, cons35, cons36)
rule23 = ReplacementRule(pattern23, With23)
def With24(e, b, c, f, n, a, p, x, u, d, q, m):
k = Symbol("k")
r = Numerator(Rt(-a/b, n))
s = Denominator(Rt(-a/b, n))
return Sum_doit((c*r + (-1)**(-2*k*q/n)*f*r*(r/s)**q + (-1)**(-2*k*p/n)*e*r*(r/s)**p + (-1)**(-2*k*m/n)*d*r*(r/s)**m)/(a*n*(-(-1)**(2*k/n)*s*u + r)), List(k, 1, n))
pattern24 = Pattern(UtilityOperator((u_**p_*WC('e', S(1)) + u_**q_*WC('f', S(1)) + u_**WC('m', S(1))*WC('d', S(1)) + WC('c', S(0)))/(a_ + u_**n_*WC('b', S(1))), x_), cons3, cons4, cons5, cons6, cons7, cons8, cons37, cons38)
rule24 = ReplacementRule(pattern24, With24)
def With25(c, n, a, p, x, u):
q = Symbol('q')
return ReplaceAll(ExpandIntegrand(c**(-p), (c*x - q)**p*(c*x + q)**p, x), List(Rule(q, Rt(-a*c, S(2))), Rule(x, u**(n/S(2)))))
pattern25 = Pattern(UtilityOperator((a_ + u_**WC('n', S(1))*WC('c', S(1)))**p_, x_), cons3, cons5, cons39, cons40)
rule25 = ReplacementRule(pattern25, With25)
def With26(c, n, a, p, x, u, m):
q = Symbol('q')
return ReplaceAll(ExpandIntegrand(c**(-p), x**m*(c*x**(n/S(2)) - q)**p*(c*x**(n/S(2)) + q)**p, x), List(Rule(q, Rt(-a*c, S(2))), Rule(x, u)))
pattern26 = Pattern(UtilityOperator(u_**WC('m', S(1))*(u_**WC('n', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_, x_), cons3, cons5, cons41, cons40, cons32, cons42)
rule26 = ReplacementRule(pattern26, With26)
def With27(b, c, n, a, p, x, u, j):
q = Symbol('q')
return ReplaceAll(ExpandIntegrand(S(4)**(-p)*c**(-p), (b + S(2)*c*x - q)**p*(b + S(2)*c*x + q)**p, x), List(Rule(q, Rt(-S(4)*a*c + b**S(2), S(2))), Rule(x, u**n)))
pattern27 = Pattern(UtilityOperator((u_**WC('j', S(1))*WC('c', S(1)) + u_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons3, cons4, cons5, cons30, cons23, cons40, cons43)
rule27 = ReplacementRule(pattern27, With27)
def With28(b, c, n, a, p, x, u, j, m):
q = Symbol('q')
return ReplaceAll(ExpandIntegrand(S(4)**(-p)*c**(-p), x**m*(b + S(2)*c*x**n - q)**p*(b + S(2)*c*x**n + q)**p, x), List(Rule(q, Rt(-S(4)*a*c + b**S(2), S(2))), Rule(x, u)))
pattern28 = Pattern(UtilityOperator(u_**WC('m', S(1))*(u_**WC('j', S(1))*WC('c', S(1)) + u_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons3, cons4, cons5, cons44, cons23, cons40, cons45, cons46, cons43)
rule28 = ReplacementRule(pattern28, With28)
def With29(b, c, n, a, x, u, d, j):
q = Rt(-a/b, S(2))
return -(c - d*q)/(S(2)*b*q*(q + u**n)) - (c + d*q)/(S(2)*b*q*(q - u**n))
pattern29 = Pattern(UtilityOperator((u_**WC('n', S(1))*WC('d', S(1)) + WC('c', S(0)))/(a_ + u_**WC('j', S(1))*WC('b', S(1))), x_), cons3, cons4, cons5, cons6, cons14, cons23)
rule29 = ReplacementRule(pattern29, With29)
def With30(e, b, c, f, n, a, g, x, u, d, j):
q = Rt(-S(4)*a*c + b**S(2), S(2))
r = TogetherSimplify((-b*e*g + S(2)*c*(d + e*f))/q)
return (e*g - r)/(b + 2*c*u**n + q) + (e*g + r)/(b + 2*c*u**n - q)
pattern30 = Pattern(UtilityOperator(((u_**WC('n', S(1))*WC('g', S(1)) + WC('f', S(0)))*WC('e', S(1)) + WC('d', S(0)))/(u_**WC('j', S(1))*WC('c', S(1)) + u_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))), x_), cons3, cons4, cons5, cons6, cons7, cons8, cons9, cons14, cons23, cons43)
rule30 = ReplacementRule(pattern30, With30)
def With31(v, x, u):
lst = CoefficientList(u, x)
i = Symbol('i')
return x**Exponent(u, x)*lst[-1]/v + Sum_doit(x**(i - 1)*Part(lst, i), List(i, 1, Exponent(u, x)))/v
pattern31 = Pattern(UtilityOperator(u_/v_, x_), cons19, cons47, cons48, cons49)
rule31 = ReplacementRule(pattern31, With31)
pattern32 = Pattern(UtilityOperator(u_/v_, x_), cons19, cons47, cons50)
def replacement32(v, x, u):
return PolynomialDivide(u, v, x)
rule32 = ReplacementRule(pattern32, replacement32)
pattern33 = Pattern(UtilityOperator(u_*(x_*WC('a', S(1)))**p_, x_), cons51, cons19)
def replacement33(x, a, u, p):
return ExpandToSum((a*x)**p, u, x)
rule33 = ReplacementRule(pattern33, replacement33)
pattern34 = Pattern(UtilityOperator(v_**p_*WC('u', S(1)), x_), cons51)
def replacement34(v, x, u, p):
return ExpandIntegrand(NormalizeIntegrand(v**p, x), u, x)
rule34 = ReplacementRule(pattern34, replacement34)
pattern35 = Pattern(UtilityOperator(u_, x_))
def replacement35(x, u):
return ExpandExpression(u, x)
rule35 = ReplacementRule(pattern35, replacement35)
return [ rule2,rule3, rule4, rule5, rule6, rule7, rule8, rule10, rule12, rule13, rule14, rule15, rule16, rule17, rule18, rule19, rule20, rule21, rule22, rule23, rule24, rule25, rule26, rule27, rule28, rule29, rule30, rule31, rule32, rule33, rule34, rule35]
def _RemoveContentAux():
def cons_f1(b, a):
return IntegersQ(a, b)
cons1 = CustomConstraint(cons_f1)
def cons_f2(b, a):
return Equal(a + b, S(0))
cons2 = CustomConstraint(cons_f2)
def cons_f3(m):
return RationalQ(m)
cons3 = CustomConstraint(cons_f3)
def cons_f4(m, n):
return RationalQ(m, n)
cons4 = CustomConstraint(cons_f4)
def cons_f5(m, n):
return GreaterEqual(-m + n, S(0))
cons5 = CustomConstraint(cons_f5)
def cons_f6(a, x):
return FreeQ(a, x)
cons6 = CustomConstraint(cons_f6)
def cons_f7(m, n, p):
return RationalQ(m, n, p)
cons7 = CustomConstraint(cons_f7)
def cons_f8(m, p):
return GreaterEqual(-m + p, S(0))
cons8 = CustomConstraint(cons_f8)
pattern1 = Pattern(UtilityOperator(a_**m_*WC('u', S(1)) + b_*WC('v', S(1)), x_), cons1, cons2, cons3)
def replacement1(v, x, a, u, m, b):
return If(Greater(m, S(1)), RemoveContentAux(a**(m + S(-1))*u - v, x), RemoveContentAux(-a**(-m + S(1))*v + u, x))
rule1 = ReplacementRule(pattern1, replacement1)
pattern2 = Pattern(UtilityOperator(a_**WC('m', S(1))*WC('u', S(1)) + a_**WC('n', S(1))*WC('v', S(1)), x_), cons6, cons4, cons5)
def replacement2(n, v, x, u, m, a):
return RemoveContentAux(a**(-m + n)*v + u, x)
rule2 = ReplacementRule(pattern2, replacement2)
pattern3 = Pattern(UtilityOperator(a_**WC('m', S(1))*WC('u', S(1)) + a_**WC('n', S(1))*WC('v', S(1)) + a_**WC('p', S(1))*WC('w', S(1)), x_), cons6, cons7, cons5, cons8)
def replacement3(n, v, x, p, u, w, m, a):
return RemoveContentAux(a**(-m + n)*v + a**(-m + p)*w + u, x)
rule3 = ReplacementRule(pattern3, replacement3)
pattern4 = Pattern(UtilityOperator(u_, x_))
def replacement4(u, x):
return If(And(SumQ(u), NegQ(First(u))), -u, u)
rule4 = ReplacementRule(pattern4, replacement4)
return [rule1, rule2, rule3, rule4, ]
IntHide = Int
Log = log
Null = None
if matchpy:
RemoveContentAux_replacer = ManyToOneReplacer(* _RemoveContentAux())
ExpandIntegrand_rules = _ExpandIntegrand()
TrigSimplifyAux_replacer = _TrigSimplifyAux()
SimplifyAntiderivative_replacer = _SimplifyAntiderivative()
SimplifyAntiderivativeSum_replacer = _SimplifyAntiderivativeSum()
FixSimplify_rules = _FixSimplify()
SimpFixFactor_replacer = _SimpFixFactor()
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df0ff2afb62714e516dbb21f2b90d87609e087058048edfc936b77ced1d6a6e9
|
"""Most of these tests come from the examples in Bronstein's book."""
from sympy import Poly, S, symbols, oo, I
from sympy.core.compatibility import PY3
from sympy.integrals.risch import (DifferentialExtension,
NonElementaryIntegralException)
from sympy.integrals.rde import (order_at, order_at_oo, weak_normalizer,
normal_denom, special_denom, bound_degree, spde, solve_poly_rde,
no_cancel_equal, cancel_primitive, cancel_exp, rischDE)
from sympy.utilities.pytest import raises, XFAIL
from sympy.abc import x, t, z, n
t0, t1, t2, k = symbols('t:3 k')
def test_order_at():
a = Poly(t**4, t)
b = Poly((t**2 + 1)**3*t, t)
c = Poly((t**2 + 1)**6*t, t)
d = Poly((t**2 + 1)**10*t**10, t)
e = Poly((t**2 + 1)**100*t**37, t)
p1 = Poly(t, t)
p2 = Poly(1 + t**2, t)
assert order_at(a, p1, t) == 4
assert order_at(b, p1, t) == 1
assert order_at(c, p1, t) == 1
assert order_at(d, p1, t) == 10
assert order_at(e, p1, t) == 37
assert order_at(a, p2, t) == 0
assert order_at(b, p2, t) == 3
assert order_at(c, p2, t) == 6
assert order_at(d, p1, t) == 10
assert order_at(e, p2, t) == 100
assert order_at(Poly(0, t), Poly(t, t), t) == oo
assert order_at_oo(Poly(t**2 - 1, t), Poly(t + 1), t) == \
order_at_oo(Poly(t - 1, t), Poly(1, t), t) == -1
assert order_at_oo(Poly(0, t), Poly(1, t), t) == oo
def test_weak_normalizer():
a = Poly((1 + x)*t**5 + 4*t**4 + (-1 - 3*x)*t**3 - 4*t**2 + (-2 + 2*x)*t, t)
d = Poly(t**4 - 3*t**2 + 2, t)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
r = weak_normalizer(a, d, DE, z)
assert r == (Poly(t**5 - t**4 - 4*t**3 + 4*t**2 + 4*t - 4, t),
(Poly((1 + x)*t**2 + x*t, t), Poly(t + 1, t)))
assert weak_normalizer(r[1][0], r[1][1], DE) == (Poly(1, t), r[1])
r = weak_normalizer(Poly(1 + t**2), Poly(t**2 - 1, t), DE, z)
assert r == (Poly(t**4 - 2*t**2 + 1, t), (Poly(-3*t**2 + 1, t), Poly(t**2 - 1, t)))
assert weak_normalizer(r[1][0], r[1][1], DE, z) == (Poly(1, t), r[1])
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2)]})
r = weak_normalizer(Poly(1 + t**2), Poly(t, t), DE, z)
assert r == (Poly(t, t), (Poly(0, t), Poly(1, t)))
assert weak_normalizer(r[1][0], r[1][1], DE, z) == (Poly(1, t), r[1])
def test_normal_denom():
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
raises(NonElementaryIntegralException, lambda: normal_denom(Poly(1, x), Poly(1, x),
Poly(1, x), Poly(x, x), DE))
fa, fd = Poly(t**2 + 1, t), Poly(1, t)
ga, gd = Poly(1, t), Poly(t**2, t)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
assert normal_denom(fa, fd, ga, gd, DE) == \
(Poly(t, t), (Poly(t**3 - t**2 + t - 1, t), Poly(1, t)), (Poly(1, t),
Poly(1, t)), Poly(t, t))
def test_special_denom():
# TODO: add more tests here
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
assert special_denom(Poly(1, t), Poly(t**2, t), Poly(1, t), Poly(t**2 - 1, t),
Poly(t, t), DE) == \
(Poly(1, t), Poly(t**2 - 1, t), Poly(t**2 - 1, t), Poly(t, t))
# assert special_denom(Poly(1, t), Poly(2*x, t), Poly((1 + 2*x)*t, t), DE) == 1
# issue 3940
# Note, this isn't a very good test, because the denominator is just 1,
# but at least it tests the exp cancellation case
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-2*x*t0, t0),
Poly(I*k*t1, t1)]})
DE.decrement_level()
assert special_denom(Poly(1, t0), Poly(I*k, t0), Poly(1, t0), Poly(t0, t0),
Poly(1, t0), DE) == \
(Poly(1, t0), Poly(I*k, t0), Poly(t0, t0), Poly(1, t0))
# @XFAIL
# Probably only fails in Python 2.7
def test_bound_degree_fail():
# Primitive
DE = DifferentialExtension(extension={'D': [Poly(1, x),
Poly(t0/x**2, t0), Poly(1/x, t)]})
assert bound_degree(Poly(t**2, t), Poly(-(1/x**2*t**2 + 1/x), t),
Poly((2*x - 1)*t**4 + (t0 + x)/x*t**3 - (t0 + 4*x**2)/2*x*t**2 + x*t,
t), DE) == 3
if not PY3:
test_bound_degree_fail = XFAIL(test_bound_degree_fail)
def test_bound_degree():
# Base
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
assert bound_degree(Poly(1, x), Poly(-2*x, x), Poly(1, x), DE) == 0
# Primitive (see above test_bound_degree_fail)
# TODO: Add test for when the degree bound becomes larger after limited_integrate
# TODO: Add test for db == da - 1 case
# Exp
# TODO: Add tests
# TODO: Add test for when the degree becomes larger after parametric_log_deriv()
# Nonlinear
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
assert bound_degree(Poly(t, t), Poly((t - 1)*(t**2 + 1), t), Poly(1, t), DE) == 0
def test_spde():
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
raises(NonElementaryIntegralException, lambda: spde(Poly(t, t), Poly((t - 1)*(t**2 + 1), t), Poly(1, t), 0, DE))
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
assert spde(Poly(t**2 + x*t*2 + x**2, t), Poly(t**2/x**2 + (2/x - 1)*t, t),
Poly(t**2/x**2 + (2/x - 1)*t, t), 0, DE) == \
(Poly(0, t), Poly(0, t), 0, Poly(0, t), Poly(1, t))
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t0/x**2, t0), Poly(1/x, t)]})
assert spde(Poly(t**2, t), Poly(-t**2/x**2 - 1/x, t),
Poly((2*x - 1)*t**4 + (t0 + x)/x*t**3 - (t0 + 4*x**2)/(2*x)*t**2 + x*t, t), 3, DE) == \
(Poly(0, t), Poly(0, t), 0, Poly(0, t),
Poly(t0*t**2/2 + x**2*t**2 - x**2*t, t))
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
assert spde(Poly(x**2 + x + 1, x), Poly(-2*x - 1, x), Poly(x**5/2 +
3*x**4/4 + x**3 - x**2 + 1, x), 4, DE) == \
(Poly(0, x), Poly(x/2 - S(1)/4, x), 2, Poly(x**2 + x + 1, x), Poly(5*x/4, x))
assert spde(Poly(x**2 + x + 1, x), Poly(-2*x - 1, x), Poly(x**5/2 +
3*x**4/4 + x**3 - x**2 + 1, x), n, DE) == \
(Poly(0, x), Poly(x/2 - S(1)/4, x), -2 + n, Poly(x**2 + x + 1, x), Poly(5*x/4, x))
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1, t)]})
raises(NonElementaryIntegralException, lambda: spde(Poly((t - 1)*(t**2 + 1)**2, t), Poly((t - 1)*(t**2 + 1), t), Poly(1, t), 0, DE))
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
assert spde(Poly(x**2 - x, x), Poly(1, x), Poly(9*x**4 - 10*x**3 + 2*x**2, x), 4, DE) == (Poly(0, x), Poly(0, x), 0, Poly(0, x), Poly(3*x**3 - 2*x**2, x))
assert spde(Poly(x**2 - x, x), Poly(x**2 - 5*x + 3, x), Poly(x**7 - x**6 - 2*x**4 + 3*x**3 - x**2, x), 5, DE) == \
(Poly(1, x), Poly(x + 1, x), 1, Poly(x**4 - x**3, x), Poly(x**3 - x**2, x))
def test_solve_poly_rde_no_cancel():
# deg(b) large
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
assert solve_poly_rde(Poly(t**2 + 1, t), Poly(t**3 + (x + 1)*t**2 + t + x + 2, t),
oo, DE) == Poly(t + x, t)
# deg(b) small
DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
assert solve_poly_rde(Poly(0, x), Poly(x/2 - S(1)/4, x), oo, DE) == \
Poly(x**2/4 - x/4, x)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
assert solve_poly_rde(Poly(2, t), Poly(t**2 + 2*t + 3, t), 1, DE) == \
Poly(t + 1, t, x)
# deg(b) == deg(D) - 1
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
assert no_cancel_equal(Poly(1 - t, t),
Poly(t**3 + t**2 - 2*x*t - 2*x, t), oo, DE) == \
(Poly(t**2, t), 1, Poly((-2 - 2*x)*t - 2*x, t))
def test_solve_poly_rde_cancel():
# exp
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
assert cancel_exp(Poly(2*x, t), Poly(2*x, t), 0, DE) == \
Poly(1, t)
assert cancel_exp(Poly(2*x, t), Poly((1 + 2*x)*t, t), 1, DE) == \
Poly(t, t)
# TODO: Add more exp tests, including tests that require is_deriv_in_field()
# primitive
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
# If the DecrementLevel context manager is working correctly, this shouldn't
# cause any problems with the further tests.
raises(NonElementaryIntegralException, lambda: cancel_primitive(Poly(1, t), Poly(t, t), oo, DE))
assert cancel_primitive(Poly(1, t), Poly(t + 1/x, t), 2, DE) == \
Poly(t, t)
assert cancel_primitive(Poly(4*x, t), Poly(4*x*t**2 + 2*t/x, t), 3, DE) == \
Poly(t**2, t)
# TODO: Add more primitive tests, including tests that require is_deriv_in_field()
def test_rischDE():
# TODO: Add more tests for rischDE, including ones from the text
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
DE.decrement_level()
assert rischDE(Poly(-2*x, x), Poly(1, x), Poly(1 - 2*x - 2*x**2, x),
Poly(1, x), DE) == \
(Poly(x + 1, x), Poly(1, x))
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60bf7497493a8b8ef810393ccb5d2433e6c9b4dc184bc7d107bb889be01ee22e
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from sympy import sqrt, Abs
from sympy.core import S
from sympy.integrals.intpoly import (decompose, best_origin, distance_to_side,
polytope_integrate, point_sort,
hyperplane_parameters, main_integrate3d,
main_integrate, polygon_integrate,
lineseg_integrate, integration_reduction,
integration_reduction_dynamic, is_vertex)
from sympy.geometry.line import Segment2D
from sympy.geometry.polygon import Polygon
from sympy.geometry.point import Point, Point2D
from sympy.abc import x, y, z
from sympy.utilities.pytest import slow
def test_decompose():
assert decompose(x) == {1: x}
assert decompose(x**2) == {2: x**2}
assert decompose(x*y) == {2: x*y}
assert decompose(x + y) == {1: x + y}
assert decompose(x**2 + y) == {1: y, 2: x**2}
assert decompose(8*x**2 + 4*y + 7) == {0: 7, 1: 4*y, 2: 8*x**2}
assert decompose(x**2 + 3*y*x) == {2: x**2 + 3*x*y}
assert decompose(9*x**2 + y + 4*x + x**3 + y**2*x + 3) ==\
{0: 3, 1: 4*x + y, 2: 9*x**2, 3: x**3 + x*y**2}
assert decompose(x, True) == {x}
assert decompose(x ** 2, True) == {x**2}
assert decompose(x * y, True) == {x * y}
assert decompose(x + y, True) == {x, y}
assert decompose(x ** 2 + y, True) == {y, x ** 2}
assert decompose(8 * x ** 2 + 4 * y + 7, True) == {7, 4*y, 8*x**2}
assert decompose(x ** 2 + 3 * y * x, True) == {x ** 2, 3 * x * y}
assert decompose(9 * x ** 2 + y + 4 * x + x ** 3 + y ** 2 * x + 3, True) == \
{3, y, 4*x, 9*x**2, x*y**2, x**3}
def test_best_origin():
expr1 = y ** 2 * x ** 5 + y ** 5 * x ** 7 + 7 * x + x ** 12 + y ** 7 * x
l1 = Segment2D(Point(0, 3), Point(1, 1))
l2 = Segment2D(Point(S(3) / 2, 0), Point(S(3) / 2, 3))
l3 = Segment2D(Point(0, S(3) / 2), Point(3, S(3) / 2))
l4 = Segment2D(Point(0, 2), Point(2, 0))
l5 = Segment2D(Point(0, 2), Point(1, 1))
l6 = Segment2D(Point(2, 0), Point(1, 1))
assert best_origin((2, 1), 3, l1, expr1) == (0, 3)
assert best_origin((2, 0), 3, l2, x ** 7) == (S(3) / 2, 0)
assert best_origin((0, 2), 3, l3, x ** 7) == (0, S(3) / 2)
assert best_origin((1, 1), 2, l4, x ** 7 * y ** 3) == (0, 2)
assert best_origin((1, 1), 2, l4, x ** 3 * y ** 7) == (2, 0)
assert best_origin((1, 1), 2, l5, x ** 2 * y ** 9) == (0, 2)
assert best_origin((1, 1), 2, l6, x ** 9 * y ** 2) == (2, 0)
@slow
def test_polytope_integrate():
# Convex 2-Polytopes
# Vertex representation
assert polytope_integrate(Polygon(Point(0, 0), Point(0, 2),
Point(4, 0)), 1, dims=(x, y)) == 4
assert polytope_integrate(Polygon(Point(0, 0), Point(0, 1),
Point(1, 1), Point(1, 0)), x * y) ==\
S(1)/4
assert polytope_integrate(Polygon(Point(0, 3), Point(5, 3), Point(1, 1)),
6*x**2 - 40*y) == S(-935)/3
assert polytope_integrate(Polygon(Point(0, 0), Point(0, sqrt(3)),
Point(sqrt(3), sqrt(3)),
Point(sqrt(3), 0)), 1) == 3
hexagon = Polygon(Point(0, 0), Point(-sqrt(3) / 2, S(1)/2),
Point(-sqrt(3) / 2, S(3) / 2), Point(0, 2),
Point(sqrt(3) / 2, S(3) / 2), Point(sqrt(3) / 2, S(1)/2))
assert polytope_integrate(hexagon, 1) == S(3*sqrt(3)) / 2
# Hyperplane representation
assert polytope_integrate([((-1, 0), 0), ((1, 2), 4),
((0, -1), 0)], 1, dims=(x, y)) == 4
assert polytope_integrate([((-1, 0), 0), ((0, 1), 1),
((1, 0), 1), ((0, -1), 0)], x * y) == S(1)/4
assert polytope_integrate([((0, 1), 3), ((1, -2), -1),
((-2, -1), -3)], 6*x**2 - 40*y) == S(-935)/3
assert polytope_integrate([((-1, 0), 0), ((0, sqrt(3)), 3),
((sqrt(3), 0), 3), ((0, -1), 0)], 1) == 3
hexagon = [((-S(1) / 2, -sqrt(3) / 2), 0),
((-1, 0), sqrt(3) / 2),
((-S(1) / 2, sqrt(3) / 2), sqrt(3)),
((S(1) / 2, sqrt(3) / 2), sqrt(3)),
((1, 0), sqrt(3) / 2),
((S(1) / 2, -sqrt(3) / 2), 0)]
assert polytope_integrate(hexagon, 1) == S(3*sqrt(3)) / 2
# Non-convex polytopes
# Vertex representation
assert polytope_integrate(Polygon(Point(-1, -1), Point(-1, 1),
Point(1, 1), Point(0, 0),
Point(1, -1)), 1) == 3
assert polytope_integrate(Polygon(Point(-1, -1), Point(-1, 1),
Point(0, 0), Point(1, 1),
Point(1, -1), Point(0, 0)), 1) == 2
# Hyperplane representation
assert polytope_integrate([((-1, 0), 1), ((0, 1), 1), ((1, -1), 0),
((1, 1), 0), ((0, -1), 1)], 1) == 3
assert polytope_integrate([((-1, 0), 1), ((1, 1), 0), ((-1, 1), 0),
((1, 0), 1), ((-1, -1), 0),
((1, -1), 0)], 1) == 2
# Tests for 2D polytopes mentioned in Chin et al(Page 10):
# http://dilbert.engr.ucdavis.edu/~suku/quadrature/cls-integration.pdf
fig1 = Polygon(Point(1.220, -0.827), Point(-1.490, -4.503),
Point(-3.766, -1.622), Point(-4.240, -0.091),
Point(-3.160, 4), Point(-0.981, 4.447),
Point(0.132, 4.027))
assert polytope_integrate(fig1, x**2 + x*y + y**2) ==\
S(2031627344735367)/(8*10**12)
fig2 = Polygon(Point(4.561, 2.317), Point(1.491, -1.315),
Point(-3.310, -3.164), Point(-4.845, -3.110),
Point(-4.569, 1.867))
assert polytope_integrate(fig2, x**2 + x*y + y**2) ==\
S(517091313866043)/(16*10**11)
fig3 = Polygon(Point(-2.740, -1.888), Point(-3.292, 4.233),
Point(-2.723, -0.697), Point(-0.643, -3.151))
assert polytope_integrate(fig3, x**2 + x*y + y**2) ==\
S(147449361647041)/(8*10**12)
fig4 = Polygon(Point(0.211, -4.622), Point(-2.684, 3.851),
Point(0.468, 4.879), Point(4.630, -1.325),
Point(-0.411, -1.044))
assert polytope_integrate(fig4, x**2 + x*y + y**2) ==\
S(180742845225803)/(10**12)
# Tests for many polynomials with maximum degree given(2D case).
tri = Polygon(Point(0, 3), Point(5, 3), Point(1, 1))
polys = []
expr1 = x**9*y + x**7*y**3 + 2*x**2*y**8
expr2 = x**6*y**4 + x**5*y**5 + 2*y**10
expr3 = x**10 + x**9*y + x**8*y**2 + x**5*y**5
polys.extend((expr1, expr2, expr3))
result_dict = polytope_integrate(tri, polys, max_degree=10)
assert result_dict[expr1] == S(615780107)/594
assert result_dict[expr2] == S(13062161)/27
assert result_dict[expr3] == S(1946257153)/924
# Tests when all integral of all monomials up to a max_degree is to be
# calculated.
assert polytope_integrate(Polygon(Point(0, 0), Point(0, 1),
Point(1, 1), Point(1, 0)),
max_degree=4) == {0: 0, 1: 1, x: S(1) / 2,
x ** 2 * y ** 2: S(1) / 9,
x ** 4: S(1) / 5,
y ** 4: S(1) / 5,
y: S(1) / 2,
x * y ** 2: S(1) / 6,
y ** 2: S(1) / 3,
x ** 3: S(1) / 4,
x ** 2 * y: S(1) / 6,
x ** 3 * y: S(1) / 8,
x * y: S(1) / 4,
y ** 3: S(1) / 4,
x ** 2: S(1) / 3,
x * y ** 3: S(1) / 8}
# Tests for 3D polytopes
cube1 = [[(0, 0, 0), (0, 6, 6), (6, 6, 6), (3, 6, 0),
(0, 6, 0), (6, 0, 6), (3, 0, 0), (0, 0, 6)],
[1, 2, 3, 4], [3, 2, 5, 6], [1, 7, 5, 2], [0, 6, 5, 7],
[1, 4, 0, 7], [0, 4, 3, 6]]
assert polytope_integrate(cube1, 1) == S(162)
# 3D Test cases in Chin et al(2015)
cube2 = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0),
(5, 0, 5), (5, 5, 0), (5, 5, 5)],
[3, 7, 6, 2], [1, 5, 7, 3], [5, 4, 6, 7], [0, 4, 5, 1],
[2, 0, 1, 3], [2, 6, 4, 0]]
cube3 = [[(0, 0, 0), (5, 0, 0), (5, 4, 0), (3, 2, 0), (3, 5, 0),
(0, 5, 0), (0, 0, 5), (5, 0, 5), (5, 4, 5), (3, 2, 5),
(3, 5, 5), (0, 5, 5)],
[6, 11, 5, 0], [1, 7, 6, 0], [5, 4, 3, 2, 1, 0], [11, 10, 4, 5],
[10, 9, 3, 4], [9, 8, 2, 3], [8, 7, 1, 2], [7, 8, 9, 10, 11, 6]]
cube4 = [[(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1),
(S(1) / 4, S(1) / 4, S(1) / 4)],
[0, 2, 1], [1, 3, 0], [4, 2, 3], [4, 3, 1],
[0, 1, 2], [2, 4, 1], [0, 3, 2]]
assert polytope_integrate(cube2, x ** 2 + y ** 2 + x * y + z ** 2) ==\
S(15625)/4
assert polytope_integrate(cube3, x ** 2 + y ** 2 + x * y + z ** 2) ==\
S(33835) / 12
assert polytope_integrate(cube4, x ** 2 + y ** 2 + x * y + z ** 2) ==\
S(37) / 960
# Test cases from Mathematica's PolyhedronData library
octahedron = [[(S(-1) / sqrt(2), 0, 0), (0, S(1) / sqrt(2), 0),
(0, 0, S(-1) / sqrt(2)), (0, 0, S(1) / sqrt(2)),
(0, S(-1) / sqrt(2), 0), (S(1) / sqrt(2), 0, 0)],
[3, 4, 5], [3, 5, 1], [3, 1, 0], [3, 0, 4], [4, 0, 2],
[4, 2, 5], [2, 0, 1], [5, 2, 1]]
assert polytope_integrate(octahedron, 1) == sqrt(2) / 3
great_stellated_dodecahedron =\
[[(-0.32491969623290634095, 0, 0.42532540417601993887),
(0.32491969623290634095, 0, -0.42532540417601993887),
(-0.52573111211913359231, 0, 0.10040570794311363956),
(0.52573111211913359231, 0, -0.10040570794311363956),
(-0.10040570794311363956, -0.3090169943749474241, 0.42532540417601993887),
(-0.10040570794311363956, 0.30901699437494742410, 0.42532540417601993887),
(0.10040570794311363956, -0.3090169943749474241, -0.42532540417601993887),
(0.10040570794311363956, 0.30901699437494742410, -0.42532540417601993887),
(-0.16245984811645317047, -0.5, 0.10040570794311363956),
(-0.16245984811645317047, 0.5, 0.10040570794311363956),
(0.16245984811645317047, -0.5, -0.10040570794311363956),
(0.16245984811645317047, 0.5, -0.10040570794311363956),
(-0.42532540417601993887, -0.3090169943749474241, -0.10040570794311363956),
(-0.42532540417601993887, 0.30901699437494742410, -0.10040570794311363956),
(-0.26286555605956679615, 0.1909830056250525759, -0.42532540417601993887),
(-0.26286555605956679615, -0.1909830056250525759, -0.42532540417601993887),
(0.26286555605956679615, 0.1909830056250525759, 0.42532540417601993887),
(0.26286555605956679615, -0.1909830056250525759, 0.42532540417601993887),
(0.42532540417601993887, -0.3090169943749474241, 0.10040570794311363956),
(0.42532540417601993887, 0.30901699437494742410, 0.10040570794311363956)],
[12, 3, 0, 6, 16], [17, 7, 0, 3, 13],
[9, 6, 0, 7, 8], [18, 2, 1, 4, 14],
[15, 5, 1, 2, 19], [11, 4, 1, 5, 10],
[8, 19, 2, 18, 9], [10, 13, 3, 12, 11],
[16, 14, 4, 11, 12], [13, 10, 5, 15, 17],
[14, 16, 6, 9, 18], [19, 8, 7, 17, 15]]
# Actual volume is : 0.163118960624632
assert Abs(polytope_integrate(great_stellated_dodecahedron, 1) -\
0.163118960624632) < 1e-12
expr = x **2 + y ** 2 + z ** 2
octahedron_five_compound = [[(0, -0.7071067811865475244, 0),
(0, 0.70710678118654752440, 0),
(0.1148764602736805918,
-0.35355339059327376220, -0.60150095500754567366),
(0.1148764602736805918, 0.35355339059327376220,
-0.60150095500754567366),
(0.18587401723009224507,
-0.57206140281768429760, 0.37174803446018449013),
(0.18587401723009224507, 0.57206140281768429760,
0.37174803446018449013),
(0.30075047750377283683, -0.21850801222441053540,
0.60150095500754567366),
(0.30075047750377283683, 0.21850801222441053540,
0.60150095500754567366),
(0.48662449473386508189, -0.35355339059327376220,
-0.37174803446018449013),
(0.48662449473386508189, 0.35355339059327376220,
-0.37174803446018449013),
(-0.60150095500754567366, 0, -0.37174803446018449013),
(-0.30075047750377283683, -0.21850801222441053540,
-0.60150095500754567366),
(-0.30075047750377283683, 0.21850801222441053540,
-0.60150095500754567366),
(0.60150095500754567366, 0, 0.37174803446018449013),
(0.4156269377774534286, -0.57206140281768429760, 0),
(0.4156269377774534286, 0.57206140281768429760, 0),
(0.37174803446018449013, 0, -0.60150095500754567366),
(-0.4156269377774534286, -0.57206140281768429760, 0),
(-0.4156269377774534286, 0.57206140281768429760, 0),
(-0.67249851196395732696, -0.21850801222441053540, 0),
(-0.67249851196395732696, 0.21850801222441053540, 0),
(0.67249851196395732696, -0.21850801222441053540, 0),
(0.67249851196395732696, 0.21850801222441053540, 0),
(-0.37174803446018449013, 0, 0.60150095500754567366),
(-0.48662449473386508189, -0.35355339059327376220,
0.37174803446018449013),
(-0.48662449473386508189, 0.35355339059327376220,
0.37174803446018449013),
(-0.18587401723009224507, -0.57206140281768429760,
-0.37174803446018449013),
(-0.18587401723009224507, 0.57206140281768429760,
-0.37174803446018449013),
(-0.11487646027368059176, -0.35355339059327376220,
0.60150095500754567366),
(-0.11487646027368059176, 0.35355339059327376220,
0.60150095500754567366)],
[0, 10, 16], [23, 10, 0], [16, 13, 0],
[0, 13, 23], [16, 10, 1], [1, 10, 23],
[1, 13, 16], [23, 13, 1], [2, 4, 19],
[22, 4, 2], [2, 19, 27], [27, 22, 2],
[20, 5, 3], [3, 5, 21], [26, 20, 3],
[3, 21, 26], [29, 19, 4], [4, 22, 29],
[5, 20, 28], [28, 21, 5], [6, 8, 15],
[17, 8, 6], [6, 15, 25], [25, 17, 6],
[14, 9, 7], [7, 9, 18], [24, 14, 7],
[7, 18, 24], [8, 12, 15], [17, 12, 8],
[14, 11, 9], [9, 11, 18], [11, 14, 24],
[24, 18, 11], [25, 15, 12], [12, 17, 25],
[29, 27, 19], [20, 26, 28], [28, 26, 21],
[22, 27, 29]]
assert Abs(polytope_integrate(octahedron_five_compound, expr)) - 0.353553\
< 1e-6
cube_five_compound = [[(-0.1624598481164531631, -0.5, -0.6881909602355867691),
(-0.1624598481164531631, 0.5, -0.6881909602355867691),
(0.1624598481164531631, -0.5, 0.68819096023558676910),
(0.1624598481164531631, 0.5, 0.68819096023558676910),
(-0.52573111211913359231, 0, -0.6881909602355867691),
(0.52573111211913359231, 0, 0.68819096023558676910),
(-0.26286555605956679615, -0.8090169943749474241,
-0.1624598481164531631),
(-0.26286555605956679615, 0.8090169943749474241,
-0.1624598481164531631),
(0.26286555605956680301, -0.8090169943749474241,
0.1624598481164531631),
(0.26286555605956680301, 0.8090169943749474241,
0.1624598481164531631),
(-0.42532540417601993887, -0.3090169943749474241,
0.68819096023558676910),
(-0.42532540417601993887, 0.30901699437494742410,
0.68819096023558676910),
(0.42532540417601996609, -0.3090169943749474241,
-0.6881909602355867691),
(0.42532540417601996609, 0.30901699437494742410,
-0.6881909602355867691),
(-0.6881909602355867691, -0.5, 0.1624598481164531631),
(-0.6881909602355867691, 0.5, 0.1624598481164531631),
(0.68819096023558676910, -0.5, -0.1624598481164531631),
(0.68819096023558676910, 0.5, -0.1624598481164531631),
(-0.85065080835203998877, 0, -0.1624598481164531631),
(0.85065080835203993218, 0, 0.1624598481164531631)],
[18, 10, 3, 7], [13, 19, 8, 0], [18, 0, 8, 10],
[3, 19, 13, 7], [18, 7, 13, 0], [8, 19, 3, 10],
[6, 2, 11, 18], [1, 9, 19, 12], [11, 9, 1, 18],
[6, 12, 19, 2], [1, 12, 6, 18], [11, 2, 19, 9],
[4, 14, 11, 7], [17, 5, 8, 12], [4, 12, 8, 14],
[11, 5, 17, 7], [4, 7, 17, 12], [8, 5, 11, 14],
[6, 10, 15, 4], [13, 9, 5, 16], [15, 9, 13, 4],
[6, 16, 5, 10], [13, 16, 6, 4], [15, 10, 5, 9],
[14, 15, 1, 0], [16, 17, 3, 2], [14, 2, 3, 15],
[1, 17, 16, 0], [14, 0, 16, 2], [3, 17, 1, 15]]
assert Abs(polytope_integrate(cube_five_compound, expr) - 1.25) < 1e-12
echidnahedron = [[(0, 0, -2.4898982848827801995),
(0, 0, 2.4898982848827802734),
(0, -4.2360679774997896964, -2.4898982848827801995),
(0, -4.2360679774997896964, 2.4898982848827802734),
(0, 4.2360679774997896964, -2.4898982848827801995),
(0, 4.2360679774997896964, 2.4898982848827802734),
(-4.0287400534704067567, -1.3090169943749474241, -2.4898982848827801995),
(-4.0287400534704067567, -1.3090169943749474241, 2.4898982848827802734),
(-4.0287400534704067567, 1.3090169943749474241, -2.4898982848827801995),
(-4.0287400534704067567, 1.3090169943749474241, 2.4898982848827802734),
(4.0287400534704069747, -1.3090169943749474241, -2.4898982848827801995),
(4.0287400534704069747, -1.3090169943749474241, 2.4898982848827802734),
(4.0287400534704069747, 1.3090169943749474241, -2.4898982848827801995),
(4.0287400534704069747, 1.3090169943749474241, 2.4898982848827802734),
(-2.4898982848827801995, -3.4270509831248422723, -2.4898982848827801995),
(-2.4898982848827801995, -3.4270509831248422723, 2.4898982848827802734),
(-2.4898982848827801995, 3.4270509831248422723, -2.4898982848827801995),
(-2.4898982848827801995, 3.4270509831248422723, 2.4898982848827802734),
(2.4898982848827802734, -3.4270509831248422723, -2.4898982848827801995),
(2.4898982848827802734, -3.4270509831248422723, 2.4898982848827802734),
(2.4898982848827802734, 3.4270509831248422723, -2.4898982848827801995),
(2.4898982848827802734, 3.4270509831248422723, 2.4898982848827802734),
(-4.7169310137059934362, -0.8090169943749474241, -1.1135163644116066184),
(-4.7169310137059934362, 0.8090169943749474241, -1.1135163644116066184),
(4.7169310137059937438, -0.8090169943749474241, 1.11351636441160673519),
(4.7169310137059937438, 0.8090169943749474241, 1.11351636441160673519),
(-4.2916056095299737777, -2.1180339887498948482, 1.11351636441160673519),
(-4.2916056095299737777, 2.1180339887498948482, 1.11351636441160673519),
(4.2916056095299737777, -2.1180339887498948482, -1.1135163644116066184),
(4.2916056095299737777, 2.1180339887498948482, -1.1135163644116066184),
(-3.6034146492943870399, 0, -3.3405490932348205213),
(3.6034146492943870399, 0, 3.3405490932348202056),
(-3.3405490932348205213, -3.4270509831248422723, 1.11351636441160673519),
(-3.3405490932348205213, 3.4270509831248422723, 1.11351636441160673519),
(3.3405490932348202056, -3.4270509831248422723, -1.1135163644116066184),
(3.3405490932348202056, 3.4270509831248422723, -1.1135163644116066184),
(-2.9152236890588002395, -2.1180339887498948482, 3.3405490932348202056),
(-2.9152236890588002395, 2.1180339887498948482, 3.3405490932348202056),
(2.9152236890588002395, -2.1180339887498948482, -3.3405490932348205213),
(2.9152236890588002395, 2.1180339887498948482, -3.3405490932348205213),
(-2.2270327288232132368, 0, -1.1135163644116066184),
(-2.2270327288232132368, -4.2360679774997896964, -1.1135163644116066184),
(-2.2270327288232132368, 4.2360679774997896964, -1.1135163644116066184),
(2.2270327288232134704, 0, 1.11351636441160673519),
(2.2270327288232134704, -4.2360679774997896964, 1.11351636441160673519),
(2.2270327288232134704, 4.2360679774997896964, 1.11351636441160673519),
(-1.8017073246471935200, -1.3090169943749474241, 1.11351636441160673519),
(-1.8017073246471935200, 1.3090169943749474241, 1.11351636441160673519),
(1.8017073246471935043, -1.3090169943749474241, -1.1135163644116066184),
(1.8017073246471935043, 1.3090169943749474241, -1.1135163644116066184),
(-1.3763819204711735382, 0, -4.7169310137059934362),
(-1.3763819204711735382, 0, 0.26286555605956679615),
(1.37638192047117353821, 0, 4.7169310137059937438),
(1.37638192047117353821, 0, -0.26286555605956679615),
(-1.1135163644116066184, -3.4270509831248422723, -3.3405490932348205213),
(-1.1135163644116066184, -0.8090169943749474241, 4.7169310137059937438),
(-1.1135163644116066184, -0.8090169943749474241, -0.26286555605956679615),
(-1.1135163644116066184, 0.8090169943749474241, 4.7169310137059937438),
(-1.1135163644116066184, 0.8090169943749474241, -0.26286555605956679615),
(-1.1135163644116066184, 3.4270509831248422723, -3.3405490932348205213),
(1.11351636441160673519, -3.4270509831248422723, 3.3405490932348202056),
(1.11351636441160673519, -0.8090169943749474241, -4.7169310137059934362),
(1.11351636441160673519, -0.8090169943749474241, 0.26286555605956679615),
(1.11351636441160673519, 0.8090169943749474241, -4.7169310137059934362),
(1.11351636441160673519, 0.8090169943749474241, 0.26286555605956679615),
(1.11351636441160673519, 3.4270509831248422723, 3.3405490932348202056),
(-0.85065080835203998877, 0, 1.11351636441160673519),
(0.85065080835203993218, 0, -1.1135163644116066184),
(-0.6881909602355867691, -0.5, -1.1135163644116066184),
(-0.6881909602355867691, 0.5, -1.1135163644116066184),
(-0.6881909602355867691, -4.7360679774997896964, -1.1135163644116066184),
(-0.6881909602355867691, -2.1180339887498948482, -1.1135163644116066184),
(-0.6881909602355867691, 2.1180339887498948482, -1.1135163644116066184),
(-0.6881909602355867691, 4.7360679774997896964, -1.1135163644116066184),
(0.68819096023558676910, -0.5, 1.11351636441160673519),
(0.68819096023558676910, 0.5, 1.11351636441160673519),
(0.68819096023558676910, -4.7360679774997896964, 1.11351636441160673519),
(0.68819096023558676910, -2.1180339887498948482, 1.11351636441160673519),
(0.68819096023558676910, 2.1180339887498948482, 1.11351636441160673519),
(0.68819096023558676910, 4.7360679774997896964, 1.11351636441160673519),
(-0.42532540417601993887, -1.3090169943749474241, -4.7169310137059934362),
(-0.42532540417601993887, -1.3090169943749474241, 0.26286555605956679615),
(-0.42532540417601993887, 1.3090169943749474241, -4.7169310137059934362),
(-0.42532540417601993887, 1.3090169943749474241, 0.26286555605956679615),
(-0.26286555605956679615, -0.8090169943749474241, 1.11351636441160673519),
(-0.26286555605956679615, 0.8090169943749474241, 1.11351636441160673519),
(0.26286555605956679615, -0.8090169943749474241, -1.1135163644116066184),
(0.26286555605956679615, 0.8090169943749474241, -1.1135163644116066184),
(0.42532540417601996609, -1.3090169943749474241, 4.7169310137059937438),
(0.42532540417601996609, -1.3090169943749474241, -0.26286555605956679615),
(0.42532540417601996609, 1.3090169943749474241, 4.7169310137059937438),
(0.42532540417601996609, 1.3090169943749474241, -0.26286555605956679615)],
[9, 66, 47], [44, 62, 77], [20, 91, 49], [33, 47, 83],
[3, 77, 84], [12, 49, 53], [36, 84, 66], [28, 53, 62],
[73, 83, 91], [15, 84, 46], [25, 64, 43], [16, 58, 72],
[26, 46, 51], [11, 43, 74], [4, 72, 91], [60, 74, 84],
[35, 91, 64], [23, 51, 58], [19, 74, 77], [79, 83, 78],
[6, 56, 40], [76, 77, 81], [21, 78, 75], [8, 40, 58],
[31, 75, 74], [42, 58, 83], [41, 81, 56], [13, 75, 43],
[27, 51, 47], [2, 89, 71], [24, 43, 62], [17, 47, 85],
[14, 71, 56], [65, 85, 75], [22, 56, 51], [34, 62, 89],
[5, 85, 78], [32, 81, 46], [10, 53, 48], [45, 78, 64],
[7, 46, 66], [18, 48, 89], [37, 66, 85], [70, 89, 81],
[29, 64, 53], [88, 74, 1], [38, 67, 48], [42, 83, 72],
[57, 1, 85], [34, 48, 62], [59, 72, 87], [19, 62, 74],
[63, 87, 67], [17, 85, 83], [52, 75, 1], [39, 87, 49],
[22, 51, 40], [55, 1, 66], [29, 49, 64], [30, 40, 69],
[13, 64, 75], [82, 69, 87], [7, 66, 51], [90, 85, 1],
[59, 69, 72], [70, 81, 71], [88, 1, 84], [73, 72, 83],
[54, 71, 68], [5, 83, 85], [50, 68, 69], [3, 84, 81],
[57, 66, 1], [30, 68, 40], [28, 62, 48], [52, 1, 74],
[23, 40, 51], [38, 48, 86], [9, 51, 66], [80, 86, 68],
[11, 74, 62], [55, 84, 1], [54, 86, 71], [35, 64, 49],
[90, 1, 75], [41, 71, 81], [39, 49, 67], [15, 81, 84],
[61, 67, 86], [21, 75, 64], [24, 53, 43], [50, 69, 0],
[37, 85, 47], [31, 43, 75], [61, 0, 67], [27, 47, 58],
[10, 67, 53], [8, 58, 69], [90, 75, 85], [45, 91, 78],
[80, 68, 0], [36, 66, 46], [65, 78, 85], [63, 0, 87],
[32, 46, 56], [20, 87, 91], [14, 56, 68], [57, 85, 66],
[33, 58, 47], [61, 86, 0], [60, 84, 77], [37, 47, 66],
[82, 0, 69], [44, 77, 89], [16, 69, 58], [18, 89, 86],
[55, 66, 84], [26, 56, 46], [63, 67, 0], [31, 74, 43],
[36, 46, 84], [50, 0, 68], [25, 43, 53], [6, 68, 56],
[12, 53, 67], [88, 84, 74], [76, 89, 77], [82, 87, 0],
[65, 75, 78], [60, 77, 74], [80, 0, 86], [79, 78, 91],
[2, 86, 89], [4, 91, 87], [52, 74, 75], [21, 64, 78],
[18, 86, 48], [23, 58, 40], [5, 78, 83], [28, 48, 53],
[6, 40, 68], [25, 53, 64], [54, 68, 86], [33, 83, 58],
[17, 83, 47], [12, 67, 49], [41, 56, 71], [9, 47, 51],
[35, 49, 91], [2, 71, 86], [79, 91, 83], [38, 86, 67],
[26, 51, 56], [7, 51, 46], [4, 87, 72], [34, 89, 48],
[15, 46, 81], [42, 72, 58], [10, 48, 67], [27, 58, 51],
[39, 67, 87], [76, 81, 89], [3, 81, 77], [8, 69, 40],
[29, 53, 49], [19, 77, 62], [22, 40, 56], [20, 49, 87],
[32, 56, 81], [59, 87, 69], [24, 62, 53], [11, 62, 43],
[14, 68, 71], [73, 91, 72], [13, 43, 64], [70, 71, 89],
[16, 72, 69], [44, 89, 62], [30, 69, 68], [45, 64, 91]]
# Actual volume is : 51.405764746872634
assert Abs(polytope_integrate(echidnahedron, 1) - 51.4057647468726) < 1e-12
assert Abs(polytope_integrate(echidnahedron, expr) - 253.569603474519) <\
1e-12
# Tests for many polynomials with maximum degree given(2D case).
assert polytope_integrate(cube2, [x**2, y*z], max_degree=2) == \
{y * z: 3125 / S(4), x ** 2: 3125 / S(3)}
assert polytope_integrate(cube2, max_degree=2) == \
{1: 125, x: 625 / S(2), x * z: 3125 / S(4), y: 625 / S(2),
y * z: 3125 / S(4), z ** 2: 3125 / S(3), y ** 2: 3125 / S(3),
z: 625 / S(2), x * y: 3125 / S(4), x ** 2: 3125 / S(3)}
def test_point_sort():
assert point_sort([Point(0, 0), Point(1, 0), Point(1, 1)]) == \
[Point2D(1, 1), Point2D(1, 0), Point2D(0, 0)]
fig6 = Polygon((0, 0), (1, 0), (1, 1))
assert polytope_integrate(fig6, x*y) == S(-1)/8
assert polytope_integrate(fig6, x*y, clockwise = True) == S(1)/8
def test_polytopes_intersecting_sides():
fig5 = Polygon(Point(-4.165, -0.832), Point(-3.668, 1.568),
Point(-3.266, 1.279), Point(-1.090, -2.080),
Point(3.313, -0.683), Point(3.033, -4.845),
Point(-4.395, 4.840), Point(-1.007, -3.328))
assert polytope_integrate(fig5, x**2 + x*y + y**2) ==\
S(1633405224899363)/(24*10**12)
fig6 = Polygon(Point(-3.018, -4.473), Point(-0.103, 2.378),
Point(-1.605, -2.308), Point(4.516, -0.771),
Point(4.203, 0.478))
assert polytope_integrate(fig6, x**2 + x*y + y**2) ==\
S(88161333955921)/(3*10**12)
def test_max_degree():
polygon = Polygon((0, 0), (0, 1), (1, 1), (1, 0))
polys = [1, x, y, x*y, x**2*y, x*y**2]
assert polytope_integrate(polygon, polys, max_degree=3) == \
{1: 1, x: S(1)/2, y: S(1)/2, x*y: S(1)/4, x**2*y: S(1)/6, x*y**2: S(1)/6}
def test_main_integrate3d():
cube = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0),\
(5, 0, 5), (5, 5, 0), (5, 5, 5)],\
[2, 6, 7, 3], [3, 7, 5, 1], [7, 6, 4, 5], [1, 5, 4, 0],\
[3, 1, 0, 2], [0, 4, 6, 2]]
vertices = cube[0]
faces = cube[1:]
hp_params = hyperplane_parameters(faces, vertices)
assert main_integrate3d(1, faces, vertices, hp_params) == -125
assert main_integrate3d(1, faces, vertices, hp_params, max_degree=1) == \
{1: -125, y: -S(625)/2, z: -S(625)/2, x: -S(625)/2}
def test_main_integrate():
triangle = Polygon((0, 3), (5, 3), (1, 1))
facets = triangle.sides
hp_params = hyperplane_parameters(triangle)
assert main_integrate(x**2 + y**2, facets, hp_params) == S(325)/6
assert main_integrate(x**2 + y**2, facets, hp_params, max_degree=1) == \
{0: 0, 1: 5, y: S(35)/3, x: 10}
def test_polygon_integrate():
cube = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0),\
(5, 0, 5), (5, 5, 0), (5, 5, 5)],\
[2, 6, 7, 3], [3, 7, 5, 1], [7, 6, 4, 5], [1, 5, 4, 0],\
[3, 1, 0, 2], [0, 4, 6, 2]]
facet = cube[1]
facets = cube[1:]
vertices = cube[0]
assert polygon_integrate(facet, [(0, 1, 0), 5], 0, facets, vertices, 1, 0) == -25
def test_distance_to_side():
point = (0, 0, 0)
assert distance_to_side(point, [(0, 0, 1), (0, 1, 0)], (1, 0, 0)) == -sqrt(2)/2
def test_lineseg_integrate():
polygon = [(0, 5, 0), (5, 5, 0), (5, 5, 5), (0, 5, 5)]
line_seg = [(0, 5, 0), (5, 5, 0)]
assert lineseg_integrate(polygon, 0, line_seg, 1, 0) == 5
assert lineseg_integrate(polygon, 0, line_seg, 0, 0) == 0
def test_integration_reduction():
triangle = Polygon(Point(0, 3), Point(5, 3), Point(1, 1))
facets = triangle.sides
a, b = hyperplane_parameters(triangle)[0]
assert integration_reduction(facets, 0, a, b, 1, (x, y), 0) == 5
assert integration_reduction(facets, 0, a, b, 0, (x, y), 0) == 0
def test_integration_reduction_dynamic():
triangle = Polygon(Point(0, 3), Point(5, 3), Point(1, 1))
facets = triangle.sides
a, b = hyperplane_parameters(triangle)[0]
x0 = facets[0].points[0]
monomial_values = [[0, 0, 0, 0], [1, 0, 0, 5],\
[y, 0, 1, 15], [x, 1, 0, None]]
assert integration_reduction_dynamic(facets, 0, a, b, x, 1, (x, y), 1,\
0, 1, x0, monomial_values, 3) == S(25)/2
assert integration_reduction_dynamic(facets, 0, a, b, 0, 1, (x, y), 1,\
0, 1, x0, monomial_values, 3) == 0
def test_is_vertex():
assert is_vertex(2) is False
assert is_vertex((2, 3)) is True
assert is_vertex(Point(2, 3)) is True
assert is_vertex((2, 3, 4)) is True
assert is_vertex((2, 3, 4, 5)) is False
|
f3770dd0a8d2a29f5e3700070926c2a3c9bb65d8a1b0abbbc8eda5d1e271e382
|
from sympy import Rational, sqrt, symbols, sin, exp, log, sinh, cosh, cos, pi, \
I, erf, tan, asin, asinh, acos, atan, Function, Derivative, diff, simplify, \
LambertW, Eq, Ne, Piecewise, Symbol, Add, ratsimp, Integral, Sum, \
besselj, besselk, bessely, jn, tanh
from sympy.integrals.heurisch import components, heurisch, heurisch_wrapper
from sympy.utilities.pytest import XFAIL, skip, slow, ON_TRAVIS
from sympy.integrals.integrals import integrate
x, y, z, nu = symbols('x,y,z,nu')
f = Function('f')
def test_components():
assert components(x*y, x) == {x}
assert components(1/(x + y), x) == {x}
assert components(sin(x), x) == {sin(x), x}
assert components(sin(x)*sqrt(log(x)), x) == \
{log(x), sin(x), sqrt(log(x)), x}
assert components(x*sin(exp(x)*y), x) == \
{sin(y*exp(x)), x, exp(x)}
assert components(x**Rational(17, 54)/sqrt(sin(x)), x) == \
{sin(x), x**Rational(1, 54), sqrt(sin(x)), x}
assert components(f(x), x) == \
{x, f(x)}
assert components(Derivative(f(x), x), x) == \
{x, f(x), Derivative(f(x), x)}
assert components(f(x)*diff(f(x), x), x) == \
{x, f(x), Derivative(f(x), x), Derivative(f(x), x)}
def test_issue_10680():
assert isinstance(integrate(x**log(x**log(x**log(x))),x), Integral)
def test_heurisch_polynomials():
assert heurisch(1, x) == x
assert heurisch(x, x) == x**2/2
assert heurisch(x**17, x) == x**18/18
def test_heurisch_fractions():
assert heurisch(1/x, x) == log(x)
assert heurisch(1/(2 + x), x) == log(x + 2)
assert heurisch(1/(x + sin(y)), x) == log(x + sin(y))
# Up to a constant, where C = 5*pi*I/12, Mathematica gives identical
# result in the first case. The difference is because sympy changes
# signs of expressions without any care.
# XXX ^ ^ ^ is this still correct?
assert heurisch(5*x**5/(
2*x**6 - 5), x) in [5*log(2*x**6 - 5) / 12, 5*log(-2*x**6 + 5) / 12]
assert heurisch(5*x**5/(2*x**6 + 5), x) == 5*log(2*x**6 + 5) / 12
assert heurisch(1/x**2, x) == -1/x
assert heurisch(-1/x**5, x) == 1/(4*x**4)
def test_heurisch_log():
assert heurisch(log(x), x) == x*log(x) - x
assert heurisch(log(3*x), x) == -x + x*log(3) + x*log(x)
assert heurisch(log(x**2), x) in [x*log(x**2) - 2*x, 2*x*log(x) - 2*x]
def test_heurisch_exp():
assert heurisch(exp(x), x) == exp(x)
assert heurisch(exp(-x), x) == -exp(-x)
assert heurisch(exp(17*x), x) == exp(17*x) / 17
assert heurisch(x*exp(x), x) == x*exp(x) - exp(x)
assert heurisch(x*exp(x**2), x) == exp(x**2) / 2
assert heurisch(exp(-x**2), x) is None
assert heurisch(2**x, x) == 2**x/log(2)
assert heurisch(x*2**x, x) == x*2**x/log(2) - 2**x*log(2)**(-2)
assert heurisch(Integral(x**z*y, (y, 1, 2), (z, 2, 3)).function, x) == (x*x**z*y)/(z+1)
assert heurisch(Sum(x**z, (z, 1, 2)).function, z) == x**z/log(x)
def test_heurisch_trigonometric():
assert heurisch(sin(x), x) == -cos(x)
assert heurisch(pi*sin(x) + 1, x) == x - pi*cos(x)
assert heurisch(cos(x), x) == sin(x)
assert heurisch(tan(x), x) in [
log(1 + tan(x)**2)/2,
log(tan(x) + I) + I*x,
log(tan(x) - I) - I*x,
]
assert heurisch(sin(x)*sin(y), x) == -cos(x)*sin(y)
assert heurisch(sin(x)*sin(y), y) == -cos(y)*sin(x)
# gives sin(x) in answer when run via setup.py and cos(x) when run via py.test
assert heurisch(sin(x)*cos(x), x) in [sin(x)**2 / 2, -cos(x)**2 / 2]
assert heurisch(cos(x)/sin(x), x) == log(sin(x))
assert heurisch(x*sin(7*x), x) == sin(7*x) / 49 - x*cos(7*x) / 7
assert heurisch(1/pi/4 * x**2*cos(x), x) == 1/pi/4*(x**2*sin(x) -
2*sin(x) + 2*x*cos(x))
assert heurisch(acos(x/4) * asin(x/4), x) == 2*x - (sqrt(16 - x**2))*asin(x/4) \
+ (sqrt(16 - x**2))*acos(x/4) + x*asin(x/4)*acos(x/4)
assert heurisch(sin(x)/(cos(x)**2+1), x) == -atan(cos(x)) #fixes issue 13723
assert heurisch(1/(cos(x)+2), x) == 2*sqrt(3)*atan(sqrt(3)*tan(x/2)/3)/3
assert heurisch(2*sin(x)*cos(x)/(sin(x)**4 + 1), x) == atan(sqrt(2)*sin(x)
- 1) - atan(sqrt(2)*sin(x) + 1)
assert heurisch(1/cosh(x), x) == 2*atan(tanh(x/2))
def test_heurisch_hyperbolic():
assert heurisch(sinh(x), x) == cosh(x)
assert heurisch(cosh(x), x) == sinh(x)
assert heurisch(x*sinh(x), x) == x*cosh(x) - sinh(x)
assert heurisch(x*cosh(x), x) == x*sinh(x) - cosh(x)
assert heurisch(
x*asinh(x/2), x) == x**2*asinh(x/2)/2 + asinh(x/2) - x*sqrt(4 + x**2)/4
def test_heurisch_mixed():
assert heurisch(sin(x)*exp(x), x) == exp(x)*sin(x)/2 - exp(x)*cos(x)/2
def test_heurisch_radicals():
assert heurisch(1/sqrt(x), x) == 2*sqrt(x)
assert heurisch(1/sqrt(x)**3, x) == -2/sqrt(x)
assert heurisch(sqrt(x)**3, x) == 2*sqrt(x)**5/5
assert heurisch(sin(x)*sqrt(cos(x)), x) == -2*sqrt(cos(x))**3/3
y = Symbol('y')
assert heurisch(sin(y*sqrt(x)), x) == 2/y**2*sin(y*sqrt(x)) - \
2*sqrt(x)*cos(y*sqrt(x))/y
assert heurisch_wrapper(sin(y*sqrt(x)), x) == Piecewise(
(-2*sqrt(x)*cos(sqrt(x)*y)/y + 2*sin(sqrt(x)*y)/y**2, Ne(y, 0)),
(0, True))
y = Symbol('y', positive=True)
assert heurisch_wrapper(sin(y*sqrt(x)), x) == 2/y**2*sin(y*sqrt(x)) - \
2*sqrt(x)*cos(y*sqrt(x))/y
def test_heurisch_special():
assert heurisch(erf(x), x) == x*erf(x) + exp(-x**2)/sqrt(pi)
assert heurisch(exp(-x**2)*erf(x), x) == sqrt(pi)*erf(x)**2 / 4
def test_heurisch_symbolic_coeffs():
assert heurisch(1/(x + y), x) == log(x + y)
assert heurisch(1/(x + sqrt(2)), x) == log(x + sqrt(2))
assert simplify(diff(heurisch(log(x + y + z), y), y)) == log(x + y + z)
def test_heurisch_symbolic_coeffs_1130():
y = Symbol('y')
assert heurisch_wrapper(1/(x**2 + y), x) == Piecewise(
(-I*log(x - I*sqrt(y))/(2*sqrt(y))
+ I*log(x + I*sqrt(y))/(2*sqrt(y)), Ne(y, 0)),
(-1/x, True))
y = Symbol('y', positive=True)
assert heurisch_wrapper(1/(x**2 + y), x) == (atan(x/sqrt(y))/sqrt(y))
def test_heurisch_hacking():
assert heurisch(sqrt(1 + 7*x**2), x, hints=[]) == \
x*sqrt(1 + 7*x**2)/2 + sqrt(7)*asinh(sqrt(7)*x)/14
assert heurisch(sqrt(1 - 7*x**2), x, hints=[]) == \
x*sqrt(1 - 7*x**2)/2 + sqrt(7)*asin(sqrt(7)*x)/14
assert heurisch(1/sqrt(1 + 7*x**2), x, hints=[]) == \
sqrt(7)*asinh(sqrt(7)*x)/7
assert heurisch(1/sqrt(1 - 7*x**2), x, hints=[]) == \
sqrt(7)*asin(sqrt(7)*x)/7
assert heurisch(exp(-7*x**2), x, hints=[]) == \
sqrt(7*pi)*erf(sqrt(7)*x)/14
assert heurisch(1/sqrt(9 - 4*x**2), x, hints=[]) == \
asin(2*x/3)/2
assert heurisch(1/sqrt(9 + 4*x**2), x, hints=[]) == \
asinh(2*x/3)/2
def test_heurisch_function():
assert heurisch(f(x), x) is None
@XFAIL
def test_heurisch_function_derivative():
# TODO: it looks like this used to work just by coincindence and
# thanks to sloppy implementation. Investigate why this used to
# work at all and if support for this can be restored.
df = diff(f(x), x)
assert heurisch(f(x)*df, x) == f(x)**2/2
assert heurisch(f(x)**2*df, x) == f(x)**3/3
assert heurisch(df/f(x), x) == log(f(x))
def test_heurisch_wrapper():
f = 1/(y + x)
assert heurisch_wrapper(f, x) == log(x + y)
f = 1/(y - x)
assert heurisch_wrapper(f, x) == -log(x - y)
f = 1/((y - x)*(y + x))
assert heurisch_wrapper(f, x) == Piecewise(
(-log(x - y)/(2*y) + log(x + y)/(2*y), Ne(y, 0)), (1/x, True))
# issue 6926
f = sqrt(x**2/((y - x)*(y + x)))
assert heurisch_wrapper(f, x) == x*sqrt(x**2)*sqrt(1/(-x**2 + y**2)) \
- y**2*sqrt(x**2)*sqrt(1/(-x**2 + y**2))/x
def test_issue_3609():
assert heurisch(1/(x * (1 + log(x)**2)), x) == atan(log(x))
### These are examples from the Poor Man's Integrator
### http://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/examples/
def test_pmint_rat():
# TODO: heurisch() is off by a constant: -3/4. Possibly different permutation
# would give the optimal result?
def drop_const(expr, x):
if expr.is_Add:
return Add(*[ arg for arg in expr.args if arg.has(x) ])
else:
return expr
f = (x**7 - 24*x**4 - 4*x**2 + 8*x - 8)/(x**8 + 6*x**6 + 12*x**4 + 8*x**2)
g = (4 + 8*x**2 + 6*x + 3*x**3)/(x**5 + 4*x**3 + 4*x) + log(x)
assert drop_const(ratsimp(heurisch(f, x)), x) == g
def test_pmint_trig():
f = (x - tan(x)) / tan(x)**2 + tan(x)
g = -x**2/2 - x/tan(x) + log(tan(x)**2 + 1)/2
assert heurisch(f, x) == g
@slow # 8 seconds on 3.4 GHz
def test_pmint_logexp():
if ON_TRAVIS:
# See https://github.com/sympy/sympy/pull/12795
skip("Too slow for travis.")
f = (1 + x + x*exp(x))*(x + log(x) + exp(x) - 1)/(x + log(x) + exp(x))**2/x
g = log(x + exp(x) + log(x)) + 1/(x + exp(x) + log(x))
assert ratsimp(heurisch(f, x)) == g
@XFAIL # there's a hash dependent failure lurking here
def test_pmint_erf():
f = exp(-x**2)*erf(x)/(erf(x)**3 - erf(x)**2 - erf(x) + 1)
g = sqrt(pi)*log(erf(x) - 1)/8 - sqrt(pi)*log(erf(x) + 1)/8 - sqrt(pi)/(4*erf(x) - 4)
assert ratsimp(heurisch(f, x)) == g
def test_pmint_LambertW():
f = LambertW(x)
g = x*LambertW(x) - x + x/LambertW(x)
assert heurisch(f, x) == g
def test_pmint_besselj():
f = besselj(nu + 1, x)/besselj(nu, x)
g = nu*log(x) - log(besselj(nu, x))
assert heurisch(f, x) == g
f = (nu*besselj(nu, x) - x*besselj(nu + 1, x))/x
g = besselj(nu, x)
assert heurisch(f, x) == g
f = jn(nu + 1, x)/jn(nu, x)
g = nu*log(x) - log(jn(nu, x))
assert heurisch(f, x) == g
@slow
def test_pmint_bessel_products():
# Note: Derivatives of Bessel functions have many forms.
# Recurrence relations are needed for comparisons.
if ON_TRAVIS:
skip("Too slow for travis.")
f = x*besselj(nu, x)*bessely(nu, 2*x)
g = -2*x*besselj(nu, x)*bessely(nu - 1, 2*x)/3 + x*besselj(nu - 1, x)*bessely(nu, 2*x)/3
assert heurisch(f, x) == g
f = x*besselj(nu, x)*besselk(nu, 2*x)
g = -2*x*besselj(nu, x)*besselk(nu - 1, 2*x)/5 - x*besselj(nu - 1, x)*besselk(nu, 2*x)/5
assert heurisch(f, x) == g
@slow # 110 seconds on 3.4 GHz
def test_pmint_WrightOmega():
if ON_TRAVIS:
skip("Too slow for travis.")
def omega(x):
return LambertW(exp(x))
f = (1 + omega(x) * (2 + cos(omega(x)) * (x + omega(x))))/(1 + omega(x))/(x + omega(x))
g = log(x + LambertW(exp(x))) + sin(LambertW(exp(x)))
assert heurisch(f, x) == g
def test_RR():
# Make sure the algorithm does the right thing if the ring is RR. See
# issue 8685.
assert heurisch(sqrt(1 + 0.25*x**2), x, hints=[]) == \
0.5*x*sqrt(0.25*x**2 + 1) + 1.0*asinh(0.5*x)
# TODO: convert the rest of PMINT tests:
# Airy functions
# f = (x - AiryAi(x)*AiryAi(1, x)) / (x**2 - AiryAi(x)**2)
# g = Rational(1,2)*ln(x + AiryAi(x)) + Rational(1,2)*ln(x - AiryAi(x))
# f = x**2 * AiryAi(x)
# g = -AiryAi(x) + AiryAi(1, x)*x
# Whittaker functions
# f = WhittakerW(mu + 1, nu, x) / (WhittakerW(mu, nu, x) * x)
# g = x/2 - mu*ln(x) - ln(WhittakerW(mu, nu, x))
|
d17533dbc2eb5baee428cedacc2a670fb6a241421e0f42ed495f55ec0a1fcfa2
|
from sympy import (
Abs, acos, acosh, Add, And, asin, asinh, atan, Ci, cos, sinh, cosh,
tanh, Derivative, diff, DiracDelta, E, Ei, Eq, exp, erf, erfc, erfi,
EulerGamma, Expr, factor, Function, gamma, gammasimp, I, Idx, im, IndexedBase,
Integral, integrate, Interval, Lambda, LambertW, log, Matrix, Max, meijerg, Min, nan,
Ne, O, oo, pi, Piecewise, polar_lift, Poly, polygamma, Rational, re, S, Si, sign,
simplify, sin, sinc, SingularityFunction, sqrt, sstr, Sum, Symbol,
symbols, sympify, tan, trigsimp, Tuple
)
from sympy.functions.elementary.complexes import periodic_argument
from sympy.functions.elementary.integers import floor
from sympy.integrals.risch import NonElementaryIntegral
from sympy.physics import units
from sympy.core.compatibility import range
from sympy.utilities.pytest import XFAIL, raises, slow, skip, ON_TRAVIS
from sympy.utilities.randtest import verify_numerically
from sympy.integrals.integrals import Integral
x, y, a, t, x_1, x_2, z, s, b= symbols('x y a t x_1 x_2 z s b')
n = Symbol('n', integer=True)
f = Function('f')
def test_principal_value():
g = 1 / x
assert Integral(g, (x, -oo, oo)).principal_value() == 0
assert Integral(g, (y, -oo, oo)).principal_value() == oo * sign(1 / x)
raises(ValueError, lambda: Integral(g, (x)).principal_value())
raises(ValueError, lambda: Integral(g).principal_value())
l = 1 / ((x ** 3) - 1)
assert Integral(l, (x, -oo, oo)).principal_value() == -sqrt(3)*pi/3
raises(ValueError, lambda: Integral(l, (x, -oo, 1)).principal_value())
d = 1 / (x ** 2 - 1)
assert Integral(d, (x, -oo, oo)).principal_value() == 0
assert Integral(d, (x, -2, 2)).principal_value() == -log(3)
v = x / (x ** 2 - 1)
assert Integral(v, (x, -oo, oo)).principal_value() == 0
assert Integral(v, (x, -2, 2)).principal_value() == 0
s = x ** 2 / (x ** 2 - 1)
assert Integral(s, (x, -oo, oo)).principal_value() == oo
assert Integral(s, (x, -2, 2)).principal_value() == -log(3) + 4
f = 1 / ((x ** 2 - 1) * (1 + x ** 2))
assert Integral(f, (x, -oo, oo)).principal_value() == -pi / 2
assert Integral(f, (x, -2, 2)).principal_value() == -atan(2) - log(3) / 2
def diff_test(i):
"""Return the set of symbols, s, which were used in testing that
i.diff(s) agrees with i.doit().diff(s). If there is an error then
the assertion will fail, causing the test to fail."""
syms = i.free_symbols
for s in syms:
assert (i.diff(s).doit() - i.doit().diff(s)).expand() == 0
return syms
def test_improper_integral():
assert integrate(log(x), (x, 0, 1)) == -1
assert integrate(x**(-2), (x, 1, oo)) == 1
assert integrate(1/(1 + exp(x)), (x, 0, oo)) == log(2)
def test_constructor():
# this is shared by Sum, so testing Integral's constructor
# is equivalent to testing Sum's
s1 = Integral(n, n)
assert s1.limits == (Tuple(n),)
s2 = Integral(n, (n,))
assert s2.limits == (Tuple(n),)
s3 = Integral(Sum(x, (x, 1, y)))
assert s3.limits == (Tuple(y),)
s4 = Integral(n, Tuple(n,))
assert s4.limits == (Tuple(n),)
s5 = Integral(n, (n, Interval(1, 2)))
assert s5.limits == (Tuple(n, 1, 2),)
# Testing constructor with inequalities:
s6 = Integral(n, n > 10)
assert s6.limits == (Tuple(n, 10, oo),)
s7 = Integral(n, (n > 2) & (n < 5))
assert s7.limits == (Tuple(n, 2, 5),)
def test_basics():
assert Integral(0, x) != 0
assert Integral(x, (x, 1, 1)) != 0
assert Integral(oo, x) != oo
assert Integral(S.NaN, x) == S.NaN
assert diff(Integral(y, y), x) == 0
assert diff(Integral(x, (x, 0, 1)), x) == 0
assert diff(Integral(x, x), x) == x
assert diff(Integral(t, (t, 0, x)), x) == x
e = (t + 1)**2
assert diff(integrate(e, (t, 0, x)), x) == \
diff(Integral(e, (t, 0, x)), x).doit().expand() == \
((1 + x)**2).expand()
assert diff(integrate(e, (t, 0, x)), t) == \
diff(Integral(e, (t, 0, x)), t) == 0
assert diff(integrate(e, (t, 0, x)), a) == \
diff(Integral(e, (t, 0, x)), a) == 0
assert diff(integrate(e, t), a) == diff(Integral(e, t), a) == 0
assert integrate(e, (t, a, x)).diff(x) == \
Integral(e, (t, a, x)).diff(x).doit().expand()
assert Integral(e, (t, a, x)).diff(x).doit() == ((1 + x)**2)
assert integrate(e, (t, x, a)).diff(x).doit() == (-(1 + x)**2).expand()
assert integrate(t**2, (t, x, 2*x)).diff(x) == 7*x**2
assert Integral(x, x).atoms() == {x}
assert Integral(f(x), (x, 0, 1)).atoms() == {S(0), S(1), x}
assert diff_test(Integral(x, (x, 3*y))) == {y}
assert diff_test(Integral(x, (a, 3*y))) == {x, y}
assert integrate(x, (x, oo, oo)) == 0 #issue 8171
assert integrate(x, (x, -oo, -oo)) == 0
# sum integral of terms
assert integrate(y + x + exp(x), x) == x*y + x**2/2 + exp(x)
assert Integral(x).is_commutative
n = Symbol('n', commutative=False)
assert Integral(n + x, x).is_commutative is False
def test_diff_wrt():
class Test(Expr):
_diff_wrt = True
is_commutative = True
t = Test()
assert integrate(t + 1, t) == t**2/2 + t
assert integrate(t + 1, (t, 0, 1)) == S(3)/2
raises(ValueError, lambda: integrate(x + 1, x + 1))
raises(ValueError, lambda: integrate(x + 1, (x + 1, 0, 1)))
def test_basics_multiple():
assert diff_test(Integral(x, (x, 3*x, 5*y), (y, x, 2*x))) == {x}
assert diff_test(Integral(x, (x, 5*y), (y, x, 2*x))) == {x}
assert diff_test(Integral(x, (x, 5*y), (y, y, 2*x))) == {x, y}
assert diff_test(Integral(y, y, x)) == {x, y}
assert diff_test(Integral(y*x, x, y)) == {x, y}
assert diff_test(Integral(x + y, y, (y, 1, x))) == {x}
assert diff_test(Integral(x + y, (x, x, y), (y, y, x))) == {x, y}
def test_conjugate_transpose():
A, B = symbols("A B", commutative=False)
x = Symbol("x", complex=True)
p = Integral(A*B, (x,))
assert p.adjoint().doit() == p.doit().adjoint()
assert p.conjugate().doit() == p.doit().conjugate()
assert p.transpose().doit() == p.doit().transpose()
x = Symbol("x", real=True)
p = Integral(A*B, (x,))
assert p.adjoint().doit() == p.doit().adjoint()
assert p.conjugate().doit() == p.doit().conjugate()
assert p.transpose().doit() == p.doit().transpose()
def test_integration():
assert integrate(0, (t, 0, x)) == 0
assert integrate(3, (t, 0, x)) == 3*x
assert integrate(t, (t, 0, x)) == x**2/2
assert integrate(3*t, (t, 0, x)) == 3*x**2/2
assert integrate(3*t**2, (t, 0, x)) == x**3
assert integrate(1/t, (t, 1, x)) == log(x)
assert integrate(-1/t**2, (t, 1, x)) == 1/x - 1
assert integrate(t**2 + 5*t - 8, (t, 0, x)) == x**3/3 + 5*x**2/2 - 8*x
assert integrate(x**2, x) == x**3/3
assert integrate((3*t*x)**5, x) == (3*t)**5 * x**6 / 6
b = Symbol("b")
c = Symbol("c")
assert integrate(a*t, (t, 0, x)) == a*x**2/2
assert integrate(a*t**4, (t, 0, x)) == a*x**5/5
assert integrate(a*t**2 + b*t + c, (t, 0, x)) == a*x**3/3 + b*x**2/2 + c*x
def test_multiple_integration():
assert integrate((x**2)*(y**2), (x, 0, 1), (y, -1, 2)) == Rational(1)
assert integrate((y**2)*(x**2), x, y) == Rational(1, 9)*(x**3)*(y**3)
assert integrate(1/(x + 3)/(1 + x)**3, x) == \
-S(1)/8*log(3 + x) + S(1)/8*log(1 + x) + x/(4 + 8*x + 4*x**2)
assert integrate(sin(x*y)*y, (x, 0, 1), (y, 0, 1)) == -sin(1) + 1
def test_issue_3532():
assert integrate(exp(-x), (x, 0, oo)) == 1
def test_issue_3560():
assert integrate(sqrt(x)**3, x) == 2*sqrt(x)**5/5
assert integrate(sqrt(x), x) == 2*sqrt(x)**3/3
assert integrate(1/sqrt(x)**3, x) == -2/sqrt(x)
def test_integrate_poly():
p = Poly(x + x**2*y + y**3, x, y)
qx = integrate(p, x)
qy = integrate(p, y)
assert isinstance(qx, Poly) is True
assert isinstance(qy, Poly) is True
assert qx.gens == (x, y)
assert qy.gens == (x, y)
assert qx.as_expr() == x**2/2 + x**3*y/3 + x*y**3
assert qy.as_expr() == x*y + x**2*y**2/2 + y**4/4
def test_integrate_poly_defined():
p = Poly(x + x**2*y + y**3, x, y)
Qx = integrate(p, (x, 0, 1))
Qy = integrate(p, (y, 0, pi))
assert isinstance(Qx, Poly) is True
assert isinstance(Qy, Poly) is True
assert Qx.gens == (y,)
assert Qy.gens == (x,)
assert Qx.as_expr() == Rational(1, 2) + y/3 + y**3
assert Qy.as_expr() == pi**4/4 + pi*x + pi**2*x**2/2
def test_integrate_omit_var():
y = Symbol('y')
assert integrate(x) == x**2/2
raises(ValueError, lambda: integrate(2))
raises(ValueError, lambda: integrate(x*y))
def test_integrate_poly_accurately():
y = Symbol('y')
assert integrate(x*sin(y), x) == x**2*sin(y)/2
# when passed to risch_norman, this will be a CPU hog, so this really
# checks, that integrated function is recognized as polynomial
assert integrate(x**1000*sin(y), x) == x**1001*sin(y)/1001
def test_issue_3635():
y = Symbol('y')
assert integrate(x**2, y) == x**2*y
assert integrate(x**2, (y, -1, 1)) == 2*x**2
# works in sympy and py.test but hangs in `setup.py test`
def test_integrate_linearterm_pow():
# check integrate((a*x+b)^c, x) -- issue 3499
y = Symbol('y', positive=True)
# TODO: Remove conds='none' below, let the assumption take care of it.
assert integrate(x**y, x, conds='none') == x**(y + 1)/(y + 1)
assert integrate((exp(y)*x + 1/y)**(1 + sin(y)), x, conds='none') == \
exp(-y)*(exp(y)*x + 1/y)**(2 + sin(y)) / (2 + sin(y))
def test_issue_3618():
assert integrate(pi*sqrt(x), x) == 2*pi*sqrt(x)**3/3
assert integrate(pi*sqrt(x) + E*sqrt(x)**3, x) == \
2*pi*sqrt(x)**3/3 + 2*E *sqrt(x)**5/5
def test_issue_3623():
assert integrate(cos((n + 1)*x), x) == Piecewise(
(sin(x*(n + 1))/(n + 1), Ne(n + 1, 0)), (x, True))
assert integrate(cos((n - 1)*x), x) == Piecewise(
(sin(x*(n - 1))/(n - 1), Ne(n - 1, 0)), (x, True))
assert integrate(cos((n + 1)*x) + cos((n - 1)*x), x) == \
Piecewise((sin(x*(n - 1))/(n - 1), Ne(n - 1, 0)), (x, True)) + \
Piecewise((sin(x*(n + 1))/(n + 1), Ne(n + 1, 0)), (x, True))
def test_issue_3664():
n = Symbol('n', integer=True, nonzero=True)
assert integrate(-1./2 * x * sin(n * pi * x/2), [x, -2, 0]) == \
2*cos(pi*n)/(pi*n)
assert integrate(-Rational(1)/2 * x * sin(n * pi * x/2), [x, -2, 0]) == \
2*cos(pi*n)/(pi*n)
def test_issue_3679():
# definite integration of rational functions gives wrong answers
assert NS(Integral(1/(x**2 - 8*x + 17), (x, 2, 4))) == '1.10714871779409'
def test_issue_3686(): # remove this when fresnel itegrals are implemented
from sympy import expand_func, fresnels
assert expand_func(integrate(sin(x**2), x)) == \
sqrt(2)*sqrt(pi)*fresnels(sqrt(2)*x/sqrt(pi))/2
def test_integrate_units():
m = units.m
s = units.s
assert integrate(x * m/s, (x, 1*s, 5*s)) == 12*m*s
def test_transcendental_functions():
assert integrate(LambertW(2*x), x) == \
-x + x*LambertW(2*x) + x/LambertW(2*x)
def test_log_polylog():
assert integrate(log(1 - x)/x, (x, 0, 1)) == -pi**2/6
assert integrate(log(x)*(1 - x)**(-1), (x, 0, 1)) == -pi**2/6
def test_issue_3740():
f = 4*log(x) - 2*log(x)**2
fid = diff(integrate(f, x), x)
assert abs(f.subs(x, 42).evalf() - fid.subs(x, 42).evalf()) < 1e-10
def test_issue_3788():
assert integrate(1/(1 + x**2), x) == atan(x)
def test_issue_3952():
f = sin(x)
assert integrate(f, x) == -cos(x)
raises(ValueError, lambda: integrate(f, 2*x))
def test_issue_4516():
assert integrate(2**x - 2*x, x) == 2**x/log(2) - x**2
def test_issue_7450():
ans = integrate(exp(-(1 + I)*x), (x, 0, oo))
assert re(ans) == S.Half and im(ans) == -S.Half
def test_issue_8623():
assert integrate((1 + cos(2*x)) / (3 - 2*cos(2*x)), (x, 0, pi)) == -pi/2 + sqrt(5)*pi/2
assert integrate((1 + cos(2*x))/(3 - 2*cos(2*x))) == -x/2 + sqrt(5)*(atan(sqrt(5)*tan(x)) + \
pi*floor((x - pi/2)/pi))/2
def test_issue_9569():
assert integrate(1 / (2 - cos(x)), (x, 0, pi)) == pi/sqrt(3)
assert integrate(1/(2 - cos(x))) == 2*sqrt(3)*(atan(sqrt(3)*tan(x/2)) + pi*floor((x/2 - pi/2)/pi))/3
def test_issue_13749():
assert integrate(1 / (2 + cos(x)), (x, 0, pi)) == pi/sqrt(3)
assert integrate(1/(2 + cos(x))) == 2*sqrt(3)*(atan(sqrt(3)*tan(x/2)/3) + pi*floor((x/2 - pi/2)/pi))/3
def test_matrices():
M = Matrix(2, 2, lambda i, j: (i + j + 1)*sin((i + j + 1)*x))
assert integrate(M, x) == Matrix([
[-cos(x), -cos(2*x)],
[-cos(2*x), -cos(3*x)],
])
def test_integrate_functions():
# issue 4111
assert integrate(f(x), x) == Integral(f(x), x)
assert integrate(f(x), (x, 0, 1)) == Integral(f(x), (x, 0, 1))
assert integrate(f(x)*diff(f(x), x), x) == f(x)**2/2
assert integrate(diff(f(x), x) / f(x), x) == log(f(x))
def test_integrate_derivatives():
assert integrate(Derivative(f(x), x), x) == f(x)
assert integrate(Derivative(f(y), y), x) == x*Derivative(f(y), y)
assert integrate(Derivative(f(x), x)**2, x) == \
Integral(Derivative(f(x), x)**2, x)
def test_transform():
a = Integral(x**2 + 1, (x, -1, 2))
fx = x
fy = 3*y + 1
assert a.doit() == a.transform(fx, fy).doit()
assert a.transform(fx, fy).transform(fy, fx) == a
fx = 3*x + 1
fy = y
assert a.transform(fx, fy).transform(fy, fx) == a
a = Integral(sin(1/x), (x, 0, 1))
assert a.transform(x, 1/y) == Integral(sin(y)/y**2, (y, 1, oo))
assert a.transform(x, 1/y).transform(y, 1/x) == a
a = Integral(exp(-x**2), (x, -oo, oo))
assert a.transform(x, 2*y) == Integral(2*exp(-4*y**2), (y, -oo, oo))
# < 3 arg limit handled properly
assert Integral(x, x).transform(x, a*y).doit() == \
Integral(y*a**2, y).doit()
_3 = S(3)
assert Integral(x, (x, 0, -_3)).transform(x, 1/y).doit() == \
Integral(-1/x**3, (x, -oo, -1/_3)).doit()
assert Integral(x, (x, 0, _3)).transform(x, 1/y) == \
Integral(y**(-3), (y, 1/_3, oo))
# issue 8400
i = Integral(x + y, (x, 1, 2), (y, 1, 2))
assert i.transform(x, (x + 2*y, x)).doit() == \
i.transform(x, (x + 2*z, x)).doit() == 3
def test_issue_4052():
f = S(1)/2*asin(x) + x*sqrt(1 - x**2)/2
assert integrate(cos(asin(x)), x) == f
assert integrate(sin(acos(x)), x) == f
def NS(e, n=15, **options):
return sstr(sympify(e).evalf(n, **options), full_prec=True)
@slow
def test_evalf_integrals():
assert NS(Integral(x, (x, 2, 5)), 15) == '10.5000000000000'
gauss = Integral(exp(-x**2), (x, -oo, oo))
assert NS(gauss, 15) == '1.77245385090552'
assert NS(gauss**2 - pi + E*Rational(
1, 10**20), 15) in ('2.71828182845904e-20', '2.71828182845905e-20')
# A monster of an integral from http://mathworld.wolfram.com/DefiniteIntegral.html
t = Symbol('t')
a = 8*sqrt(3)/(1 + 3*t**2)
b = 16*sqrt(2)*(3*t + 1)*sqrt(4*t**2 + t + 1)**3
c = (3*t**2 + 1)*(11*t**2 + 2*t + 3)**2
d = sqrt(2)*(249*t**2 + 54*t + 65)/(11*t**2 + 2*t + 3)**2
f = a - b/c - d
assert NS(Integral(f, (t, 0, 1)), 50) == \
NS((3*sqrt(2) - 49*pi + 162*atan(sqrt(2)))/12, 50)
# http://mathworld.wolfram.com/VardisIntegral.html
assert NS(Integral(log(log(1/x))/(1 + x + x**2), (x, 0, 1)), 15) == \
NS('pi/sqrt(3) * log(2*pi**(5/6) / gamma(1/6))', 15)
# http://mathworld.wolfram.com/AhmedsIntegral.html
assert NS(Integral(atan(sqrt(x**2 + 2))/(sqrt(x**2 + 2)*(x**2 + 1)), (x,
0, 1)), 15) == NS(5*pi**2/96, 15)
# http://mathworld.wolfram.com/AbelsIntegral.html
assert NS(Integral(x/((exp(pi*x) - exp(
-pi*x))*(x**2 + 1)), (x, 0, oo)), 15) == NS('log(2)/2-1/4', 15)
# Complex part trimming
# http://mathworld.wolfram.com/VardisIntegral.html
assert NS(Integral(log(log(sin(x)/cos(x))), (x, pi/4, pi/2)), 15, chop=True) == \
NS('pi/4*log(4*pi**3/gamma(1/4)**4)', 15)
#
# Endpoints causing trouble (rounding error in integration points -> complex log)
assert NS(
2 + Integral(log(2*cos(x/2)), (x, -pi, pi)), 17, chop=True) == NS(2, 17)
assert NS(
2 + Integral(log(2*cos(x/2)), (x, -pi, pi)), 20, chop=True) == NS(2, 20)
assert NS(
2 + Integral(log(2*cos(x/2)), (x, -pi, pi)), 22, chop=True) == NS(2, 22)
# Needs zero handling
assert NS(pi - 4*Integral(
'sqrt(1-x**2)', (x, 0, 1)), 15, maxn=30, chop=True) in ('0.0', '0')
# Oscillatory quadrature
a = Integral(sin(x)/x**2, (x, 1, oo)).evalf(maxn=15)
assert 0.49 < a < 0.51
assert NS(
Integral(sin(x)/x**2, (x, 1, oo)), quad='osc') == '0.504067061906928'
assert NS(Integral(
cos(pi*x + 1)/x, (x, -oo, -1)), quad='osc') == '0.276374705640365'
# indefinite integrals aren't evaluated
assert NS(Integral(x, x)) == 'Integral(x, x)'
assert NS(Integral(x, (x, y))) == 'Integral(x, (x, y))'
def test_evalf_issue_939():
# https://github.com/sympy/sympy/issues/4038
# The output form of an integral may differ by a step function between
# revisions, making this test a bit useless. This can't be said about
# other two tests. For now, all values of this evaluation are used here,
# but in future this should be reconsidered.
assert NS(integrate(1/(x**5 + 1), x).subs(x, 4), chop=True) in \
['-0.000976138910649103', '0.965906660135753', '1.93278945918216']
assert NS(Integral(1/(x**5 + 1), (x, 2, 4))) == '0.0144361088886740'
assert NS(
integrate(1/(x**5 + 1), (x, 2, 4)), chop=True) == '0.0144361088886740'
@XFAIL
def test_failing_integrals():
#---
# Double integrals not implemented
assert NS(Integral(
sqrt(x) + x*y, (x, 1, 2), (y, -1, 1)), 15) == '2.43790283299492'
# double integral + zero detection
assert NS(Integral(sin(x + x*y), (x, -1, 1), (y, -1, 1)), 15) == '0.0'
def test_integrate_SingularityFunction():
in_1 = SingularityFunction(x, a, 3) + SingularityFunction(x, 5, -1)
out_1 = SingularityFunction(x, a, 4)/4 + SingularityFunction(x, 5, 0)
assert integrate(in_1, x) == out_1
in_2 = 10*SingularityFunction(x, 4, 0) - 5*SingularityFunction(x, -6, -2)
out_2 = 10*SingularityFunction(x, 4, 1) - 5*SingularityFunction(x, -6, -1)
assert integrate(in_2, x) == out_2
in_3 = 2*x**2*y -10*SingularityFunction(x, -4, 7) - 2*SingularityFunction(y, 10, -2)
out_3_1 = 2*x**3*y/3 - 2*x*SingularityFunction(y, 10, -2) - 5*SingularityFunction(x, -4, 8)/4
out_3_2 = x**2*y**2 - 10*y*SingularityFunction(x, -4, 7) - 2*SingularityFunction(y, 10, -1)
assert integrate(in_3, x) == out_3_1
assert integrate(in_3, y) == out_3_2
assert Integral(in_3, x) == Integral(in_3, x)
assert Integral(in_3, x).doit() == out_3_1
in_4 = 10*SingularityFunction(x, -4, 7) - 2*SingularityFunction(x, 10, -2)
out_4 = 5*SingularityFunction(x, -4, 8)/4 - 2*SingularityFunction(x, 10, -1)
assert integrate(in_4, (x, -oo, x)) == out_4
assert integrate(SingularityFunction(x, 5, -1), x) == SingularityFunction(x, 5, 0)
assert integrate(SingularityFunction(x, 0, -1), (x, -oo, oo)) == 1
assert integrate(5*SingularityFunction(x, 5, -1), (x, -oo, oo)) == 5
assert integrate(SingularityFunction(x, 5, -1) * f(x), (x, -oo, oo)) == f(5)
def test_integrate_DiracDelta():
# This is here to check that deltaintegrate is being called, but also
# to test definite integrals. More tests are in test_deltafunctions.py
assert integrate(DiracDelta(x) * f(x), (x, -oo, oo)) == f(0)
assert integrate(DiracDelta(x)**2, (x, -oo, oo)) == DiracDelta(0)
# issue 4522
assert integrate(integrate((4 - 4*x + x*y - 4*y) * \
DiracDelta(x)*DiracDelta(y - 1), (x, 0, 1)), (y, 0, 1)) == 0
# issue 5729
p = exp(-(x**2 + y**2))/pi
assert integrate(p*DiracDelta(x - 10*y), (x, -oo, oo), (y, -oo, oo)) == \
integrate(p*DiracDelta(x - 10*y), (y, -oo, oo), (x, -oo, oo)) == \
integrate(p*DiracDelta(10*x - y), (x, -oo, oo), (y, -oo, oo)) == \
integrate(p*DiracDelta(10*x - y), (y, -oo, oo), (x, -oo, oo)) == \
1/sqrt(101*pi)
@XFAIL
def test_integrate_DiracDelta_fails():
# issue 6427
assert integrate(integrate(integrate(
DiracDelta(x - y - z), (z, 0, oo)), (y, 0, 1)), (x, 0, 1)) == S(1)/2
def test_integrate_returns_piecewise():
assert integrate(x**y, x) == Piecewise(
(x**(y + 1)/(y + 1), Ne(y, -1)), (log(x), True))
assert integrate(x**y, y) == Piecewise(
(x**y/log(x), Ne(log(x), 0)), (y, True))
assert integrate(exp(n*x), x) == Piecewise(
(exp(n*x)/n, Ne(n, 0)), (x, True))
assert integrate(x*exp(n*x), x) == Piecewise(
((n*x - 1)*exp(n*x)/n**2, Ne(n**2, 0)), (x**2/2, True))
assert integrate(x**(n*y), x) == Piecewise(
(x**(n*y + 1)/(n*y + 1), Ne(n*y, -1)), (log(x), True))
assert integrate(x**(n*y), y) == Piecewise(
(x**(n*y)/(n*log(x)), Ne(n*log(x), 0)), (y, True))
assert integrate(cos(n*x), x) == Piecewise(
(sin(n*x)/n, Ne(n, 0)), (x, True))
assert integrate(cos(n*x)**2, x) == Piecewise(
((n*x/2 + sin(n*x)*cos(n*x)/2)/n, Ne(n, 0)), (x, True))
assert integrate(x*cos(n*x), x) == Piecewise(
(x*sin(n*x)/n + cos(n*x)/n**2, Ne(n, 0)), (x**2/2, True))
assert integrate(sin(n*x), x) == Piecewise(
(-cos(n*x)/n, Ne(n, 0)), (0, True))
assert integrate(sin(n*x)**2, x) == Piecewise(
((n*x/2 - sin(n*x)*cos(n*x)/2)/n, Ne(n, 0)), (0, True))
assert integrate(x*sin(n*x), x) == Piecewise(
(-x*cos(n*x)/n + sin(n*x)/n**2, Ne(n, 0)), (0, True))
assert integrate(exp(x*y), (x, 0, z)) == Piecewise(
(exp(y*z)/y - 1/y, (y > -oo) & (y < oo) & Ne(y, 0)), (z, True))
def test_integrate_max_min():
x = symbols('x', real=True)
assert integrate(Min(x, 2), (x, 0, 3)) == 4
assert integrate(Max(x**2, x**3), (x, 0, 2)) == S(49)/12
assert integrate(Min(exp(x), exp(-x))**2, x) == Piecewise( \
(exp(2*x)/2, x <= 0), (1 - exp(-2*x)/2, True))
# issue 7907
c = symbols('c', real=True)
int1 = integrate(Max(c, x)*exp(-x**2), (x, -oo, oo))
int2 = integrate(c*exp(-x**2), (x, -oo, c))
int3 = integrate(x*exp(-x**2), (x, c, oo))
assert int1 == int2 + int3 == sqrt(pi)*c*erf(c)/2 + \
sqrt(pi)*c/2 + exp(-c**2)/2
def test_integrate_Abs_sign():
assert integrate(Abs(x), (x, -2, 1)) == S(5)/2
assert integrate(Abs(x), (x, 0, 1)) == S(1)/2
assert integrate(Abs(x + 1), (x, 0, 1)) == S(3)/2
assert integrate(Abs(x**2 - 1), (x, -2, 2)) == 4
assert integrate(Abs(x**2 - 3*x), (x, -15, 15)) == 2259
assert integrate(sign(x), (x, -1, 2)) == 1
assert integrate(sign(x)*sin(x), (x, -pi, pi)) == 4
assert integrate(sign(x - 2) * x**2, (x, 0, 3)) == S(11)/3
t, s = symbols('t s', real=True)
assert integrate(Abs(t), t) == Piecewise(
(-t**2/2, t <= 0), (t**2/2, True))
assert integrate(Abs(2*t - 6), t) == Piecewise(
(-t**2 + 6*t, t <= 3), (t**2 - 6*t + 18, True))
assert (integrate(abs(t - s**2), (t, 0, 2)) ==
2*s**2*Min(2, s**2) - 2*s**2 - Min(2, s**2)**2 + 2)
assert integrate(exp(-Abs(t)), t) == Piecewise(
(exp(t), t <= 0), (2 - exp(-t), True))
assert integrate(sign(2*t - 6), t) == Piecewise(
(-t, t < 3), (t - 6, True))
assert integrate(2*t*sign(t**2 - 1), t) == Piecewise(
(t**2, t < -1), (-t**2 + 2, t < 1), (t**2, True))
assert integrate(sign(t), (t, s + 1)) == Piecewise(
(s + 1, s + 1 > 0), (-s - 1, s + 1 < 0), (0, True))
def test_subs1():
e = Integral(exp(x - y), x)
assert e.subs(y, 3) == Integral(exp(x - 3), x)
e = Integral(exp(x - y), (x, 0, 1))
assert e.subs(y, 3) == Integral(exp(x - 3), (x, 0, 1))
f = Lambda(x, exp(-x**2))
conv = Integral(f(x - y)*f(y), (y, -oo, oo))
assert conv.subs({x: 0}) == Integral(exp(-2*y**2), (y, -oo, oo))
def test_subs2():
e = Integral(exp(x - y), x, t)
assert e.subs(y, 3) == Integral(exp(x - 3), x, t)
e = Integral(exp(x - y), (x, 0, 1), (t, 0, 1))
assert e.subs(y, 3) == Integral(exp(x - 3), (x, 0, 1), (t, 0, 1))
f = Lambda(x, exp(-x**2))
conv = Integral(f(x - y)*f(y), (y, -oo, oo), (t, 0, 1))
assert conv.subs({x: 0}) == Integral(exp(-2*y**2), (y, -oo, oo), (t, 0, 1))
def test_subs3():
e = Integral(exp(x - y), (x, 0, y), (t, y, 1))
assert e.subs(y, 3) == Integral(exp(x - 3), (x, 0, 3), (t, 3, 1))
f = Lambda(x, exp(-x**2))
conv = Integral(f(x - y)*f(y), (y, -oo, oo), (t, x, 1))
assert conv.subs({x: 0}) == Integral(exp(-2*y**2), (y, -oo, oo), (t, 0, 1))
def test_subs4():
e = Integral(exp(x), (x, 0, y), (t, y, 1))
assert e.subs(y, 3) == Integral(exp(x), (x, 0, 3), (t, 3, 1))
f = Lambda(x, exp(-x**2))
conv = Integral(f(y)*f(y), (y, -oo, oo), (t, x, 1))
assert conv.subs({x: 0}) == Integral(exp(-2*y**2), (y, -oo, oo), (t, 0, 1))
def test_subs5():
e = Integral(exp(-x**2), (x, -oo, oo))
assert e.subs(x, 5) == e
e = Integral(exp(-x**2 + y), x)
assert e.subs(y, 5) == Integral(exp(-x**2 + 5), x)
e = Integral(exp(-x**2 + y), (x, x))
assert e.subs(x, 5) == Integral(exp(y - x**2), (x, 5))
assert e.subs(y, 5) == Integral(exp(-x**2 + 5), x)
e = Integral(exp(-x**2 + y), (y, -oo, oo), (x, -oo, oo))
assert e.subs(x, 5) == e
assert e.subs(y, 5) == e
# Test evaluation of antiderivatives
e = Integral(exp(-x**2), (x, x))
assert e.subs(x, 5) == Integral(exp(-x**2), (x, 5))
e = Integral(exp(x), x)
assert (e.subs(x,1) - e.subs(x,0) - Integral(exp(x), (x, 0, 1))
).doit().is_zero
def test_subs6():
a, b = symbols('a b')
e = Integral(x*y, (x, f(x), f(y)))
assert e.subs(x, 1) == Integral(x*y, (x, f(1), f(y)))
assert e.subs(y, 1) == Integral(x, (x, f(x), f(1)))
e = Integral(x*y, (x, f(x), f(y)), (y, f(x), f(y)))
assert e.subs(x, 1) == Integral(x*y, (x, f(1), f(y)), (y, f(1), f(y)))
assert e.subs(y, 1) == Integral(x*y, (x, f(x), f(y)), (y, f(x), f(1)))
e = Integral(x*y, (x, f(x), f(a)), (y, f(x), f(a)))
assert e.subs(a, 1) == Integral(x*y, (x, f(x), f(1)), (y, f(x), f(1)))
def test_subs7():
e = Integral(x, (x, 1, y), (y, 1, 2))
assert e.subs({x: 1, y: 2}) == e
e = Integral(sin(x) + sin(y), (x, sin(x), sin(y)),
(y, 1, 2))
assert e.subs(sin(y), 1) == e
assert e.subs(sin(x), 1) == Integral(sin(x) + sin(y), (x, 1, sin(y)),
(y, 1, 2))
def test_expand():
e = Integral(f(x)+f(x**2), (x, 1, y))
assert e.expand() == Integral(f(x), (x, 1, y)) + Integral(f(x**2), (x, 1, y))
def test_integration_variable():
raises(ValueError, lambda: Integral(exp(-x**2), 3))
raises(ValueError, lambda: Integral(exp(-x**2), (3, -oo, oo)))
def test_expand_integral():
assert Integral(cos(x**2)*(sin(x**2) + 1), (x, 0, 1)).expand() == \
Integral(cos(x**2)*sin(x**2), (x, 0, 1)) + \
Integral(cos(x**2), (x, 0, 1))
assert Integral(cos(x**2)*(sin(x**2) + 1), x).expand() == \
Integral(cos(x**2)*sin(x**2), x) + \
Integral(cos(x**2), x)
def test_as_sum_midpoint1():
e = Integral(sqrt(x**3 + 1), (x, 2, 10))
assert e.as_sum(1, method="midpoint") == 8*sqrt(217)
assert e.as_sum(2, method="midpoint") == 4*sqrt(65) + 12*sqrt(57)
assert e.as_sum(3, method="midpoint") == 8*sqrt(217)/3 + \
8*sqrt(3081)/27 + 8*sqrt(52809)/27
assert e.as_sum(4, method="midpoint") == 2*sqrt(730) + \
4*sqrt(7) + 4*sqrt(86) + 6*sqrt(14)
assert abs(e.as_sum(4, method="midpoint").n() - e.n()) < 0.5
e = Integral(sqrt(x**3 + y**3), (x, 2, 10), (y, 0, 10))
raises(NotImplementedError, lambda: e.as_sum(4))
def test_as_sum_midpoint2():
e = Integral((x + y)**2, (x, 0, 1))
n = Symbol('n', positive=True, integer=True)
assert e.as_sum(1, method="midpoint").expand() == S(1)/4 + y + y**2
assert e.as_sum(2, method="midpoint").expand() == S(5)/16 + y + y**2
assert e.as_sum(3, method="midpoint").expand() == S(35)/108 + y + y**2
assert e.as_sum(4, method="midpoint").expand() == S(21)/64 + y + y**2
assert e.as_sum(n, method="midpoint").expand() == \
y**2 + y + S(1)/3 - 1/(12*n**2)
def test_as_sum_left():
e = Integral((x + y)**2, (x, 0, 1))
assert e.as_sum(1, method="left").expand() == y**2
assert e.as_sum(2, method="left").expand() == S(1)/8 + y/2 + y**2
assert e.as_sum(3, method="left").expand() == S(5)/27 + 2*y/3 + y**2
assert e.as_sum(4, method="left").expand() == S(7)/32 + 3*y/4 + y**2
assert e.as_sum(n, method="left").expand() == \
y**2 + y + S(1)/3 - y/n - 1/(2*n) + 1/(6*n**2)
assert e.as_sum(10, method="left", evaluate=False).has(Sum)
def test_as_sum_right():
e = Integral((x + y)**2, (x, 0, 1))
assert e.as_sum(1, method="right").expand() == 1 + 2*y + y**2
assert e.as_sum(2, method="right").expand() == S(5)/8 + 3*y/2 + y**2
assert e.as_sum(3, method="right").expand() == S(14)/27 + 4*y/3 + y**2
assert e.as_sum(4, method="right").expand() == S(15)/32 + 5*y/4 + y**2
assert e.as_sum(n, method="right").expand() == \
y**2 + y + S(1)/3 + y/n + 1/(2*n) + 1/(6*n**2)
def test_as_sum_trapezoid():
e = Integral((x + y)**2, (x, 0, 1))
assert e.as_sum(1, method="trapezoid").expand() == y**2 + y + S(1)/2
assert e.as_sum(2, method="trapezoid").expand() == y**2 + y + S(3)/8
assert e.as_sum(3, method="trapezoid").expand() == y**2 + y + S(19)/54
assert e.as_sum(4, method="trapezoid").expand() == y**2 + y + S(11)/32
assert e.as_sum(n, method="trapezoid").expand() == \
y**2 + y + S(1)/3 + 1/(6*n**2)
assert Integral(sign(x), (x, 0, 1)).as_sum(1, 'trapezoid') == S(1)/2
def test_as_sum_raises():
e = Integral((x + y)**2, (x, 0, 1))
raises(ValueError, lambda: e.as_sum(-1))
raises(ValueError, lambda: e.as_sum(0))
raises(ValueError, lambda: Integral(x).as_sum(3))
raises(ValueError, lambda: e.as_sum(oo))
raises(ValueError, lambda: e.as_sum(3, method='xxxx2'))
def test_nested_doit():
e = Integral(Integral(x, x), x)
f = Integral(x, x, x)
assert e.doit() == f.doit()
def test_issue_4665():
# Allow only upper or lower limit evaluation
e = Integral(x**2, (x, None, 1))
f = Integral(x**2, (x, 1, None))
assert e.doit() == Rational(1, 3)
assert f.doit() == Rational(-1, 3)
assert Integral(x*y, (x, None, y)).subs(y, t) == Integral(x*t, (x, None, t))
assert Integral(x*y, (x, y, None)).subs(y, t) == Integral(x*t, (x, t, None))
assert integrate(x**2, (x, None, 1)) == Rational(1, 3)
assert integrate(x**2, (x, 1, None)) == Rational(-1, 3)
assert integrate("x**2", ("x", "1", None)) == Rational(-1, 3)
def test_integral_reconstruct():
e = Integral(x**2, (x, -1, 1))
assert e == Integral(*e.args)
def test_doit_integrals():
e = Integral(Integral(2*x), (x, 0, 1))
assert e.doit() == Rational(1, 3)
assert e.doit(deep=False) == Rational(1, 3)
f = Function('f')
# doesn't matter if the integral can't be performed
assert Integral(f(x), (x, 1, 1)).doit() == 0
# doesn't matter if the limits can't be evaluated
assert Integral(0, (x, 1, Integral(f(x), x))).doit() == 0
assert Integral(x, (a, 0)).doit() == 0
limits = ((a, 1, exp(x)), (x, 0))
assert Integral(a, *limits).doit() == S(1)/4
assert Integral(a, *list(reversed(limits))).doit() == 0
def test_issue_4884():
assert integrate(sqrt(x)*(1 + x)) == \
Piecewise(
(2*sqrt(x)*(x + 1)**2/5 - 2*sqrt(x)*(x + 1)/15 - 4*sqrt(x)/15,
Abs(x + 1) > 1),
(2*I*sqrt(-x)*(x + 1)**2/5 - 2*I*sqrt(-x)*(x + 1)/15 -
4*I*sqrt(-x)/15, True))
assert integrate(x**x*(1 + log(x))) == x**x
def test_is_number():
from sympy.abc import x, y, z
from sympy import cos, sin
assert Integral(x).is_number is False
assert Integral(1, x).is_number is False
assert Integral(1, (x, 1)).is_number is True
assert Integral(1, (x, 1, 2)).is_number is True
assert Integral(1, (x, 1, y)).is_number is False
assert Integral(1, (x, y)).is_number is False
assert Integral(x, y).is_number is False
assert Integral(x, (y, 1, x)).is_number is False
assert Integral(x, (y, 1, 2)).is_number is False
assert Integral(x, (x, 1, 2)).is_number is True
# `foo.is_number` should always be equivalent to `not foo.free_symbols`
# in each of these cases, there are pseudo-free symbols
i = Integral(x, (y, 1, 1))
assert i.is_number is False and i.n() == 0
i = Integral(x, (y, z, z))
assert i.is_number is False and i.n() == 0
i = Integral(1, (y, z, z + 2))
assert i.is_number is False and i.n() == 2
assert Integral(x*y, (x, 1, 2), (y, 1, 3)).is_number is True
assert Integral(x*y, (x, 1, 2), (y, 1, z)).is_number is False
assert Integral(x, (x, 1)).is_number is True
assert Integral(x, (x, 1, Integral(y, (y, 1, 2)))).is_number is True
assert Integral(Sum(z, (z, 1, 2)), (x, 1, 2)).is_number is True
# it is possible to get a false negative if the integrand is
# actually an unsimplified zero, but this is true of is_number in general.
assert Integral(sin(x)**2 + cos(x)**2 - 1, x).is_number is False
assert Integral(f(x), (x, 0, 1)).is_number is True
def test_symbols():
from sympy.abc import x, y, z
assert Integral(0, x).free_symbols == {x}
assert Integral(x).free_symbols == {x}
assert Integral(x, (x, None, y)).free_symbols == {y}
assert Integral(x, (x, y, None)).free_symbols == {y}
assert Integral(x, (x, 1, y)).free_symbols == {y}
assert Integral(x, (x, y, 1)).free_symbols == {y}
assert Integral(x, (x, x, y)).free_symbols == {x, y}
assert Integral(x, x, y).free_symbols == {x, y}
assert Integral(x, (x, 1, 2)).free_symbols == set()
assert Integral(x, (y, 1, 2)).free_symbols == {x}
# pseudo-free in this case
assert Integral(x, (y, z, z)).free_symbols == {x, z}
assert Integral(x, (y, 1, 2), (y, None, None)).free_symbols == {x, y}
assert Integral(x, (y, 1, 2), (x, 1, y)).free_symbols == {y}
assert Integral(2, (y, 1, 2), (y, 1, x), (x, 1, 2)).free_symbols == set()
assert Integral(2, (y, x, 2), (y, 1, x), (x, 1, 2)).free_symbols == set()
assert Integral(2, (x, 1, 2), (y, x, 2), (y, 1, 2)).free_symbols == \
{x}
def test_is_zero():
from sympy.abc import x, m
assert Integral(0, (x, 1, x)).is_zero
assert Integral(1, (x, 1, 1)).is_zero
assert Integral(1, (x, 1, 2), (y, 2)).is_zero is False
assert Integral(x, (m, 0)).is_zero
assert Integral(x + m, (m, 0)).is_zero is None
i = Integral(m, (m, 1, exp(x)), (x, 0))
assert i.is_zero is None
assert Integral(m, (x, 0), (m, 1, exp(x))).is_zero is True
assert Integral(x, (x, oo, oo)).is_zero # issue 8171
assert Integral(x, (x, -oo, -oo)).is_zero
# this is zero but is beyond the scope of what is_zero
# should be doing
assert Integral(sin(x), (x, 0, 2*pi)).is_zero is None
def test_series():
from sympy.abc import x
i = Integral(cos(x), (x, x))
e = i.lseries(x)
assert i.nseries(x, n=8).removeO() == Add(*[next(e) for j in range(4)])
def test_trig_nonelementary_integrals():
x = Symbol('x')
assert integrate((1 + sin(x))/x, x) == log(x) + Si(x)
# next one comes out as log(x) + log(x**2)/2 + Ci(x)
# so not hardcoding this log ugliness
assert integrate((cos(x) + 2)/x, x).has(Ci)
def test_issue_4403():
x = Symbol('x')
y = Symbol('y')
z = Symbol('z', positive=True)
assert integrate(sqrt(x**2 + z**2), x) == \
z**2*asinh(x/z)/2 + x*sqrt(x**2 + z**2)/2
assert integrate(sqrt(x**2 - z**2), x) == \
-z**2*acosh(x/z)/2 + x*sqrt(x**2 - z**2)/2
x = Symbol('x', real=True)
y = Symbol('y', positive=True)
assert integrate(1/(x**2 + y**2)**S('3/2'), x) == \
x/(y**2*sqrt(x**2 + y**2))
# If y is real and nonzero, we get x*Abs(y)/(y**3*sqrt(x**2 + y**2)),
# which results from sqrt(1 + x**2/y**2) = sqrt(x**2 + y**2)/|y|.
def test_issue_4403_2():
assert integrate(sqrt(-x**2 - 4), x) == \
-2*atan(x/sqrt(-4 - x**2)) + x*sqrt(-4 - x**2)/2
def test_issue_4100():
R = Symbol('R', positive=True)
assert integrate(sqrt(R**2 - x**2), (x, 0, R)) == pi*R**2/4
def test_issue_5167():
from sympy.abc import w, x, y, z
f = Function('f')
assert Integral(Integral(f(x), x), x) == Integral(f(x), x, x)
assert Integral(f(x)).args == (f(x), Tuple(x))
assert Integral(Integral(f(x))).args == (f(x), Tuple(x), Tuple(x))
assert Integral(Integral(f(x)), y).args == (f(x), Tuple(x), Tuple(y))
assert Integral(Integral(f(x), z), y).args == (f(x), Tuple(z), Tuple(y))
assert Integral(Integral(Integral(f(x), x), y), z).args == \
(f(x), Tuple(x), Tuple(y), Tuple(z))
assert integrate(Integral(f(x), x), x) == Integral(f(x), x, x)
assert integrate(Integral(f(x), y), x) == y*Integral(f(x), x)
assert integrate(Integral(f(x), x), y) in [Integral(y*f(x), x), y*Integral(f(x), x)]
assert integrate(Integral(2, x), x) == x**2
assert integrate(Integral(2, x), y) == 2*x*y
# don't re-order given limits
assert Integral(1, x, y).args != Integral(1, y, x).args
# do as many as possible
assert Integral(f(x), y, x, y, x).doit() == y**2*Integral(f(x), x, x)/2
assert Integral(f(x), (x, 1, 2), (w, 1, x), (z, 1, y)).doit() == \
y*(x - 1)*Integral(f(x), (x, 1, 2)) - (x - 1)*Integral(f(x), (x, 1, 2))
def test_issue_4890():
z = Symbol('z', positive=True)
assert integrate(exp(-log(x)**2), x) == \
sqrt(pi)*exp(S(1)/4)*erf(log(x)-S(1)/2)/2
assert integrate(exp(log(x)**2), x) == \
sqrt(pi)*exp(-S(1)/4)*erfi(log(x)+S(1)/2)/2
assert integrate(exp(-z*log(x)**2), x) == \
sqrt(pi)*exp(1/(4*z))*erf(sqrt(z)*log(x) - 1/(2*sqrt(z)))/(2*sqrt(z))
def test_issue_4376():
n = Symbol('n', integer=True, positive=True)
assert simplify(integrate(n*(x**(1/n) - 1), (x, 0, S.Half)) -
(n**2 - 2**(1/n)*n**2 - n*2**(1/n))/(2**(1 + 1/n) + n*2**(1 + 1/n))) == 0
def test_issue_4517():
assert integrate((sqrt(x) - x**3)/x**Rational(1, 3), x) == \
6*x**Rational(7, 6)/7 - 3*x**Rational(11, 3)/11
def test_issue_4527():
k, m = symbols('k m', integer=True)
ans = integrate(sin(k*x)*sin(m*x), (x, 0, pi)
).simplify() == Piecewise(
(0, Eq(k, 0) | Eq(m, 0)),
(-pi/2, Eq(k, -m)),
(pi/2, Eq(k, m)),
(0, True))
assert integrate(sin(k*x)*sin(m*x), (x,)) == Piecewise(
(0, And(Eq(k, 0), Eq(m, 0))),
(-x*sin(m*x)**2/2 - x*cos(m*x)**2/2 + sin(m*x)*cos(m*x)/(2*m), Eq(k, -m)),
(x*sin(m*x)**2/2 + x*cos(m*x)**2/2 - sin(m*x)*cos(m*x)/(2*m), Eq(k, m)),
(m*sin(k*x)*cos(m*x)/(k**2 - m**2) -
k*sin(m*x)*cos(k*x)/(k**2 - m**2), True))
def test_issue_4199():
ypos = Symbol('y', positive=True)
# TODO: Remove conds='none' below, let the assumption take care of it.
assert integrate(exp(-I*2*pi*ypos*x)*x, (x, -oo, oo), conds='none') == \
Integral(exp(-I*2*pi*ypos*x)*x, (x, -oo, oo))
@slow
def test_issue_3940():
a, b, c, d = symbols('a:d', positive=True, finite=True)
assert integrate(exp(-x**2 + I*c*x), x) == \
-sqrt(pi)*exp(-c**2/4)*erf(I*c/2 - x)/2
assert integrate(exp(a*x**2 + b*x + c), x) == \
sqrt(pi)*exp(c)*exp(-b**2/(4*a))*erfi(sqrt(a)*x + b/(2*sqrt(a)))/(2*sqrt(a))
from sympy import expand_mul
from sympy.abc import k
assert expand_mul(integrate(exp(-x**2)*exp(I*k*x), (x, -oo, oo))) == \
sqrt(pi)*exp(-k**2/4)
a, d = symbols('a d', positive=True)
assert expand_mul(integrate(exp(-a*x**2 + 2*d*x), (x, -oo, oo))) == \
sqrt(pi)*exp(d**2/a)/sqrt(a)
def test_issue_5413():
# Note that this is not the same as testing ratint() because integrate()
# pulls out the coefficient.
assert integrate(-a/(a**2 + x**2), x) == I*log(-I*a + x)/2 - I*log(I*a + x)/2
def test_issue_4892a():
A, z = symbols('A z')
c = Symbol('c', nonzero=True)
P1 = -A*exp(-z)
P2 = -A/(c*t)*(sin(x)**2 + cos(y)**2)
h1 = -sin(x)**2 - cos(y)**2
h2 = -sin(x)**2 + sin(y)**2 - 1
# there is still some non-deterministic behavior in integrate
# or trigsimp which permits one of the following
assert integrate(c*(P2 - P1), t) in [
c*(-A*(-h1)*log(c*t)/c + A*t*exp(-z)),
c*(-A*(-h2)*log(c*t)/c + A*t*exp(-z)),
c*( A* h1 *log(c*t)/c + A*t*exp(-z)),
c*( A* h2 *log(c*t)/c + A*t*exp(-z)),
(A*c*t - A*(-h1)*log(t)*exp(z))*exp(-z),
(A*c*t - A*(-h2)*log(t)*exp(z))*exp(-z),
]
def test_issue_4892b():
# Issues relating to issue 4596 are making the actual result of this hard
# to test. The answer should be something like
#
# (-sin(y) + sqrt(-72 + 48*cos(y) - 8*cos(y)**2)/2)*log(x + sqrt(-72 +
# 48*cos(y) - 8*cos(y)**2)/(2*(3 - cos(y)))) + (-sin(y) - sqrt(-72 +
# 48*cos(y) - 8*cos(y)**2)/2)*log(x - sqrt(-72 + 48*cos(y) -
# 8*cos(y)**2)/(2*(3 - cos(y)))) + x**2*sin(y)/2 + 2*x*cos(y)
expr = (sin(y)*x**3 + 2*cos(y)*x**2 + 12)/(x**2 + 2)
assert trigsimp(factor(integrate(expr, x).diff(x) - expr)) == 0
def test_issue_5178():
assert integrate(sin(x)*f(y, z), (x, 0, pi), (y, 0, pi), (z, 0, pi)) == \
2*Integral(f(y, z), (y, 0, pi), (z, 0, pi))
def test_integrate_series():
f = sin(x).series(x, 0, 10)
g = x**2/2 - x**4/24 + x**6/720 - x**8/40320 + x**10/3628800 + O(x**11)
assert integrate(f, x) == g
assert diff(integrate(f, x), x) == f
assert integrate(O(x**5), x) == O(x**6)
def test_atom_bug():
from sympy import meijerg
from sympy.integrals.heurisch import heurisch
assert heurisch(meijerg([], [], [1], [], x), x) is None
def test_limit_bug():
z = Symbol('z', zero=False)
assert integrate(sin(x*y*z), (x, 0, pi), (y, 0, pi)) == \
(log(z**2) + 2*EulerGamma + 2*log(pi))/(2*z) - \
(-log(pi*z) + log(pi**2*z**2)/2 + Ci(pi**2*z))/z + log(pi)/z
def test_issue_4703():
g = Function('g')
assert integrate(exp(x)*g(x), x).has(Integral)
def test_issue_1888():
f = Function('f')
assert integrate(f(x).diff(x)**2, x).has(Integral)
# The following tests work using meijerint.
def test_issue_3558():
from sympy import Si
assert integrate(cos(x*y), (x, -pi/2, pi/2), (y, 0, pi)) == 2*Si(pi**2/2)
def test_issue_4422():
assert integrate(1/sqrt(16 + 4*x**2), x) == asinh(x/2) / 2
def test_issue_4493():
from sympy import simplify
assert simplify(integrate(x*sqrt(1 + 2*x), x)) == \
sqrt(2*x + 1)*(6*x**2 + x - 1)/15
def test_issue_4737():
assert integrate(sin(x)/x, (x, -oo, oo)) == pi
assert integrate(sin(x)/x, (x, 0, oo)) == pi/2
def test_issue_4992():
# Note: psi in _check_antecedents becomes NaN.
from sympy import simplify, expand_func, polygamma, gamma
a = Symbol('a', positive=True)
assert simplify(expand_func(integrate(exp(-x)*log(x)*x**a, (x, 0, oo)))) == \
(a*polygamma(0, a) + 1)*gamma(a)
def test_issue_4487():
from sympy import lowergamma, simplify
assert simplify(integrate(exp(-x)*x**y, x)) == lowergamma(y + 1, x)
def test_issue_4215():
x = Symbol("x")
assert integrate(1/(x**2), (x, -1, 1)) == oo
def test_issue_4400():
n = Symbol('n', integer=True, positive=True)
assert integrate((x**n)*log(x), x) == \
n*x*x**n*log(x)/(n**2 + 2*n + 1) + x*x**n*log(x)/(n**2 + 2*n + 1) - \
x*x**n/(n**2 + 2*n + 1)
def test_issue_6253():
# Note: this used to raise NotImplementedError
# Note: psi in _check_antecedents becomes NaN.
assert integrate((sqrt(1 - x) + sqrt(1 + x))**2/x, x, meijerg=True) == \
Integral((sqrt(-x + 1) + sqrt(x + 1))**2/x, x)
def test_issue_4153():
assert integrate(1/(1 + x + y + z), (x, 0, 1), (y, 0, 1), (z, 0, 1)) in [
-12*log(3) - 3*log(6)/2 + 3*log(8)/2 + 5*log(2) + 7*log(4),
6*log(2) + 8*log(4) - 27*log(3)/2, 22*log(2) - 27*log(3)/2,
-12*log(3) - 3*log(6)/2 + 47*log(2)/2]
def test_issue_4326():
R, b, h = symbols('R b h')
# It doesn't matter if we can do the integral. Just make sure the result
# doesn't contain nan. This is really a test against _eval_interval.
assert not integrate(((h*(x - R + b))/b)*sqrt(R**2 - x**2), (x, R - b, R)).has(nan)
def test_powers():
assert integrate(2**x + 3**x, x) == 2**x/log(2) + 3**x/log(3)
def test_manual_option():
raises(ValueError, lambda: integrate(1/x, x, manual=True, meijerg=True))
# an example of a function that manual integration cannot handle
assert integrate(log(1+x)/x, (x, 0, 1), manual=True).has(Integral)
def test_meijerg_option():
raises(ValueError, lambda: integrate(1/x, x, meijerg=True, risch=True))
# an example of a function that meijerg integration cannot handle
assert integrate(tan(x), x, meijerg=True) == Integral(tan(x), x)
def test_risch_option():
# risch=True only allowed on indefinite integrals
raises(ValueError, lambda: integrate(1/log(x), (x, 0, oo), risch=True))
assert integrate(exp(-x**2), x, risch=True) == NonElementaryIntegral(exp(-x**2), x)
assert integrate(log(1/x)*y, x, y, risch=True) == y**2*(x*log(1/x)/2 + x/2)
assert integrate(erf(x), x, risch=True) == Integral(erf(x), x)
# TODO: How to test risch=False?
def test_heurisch_option():
raises(ValueError, lambda: integrate(1/x, x, risch=True, heurisch=True))
# an integral that heurisch can handle
assert integrate(exp(x**2), x, heurisch=True) == sqrt(pi)*erfi(x)/2
# an integral that heurisch currently cannot handle
assert integrate(exp(x)/x, x, heurisch=True) == Integral(exp(x)/x, x)
# an integral where heurisch currently hangs, issue 15471
assert integrate(log(x)*cos(log(x))/x**(S(3)/4), x, heurisch=False) == (
-128*x**(S(1)/4)*sin(log(x))/289 + 240*x**(S(1)/4)*cos(log(x))/289 +
(16*x**(S(1)/4)*sin(log(x))/17 + 4*x**(S(1)/4)*cos(log(x))/17)*log(x))
def test_issue_6828():
f = 1/(1.08*x**2 - 4.3)
g = integrate(f, x).diff(x)
assert verify_numerically(f, g, tol=1e-12)
@XFAIL
def test_integrate_Piecewise_rational_over_reals():
f = Piecewise(
(0, t - 478.515625*pi < 0),
(13.2075145209219*pi/(0.000871222*t + 0.995)**2, t - 478.515625*pi >= 0))
assert integrate(f, (t, 0, oo)) == 15235.9375*pi
def test_issue_4803():
x_max = Symbol("x_max")
assert integrate(y/pi*exp(-(x_max - x)/cos(a)), x) == \
y*exp((x - x_max)/cos(a))*cos(a)/pi
def test_issue_4234():
assert integrate(1/sqrt(1 + tan(x)**2)) == tan(x)/sqrt(1 + tan(x)**2)
def test_issue_4492():
assert simplify(integrate(x**2 * sqrt(5 - x**2), x)) == Piecewise(
(I*(2*x**5 - 15*x**3 + 25*x - 25*sqrt(x**2 - 5)*acosh(sqrt(5)*x/5)) /
(8*sqrt(x**2 - 5)), 1 < Abs(x**2)/5),
((-2*x**5 + 15*x**3 - 25*x + 25*sqrt(-x**2 + 5)*asin(sqrt(5)*x/5)) /
(8*sqrt(-x**2 + 5)), True))
def test_issue_2708():
# This test needs to use an integration function that can
# not be evaluated in closed form. Update as needed.
f = 1/(a + z + log(z))
integral_f = NonElementaryIntegral(f, (z, 2, 3))
assert Integral(f, (z, 2, 3)).doit() == integral_f
assert integrate(f + exp(z), (z, 2, 3)) == integral_f - exp(2) + exp(3)
assert integrate(2*f + exp(z), (z, 2, 3)) == \
2*integral_f - exp(2) + exp(3)
assert integrate(exp(1.2*n*s*z*(-t + z)/t), (z, 0, x)) == \
NonElementaryIntegral(exp(-1.2*n*s*z)*exp(1.2*n*s*z**2/t),
(z, 0, x))
def test_issue_8368():
assert integrate(exp(-s*x)*cosh(x), (x, 0, oo)) == \
Piecewise(
( pi*Piecewise(
( -s/(pi*(-s**2 + 1)),
Abs(s**2) < 1),
( 1/(pi*s*(1 - 1/s**2)),
Abs(s**(-2)) < 1),
( meijerg(
((S(1)/2,), (0, 0)),
((0, S(1)/2), (0,)),
polar_lift(s)**2),
True)
),
And(
Abs(periodic_argument(polar_lift(s)**2, oo)) < pi,
cos(Abs(periodic_argument(polar_lift(s)**2, oo))/2)*sqrt(Abs(s**2)) - 1 > 0,
Ne(s**2, 1))
),
(
Integral(exp(-s*x)*cosh(x), (x, 0, oo)),
True))
assert integrate(exp(-s*x)*sinh(x), (x, 0, oo)) == \
Piecewise(
( -1/(s + 1)/2 - 1/(-s + 1)/2,
And(
Ne(1/s, 1),
Abs(periodic_argument(s, oo)) < pi/2,
Abs(periodic_argument(s, oo)) <= pi/2,
cos(Abs(periodic_argument(s, oo)))*Abs(s) - 1 > 0)),
( Integral(exp(-s*x)*sinh(x), (x, 0, oo)),
True))
def test_issue_8901():
assert integrate(sinh(1.0*x)) == 1.0*cosh(1.0*x)
assert integrate(tanh(1.0*x)) == 1.0*x - 1.0*log(tanh(1.0*x) + 1)
assert integrate(tanh(x)) == x - log(tanh(x) + 1)
@slow
def test_issue_8945():
assert integrate(sin(x)**3/x, (x, 0, 1)) == -Si(3)/4 + 3*Si(1)/4
assert integrate(sin(x)**3/x, (x, 0, oo)) == pi/4
assert integrate(cos(x)**2/x**2, x) == -Si(2*x) - cos(2*x)/(2*x) - 1/(2*x)
@slow
def test_issue_7130():
if ON_TRAVIS:
skip("Too slow for travis.")
i, L, a, b = symbols('i L a b')
integrand = (cos(pi*i*x/L)**2 / (a + b*x)).rewrite(exp)
assert x not in integrate(integrand, (x, 0, L)).free_symbols
def test_issue_10567():
a, b, c, t = symbols('a b c t')
vt = Matrix([a*t, b, c])
assert integrate(vt, t) == Integral(vt, t).doit()
assert integrate(vt, t) == Matrix([[a*t**2/2], [b*t], [c*t]])
def test_issue_11856():
t = symbols('t')
assert integrate(sinc(pi*t), t) == Si(pi*t)/pi
def test_issue_4950():
assert integrate((-60*exp(x) - 19.2*exp(4*x))*exp(4*x), x) ==\
-2.4*exp(8*x) - 12.0*exp(5*x)
def test_issue_4968():
assert integrate(sin(log(x**2))) == x*sin(2*log(x))/5 - 2*x*cos(2*log(x))/5
def test_singularities():
assert integrate(1/x**2, (x, -oo, oo)) == oo
assert integrate(1/x**2, (x, -1, 1)) == oo
assert integrate(1/(x - 1)**2, (x, -2, 2)) == oo
assert integrate(1/x**2, (x, 1, -1)) == -oo
assert integrate(1/(x - 1)**2, (x, 2, -2)) == -oo
def test_issue_12645():
x, y = symbols('x y', real=True)
assert (integrate(sin(x*x*x + y*y),
(x, -sqrt(pi - y*y), sqrt(pi - y*y)),
(y, -sqrt(pi), sqrt(pi)))
== Integral(sin(x**3 + y**2),
(x, -sqrt(-y**2 + pi), sqrt(-y**2 + pi)),
(y, -sqrt(pi), sqrt(pi))))
def test_issue_12677():
assert integrate(sin(x) / (cos(x)**3) , (x, 0, pi/6)) == Rational(1,6)
def test_issue_14064():
assert integrate(1/cosh(x), (x, 0, oo)) == pi/2
def test_issue_14027():
assert integrate(1/(1 + exp(x - S(1)/2)/(1 + exp(x))), x) == \
x - exp(S(1)/2)*log(exp(x) + exp(S(1)/2)/(1 + exp(S(1)/2)))/(exp(S(1)/2) + E)
def test_issue_8170():
assert integrate(tan(x), (x, 0, pi/2)) == S.Infinity
def test_issue_8440_14040():
assert integrate(1/x, (x, -1, 1)) == S.NaN
assert integrate(1/(x + 1), (x, -2, 3)) == S.NaN
def test_issue_14096():
assert integrate(1/(x + y)**2, (x, 0, 1)) == -1/(y + 1) + 1/y
assert integrate(1/(1 + x + y + z)**2, (x, 0, 1), (y, 0, 1), (z, 0, 1)) == \
-4*log(4) - 6*log(2) + 9*log(3)
def test_issue_14144():
assert Abs(integrate(1/sqrt(1 - x**3), (x, 0, 1)).n() - 1.402182) < 1e-6
assert Abs(integrate(sqrt(1 - x**3), (x, 0, 1)).n() - 0.841309) < 1e-6
def test_issue_14375():
# This raised a TypeError. The antiderivative has exp_polar, which
# may be possible to unpolarify, so the exact output is not asserted here.
assert integrate(exp(I*x)*log(x), x).has(Ei)
def test_issue_14437():
f = Function('f')(x, y, z)
assert integrate(f, (x, 0, 1), (y, 0, 2), (z, 0, 3)) == \
Integral(f, (x, 0, 1), (y, 0, 2), (z, 0, 3))
def test_issue_14470():
assert integrate(1/sqrt(exp(x) + 1), x) == \
log(-1 + 1/sqrt(exp(x) + 1)) - log(1 + 1/sqrt(exp(x) + 1))
def test_issue_14877():
f = exp(1 - exp(x**2)*x + 2*x**2)*(2*x**3 + x)/(1 - exp(x**2)*x)**2
assert integrate(f, x) == \
-exp(2*x**2 - x*exp(x**2) + 1)/(x*exp(3*x**2) - exp(2*x**2))
def test_issue_14782():
f = sqrt(-x**2 + 1)*(-x**2 + x)
assert integrate(f, [x, -1, 1]) == - pi / 8
assert integrate(f, [x, 0, 1]) == S(1) / 3 - pi / 16
def test_issue_12081():
f = x**(-S(3)/2)*exp(-x)
assert integrate(f, [x, 0, oo]) == oo
def test_issue_15285():
y = 1/x - 1
f = 4*y*exp(-2*y)/x**2
assert integrate(f, [x, 0, 1]) == 1
def test_issue_15432():
assert integrate(x**n * exp(-x) * log(x), (x, 0, oo)).gammasimp() == Piecewise(
(gamma(n + 1)*polygamma(0, n) + gamma(n + 1)/n, re(n) + 1 > 0),
(Integral(x**n*exp(-x)*log(x), (x, 0, oo)), True))
def test_issue_15124():
omega = IndexedBase('omega')
m, p = symbols('m p', cls=Idx)
assert integrate(exp(x*I*(omega[m] + omega[p])), x, conds='none') == \
-I*exp(I*x*omega[m])*exp(I*x*omega[p])/(omega[m] + omega[p])
def test_issue_15218():
assert Eq(x, y).integrate(x) == Eq(x**2/2, x*y)
assert Integral(Eq(x, y), x) == Eq(Integral(x, x), Integral(y, x))
assert Integral(Eq(x, y), x).doit() == Eq(x**2/2, x*y)
def test_issue_15292():
res = integrate(exp(-x**2*cos(2*t)) * cos(x**2*sin(2*t)), (x, 0, oo))
assert isinstance(res, Piecewise)
assert gammasimp((res - sqrt(pi)/2 * cos(t)).subs(t, pi/6)) == 0
def test_issue_4514():
assert integrate(sin(2*x)/sin(x), x) == 2*sin(x)
def test_issue_15457():
x, a, b = symbols('x a b', real=True)
definite = integrate(exp(Abs(x-2)), (x, a, b))
indefinite = integrate(exp(Abs(x-2)), x)
assert definite.subs({a: 1, b: 3}) == -2 + 2*E
assert indefinite.subs(x, 3) - indefinite.subs(x, 1) == -2 + 2*E
assert definite.subs({a: -3, b: -1}) == -exp(3) + exp(5)
assert indefinite.subs(x, -1) - indefinite.subs(x, -3) == -exp(3) + exp(5)
def test_issue_15431():
assert integrate(x*exp(x)*log(x), x) == \
(x*exp(x) - exp(x))*log(x) - exp(x) + Ei(x)
def test_issue_15640_log_substitutions():
f = x/log(x)
F = Ei(2*log(x))
assert integrate(f, x) == F and F.diff(x) == f
f = x**3/log(x)**2
F = -x**4/log(x) + 4*Ei(4*log(x))
assert integrate(f, x) == F and F.diff(x) == f
f = sqrt(log(x))/x**2
F = -sqrt(pi)*erfc(sqrt(log(x)))/2 - sqrt(log(x))/x
assert integrate(f, x) == F and F.diff(x) == f
def test_issue_15509():
from sympy.vector import CoordSys3D
N = CoordSys3D('N')
x = N.x
assert integrate(cos(a*x + b), (x, x_1, x_2), heurisch=True) == Piecewise(
(-sin(a*x_1 + b)/a + sin(a*x_2 + b)/a, (a > -oo) & (a < oo) & Ne(a, 0)), \
(-x_1*cos(b) + x_2*cos(b), True))
@slow
def test_issue_4311():
x = symbols('x')
assert integrate(x*abs(9-x**2), x) == Integral(x*abs(9-x**2), x)
x = symbols('x', real=True)
assert integrate(x*abs(9-x**2), x) == Piecewise(
(x**4/4 - 9*x**2/2, x <= -3),
(-x**4/4 + 9*x**2/2 - S(81)/2, x <= 3),
(x**4/4 - 9*x**2/2, True))
|
fe87109dd1b1125eb8aca103fef9a2ba9cb2f6754c099df8aafa8878107ce724
|
from sympy import (sin, cos, tan, sec, csc, cot, log, exp, atan, asin, acos,
Symbol, Integral, integrate, pi, Dummy, Derivative,
diff, I, sqrt, erf, Piecewise, Eq, Ne, symbols, Rational,
And, Heaviside, S, asinh, acosh, atanh, acoth, expand,
Function, jacobi, gegenbauer, chebyshevt, chebyshevu,
legendre, hermite, laguerre, assoc_laguerre, uppergamma, li,
Ei, Ci, Si, Chi, Shi, fresnels, fresnelc, polylog, erf, erfi,
sinh, cosh, elliptic_f, elliptic_e)
from sympy.integrals.manualintegrate import (manualintegrate, find_substitutions,
_parts_rule, integral_steps, contains_dont_know, manual_subs)
from sympy.utilities.pytest import slow
x, y, z, u, n, a, b, c = symbols('x y z u n a b c')
f = Function('f')
def test_find_substitutions():
assert find_substitutions((cot(x)**2 + 1)**2*csc(x)**2*cot(x)**2, x, u) == \
[(cot(x), 1, -u**6 - 2*u**4 - u**2)]
assert find_substitutions((sec(x)**2 + tan(x) * sec(x)) / (sec(x) + tan(x)),
x, u) == [(sec(x) + tan(x), 1, 1/u)]
assert find_substitutions(x * exp(-x**2), x, u) == [(-x**2, -S.Half, exp(u))]
def test_manualintegrate_polynomials():
assert manualintegrate(y, x) == x*y
assert manualintegrate(exp(2), x) == x * exp(2)
assert manualintegrate(x**2, x) == x**3 / 3
assert manualintegrate(3 * x**2 + 4 * x**3, x) == x**3 + x**4
assert manualintegrate((x + 2)**3, x) == (x + 2)**4 / 4
assert manualintegrate((3*x + 4)**2, x) == (3*x + 4)**3 / 9
assert manualintegrate((u + 2)**3, u) == (u + 2)**4 / 4
assert manualintegrate((3*u + 4)**2, u) == (3*u + 4)**3 / 9
def test_manualintegrate_exponentials():
assert manualintegrate(exp(2*x), x) == exp(2*x) / 2
assert manualintegrate(2**x, x) == (2 ** x) / log(2)
assert manualintegrate(1 / x, x) == log(x)
assert manualintegrate(1 / (2*x + 3), x) == log(2*x + 3) / 2
assert manualintegrate(log(x)**2 / x, x) == log(x)**3 / 3
def test_manualintegrate_parts():
assert manualintegrate(exp(x) * sin(x), x) == \
(exp(x) * sin(x)) / 2 - (exp(x) * cos(x)) / 2
assert manualintegrate(2*x*cos(x), x) == 2*x*sin(x) + 2*cos(x)
assert manualintegrate(x * log(x), x) == x**2*log(x)/2 - x**2/4
assert manualintegrate(log(x), x) == x * log(x) - x
assert manualintegrate((3*x**2 + 5) * exp(x), x) == \
3*x**2*exp(x) - 6*x*exp(x) + 11*exp(x)
assert manualintegrate(atan(x), x) == x*atan(x) - log(x**2 + 1)/2
# Make sure _parts_rule does not go into an infinite loop here
assert manualintegrate(log(1/x)/(x + 1), x).has(Integral)
# Make sure _parts_rule doesn't pick u = constant but can pick dv =
# constant if necessary, e.g. for integrate(atan(x))
assert _parts_rule(cos(x), x) == None
assert _parts_rule(exp(x), x) == None
assert _parts_rule(x**2, x) == None
result = _parts_rule(atan(x), x)
assert result[0] == atan(x) and result[1] == 1
def test_manualintegrate_trigonometry():
assert manualintegrate(sin(x), x) == -cos(x)
assert manualintegrate(tan(x), x) == -log(cos(x))
assert manualintegrate(sec(x), x) == log(sec(x) + tan(x))
assert manualintegrate(csc(x), x) == -log(csc(x) + cot(x))
assert manualintegrate(sin(x) * cos(x), x) in [sin(x) ** 2 / 2, -cos(x)**2 / 2]
assert manualintegrate(-sec(x) * tan(x), x) == -sec(x)
assert manualintegrate(csc(x) * cot(x), x) == -csc(x)
assert manualintegrate(sec(x)**2, x) == tan(x)
assert manualintegrate(csc(x)**2, x) == -cot(x)
assert manualintegrate(x * sec(x**2), x) == log(tan(x**2) + sec(x**2))/2
assert manualintegrate(cos(x)*csc(sin(x)), x) == -log(cot(sin(x)) + csc(sin(x)))
assert manualintegrate(cos(3*x)*sec(x), x) == -x + sin(2*x)
assert manualintegrate(sin(3*x)*sec(x), x) == \
-3*log(cos(x)) + 2*log(cos(x)**2) - 2*cos(x)**2
def test_manualintegrate_trigpowers():
assert manualintegrate(sin(x)**2 * cos(x), x) == sin(x)**3 / 3
assert manualintegrate(sin(x)**2 * cos(x) **2, x) == \
x / 8 - sin(4*x) / 32
assert manualintegrate(sin(x) * cos(x)**3, x) == -cos(x)**4 / 4
assert manualintegrate(sin(x)**3 * cos(x)**2, x) == \
cos(x)**5 / 5 - cos(x)**3 / 3
assert manualintegrate(tan(x)**3 * sec(x), x) == sec(x)**3/3 - sec(x)
assert manualintegrate(tan(x) * sec(x) **2, x) == sec(x)**2/2
assert manualintegrate(cot(x)**5 * csc(x), x) == \
-csc(x)**5/5 + 2*csc(x)**3/3 - csc(x)
assert manualintegrate(cot(x)**2 * csc(x)**6, x) == \
-cot(x)**7/7 - 2*cot(x)**5/5 - cot(x)**3/3
def test_manualintegrate_inversetrig():
# atan
assert manualintegrate(exp(x) / (1 + exp(2*x)), x) == atan(exp(x))
assert manualintegrate(1 / (4 + 9 * x**2), x) == atan(3 * x/2) / 6
assert manualintegrate(1 / (16 + 16 * x**2), x) == atan(x) / 16
assert manualintegrate(1 / (4 + x**2), x) == atan(x / 2) / 2
assert manualintegrate(1 / (1 + 4 * x**2), x) == atan(2*x) / 2
assert manualintegrate(1/(a + b*x**2), x) == \
Piecewise((atan(x/sqrt(a/b))/(b*sqrt(a/b)), a/b > 0), \
(-acoth(x/sqrt(-a/b))/(b*sqrt(-a/b)), And(a/b < 0, x**2 > -a/b)), \
(-atanh(x/sqrt(-a/b))/(b*sqrt(-a/b)), And(a/b < 0, x**2 < -a/b)))
assert manualintegrate(1/(4 + b*x**2), x) == \
Piecewise((atan(x/(2*sqrt(1/b)))/(2*b*sqrt(1/b)), 4/b > 0), \
(-acoth(x/(2*sqrt(-1/b)))/(2*b*sqrt(-1/b)), And(4/b < 0, x**2 > -4/b)), \
(-atanh(x/(2*sqrt(-1/b)))/(2*b*sqrt(-1/b)), And(4/b < 0, x**2 < -4/b)))
assert manualintegrate(1/(a + 4*x**2), x) == \
Piecewise((atan(2*x/sqrt(a))/(2*sqrt(a)), a/4 > 0), \
(-acoth(2*x/sqrt(-a))/(2*sqrt(-a)), And(a/4 < 0, x**2 > -a/4)), \
(-atanh(2*x/sqrt(-a))/(2*sqrt(-a)), And(a/4 < 0, x**2 < -a/4)))
assert manualintegrate(1/(4 + 4*x**2), x) == atan(x) / 4
# asin
assert manualintegrate(1/sqrt(1-x**2), x) == asin(x)
assert manualintegrate(1/sqrt(4-4*x**2), x) == asin(x)/2
assert manualintegrate(3/sqrt(1-9*x**2), x) == asin(3*x)
assert manualintegrate(1/sqrt(4-9*x**2), x) == asin(3*x/2)/3
# asinh
assert manualintegrate(1/sqrt(x**2 + 1), x) == \
asinh(x)
assert manualintegrate(1/sqrt(x**2 + 4), x) == \
asinh(x/2)
assert manualintegrate(1/sqrt(4*x**2 + 4), x) == \
asinh(x)/2
assert manualintegrate(1/sqrt(4*x**2 + 1), x) == \
asinh(2*x)/2
assert manualintegrate(1/sqrt(a*x**2 + 1), x) == \
Piecewise((sqrt(-1/a)*asin(x*sqrt(-a)), a < 0), (sqrt(1/a)*asinh(sqrt(a)*x), a > 0))
assert manualintegrate(1/sqrt(a + x**2), x) == \
Piecewise((asinh(x*sqrt(1/a)), a > 0), (acosh(x*sqrt(-1/a)), a < 0))
# acosh
assert manualintegrate(1/sqrt(x**2 - 1), x) == \
acosh(x)
assert manualintegrate(1/sqrt(x**2 - 4), x) == \
acosh(x/2)
assert manualintegrate(1/sqrt(4*x**2 - 4), x) == \
acosh(x)/2
assert manualintegrate(1/sqrt(9*x**2 - 1), x) == \
acosh(3*x)/3
assert manualintegrate(1/sqrt(a*x**2 - 4), x) == \
Piecewise((sqrt(1/a)*acosh(sqrt(a)*x/2), a > 0))
assert manualintegrate(1/sqrt(-a + 4*x**2), x) == \
Piecewise((asinh(2*x*sqrt(-1/a))/2, -a > 0), (acosh(2*x*sqrt(1/a))/2, -a < 0))
# piecewise
assert manualintegrate(1/sqrt(a-b*x**2), x) == \
Piecewise((sqrt(a/b)*asin(x*sqrt(b/a))/sqrt(a), And(-b < 0, a > 0)),
(sqrt(-a/b)*asinh(x*sqrt(-b/a))/sqrt(a), And(-b > 0, a > 0)),
(sqrt(a/b)*acosh(x*sqrt(b/a))/sqrt(-a), And(-b > 0, a < 0)))
assert manualintegrate(1/sqrt(a + b*x**2), x) == \
Piecewise((sqrt(-a/b)*asin(x*sqrt(-b/a))/sqrt(a), And(a > 0, b < 0)),
(sqrt(a/b)*asinh(x*sqrt(b/a))/sqrt(a), And(a > 0, b > 0)),
(sqrt(-a/b)*acosh(x*sqrt(-b/a))/sqrt(-a), And(a < 0, b > 0)))
def test_manualintegrate_trig_substitution():
assert manualintegrate(sqrt(16*x**2 - 9)/x, x) == \
Piecewise((sqrt(16*x**2 - 9) - 3*acos(3/(4*x)),
And(x < 3*S.One/4, x > -3*S.One/4)))
assert manualintegrate(1/(x**4 * sqrt(25-x**2)), x) == \
Piecewise((-sqrt(-x**2/25 + 1)/(125*x) -
(-x**2/25 + 1)**(3*S.Half)/(15*x**3), And(x < 5, x > -5)))
assert manualintegrate(x**7/(49*x**2 + 1)**(3 * S.Half), x) == \
((49*x**2 + 1)**(5*S.Half)/28824005 -
(49*x**2 + 1)**(3*S.Half)/5764801 +
3*sqrt(49*x**2 + 1)/5764801 + 1/(5764801*sqrt(49*x**2 + 1)))
def test_manualintegrate_trivial_substitution():
assert manualintegrate((exp(x) - exp(-x))/x, x) == -Ei(-x) + Ei(x)
f = Function('f')
assert manualintegrate((f(x) - f(-x))/x, x) == \
-Integral(f(-x)/x, x) + Integral(f(x)/x, x)
def test_manualintegrate_rational():
assert manualintegrate(1/(4 - x**2), x) == Piecewise((acoth(x/2)/2, x**2 > 4), (atanh(x/2)/2, x**2 < 4))
assert manualintegrate(1/(-1 + x**2), x) == Piecewise((-acoth(x), x**2 > 1), (-atanh(x), x**2 < 1))
def test_manualintegrate_special():
f, F = 4*exp(-x**2/3), 2*sqrt(3)*sqrt(pi)*erf(sqrt(3)*x/3)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = 3*exp(4*x**2), 3*sqrt(pi)*erfi(2*x)/4
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = x**(S(1)/3)*exp(-x/8), -16*uppergamma(S(4)/3, x/8)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = exp(2*x)/x, Ei(2*x)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = exp(1 + 2*x - x**2), sqrt(pi)*exp(2)*erf(x - 1)/2
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f = sin(x**2 + 4*x + 1)
F = (sqrt(2)*sqrt(pi)*(-sin(3)*fresnelc(sqrt(2)*(2*x + 4)/(2*sqrt(pi))) +
cos(3)*fresnels(sqrt(2)*(2*x + 4)/(2*sqrt(pi))))/2)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = cos(4*x**2), sqrt(2)*sqrt(pi)*fresnelc(2*sqrt(2)*x/sqrt(pi))/4
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = sin(3*x + 2)/x, sin(2)*Ci(3*x) + cos(2)*Si(3*x)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = sinh(3*x - 2)/x, -sinh(2)*Chi(3*x) + cosh(2)*Shi(3*x)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = 5*cos(2*x - 3)/x, 5*cos(3)*Ci(2*x) + 5*sin(3)*Si(2*x)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = cosh(x/2)/x, Chi(x/2)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = cos(x**2)/x, Ci(x**2)/2
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = 1/log(2*x + 1), li(2*x + 1)/2
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = polylog(2, 5*x)/x, polylog(3, 5*x)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = 5/sqrt(3 - 2*sin(x)**2), 5*sqrt(3)*elliptic_f(x, S(2)/3)/3
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
f, F = sqrt(4 + 9*sin(x)**2), 2*elliptic_e(x, -S(9)/4)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
def test_manualintegrate_derivative():
assert manualintegrate(pi * Derivative(x**2 + 2*x + 3), x) == \
pi * ((x**2 + 2*x + 3))
assert manualintegrate(Derivative(x**2 + 2*x + 3, y), x) == \
Integral(Derivative(x**2 + 2*x + 3, y))
assert manualintegrate(Derivative(sin(x), x, x, x, y), x) == \
Derivative(sin(x), x, x, y)
def test_manualintegrate_Heaviside():
assert manualintegrate(Heaviside(x), x) == x*Heaviside(x)
assert manualintegrate(x*Heaviside(2), x) == x**2/2
assert manualintegrate(x*Heaviside(-2), x) == 0
assert manualintegrate(x*Heaviside( x), x) == x**2*Heaviside( x)/2
assert manualintegrate(x*Heaviside(-x), x) == x**2*Heaviside(-x)/2
assert manualintegrate(Heaviside(2*x + 4), x) == (x+2)*Heaviside(2*x + 4)
assert manualintegrate(x*Heaviside(x), x) == x**2*Heaviside(x)/2
assert manualintegrate(Heaviside(x + 1)*Heaviside(1 - x)*x**2, x) == \
((x**3/3 + S(1)/3)*Heaviside(x + 1) - S(2)/3)*Heaviside(-x + 1)
y = Symbol('y')
assert manualintegrate(sin(7 + x)*Heaviside(3*x - 7), x) == \
(- cos(x + 7) + cos(S(28)/3))*Heaviside(3*x - S(7))
assert manualintegrate(sin(y + x)*Heaviside(3*x - y), x) == \
(cos(4*y/3) - cos(x + y))*Heaviside(3*x - y)
def test_manualintegrate_orthogonal_poly():
n = symbols('n')
a, b = 7, S(5)/3
polys = [jacobi(n, a, b, x), gegenbauer(n, a, x), chebyshevt(n, x),
chebyshevu(n, x), legendre(n, x), hermite(n, x), laguerre(n, x),
assoc_laguerre(n, a, x)]
for p in polys:
integral = manualintegrate(p, x)
for deg in [-2, -1, 0, 1, 3, 5, 8]:
# some accept negative "degree", some do not
try:
p_subbed = p.subs(n, deg)
except ValueError:
continue
assert (integral.subs(n, deg).diff(x) - p_subbed).expand() == 0
# can also integrate simple expressions with these polynomials
q = x*p.subs(x, 2*x + 1)
integral = manualintegrate(q, x)
for deg in [2, 4, 7]:
assert (integral.subs(n, deg).diff(x) - q.subs(n, deg)).expand() == 0
# cannot integrate with respect to any other parameter
t = symbols('t')
for i in range(len(p.args) - 1):
new_args = list(p.args)
new_args[i] = t
assert isinstance(manualintegrate(p.func(*new_args), t), Integral)
def test_issue_6799():
r, x, phi = map(Symbol, 'r x phi'.split())
n = Symbol('n', integer=True, positive=True)
integrand = (cos(n*(x-phi))*cos(n*x))
limits = (x, -pi, pi)
assert manualintegrate(integrand, x) == \
((n*x/2 + sin(2*n*x)/4)*cos(n*phi) - sin(n*phi)*cos(n*x)**2/2)/n
assert r * integrate(integrand, limits).trigsimp() / pi == r * cos(n * phi)
assert not integrate(integrand, limits).has(Dummy)
def test_issue_12251():
assert manualintegrate(x**y, x) == Piecewise(
(x**(y + 1)/(y + 1), Ne(y, -1)), (log(x), True))
def test_issue_3796():
assert manualintegrate(diff(exp(x + x**2)), x) == exp(x + x**2)
assert integrate(x * exp(x**4), x, risch=False) == -I*sqrt(pi)*erf(I*x**2)/4
def test_manual_true():
assert integrate(exp(x) * sin(x), x, manual=True) == \
(exp(x) * sin(x)) / 2 - (exp(x) * cos(x)) / 2
assert integrate(sin(x) * cos(x), x, manual=True) in \
[sin(x) ** 2 / 2, -cos(x)**2 / 2]
def test_issue_6746():
y = Symbol('y')
n = Symbol('n')
assert manualintegrate(y**x, x) == Piecewise(
(y**x/log(y), Ne(log(y), 0)), (x, True))
assert manualintegrate(y**(n*x), x) == Piecewise(
(Piecewise(
(y**(n*x)/log(y), Ne(log(y), 0)),
(n*x, True)
)/n, Ne(n, 0)),
(x, True))
assert manualintegrate(exp(n*x), x) == Piecewise(
(exp(n*x)/n, Ne(n, 0)), (x, True))
y = Symbol('y', positive=True)
assert manualintegrate((y + 1)**x, x) == (y + 1)**x/log(y + 1)
y = Symbol('y', zero=True)
assert manualintegrate((y + 1)**x, x) == x
y = Symbol('y')
n = Symbol('n', nonzero=True)
assert manualintegrate(y**(n*x), x) == Piecewise(
(y**(n*x)/log(y), Ne(log(y), 0)), (n*x, True))/n
y = Symbol('y', positive=True)
assert manualintegrate((y + 1)**(n*x), x) == \
(y + 1)**(n*x)/(n*log(y + 1))
a = Symbol('a', negative=True)
b = Symbol('b')
assert manualintegrate(1/(a + b*x**2), x) == \
Piecewise((atan(x/sqrt(a/b))/(b*sqrt(a/b)), a/b > 0), \
(-acoth(x/sqrt(-a/b))/(b*sqrt(-a/b)), And(a/b < 0, x**2 > -a/b)), \
(-atanh(x/sqrt(-a/b))/(b*sqrt(-a/b)), And(a/b < 0, x**2 < -a/b)))
b = Symbol('b', negative=True)
assert manualintegrate(1/(a + b*x**2), x) == \
atan(x/(sqrt(-a)*sqrt(-1/b)))/(b*sqrt(-a)*sqrt(-1/b))
assert manualintegrate(1/((x**a + y**b + 4)*sqrt(a*x**2 + 1)), x) == \
y**(-b)*Integral(x**(-a)/(y**(-b)*sqrt(a*x**2 + 1) +
x**(-a)*sqrt(a*x**2 + 1) + 4*x**(-a)*y**(-b)*sqrt(a*x**2 + 1)), x)
assert manualintegrate(1/((x**2 + 4)*sqrt(4*x**2 + 1)), x) == \
Integral(1/((x**2 + 4)*sqrt(4*x**2 + 1)), x)
assert manualintegrate(1/(x - a**x + x*b**2), x) == \
Integral(1/(-a**x + b**2*x + x), x)
@slow
def test_issue_2850():
assert manualintegrate(asin(x)*log(x), x) == -x*asin(x) - sqrt(-x**2 + 1) \
+ (x*asin(x) + sqrt(-x**2 + 1))*log(x) - Integral(sqrt(-x**2 + 1)/x, x)
assert manualintegrate(acos(x)*log(x), x) == -x*acos(x) + sqrt(-x**2 + 1) + \
(x*acos(x) - sqrt(-x**2 + 1))*log(x) + Integral(sqrt(-x**2 + 1)/x, x)
assert manualintegrate(atan(x)*log(x), x) == -x*atan(x) + (x*atan(x) - \
log(x**2 + 1)/2)*log(x) + log(x**2 + 1)/2 + Integral(log(x**2 + 1)/x, x)/2
def test_issue_9462():
assert manualintegrate(sin(2*x)*exp(x), x) == exp(x)*sin(2*x)/5 - 2*exp(x)*cos(2*x)/5
assert not contains_dont_know(integral_steps(sin(2*x)*exp(x), x))
assert manualintegrate((x - 3) / (x**2 - 2*x + 2)**2, x) == \
Integral(x/(x**4 - 4*x**3 + 8*x**2 - 8*x + 4), x) \
- 3*Integral(1/(x**4 - 4*x**3 + 8*x**2 - 8*x + 4), x)
def test_cyclic_parts():
f = cos(x)*exp(x/4)
F = 16*exp(x/4)*sin(x)/17 + 4*exp(x/4)*cos(x)/17
assert manualintegrate(f, x) == F and F.diff(x) == f
f = x*cos(x)*exp(x/4)
F = (x*(16*exp(x/4)*sin(x)/17 + 4*exp(x/4)*cos(x)/17) -
128*exp(x/4)*sin(x)/289 + 240*exp(x/4)*cos(x)/289)
assert manualintegrate(f, x) == F and F.diff(x) == f
@slow
def test_issue_10847():
assert manualintegrate(x**2 / (x**2 - c), x) == c*Piecewise((atan(x/sqrt(-c))/sqrt(-c), -c > 0), \
(-acoth(x/sqrt(c))/sqrt(c), And(-c < 0, x**2 > c)), \
(-atanh(x/sqrt(c))/sqrt(c), And(-c < 0, x**2 < c))) + x
assert manualintegrate(sqrt(x - y) * log(z / x), x) == 4*y**2*Piecewise((atan(sqrt(x - y)/sqrt(y))/sqrt(y), y > 0), \
(-acoth(sqrt(x - y)/sqrt(-y))/sqrt(-y), \
And(x - y > -y, y < 0)), \
(-atanh(sqrt(x - y)/sqrt(-y))/sqrt(-y), \
And(x - y < -y, y < 0)))/3 \
- 4*y*sqrt(x - y)/3 + 2*(x - y)**(S(3)/2)*log(z/x)/3 \
+ 4*(x - y)**(S(3)/2)/9
assert manualintegrate(sqrt(x) * log(x), x) == 2*x**(S(3)/2)*log(x)/3 - 4*x**(S(3)/2)/9
assert manualintegrate(sqrt(a*x + b) / x, x) == -2*b*Piecewise((-atan(sqrt(a*x + b)/sqrt(-b))/sqrt(-b), -b > 0), \
(acoth(sqrt(a*x + b)/sqrt(b))/sqrt(b), And(-b < 0, a*x + b > b)), \
(atanh(sqrt(a*x + b)/sqrt(b))/sqrt(b), And(-b < 0, a*x + b < b))) \
+ 2*sqrt(a*x + b)
assert expand(manualintegrate(sqrt(a*x + b) / (x + c), x)) == -2*a*c*Piecewise((atan(sqrt(a*x + b)/sqrt(a*c - b))/sqrt(a*c - b), \
a*c - b > 0), (-acoth(sqrt(a*x + b)/sqrt(-a*c + b))/sqrt(-a*c + b), And(a*c - b < 0, a*x + b > -a*c + b)), \
(-atanh(sqrt(a*x + b)/sqrt(-a*c + b))/sqrt(-a*c + b), And(a*c - b < 0, a*x + b < -a*c + b))) \
+ 2*b*Piecewise((atan(sqrt(a*x + b)/sqrt(a*c - b))/sqrt(a*c - b), a*c - b > 0), \
(-acoth(sqrt(a*x + b)/sqrt(-a*c + b))/sqrt(-a*c + b), And(a*c - b < 0, a*x + b > -a*c + b)), \
(-atanh(sqrt(a*x + b)/sqrt(-a*c + b))/sqrt(-a*c + b), And(a*c - b < 0, a*x + b < -a*c + b))) + 2*sqrt(a*x + b)
assert manualintegrate((4*x**4 + 4*x**3 + 16*x**2 + 12*x + 8) \
/ (x**6 + 2*x**5 + 3*x**4 + 4*x**3 + 3*x**2 + 2*x + 1), x) == \
2*x/(x**2 + 1) + 3*atan(x) - 1/(x**2 + 1) - 3/(x + 1)
assert manualintegrate(sqrt(2*x + 3) / (x + 1), x) == 2*sqrt(2*x + 3) - log(sqrt(2*x + 3) + 1) + log(sqrt(2*x + 3) - 1)
assert manualintegrate(sqrt(2*x + 3) / 2 * x, x) == (2*x + 3)**(S(5)/2)/20 - (2*x + 3)**(S(3)/2)/4
assert manualintegrate(x**Rational(3,2) * log(x), x) == 2*x**Rational(5,2)*log(x)/5 - 4*x**Rational(5,2)/25
assert manualintegrate(x**(-3) * log(x), x) == -log(x)/(2*x**2) - 1/(4*x**2)
assert manualintegrate(log(y)/(y**2*(1 - 1/y)), y) == \
log(y)*log(-1 + 1/y) - Integral(log(-1 + 1/y)/y, y)
def test_issue_12899():
assert manualintegrate(f(x,y).diff(x),y) == Integral(Derivative(f(x,y),x),y)
assert manualintegrate(f(x,y).diff(y).diff(x),y) == Derivative(f(x,y),x)
def test_constant_independent_of_symbol():
assert manualintegrate(Integral(y, (x, 1, 2)), x) == \
x*Integral(y, (x, 1, 2))
def test_issue_12641():
assert manualintegrate(sin(2*x), x) == -cos(2*x)/2
assert manualintegrate(cos(x)*sin(2*x), x) == -2*cos(x)**3/3
assert manualintegrate((sin(2*x)*cos(x))/(1 + cos(x)), x) == \
-2*log(cos(x) + 1) - cos(x)**2 + 2*cos(x)
def test_issue_13297():
assert manualintegrate(sin(x) * cos(x)**5, x) == -cos(x)**6 / 6
def test_issue_14470():
assert manualintegrate(1/(x*sqrt(x + 1)), x) == \
log(-1 + 1/sqrt(x + 1)) - log(1 + 1/sqrt(x + 1))
@slow
def test_issue_9858():
assert manualintegrate(exp(x)*cos(exp(x)), x) == sin(exp(x))
assert manualintegrate(exp(2*x)*cos(exp(x)), x) == \
exp(x)*sin(exp(x)) + cos(exp(x))
res = manualintegrate(exp(10*x)*sin(exp(x)), x)
assert not res.has(Integral)
assert res.diff(x) == exp(10*x)*sin(exp(x))
# an example with many similar integrations by parts
assert manualintegrate(sum([x*exp(k*x) for k in range(1, 8)]), x) == (
x*exp(7*x)/7 + x*exp(6*x)/6 + x*exp(5*x)/5 + x*exp(4*x)/4 +
x*exp(3*x)/3 + x*exp(2*x)/2 + x*exp(x) - exp(7*x)/49 -exp(6*x)/36 -
exp(5*x)/25 - exp(4*x)/16 - exp(3*x)/9 - exp(2*x)/4 - exp(x))
def test_issue_8520():
assert manualintegrate(x/(x**4 + 1), x) == atan(x**2)/2
assert manualintegrate(x**2/(x**6 + 25), x) == atan(x**3/5)/15
f = x/(9*x**4 + 4)**2
assert manualintegrate(f, x).diff(x).factor() == f
def test_manual_subs():
x, y = symbols('x y')
expr = log(x) + exp(x)
# if log(x) is y, then exp(y) is x
assert manual_subs(expr, log(x), y) == y + exp(exp(y))
# if exp(x) is y, then log(y) need not be x
assert manual_subs(expr, exp(x), y) == log(x) + y
def test_issue_15471():
f = log(x)*cos(log(x))/x**(S(3)/4)
F = -128*x**(S(1)/4)*sin(log(x))/289 + 240*x**(S(1)/4)*cos(log(x))/289 + (16*x**(S(1)/4)*sin(log(x))/17 + 4*x**(S(1)/4)*cos(log(x))/17)*log(x)
assert manualintegrate(f, x) == F and F.diff(x).equals(f)
|
2c69ce6be14c3d258389ce9ae74d421a182cd532ca5892d0d905daba688ca8b8
|
# A collection of failing integrals from the issues.
from sympy import (
integrate, Integral, exp, oo, pi, sign, sqrt, sin, cos,
tan, S, log, gamma, sinh,
)
from sympy.utilities.pytest import XFAIL, SKIP, slow, skip, ON_TRAVIS
from sympy.abc import x, k, c, y, R, b, h, a, m
@SKIP("Too slow for @slow")
@XFAIL
def test_issue_3880():
# integrate_hyperexponential(Poly(t*2*(1 - t0**2)*t0*(x**3 + x**2), t), Poly((1 + t0**2)**2*2*(x**2 + x + 1), t), [Poly(1, x), Poly(1 + t0**2, t0), Poly(t, t)], [x, t0, t], [exp, tan])
assert not integrate(exp(x)*cos(2*x)*sin(2*x) * (x**3 + x**2)/(2*(x**2 + x + 1)), x).has(Integral)
@XFAIL
def test_issue_4212():
assert not integrate(sign(x), x).has(Integral)
@XFAIL
def test_issue_4326():
assert integrate(((h*(x - R + b))/b)*sqrt(R**2 - x**2), (x, R - b, R)).has(Integral)
@XFAIL
def test_issue_4491():
assert not integrate(x*sqrt(x**2 + 2*x + 4), x).has(Integral)
@XFAIL
def test_issue_4511():
# This works, but gives a complicated answer. The correct answer is x - cos(x).
# The last one is what Maple gives. It is also quite slow.
assert integrate(cos(x)**2 / (1 - sin(x))) in [x - cos(x), 1 - cos(x) + x,
-2/(tan((S(1)/2)*x)**2 + 1) + x]
@XFAIL
def test_issue_4525():
# Warning: takes a long time
assert not integrate((x**m * (1 - x)**n * (a + b*x + c*x**2))/(1 + x**2), (x, 0, 1)).has(Integral)
@XFAIL
@slow
def test_issue_4540():
if ON_TRAVIS:
skip("Too slow for travis.")
# Note, this integral is probably nonelementary
assert not integrate(
(sin(1/x) - x*exp(x)) /
((-sin(1/x) + x*exp(x))*x + x*sin(1/x)), x).has(Integral)
@XFAIL
def test_issue_4551():
assert integrate(1/(x*sqrt(1 - x**2)), x).has(Integral)
@XFAIL
def test_issue_4737a():
# Implementation of Si()
assert integrate(sin(x)/x, x).has(Integral)
@XFAIL
def test_issue_1638b():
assert integrate(sin(x)/x, (x, -oo, oo)) == pi/2
@XFAIL
@slow
def test_issue_4891():
# Requires the hypergeometric function.
assert not integrate(cos(x)**y, x).has(Integral)
@XFAIL
@slow
def test_issue_1796a():
assert not integrate(exp(2*b*x)*exp(-a*x**2), x).has(Integral)
@XFAIL
def test_issue_4895b():
assert not integrate(exp(2*b*x)*exp(-a*x**2), (x, -oo, 0)).has(Integral)
@XFAIL
def test_issue_4895c():
assert not integrate(exp(2*b*x)*exp(-a*x**2), (x, -oo, oo)).has(Integral)
@XFAIL
def test_issue_4895d():
assert not integrate(exp(2*b*x)*exp(-a*x**2), (x, 0, oo)).has(Integral)
@XFAIL
@slow
def test_issue_4941():
if ON_TRAVIS:
skip("Too slow for travis.")
assert not integrate(sqrt(1 + sinh(x/20)**2), (x, -25, 25)).has(Integral)
@XFAIL
def test_issue_4992():
# Nonelementary integral. Requires hypergeometric/Meijer-G handling.
assert not integrate(log(x) * x**(k - 1) * exp(-x) / gamma(k), (x, 0, oo)).has(Integral)
|
7e0e9d08425767b88a987f54825c8858b98d3e8052228408d7f4ef0c731bdda5
|
from sympy import (meijerg, I, S, integrate, Integral, oo, gamma, cosh, sinc,
hyperexpand, exp, simplify, sqrt, pi, erf, erfc, sin, cos,
exp_polar, polygamma, hyper, log, expand_func)
from sympy.integrals.meijerint import (_rewrite_single, _rewrite1,
meijerint_indefinite, _inflate_g, _create_lookup_table,
meijerint_definite, meijerint_inversion)
from sympy.utilities import default_sort_key
from sympy.utilities.pytest import slow
from sympy.utilities.randtest import (verify_numerically,
random_complex_number as randcplx)
from sympy.core.compatibility import range
from sympy.abc import x, y, a, b, c, d, s, t, z
def test_rewrite_single():
def t(expr, c, m):
e = _rewrite_single(meijerg([a], [b], [c], [d], expr), x)
assert e is not None
assert isinstance(e[0][0][2], meijerg)
assert e[0][0][2].argument.as_coeff_mul(x) == (c, (m,))
def tn(expr):
assert _rewrite_single(meijerg([a], [b], [c], [d], expr), x) is None
t(x, 1, x)
t(x**2, 1, x**2)
t(x**2 + y*x**2, y + 1, x**2)
tn(x**2 + x)
tn(x**y)
def u(expr, x):
from sympy import Add, exp, exp_polar
r = _rewrite_single(expr, x)
e = Add(*[res[0]*res[2] for res in r[0]]).replace(
exp_polar, exp) # XXX Hack?
assert verify_numerically(e, expr, x)
u(exp(-x)*sin(x), x)
# The following has stopped working because hyperexpand changed slightly.
# It is probably not worth fixing
#u(exp(-x)*sin(x)*cos(x), x)
# This one cannot be done numerically, since it comes out as a g-function
# of argument 4*pi
# NOTE This also tests a bug in inverse mellin transform (which used to
# turn exp(4*pi*I*t) into a factor of exp(4*pi*I)**t instead of
# exp_polar).
#u(exp(x)*sin(x), x)
assert _rewrite_single(exp(x)*sin(x), x) == \
([(-sqrt(2)/(2*sqrt(pi)), 0,
meijerg(((-S(1)/2, 0, S(1)/4, S(1)/2, S(3)/4), (1,)),
((), (-S(1)/2, 0)), 64*exp_polar(-4*I*pi)/x**4))], True)
def test_rewrite1():
assert _rewrite1(x**3*meijerg([a], [b], [c], [d], x**2 + y*x**2)*5, x) == \
(5, x**3, [(1, 0, meijerg([a], [b], [c], [d], x**2*(y + 1)))], True)
def test_meijerint_indefinite_numerically():
def t(fac, arg):
g = meijerg([a], [b], [c], [d], arg)*fac
subs = {a: randcplx()/10, b: randcplx()/10 + I,
c: randcplx(), d: randcplx()}
integral = meijerint_indefinite(g, x)
assert integral is not None
assert verify_numerically(g.subs(subs), integral.diff(x).subs(subs), x)
t(1, x)
t(2, x)
t(1, 2*x)
t(1, x**2)
t(5, x**S('3/2'))
t(x**3, x)
t(3*x**S('3/2'), 4*x**S('7/3'))
def test_meijerint_definite():
v, b = meijerint_definite(x, x, 0, 0)
assert v.is_zero and b is True
v, b = meijerint_definite(x, x, oo, oo)
assert v.is_zero and b is True
def test_inflate():
subs = {a: randcplx()/10, b: randcplx()/10 + I, c: randcplx(),
d: randcplx(), y: randcplx()/10}
def t(a, b, arg, n):
from sympy import Mul
m1 = meijerg(a, b, arg)
m2 = Mul(*_inflate_g(m1, n))
# NOTE: (the random number)**9 must still be on the principal sheet.
# Thus make b&d small to create random numbers of small imaginary part.
return verify_numerically(m1.subs(subs), m2.subs(subs), x, b=0.1, d=-0.1)
assert t([[a], [b]], [[c], [d]], x, 3)
assert t([[a, y], [b]], [[c], [d]], x, 3)
assert t([[a], [b]], [[c, y], [d]], 2*x**3, 3)
def test_recursive():
from sympy import symbols
a, b, c = symbols('a b c', positive=True)
r = exp(-(x - a)**2)*exp(-(x - b)**2)
e = integrate(r, (x, 0, oo), meijerg=True)
assert simplify(e.expand()) == (
sqrt(2)*sqrt(pi)*(
(erf(sqrt(2)*(a + b)/2) + 1)*exp(-a**2/2 + a*b - b**2/2))/4)
e = integrate(exp(-(x - a)**2)*exp(-(x - b)**2)*exp(c*x), (x, 0, oo), meijerg=True)
assert simplify(e) == (
sqrt(2)*sqrt(pi)*(erf(sqrt(2)*(2*a + 2*b + c)/4) + 1)*exp(-a**2 - b**2
+ (2*a + 2*b + c)**2/8)/4)
assert simplify(integrate(exp(-(x - a - b - c)**2), (x, 0, oo), meijerg=True)) == \
sqrt(pi)/2*(1 + erf(a + b + c))
assert simplify(integrate(exp(-(x + a + b + c)**2), (x, 0, oo), meijerg=True)) == \
sqrt(pi)/2*(1 - erf(a + b + c))
@slow
def test_meijerint():
from sympy import symbols, expand, arg
s, t, mu = symbols('s t mu', real=True)
assert integrate(meijerg([], [], [0], [], s*t)
*meijerg([], [], [mu/2], [-mu/2], t**2/4),
(t, 0, oo)).is_Piecewise
s = symbols('s', positive=True)
assert integrate(x**s*meijerg([[], []], [[0], []], x), (x, 0, oo)) == \
gamma(s + 1)
assert integrate(x**s*meijerg([[], []], [[0], []], x), (x, 0, oo),
meijerg=True) == gamma(s + 1)
assert isinstance(integrate(x**s*meijerg([[], []], [[0], []], x),
(x, 0, oo), meijerg=False),
Integral)
assert meijerint_indefinite(exp(x), x) == exp(x)
# TODO what simplifications should be done automatically?
# This tests "extra case" for antecedents_1.
a, b = symbols('a b', positive=True)
assert simplify(meijerint_definite(x**a, x, 0, b)[0]) == \
b**(a + 1)/(a + 1)
# This tests various conditions and expansions:
meijerint_definite((x + 1)**3*exp(-x), x, 0, oo) == (16, True)
# Again, how about simplifications?
sigma, mu = symbols('sigma mu', positive=True)
i, c = meijerint_definite(exp(-((x - mu)/(2*sigma))**2), x, 0, oo)
assert simplify(i) == sqrt(pi)*sigma*(2 - erfc(mu/(2*sigma)))
assert c == True
i, _ = meijerint_definite(exp(-mu*x)*exp(sigma*x), x, 0, oo)
# TODO it would be nice to test the condition
assert simplify(i) == 1/(mu - sigma)
# Test substitutions to change limits
assert meijerint_definite(exp(x), x, -oo, 2) == (exp(2), True)
# Note: causes a NaN in _check_antecedents
assert expand(meijerint_definite(exp(x), x, 0, I)[0]) == exp(I) - 1
assert expand(meijerint_definite(exp(-x), x, 0, x)[0]) == \
1 - exp(-exp(I*arg(x))*abs(x))
# Test -oo to oo
assert meijerint_definite(exp(-x**2), x, -oo, oo) == (sqrt(pi), True)
assert meijerint_definite(exp(-abs(x)), x, -oo, oo) == (2, True)
assert meijerint_definite(exp(-(2*x - 3)**2), x, -oo, oo) == \
(sqrt(pi)/2, True)
assert meijerint_definite(exp(-abs(2*x - 3)), x, -oo, oo) == (1, True)
assert meijerint_definite(exp(-((x - mu)/sigma)**2/2)/sqrt(2*pi*sigma**2),
x, -oo, oo) == (1, True)
assert meijerint_definite(sinc(x)**2, x, -oo, oo) == (pi, True)
# Test one of the extra conditions for 2 g-functinos
assert meijerint_definite(exp(-x)*sin(x), x, 0, oo) == (S(1)/2, True)
# Test a bug
def res(n):
return (1/(1 + x**2)).diff(x, n).subs(x, 1)*(-1)**n
for n in range(6):
assert integrate(exp(-x)*sin(x)*x**n, (x, 0, oo), meijerg=True) == \
res(n)
# This used to test trigexpand... now it is done by linear substitution
assert simplify(integrate(exp(-x)*sin(x + a), (x, 0, oo), meijerg=True)
) == sqrt(2)*sin(a + pi/4)/2
# Test the condition 14 from prudnikov.
# (This is besselj*besselj in disguise, to stop the product from being
# recognised in the tables.)
a, b, s = symbols('a b s')
from sympy import And, re
assert meijerint_definite(meijerg([], [], [a/2], [-a/2], x/4)
*meijerg([], [], [b/2], [-b/2], x/4)*x**(s - 1), x, 0, oo) == \
(4*2**(2*s - 2)*gamma(-2*s + 1)*gamma(a/2 + b/2 + s)
/(gamma(-a/2 + b/2 - s + 1)*gamma(a/2 - b/2 - s + 1)
*gamma(a/2 + b/2 - s + 1)),
And(0 < -2*re(4*s) + 8, 0 < re(a/2 + b/2 + s), re(2*s) < 1))
# test a bug
assert integrate(sin(x**a)*sin(x**b), (x, 0, oo), meijerg=True) == \
Integral(sin(x**a)*sin(x**b), (x, 0, oo))
# test better hyperexpand
assert integrate(exp(-x**2)*log(x), (x, 0, oo), meijerg=True) == \
(sqrt(pi)*polygamma(0, S(1)/2)/4).expand()
# Test hyperexpand bug.
from sympy import lowergamma
n = symbols('n', integer=True)
assert simplify(integrate(exp(-x)*x**n, x, meijerg=True)) == \
lowergamma(n + 1, x)
# Test a bug with argument 1/x
alpha = symbols('alpha', positive=True)
assert meijerint_definite((2 - x)**alpha*sin(alpha/x), x, 0, 2) == \
(sqrt(pi)*alpha*gamma(alpha + 1)*meijerg(((), (alpha/2 + S(1)/2,
alpha/2 + 1)), ((0, 0, S(1)/2), (-S(1)/2,)), alpha**S(2)/16)/4, True)
# test a bug related to 3016
a, s = symbols('a s', positive=True)
assert simplify(integrate(x**s*exp(-a*x**2), (x, -oo, oo))) == \
a**(-s/2 - S(1)/2)*((-1)**s + 1)*gamma(s/2 + S(1)/2)/2
def test_bessel():
from sympy import besselj, besseli
assert simplify(integrate(besselj(a, z)*besselj(b, z)/z, (z, 0, oo),
meijerg=True, conds='none')) == \
2*sin(pi*(a/2 - b/2))/(pi*(a - b)*(a + b))
assert simplify(integrate(besselj(a, z)*besselj(a, z)/z, (z, 0, oo),
meijerg=True, conds='none')) == 1/(2*a)
# TODO more orthogonality integrals
assert simplify(integrate(sin(z*x)*(x**2 - 1)**(-(y + S(1)/2)),
(x, 1, oo), meijerg=True, conds='none')
*2/((z/2)**y*sqrt(pi)*gamma(S(1)/2 - y))) == \
besselj(y, z)
# Werner Rosenheinrich
# SOME INDEFINITE INTEGRALS OF BESSEL FUNCTIONS
assert integrate(x*besselj(0, x), x, meijerg=True) == x*besselj(1, x)
assert integrate(x*besseli(0, x), x, meijerg=True) == x*besseli(1, x)
# TODO can do higher powers, but come out as high order ... should they be
# reduced to order 0, 1?
assert integrate(besselj(1, x), x, meijerg=True) == -besselj(0, x)
assert integrate(besselj(1, x)**2/x, x, meijerg=True) == \
-(besselj(0, x)**2 + besselj(1, x)**2)/2
# TODO more besseli when tables are extended or recursive mellin works
assert integrate(besselj(0, x)**2/x**2, x, meijerg=True) == \
-2*x*besselj(0, x)**2 - 2*x*besselj(1, x)**2 \
+ 2*besselj(0, x)*besselj(1, x) - besselj(0, x)**2/x
assert integrate(besselj(0, x)*besselj(1, x), x, meijerg=True) == \
-besselj(0, x)**2/2
assert integrate(x**2*besselj(0, x)*besselj(1, x), x, meijerg=True) == \
x**2*besselj(1, x)**2/2
assert integrate(besselj(0, x)*besselj(1, x)/x, x, meijerg=True) == \
(x*besselj(0, x)**2 + x*besselj(1, x)**2 -
besselj(0, x)*besselj(1, x))
# TODO how does besselj(0, a*x)*besselj(0, b*x) work?
# TODO how does besselj(0, x)**2*besselj(1, x)**2 work?
# TODO sin(x)*besselj(0, x) etc come out a mess
# TODO can x*log(x)*besselj(0, x) be done?
# TODO how does besselj(1, x)*besselj(0, x+a) work?
# TODO more indefinite integrals when struve functions etc are implemented
# test a substitution
assert integrate(besselj(1, x**2)*x, x, meijerg=True) == \
-besselj(0, x**2)/2
def test_inversion():
from sympy import piecewise_fold, besselj, sqrt, sin, cos, Heaviside
def inv(f):
return piecewise_fold(meijerint_inversion(f, s, t))
assert inv(1/(s**2 + 1)) == sin(t)*Heaviside(t)
assert inv(s/(s**2 + 1)) == cos(t)*Heaviside(t)
assert inv(exp(-s)/s) == Heaviside(t - 1)
assert inv(1/sqrt(1 + s**2)) == besselj(0, t)*Heaviside(t)
# Test some antcedents checking.
assert meijerint_inversion(sqrt(s)/sqrt(1 + s**2), s, t) is None
assert inv(exp(s**2)) is None
assert meijerint_inversion(exp(-s**2), s, t) is None
def test_inversion_conditional_output():
from sympy import Symbol, InverseLaplaceTransform
a = Symbol('a', positive=True)
F = sqrt(pi/a)*exp(-2*sqrt(a)*sqrt(s))
f = meijerint_inversion(F, s, t)
assert not f.is_Piecewise
b = Symbol('b', real=True)
F = F.subs(a, b)
f2 = meijerint_inversion(F, s, t)
assert f2.is_Piecewise
# first piece is same as f
assert f2.args[0][0] == f.subs(a, b)
# last piece is an unevaluated transform
assert f2.args[-1][1]
ILT = InverseLaplaceTransform(F, s, t, None)
assert f2.args[-1][0] == ILT or f2.args[-1][0] == ILT.as_integral
def test_inversion_exp_real_nonreal_shift():
from sympy import Symbol, DiracDelta
r = Symbol('r', real=True)
c = Symbol('c', real=False)
a = 1 + 2*I
z = Symbol('z')
assert not meijerint_inversion(exp(r*s), s, t).is_Piecewise
assert meijerint_inversion(exp(a*s), s, t) is None
assert meijerint_inversion(exp(c*s), s, t) is None
f = meijerint_inversion(exp(z*s), s, t)
assert f.is_Piecewise
assert isinstance(f.args[0][0], DiracDelta)
@slow
def test_lookup_table():
from random import uniform, randrange
from sympy import Add
from sympy.integrals.meijerint import z as z_dummy
table = {}
_create_lookup_table(table)
for _, l in sorted(table.items()):
for formula, terms, cond, hint in sorted(l, key=default_sort_key):
subs = {}
for a in list(formula.free_symbols) + [z_dummy]:
if hasattr(a, 'properties') and a.properties:
# these Wilds match positive integers
subs[a] = randrange(1, 10)
else:
subs[a] = uniform(1.5, 2.0)
if not isinstance(terms, list):
terms = terms(subs)
# First test that hyperexpand can do this.
expanded = [hyperexpand(g) for (_, g) in terms]
assert all(x.is_Piecewise or not x.has(meijerg) for x in expanded)
# Now test that the meijer g-function is indeed as advertised.
expanded = Add(*[f*x for (f, x) in terms])
a, b = formula.n(subs=subs), expanded.n(subs=subs)
r = min(abs(a), abs(b))
if r < 1:
assert abs(a - b).n() <= 1e-10
else:
assert (abs(a - b)/r).n() <= 1e-10
def test_branch_bug():
from sympy import powdenest, lowergamma
# TODO gammasimp cannot prove that the factor is unity
assert powdenest(integrate(erf(x**3), x, meijerg=True).diff(x),
polar=True) == 2*erf(x**3)*gamma(S(2)/3)/3/gamma(S(5)/3)
assert integrate(erf(x**3), x, meijerg=True) == \
2*x*erf(x**3)*gamma(S(2)/3)/(3*gamma(S(5)/3)) \
- 2*gamma(S(2)/3)*lowergamma(S(2)/3, x**6)/(3*sqrt(pi)*gamma(S(5)/3))
def test_linear_subs():
from sympy import besselj
assert integrate(sin(x - 1), x, meijerg=True) == -cos(1 - x)
assert integrate(besselj(1, x - 1), x, meijerg=True) == -besselj(0, 1 - x)
@slow
def test_probability():
# various integrals from probability theory
from sympy.abc import x, y
from sympy import symbols, Symbol, Abs, expand_mul, gammasimp, powsimp, sin
mu1, mu2 = symbols('mu1 mu2', real=True, nonzero=True, finite=True)
sigma1, sigma2 = symbols('sigma1 sigma2', real=True, nonzero=True,
finite=True, positive=True)
rate = Symbol('lambda', real=True, positive=True, finite=True)
def normal(x, mu, sigma):
return 1/sqrt(2*pi*sigma**2)*exp(-(x - mu)**2/2/sigma**2)
def exponential(x, rate):
return rate*exp(-rate*x)
assert integrate(normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == 1
assert integrate(x*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == \
mu1
assert integrate(x**2*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) \
== mu1**2 + sigma1**2
assert integrate(x**3*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) \
== mu1**3 + 3*mu1*sigma1**2
assert integrate(normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == 1
assert integrate(x*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == mu1
assert integrate(y*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == mu2
assert integrate(x*y*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == mu1*mu2
assert integrate((x + y + 1)*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == 1 + mu1 + mu2
assert integrate((x + y - 1)*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == \
-1 + mu1 + mu2
i = integrate(x**2*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True)
assert not i.has(Abs)
assert simplify(i) == mu1**2 + sigma1**2
assert integrate(y**2*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == \
sigma2**2 + mu2**2
assert integrate(exponential(x, rate), (x, 0, oo), meijerg=True) == 1
assert integrate(x*exponential(x, rate), (x, 0, oo), meijerg=True) == \
1/rate
assert integrate(x**2*exponential(x, rate), (x, 0, oo), meijerg=True) == \
2/rate**2
def E(expr):
res1 = integrate(expr*exponential(x, rate)*normal(y, mu1, sigma1),
(x, 0, oo), (y, -oo, oo), meijerg=True)
res2 = integrate(expr*exponential(x, rate)*normal(y, mu1, sigma1),
(y, -oo, oo), (x, 0, oo), meijerg=True)
assert expand_mul(res1) == expand_mul(res2)
return res1
assert E(1) == 1
assert E(x*y) == mu1/rate
assert E(x*y**2) == mu1**2/rate + sigma1**2/rate
ans = sigma1**2 + 1/rate**2
assert simplify(E((x + y + 1)**2) - E(x + y + 1)**2) == ans
assert simplify(E((x + y - 1)**2) - E(x + y - 1)**2) == ans
assert simplify(E((x + y)**2) - E(x + y)**2) == ans
# Beta' distribution
alpha, beta = symbols('alpha beta', positive=True)
betadist = x**(alpha - 1)*(1 + x)**(-alpha - beta)*gamma(alpha + beta) \
/gamma(alpha)/gamma(beta)
assert integrate(betadist, (x, 0, oo), meijerg=True) == 1
i = integrate(x*betadist, (x, 0, oo), meijerg=True, conds='separate')
assert (gammasimp(i[0]), i[1]) == (alpha/(beta - 1), 1 < beta)
j = integrate(x**2*betadist, (x, 0, oo), meijerg=True, conds='separate')
assert j[1] == (1 < beta - 1)
assert gammasimp(j[0] - i[0]**2) == (alpha + beta - 1)*alpha \
/(beta - 2)/(beta - 1)**2
# Beta distribution
# NOTE: this is evaluated using antiderivatives. It also tests that
# meijerint_indefinite returns the simplest possible answer.
a, b = symbols('a b', positive=True)
betadist = x**(a - 1)*(-x + 1)**(b - 1)*gamma(a + b)/(gamma(a)*gamma(b))
assert simplify(integrate(betadist, (x, 0, 1), meijerg=True)) == 1
assert simplify(integrate(x*betadist, (x, 0, 1), meijerg=True)) == \
a/(a + b)
assert simplify(integrate(x**2*betadist, (x, 0, 1), meijerg=True)) == \
a*(a + 1)/(a + b)/(a + b + 1)
assert simplify(integrate(x**y*betadist, (x, 0, 1), meijerg=True)) == \
gamma(a + b)*gamma(a + y)/gamma(a)/gamma(a + b + y)
# Chi distribution
k = Symbol('k', integer=True, positive=True)
chi = 2**(1 - k/2)*x**(k - 1)*exp(-x**2/2)/gamma(k/2)
assert powsimp(integrate(chi, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x*chi, (x, 0, oo), meijerg=True)) == \
sqrt(2)*gamma((k + 1)/2)/gamma(k/2)
assert simplify(integrate(x**2*chi, (x, 0, oo), meijerg=True)) == k
# Chi^2 distribution
chisquared = 2**(-k/2)/gamma(k/2)*x**(k/2 - 1)*exp(-x/2)
assert powsimp(integrate(chisquared, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x*chisquared, (x, 0, oo), meijerg=True)) == k
assert simplify(integrate(x**2*chisquared, (x, 0, oo), meijerg=True)) == \
k*(k + 2)
assert gammasimp(integrate(((x - k)/sqrt(2*k))**3*chisquared, (x, 0, oo),
meijerg=True)) == 2*sqrt(2)/sqrt(k)
# Dagum distribution
a, b, p = symbols('a b p', positive=True)
# XXX (x/b)**a does not work
dagum = a*p/x*(x/b)**(a*p)/(1 + x**a/b**a)**(p + 1)
assert simplify(integrate(dagum, (x, 0, oo), meijerg=True)) == 1
# XXX conditions are a mess
arg = x*dagum
assert simplify(integrate(arg, (x, 0, oo), meijerg=True, conds='none')
) == a*b*gamma(1 - 1/a)*gamma(p + 1 + 1/a)/(
(a*p + 1)*gamma(p))
assert simplify(integrate(x*arg, (x, 0, oo), meijerg=True, conds='none')
) == a*b**2*gamma(1 - 2/a)*gamma(p + 1 + 2/a)/(
(a*p + 2)*gamma(p))
# F-distribution
d1, d2 = symbols('d1 d2', positive=True)
f = sqrt(((d1*x)**d1 * d2**d2)/(d1*x + d2)**(d1 + d2))/x \
/gamma(d1/2)/gamma(d2/2)*gamma((d1 + d2)/2)
assert simplify(integrate(f, (x, 0, oo), meijerg=True)) == 1
# TODO conditions are a mess
assert simplify(integrate(x*f, (x, 0, oo), meijerg=True, conds='none')
) == d2/(d2 - 2)
assert simplify(integrate(x**2*f, (x, 0, oo), meijerg=True, conds='none')
) == d2**2*(d1 + 2)/d1/(d2 - 4)/(d2 - 2)
# TODO gamma, rayleigh
# inverse gaussian
lamda, mu = symbols('lamda mu', positive=True)
dist = sqrt(lamda/2/pi)*x**(-S(3)/2)*exp(-lamda*(x - mu)**2/x/2/mu**2)
mysimp = lambda expr: simplify(expr.rewrite(exp))
assert mysimp(integrate(dist, (x, 0, oo))) == 1
assert mysimp(integrate(x*dist, (x, 0, oo))) == mu
assert mysimp(integrate((x - mu)**2*dist, (x, 0, oo))) == mu**3/lamda
assert mysimp(integrate((x - mu)**3*dist, (x, 0, oo))) == 3*mu**5/lamda**2
# Levi
c = Symbol('c', positive=True)
assert integrate(sqrt(c/2/pi)*exp(-c/2/(x - mu))/(x - mu)**S('3/2'),
(x, mu, oo)) == 1
# higher moments oo
# log-logistic
distn = (beta/alpha)*x**(beta - 1)/alpha**(beta - 1)/ \
(1 + x**beta/alpha**beta)**2
assert simplify(integrate(distn, (x, 0, oo))) == 1
# NOTE the conditions are a mess, but correctly state beta > 1
assert simplify(integrate(x*distn, (x, 0, oo), conds='none')) == \
pi*alpha/beta/sin(pi/beta)
# (similar comment for conditions applies)
assert simplify(integrate(x**y*distn, (x, 0, oo), conds='none')) == \
pi*alpha**y*y/beta/sin(pi*y/beta)
# weibull
k = Symbol('k', positive=True)
n = Symbol('n', positive=True)
distn = k/lamda*(x/lamda)**(k - 1)*exp(-(x/lamda)**k)
assert simplify(integrate(distn, (x, 0, oo))) == 1
assert simplify(integrate(x**n*distn, (x, 0, oo))) == \
lamda**n*gamma(1 + n/k)
# rice distribution
from sympy import besseli
nu, sigma = symbols('nu sigma', positive=True)
rice = x/sigma**2*exp(-(x**2 + nu**2)/2/sigma**2)*besseli(0, x*nu/sigma**2)
assert integrate(rice, (x, 0, oo), meijerg=True) == 1
# can someone verify higher moments?
# Laplace distribution
mu = Symbol('mu', real=True)
b = Symbol('b', positive=True)
laplace = exp(-abs(x - mu)/b)/2/b
assert integrate(laplace, (x, -oo, oo), meijerg=True) == 1
assert integrate(x*laplace, (x, -oo, oo), meijerg=True) == mu
assert integrate(x**2*laplace, (x, -oo, oo), meijerg=True) == \
2*b**2 + mu**2
# TODO are there other distributions supported on (-oo, oo) that we can do?
# misc tests
k = Symbol('k', positive=True)
assert gammasimp(expand_mul(integrate(log(x)*x**(k - 1)*exp(-x)/gamma(k),
(x, 0, oo)))) == polygamma(0, k)
@slow
def test_expint():
""" Test various exponential integrals. """
from sympy import (expint, unpolarify, Symbol, Ci, Si, Shi, Chi,
sin, cos, sinh, cosh, Ei)
assert simplify(unpolarify(integrate(exp(-z*x)/x**y, (x, 1, oo),
meijerg=True, conds='none'
).rewrite(expint).expand(func=True))) == expint(y, z)
assert integrate(exp(-z*x)/x, (x, 1, oo), meijerg=True,
conds='none').rewrite(expint).expand() == \
expint(1, z)
assert integrate(exp(-z*x)/x**2, (x, 1, oo), meijerg=True,
conds='none').rewrite(expint).expand() == \
expint(2, z).rewrite(Ei).rewrite(expint)
assert integrate(exp(-z*x)/x**3, (x, 1, oo), meijerg=True,
conds='none').rewrite(expint).expand() == \
expint(3, z).rewrite(Ei).rewrite(expint).expand()
t = Symbol('t', positive=True)
assert integrate(-cos(x)/x, (x, t, oo), meijerg=True).expand() == Ci(t)
assert integrate(-sin(x)/x, (x, t, oo), meijerg=True).expand() == \
Si(t) - pi/2
assert integrate(sin(x)/x, (x, 0, z), meijerg=True) == Si(z)
assert integrate(sinh(x)/x, (x, 0, z), meijerg=True) == Shi(z)
assert integrate(exp(-x)/x, x, meijerg=True).expand().rewrite(expint) == \
I*pi - expint(1, x)
assert integrate(exp(-x)/x**2, x, meijerg=True).rewrite(expint).expand() \
== expint(1, x) - exp(-x)/x - I*pi
u = Symbol('u', polar=True)
assert integrate(cos(u)/u, u, meijerg=True).expand().as_independent(u)[1] \
== Ci(u)
assert integrate(cosh(u)/u, u, meijerg=True).expand().as_independent(u)[1] \
== Chi(u)
assert integrate(expint(1, x), x, meijerg=True
).rewrite(expint).expand() == x*expint(1, x) - exp(-x)
assert integrate(expint(2, x), x, meijerg=True
).rewrite(expint).expand() == \
-x**2*expint(1, x)/2 + x*exp(-x)/2 - exp(-x)/2
assert simplify(unpolarify(integrate(expint(y, x), x,
meijerg=True).rewrite(expint).expand(func=True))) == \
-expint(y + 1, x)
assert integrate(Si(x), x, meijerg=True) == x*Si(x) + cos(x)
assert integrate(Ci(u), u, meijerg=True).expand() == u*Ci(u) - sin(u)
assert integrate(Shi(x), x, meijerg=True) == x*Shi(x) - cosh(x)
assert integrate(Chi(u), u, meijerg=True).expand() == u*Chi(u) - sinh(u)
assert integrate(Si(x)*exp(-x), (x, 0, oo), meijerg=True) == pi/4
assert integrate(expint(1, x)*sin(x), (x, 0, oo), meijerg=True) == log(2)/2
def test_messy():
from sympy import (laplace_transform, Si, Shi, Chi, atan, Piecewise,
acoth, E1, besselj, acosh, asin, And, re,
fourier_transform, sqrt)
assert laplace_transform(Si(x), x, s) == ((-atan(s) + pi/2)/s, 0, True)
assert laplace_transform(Shi(x), x, s) == (acoth(s)/s, 1, True)
# where should the logs be simplified?
assert laplace_transform(Chi(x), x, s) == \
((log(s**(-2)) - log((s**2 - 1)/s**2))/(2*s), 1, True)
# TODO maybe simplify the inequalities?
assert laplace_transform(besselj(a, x), x, s)[1:] == \
(0, And(re(a/2) + S(1)/2 > S(0), re(a/2) + 1 > S(0)))
# NOTE s < 0 can be done, but argument reduction is not good enough yet
assert fourier_transform(besselj(1, x)/x, x, s, noconds=False) == \
(Piecewise((0, 4*abs(pi**2*s**2) > 1),
(2*sqrt(-4*pi**2*s**2 + 1), True)), s > 0)
# TODO FT(besselj(0,x)) - conditions are messy (but for acceptable reasons)
# - folding could be better
assert integrate(E1(x)*besselj(0, x), (x, 0, oo), meijerg=True) == \
log(1 + sqrt(2))
assert integrate(E1(x)*besselj(1, x), (x, 0, oo), meijerg=True) == \
log(S(1)/2 + sqrt(2)/2)
assert integrate(1/x/sqrt(1 - x**2), x, meijerg=True) == \
Piecewise((-acosh(1/x), abs(x**(-2)) > 1), (I*asin(1/x), True))
def test_issue_6122():
assert integrate(exp(-I*x**2), (x, -oo, oo), meijerg=True) == \
-I*sqrt(pi)*exp(I*pi/4)
def test_issue_6252():
expr = 1/x/(a + b*x)**(S(1)/3)
anti = integrate(expr, x, meijerg=True)
assert not expr.has(hyper)
# XXX the expression is a mess, but actually upon differentiation and
# putting in numerical values seems to work...
def test_issue_6348():
assert integrate(exp(I*x)/(1 + x**2), (x, -oo, oo)).simplify().rewrite(exp) \
== pi*exp(-1)
def test_fresnel():
from sympy import fresnels, fresnelc
assert expand_func(integrate(sin(pi*x**2/2), x)) == fresnels(x)
assert expand_func(integrate(cos(pi*x**2/2), x)) == fresnelc(x)
def test_issue_6860():
assert meijerint_indefinite(x**x**x, x) is None
def test_issue_7337():
f = meijerint_indefinite(x*sqrt(2*x + 3), x).together()
assert f == sqrt(2*x + 3)*(2*x**2 + x - 3)/5
assert f._eval_interval(x, S(-1), S(1)) == S(2)/5
def test_issue_8368():
assert meijerint_indefinite(cosh(x)*exp(-x*t), x) == (
(-t - 1)*exp(x) + (-t + 1)*exp(-x))*exp(-t*x)/2/(t**2 - 1)
def test_issue_10211():
from sympy.abc import h, w
assert integrate((1/sqrt(((y-x)**2 + h**2))**3), (x,0,w), (y,0,w)) == \
2*sqrt(1 + w**2/h**2)/h - 2/h
def test_issue_11806():
from sympy import symbols
y, L = symbols('y L', positive=True)
assert integrate(1/sqrt(x**2 + y**2)**3, (x, -L, L)) == \
2*L/(y**2*sqrt(L**2 + y**2))
def test_issue_10681():
from sympy import RR
from sympy.abc import R, r
f = integrate(r**2*(R**2-r**2)**0.5, r, meijerg=True)
g = (1.0/3)*R**1.0*r**3*hyper((-0.5, S(3)/2), (S(5)/2,),
r**2*exp_polar(2*I*pi)/R**2)
assert RR.almosteq((f/g).n(), 1.0, 1e-12)
def test_issue_13536():
from sympy import Symbol
a = Symbol('a', real=True, positive=True)
assert integrate(1/x**2, (x, oo, a)) == -1/a
|
6e710fa93cef513961fc0a10b7575c3b9e9300cfeffad2df6452e0fd99f27435
|
from sympy.integrals.transforms import (mellin_transform,
inverse_mellin_transform, laplace_transform, inverse_laplace_transform,
fourier_transform, inverse_fourier_transform,
sine_transform, inverse_sine_transform,
cosine_transform, inverse_cosine_transform,
hankel_transform, inverse_hankel_transform,
LaplaceTransform, FourierTransform, SineTransform, CosineTransform,
InverseLaplaceTransform, InverseFourierTransform,
InverseSineTransform, InverseCosineTransform, IntegralTransformError)
from sympy import (
gamma, exp, oo, Heaviside, symbols, Symbol, re, factorial, pi, arg,
cos, S, Abs, And, Or, sin, sqrt, I, log, tan, hyperexpand, meijerg,
EulerGamma, erf, erfc, besselj, bessely, besseli, besselk,
exp_polar, polar_lift, unpolarify, Function, expint, expand_mul,
gammasimp, trigsimp, atan, sinh, cosh, Ne, periodic_argument, atan2, Abs)
from sympy.utilities.pytest import XFAIL, slow, skip, raises
from sympy.matrices import Matrix, eye
from sympy.abc import x, s, a, b, c, d
nu, beta, rho = symbols('nu beta rho')
def test_undefined_function():
from sympy import Function, MellinTransform
f = Function('f')
assert mellin_transform(f(x), x, s) == MellinTransform(f(x), x, s)
assert mellin_transform(f(x) + exp(-x), x, s) == \
(MellinTransform(f(x), x, s) + gamma(s), (0, oo), True)
assert laplace_transform(2*f(x), x, s) == 2*LaplaceTransform(f(x), x, s)
# TODO test derivative and other rules when implemented
def test_free_symbols():
from sympy import Function
f = Function('f')
assert mellin_transform(f(x), x, s).free_symbols == {s}
assert mellin_transform(f(x)*a, x, s).free_symbols == {s, a}
def test_as_integral():
from sympy import Function, Integral
f = Function('f')
assert mellin_transform(f(x), x, s).rewrite('Integral') == \
Integral(x**(s - 1)*f(x), (x, 0, oo))
assert fourier_transform(f(x), x, s).rewrite('Integral') == \
Integral(f(x)*exp(-2*I*pi*s*x), (x, -oo, oo))
assert laplace_transform(f(x), x, s).rewrite('Integral') == \
Integral(f(x)*exp(-s*x), (x, 0, oo))
assert str(2*pi*I*inverse_mellin_transform(f(s), s, x, (a, b)).rewrite('Integral')) \
== "Integral(x**(-s)*f(s), (s, _c - oo*I, _c + oo*I))"
assert str(2*pi*I*inverse_laplace_transform(f(s), s, x).rewrite('Integral')) == \
"Integral(f(s)*exp(s*x), (s, _c - oo*I, _c + oo*I))"
assert inverse_fourier_transform(f(s), s, x).rewrite('Integral') == \
Integral(f(s)*exp(2*I*pi*s*x), (s, -oo, oo))
# NOTE this is stuck in risch because meijerint cannot handle it
@slow
@XFAIL
def test_mellin_transform_fail():
skip("Risch takes forever.")
MT = mellin_transform
bpos = symbols('b', positive=True)
bneg = symbols('b', negative=True)
expr = (sqrt(x + b**2) + b)**a/sqrt(x + b**2)
# TODO does not work with bneg, argument wrong. Needs changes to matching.
assert MT(expr.subs(b, -bpos), x, s) == \
((-1)**(a + 1)*2**(a + 2*s)*bpos**(a + 2*s - 1)*gamma(a + s)
*gamma(1 - a - 2*s)/gamma(1 - s),
(-re(a), -re(a)/2 + S(1)/2), True)
expr = (sqrt(x + b**2) + b)**a
assert MT(expr.subs(b, -bpos), x, s) == \
(
2**(a + 2*s)*a*bpos**(a + 2*s)*gamma(-a - 2*
s)*gamma(a + s)/gamma(-s + 1),
(-re(a), -re(a)/2), True)
# Test exponent 1:
assert MT(expr.subs({b: -bpos, a: 1}), x, s) == \
(-bpos**(2*s + 1)*gamma(s)*gamma(-s - S(1)/2)/(2*sqrt(pi)),
(-1, -S(1)/2), True)
def test_mellin_transform():
from sympy import Max, Min
MT = mellin_transform
bpos = symbols('b', positive=True)
# 8.4.2
assert MT(x**nu*Heaviside(x - 1), x, s) == \
(-1/(nu + s), (-oo, -re(nu)), True)
assert MT(x**nu*Heaviside(1 - x), x, s) == \
(1/(nu + s), (-re(nu), oo), True)
assert MT((1 - x)**(beta - 1)*Heaviside(1 - x), x, s) == \
(gamma(beta)*gamma(s)/gamma(beta + s), (0, oo), re(beta) > 0)
assert MT((x - 1)**(beta - 1)*Heaviside(x - 1), x, s) == \
(gamma(beta)*gamma(1 - beta - s)/gamma(1 - s),
(-oo, -re(beta) + 1), re(beta) > 0)
assert MT((1 + x)**(-rho), x, s) == \
(gamma(s)*gamma(rho - s)/gamma(rho), (0, re(rho)), True)
# TODO also the conditions should be simplified, e.g.
# And(re(rho) - 1 < 0, re(rho) < 1) should just be
# re(rho) < 1
assert MT(abs(1 - x)**(-rho), x, s) == (
2*sin(pi*rho/2)*gamma(1 - rho)*
cos(pi*(rho/2 - s))*gamma(s)*gamma(rho-s)/pi,
(0, re(rho)), And(re(rho) - 1 < 0, re(rho) < 1))
mt = MT((1 - x)**(beta - 1)*Heaviside(1 - x)
+ a*(x - 1)**(beta - 1)*Heaviside(x - 1), x, s)
assert mt[1], mt[2] == ((0, -re(beta) + 1), re(beta) > 0)
assert MT((x**a - b**a)/(x - b), x, s)[0] == \
pi*b**(a + s - 1)*sin(pi*a)/(sin(pi*s)*sin(pi*(a + s)))
assert MT((x**a - bpos**a)/(x - bpos), x, s) == \
(pi*bpos**(a + s - 1)*sin(pi*a)/(sin(pi*s)*sin(pi*(a + s))),
(Max(-re(a), 0), Min(1 - re(a), 1)), True)
expr = (sqrt(x + b**2) + b)**a
assert MT(expr.subs(b, bpos), x, s) == \
(-a*(2*bpos)**(a + 2*s)*gamma(s)*gamma(-a - 2*s)/gamma(-a - s + 1),
(0, -re(a)/2), True)
expr = (sqrt(x + b**2) + b)**a/sqrt(x + b**2)
assert MT(expr.subs(b, bpos), x, s) == \
(2**(a + 2*s)*bpos**(a + 2*s - 1)*gamma(s)
*gamma(1 - a - 2*s)/gamma(1 - a - s),
(0, -re(a)/2 + S(1)/2), True)
# 8.4.2
assert MT(exp(-x), x, s) == (gamma(s), (0, oo), True)
assert MT(exp(-1/x), x, s) == (gamma(-s), (-oo, 0), True)
# 8.4.5
assert MT(log(x)**4*Heaviside(1 - x), x, s) == (24/s**5, (0, oo), True)
assert MT(log(x)**3*Heaviside(x - 1), x, s) == (6/s**4, (-oo, 0), True)
assert MT(log(x + 1), x, s) == (pi/(s*sin(pi*s)), (-1, 0), True)
assert MT(log(1/x + 1), x, s) == (pi/(s*sin(pi*s)), (0, 1), True)
assert MT(log(abs(1 - x)), x, s) == (pi/(s*tan(pi*s)), (-1, 0), True)
assert MT(log(abs(1 - 1/x)), x, s) == (pi/(s*tan(pi*s)), (0, 1), True)
# 8.4.14
assert MT(erf(sqrt(x)), x, s) == \
(-gamma(s + S(1)/2)/(sqrt(pi)*s), (-S(1)/2, 0), True)
@slow
def test_mellin_transform2():
MT = mellin_transform
# TODO we cannot currently do these (needs summation of 3F2(-1))
# this also implies that they cannot be written as a single g-function
# (although this is possible)
mt = MT(log(x)/(x + 1), x, s)
assert mt[1:] == ((0, 1), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
mt = MT(log(x)**2/(x + 1), x, s)
assert mt[1:] == ((0, 1), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
mt = MT(log(x)/(x + 1)**2, x, s)
assert mt[1:] == ((0, 2), True)
assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
@slow
def test_mellin_transform_bessel():
from sympy import Max
MT = mellin_transform
# 8.4.19
assert MT(besselj(a, 2*sqrt(x)), x, s) == \
(gamma(a/2 + s)/gamma(a/2 - s + 1), (-re(a)/2, S(3)/4), True)
assert MT(sin(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(2**a*gamma(-2*s + S(1)/2)*gamma(a/2 + s + S(1)/2)/(
gamma(-a/2 - s + 1)*gamma(a - 2*s + 1)), (
-re(a)/2 - S(1)/2, S(1)/4), True)
assert MT(cos(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(2**a*gamma(a/2 + s)*gamma(-2*s + S(1)/2)/(
gamma(-a/2 - s + S(1)/2)*gamma(a - 2*s + 1)), (
-re(a)/2, S(1)/4), True)
assert MT(besselj(a, sqrt(x))**2, x, s) == \
(gamma(a + s)*gamma(S(1)/2 - s)
/ (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
(-re(a), S(1)/2), True)
assert MT(besselj(a, sqrt(x))*besselj(-a, sqrt(x)), x, s) == \
(gamma(s)*gamma(S(1)/2 - s)
/ (sqrt(pi)*gamma(1 - a - s)*gamma(1 + a - s)),
(0, S(1)/2), True)
# NOTE: prudnikov gives the strip below as (1/2 - re(a), 1). As far as
# I can see this is wrong (since besselj(z) ~ 1/sqrt(z) for z large)
assert MT(besselj(a - 1, sqrt(x))*besselj(a, sqrt(x)), x, s) == \
(gamma(1 - s)*gamma(a + s - S(1)/2)
/ (sqrt(pi)*gamma(S(3)/2 - s)*gamma(a - s + S(1)/2)),
(S(1)/2 - re(a), S(1)/2), True)
assert MT(besselj(a, sqrt(x))*besselj(b, sqrt(x)), x, s) == \
(4**s*gamma(1 - 2*s)*gamma((a + b)/2 + s)
/ (gamma(1 - s + (b - a)/2)*gamma(1 - s + (a - b)/2)
*gamma( 1 - s + (a + b)/2)),
(-(re(a) + re(b))/2, S(1)/2), True)
assert MT(besselj(a, sqrt(x))**2 + besselj(-a, sqrt(x))**2, x, s)[1:] == \
((Max(re(a), -re(a)), S(1)/2), True)
# Section 8.4.20
assert MT(bessely(a, 2*sqrt(x)), x, s) == \
(-cos(pi*(a/2 - s))*gamma(s - a/2)*gamma(s + a/2)/pi,
(Max(-re(a)/2, re(a)/2), S(3)/4), True)
assert MT(sin(sqrt(x))*bessely(a, sqrt(x)), x, s) == \
(-4**s*sin(pi*(a/2 - s))*gamma(S(1)/2 - 2*s)
* gamma((1 - a)/2 + s)*gamma((1 + a)/2 + s)
/ (sqrt(pi)*gamma(1 - s - a/2)*gamma(1 - s + a/2)),
(Max(-(re(a) + 1)/2, (re(a) - 1)/2), S(1)/4), True)
assert MT(cos(sqrt(x))*bessely(a, sqrt(x)), x, s) == \
(-4**s*cos(pi*(a/2 - s))*gamma(s - a/2)*gamma(s + a/2)*gamma(S(1)/2 - 2*s)
/ (sqrt(pi)*gamma(S(1)/2 - s - a/2)*gamma(S(1)/2 - s + a/2)),
(Max(-re(a)/2, re(a)/2), S(1)/4), True)
assert MT(besselj(a, sqrt(x))*bessely(a, sqrt(x)), x, s) == \
(-cos(pi*s)*gamma(s)*gamma(a + s)*gamma(S(1)/2 - s)
/ (pi**S('3/2')*gamma(1 + a - s)),
(Max(-re(a), 0), S(1)/2), True)
assert MT(besselj(a, sqrt(x))*bessely(b, sqrt(x)), x, s) == \
(-4**s*cos(pi*(a/2 - b/2 + s))*gamma(1 - 2*s)
* gamma(a/2 - b/2 + s)*gamma(a/2 + b/2 + s)
/ (pi*gamma(a/2 - b/2 - s + 1)*gamma(a/2 + b/2 - s + 1)),
(Max((-re(a) + re(b))/2, (-re(a) - re(b))/2), S(1)/2), True)
# NOTE bessely(a, sqrt(x))**2 and bessely(a, sqrt(x))*bessely(b, sqrt(x))
# are a mess (no matter what way you look at it ...)
assert MT(bessely(a, sqrt(x))**2, x, s)[1:] == \
((Max(-re(a), 0, re(a)), S(1)/2), True)
# Section 8.4.22
# TODO we can't do any of these (delicate cancellation)
# Section 8.4.23
assert MT(besselk(a, 2*sqrt(x)), x, s) == \
(gamma(
s - a/2)*gamma(s + a/2)/2, (Max(-re(a)/2, re(a)/2), oo), True)
assert MT(besselj(a, 2*sqrt(2*sqrt(x)))*besselk(
a, 2*sqrt(2*sqrt(x))), x, s) == (4**(-s)*gamma(2*s)*
gamma(a/2 + s)/(2*gamma(a/2 - s + 1)), (Max(0, -re(a)/2), oo), True)
# TODO bessely(a, x)*besselk(a, x) is a mess
assert MT(besseli(a, sqrt(x))*besselk(a, sqrt(x)), x, s) == \
(gamma(s)*gamma(
a + s)*gamma(-s + S(1)/2)/(2*sqrt(pi)*gamma(a - s + 1)),
(Max(-re(a), 0), S(1)/2), True)
assert MT(besseli(b, sqrt(x))*besselk(a, sqrt(x)), x, s) == \
(2**(2*s - 1)*gamma(-2*s + 1)*gamma(-a/2 + b/2 + s)* \
gamma(a/2 + b/2 + s)/(gamma(-a/2 + b/2 - s + 1)* \
gamma(a/2 + b/2 - s + 1)), (Max(-re(a)/2 - re(b)/2, \
re(a)/2 - re(b)/2), S(1)/2), True)
# TODO products of besselk are a mess
mt = MT(exp(-x/2)*besselk(a, x/2), x, s)
mt0 = gammasimp((trigsimp(gammasimp(mt[0].expand(func=True)))))
assert mt0 == 2*pi**(S(3)/2)*cos(pi*s)*gamma(-s + S(1)/2)/(
(cos(2*pi*a) - cos(2*pi*s))*gamma(-a - s + 1)*gamma(a - s + 1))
assert mt[1:] == ((Max(-re(a), re(a)), oo), True)
# TODO exp(x/2)*besselk(a, x/2) [etc] cannot currently be done
# TODO various strange products of special orders
@slow
def test_expint():
from sympy import E1, expint, Max, re, lerchphi, Symbol, simplify, Si, Ci, Ei
aneg = Symbol('a', negative=True)
u = Symbol('u', polar=True)
assert mellin_transform(E1(x), x, s) == (gamma(s)/s, (0, oo), True)
assert inverse_mellin_transform(gamma(s)/s, s, x,
(0, oo)).rewrite(expint).expand() == E1(x)
assert mellin_transform(expint(a, x), x, s) == \
(gamma(s)/(a + s - 1), (Max(1 - re(a), 0), oo), True)
# XXX IMT has hickups with complicated strips ...
assert simplify(unpolarify(
inverse_mellin_transform(gamma(s)/(aneg + s - 1), s, x,
(1 - aneg, oo)).rewrite(expint).expand(func=True))) == \
expint(aneg, x)
assert mellin_transform(Si(x), x, s) == \
(-2**s*sqrt(pi)*gamma(s/2 + S(1)/2)/(
2*s*gamma(-s/2 + 1)), (-1, 0), True)
assert inverse_mellin_transform(-2**s*sqrt(pi)*gamma((s + 1)/2)
/(2*s*gamma(-s/2 + 1)), s, x, (-1, 0)) \
== Si(x)
assert mellin_transform(Ci(sqrt(x)), x, s) == \
(-2**(2*s - 1)*sqrt(pi)*gamma(s)/(s*gamma(-s + S(1)/2)), (0, 1), True)
assert inverse_mellin_transform(
-4**s*sqrt(pi)*gamma(s)/(2*s*gamma(-s + S(1)/2)),
s, u, (0, 1)).expand() == Ci(sqrt(u))
# TODO LT of Si, Shi, Chi is a mess ...
assert laplace_transform(Ci(x), x, s) == (-log(1 + s**2)/2/s, 0, True)
assert laplace_transform(expint(a, x), x, s) == \
(lerchphi(s*exp_polar(I*pi), 1, a), 0, re(a) > S(0))
assert laplace_transform(expint(1, x), x, s) == (log(s + 1)/s, 0, True)
assert laplace_transform(expint(2, x), x, s) == \
((s - log(s + 1))/s**2, 0, True)
assert inverse_laplace_transform(-log(1 + s**2)/2/s, s, u).expand() == \
Heaviside(u)*Ci(u)
assert inverse_laplace_transform(log(s + 1)/s, s, x).rewrite(expint) == \
Heaviside(x)*E1(x)
assert inverse_laplace_transform((s - log(s + 1))/s**2, s,
x).rewrite(expint).expand() == \
(expint(2, x)*Heaviside(x)).rewrite(Ei).rewrite(expint).expand()
@slow
def test_inverse_mellin_transform():
from sympy import (sin, simplify, Max, Min, expand,
powsimp, exp_polar, cos, cot)
IMT = inverse_mellin_transform
assert IMT(gamma(s), s, x, (0, oo)) == exp(-x)
assert IMT(gamma(-s), s, x, (-oo, 0)) == exp(-1/x)
assert simplify(IMT(s/(2*s**2 - 2), s, x, (2, oo))) == \
(x**2 + 1)*Heaviside(1 - x)/(4*x)
# test passing "None"
assert IMT(1/(s**2 - 1), s, x, (-1, None)) == \
-x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
assert IMT(1/(s**2 - 1), s, x, (None, 1)) == \
-x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
# test expansion of sums
assert IMT(gamma(s) + gamma(s - 1), s, x, (1, oo)) == (x + 1)*exp(-x)/x
# test factorisation of polys
r = symbols('r', real=True)
assert IMT(1/(s**2 + 1), s, exp(-x), (None, oo)
).subs(x, r).rewrite(sin).simplify() \
== sin(r)*Heaviside(1 - exp(-r))
# test multiplicative substitution
_a, _b = symbols('a b', positive=True)
assert IMT(_b**(-s/_a)*factorial(s/_a)/s, s, x, (0, oo)) == exp(-_b*x**_a)
assert IMT(factorial(_a/_b + s/_b)/(_a + s), s, x, (-_a, oo)) == x**_a*exp(-x**_b)
def simp_pows(expr):
return simplify(powsimp(expand_mul(expr, deep=False), force=True)).replace(exp_polar, exp)
# Now test the inverses of all direct transforms tested above
# Section 8.4.2
nu = symbols('nu', real=True, finite=True)
assert IMT(-1/(nu + s), s, x, (-oo, None)) == x**nu*Heaviside(x - 1)
assert IMT(1/(nu + s), s, x, (None, oo)) == x**nu*Heaviside(1 - x)
assert simp_pows(IMT(gamma(beta)*gamma(s)/gamma(s + beta), s, x, (0, oo))) \
== (1 - x)**(beta - 1)*Heaviside(1 - x)
assert simp_pows(IMT(gamma(beta)*gamma(1 - beta - s)/gamma(1 - s),
s, x, (-oo, None))) \
== (x - 1)**(beta - 1)*Heaviside(x - 1)
assert simp_pows(IMT(gamma(s)*gamma(rho - s)/gamma(rho), s, x, (0, None))) \
== (1/(x + 1))**rho
assert simp_pows(IMT(d**c*d**(s - 1)*sin(pi*c)
*gamma(s)*gamma(s + c)*gamma(1 - s)*gamma(1 - s - c)/pi,
s, x, (Max(-re(c), 0), Min(1 - re(c), 1)))) \
== (x**c - d**c)/(x - d)
assert simplify(IMT(1/sqrt(pi)*(-c/2)*gamma(s)*gamma((1 - c)/2 - s)
*gamma(-c/2 - s)/gamma(1 - c - s),
s, x, (0, -re(c)/2))) == \
(1 + sqrt(x + 1))**c
assert simplify(IMT(2**(a + 2*s)*b**(a + 2*s - 1)*gamma(s)*gamma(1 - a - 2*s)
/gamma(1 - a - s), s, x, (0, (-re(a) + 1)/2))) == \
b**(a - 1)*(sqrt(1 + x/b**2) + 1)**(a - 1)*(b**2*sqrt(1 + x/b**2) +
b**2 + x)/(b**2 + x)
assert simplify(IMT(-2**(c + 2*s)*c*b**(c + 2*s)*gamma(s)*gamma(-c - 2*s)
/ gamma(-c - s + 1), s, x, (0, -re(c)/2))) == \
b**c*(sqrt(1 + x/b**2) + 1)**c
# Section 8.4.5
assert IMT(24/s**5, s, x, (0, oo)) == log(x)**4*Heaviside(1 - x)
assert expand(IMT(6/s**4, s, x, (-oo, 0)), force=True) == \
log(x)**3*Heaviside(x - 1)
assert IMT(pi/(s*sin(pi*s)), s, x, (-1, 0)) == log(x + 1)
assert IMT(pi/(s*sin(pi*s/2)), s, x, (-2, 0)) == log(x**2 + 1)
assert IMT(pi/(s*sin(2*pi*s)), s, x, (-S(1)/2, 0)) == log(sqrt(x) + 1)
assert IMT(pi/(s*sin(pi*s)), s, x, (0, 1)) == log(1 + 1/x)
# TODO
def mysimp(expr):
from sympy import expand, logcombine, powsimp
return expand(
powsimp(logcombine(expr, force=True), force=True, deep=True),
force=True).replace(exp_polar, exp)
assert mysimp(mysimp(IMT(pi/(s*tan(pi*s)), s, x, (-1, 0)))) in [
log(1 - x)*Heaviside(1 - x) + log(x - 1)*Heaviside(x - 1),
log(x)*Heaviside(x - 1) + log(1 - 1/x)*Heaviside(x - 1) + log(-x +
1)*Heaviside(-x + 1)]
# test passing cot
assert mysimp(IMT(pi*cot(pi*s)/s, s, x, (0, 1))) in [
log(1/x - 1)*Heaviside(1 - x) + log(1 - 1/x)*Heaviside(x - 1),
-log(x)*Heaviside(-x + 1) + log(1 - 1/x)*Heaviside(x - 1) + log(-x +
1)*Heaviside(-x + 1), ]
# 8.4.14
assert IMT(-gamma(s + S(1)/2)/(sqrt(pi)*s), s, x, (-S(1)/2, 0)) == \
erf(sqrt(x))
# 8.4.19
assert simplify(IMT(gamma(a/2 + s)/gamma(a/2 - s + 1), s, x, (-re(a)/2, S(3)/4))) \
== besselj(a, 2*sqrt(x))
assert simplify(IMT(2**a*gamma(S(1)/2 - 2*s)*gamma(s + (a + 1)/2)
/ (gamma(1 - s - a/2)*gamma(1 - 2*s + a)),
s, x, (-(re(a) + 1)/2, S(1)/4))) == \
sin(sqrt(x))*besselj(a, sqrt(x))
assert simplify(IMT(2**a*gamma(a/2 + s)*gamma(S(1)/2 - 2*s)
/ (gamma(S(1)/2 - s - a/2)*gamma(1 - 2*s + a)),
s, x, (-re(a)/2, S(1)/4))) == \
cos(sqrt(x))*besselj(a, sqrt(x))
# TODO this comes out as an amazing mess, but simplifies nicely
assert simplify(IMT(gamma(a + s)*gamma(S(1)/2 - s)
/ (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
s, x, (-re(a), S(1)/2))) == \
besselj(a, sqrt(x))**2
assert simplify(IMT(gamma(s)*gamma(S(1)/2 - s)
/ (sqrt(pi)*gamma(1 - s - a)*gamma(1 + a - s)),
s, x, (0, S(1)/2))) == \
besselj(-a, sqrt(x))*besselj(a, sqrt(x))
assert simplify(IMT(4**s*gamma(-2*s + 1)*gamma(a/2 + b/2 + s)
/ (gamma(-a/2 + b/2 - s + 1)*gamma(a/2 - b/2 - s + 1)
*gamma(a/2 + b/2 - s + 1)),
s, x, (-(re(a) + re(b))/2, S(1)/2))) == \
besselj(a, sqrt(x))*besselj(b, sqrt(x))
# Section 8.4.20
# TODO this can be further simplified!
assert simplify(IMT(-2**(2*s)*cos(pi*a/2 - pi*b/2 + pi*s)*gamma(-2*s + 1) *
gamma(a/2 - b/2 + s)*gamma(a/2 + b/2 + s) /
(pi*gamma(a/2 - b/2 - s + 1)*gamma(a/2 + b/2 - s + 1)),
s, x,
(Max(-re(a)/2 - re(b)/2, -re(a)/2 + re(b)/2), S(1)/2))) == \
besselj(a, sqrt(x))*-(besselj(-b, sqrt(x)) -
besselj(b, sqrt(x))*cos(pi*b))/sin(pi*b)
# TODO more
# for coverage
assert IMT(pi/cos(pi*s), s, x, (0, S(1)/2)) == sqrt(x)/(x + 1)
@slow
def test_laplace_transform():
from sympy import fresnels, fresnelc
LT = laplace_transform
a, b, c, = symbols('a b c', positive=True)
t = symbols('t')
w = Symbol("w")
f = Function("f")
# Test unevaluated form
assert laplace_transform(f(t), t, w) == LaplaceTransform(f(t), t, w)
assert inverse_laplace_transform(
f(w), w, t, plane=0) == InverseLaplaceTransform(f(w), w, t, 0)
# test a bug
spos = symbols('s', positive=True)
assert LT(exp(t), t, spos)[:2] == (1/(spos - 1), 1)
# basic tests from wikipedia
assert LT((t - a)**b*exp(-c*(t - a))*Heaviside(t - a), t, s) == \
((s + c)**(-b - 1)*exp(-a*s)*gamma(b + 1), -c, True)
assert LT(t**a, t, s) == (s**(-a - 1)*gamma(a + 1), 0, True)
assert LT(Heaviside(t), t, s) == (1/s, 0, True)
assert LT(Heaviside(t - a), t, s) == (exp(-a*s)/s, 0, True)
assert LT(1 - exp(-a*t), t, s) == (a/(s*(a + s)), 0, True)
assert LT((exp(2*t) - 1)*exp(-b - t)*Heaviside(t)/2, t, s, noconds=True) \
== exp(-b)/(s**2 - 1)
assert LT(exp(t), t, s)[:2] == (1/(s - 1), 1)
assert LT(exp(2*t), t, s)[:2] == (1/(s - 2), 2)
assert LT(exp(a*t), t, s)[:2] == (1/(s - a), a)
assert LT(log(t/a), t, s) == ((log(a*s) + EulerGamma)/s/-1, 0, True)
assert LT(erf(t), t, s) == (erfc(s/2)*exp(s**2/4)/s, 0, True)
assert LT(sin(a*t), t, s) == (a/(a**2 + s**2), 0, True)
assert LT(cos(a*t), t, s) == (s/(a**2 + s**2), 0, True)
# TODO would be nice to have these come out better
assert LT(exp(-a*t)*sin(b*t), t, s) == (b/(b**2 + (a + s)**2), -a, True)
assert LT(exp(-a*t)*cos(b*t), t, s) == \
((a + s)/(b**2 + (a + s)**2), -a, True)
assert LT(besselj(0, t), t, s) == (1/sqrt(1 + s**2), 0, True)
assert LT(besselj(1, t), t, s) == (1 - 1/sqrt(1 + 1/s**2), 0, True)
# TODO general order works, but is a *mess*
# TODO besseli also works, but is an even greater mess
# test a bug in conditions processing
# TODO the auxiliary condition should be recognised/simplified
assert LT(exp(t)*cos(t), t, s)[:-1] in [
((s - 1)/(s**2 - 2*s + 2), -oo),
((s - 1)/((s - 1)**2 + 1), -oo),
]
# Fresnel functions
assert laplace_transform(fresnels(t), t, s) == \
((-sin(s**2/(2*pi))*fresnels(s/pi) + sin(s**2/(2*pi))/2 -
cos(s**2/(2*pi))*fresnelc(s/pi) + cos(s**2/(2*pi))/2)/s, 0, True)
assert laplace_transform(fresnelc(t), t, s) == (
((2*sin(s**2/(2*pi))*fresnelc(s/pi) - 2*cos(s**2/(2*pi))*fresnels(s/pi)
+ sqrt(2)*cos(s**2/(2*pi) + pi/4))/(2*s), 0, True))
cond = Ne(1/s, 1) & (
S(0) < cos(Abs(periodic_argument(s, oo)))*Abs(s) - 1)
assert LT(Matrix([[exp(t), t*exp(-t)], [t*exp(-t), exp(t)]]), t, s) ==\
Matrix([
[(1/(s - 1), 1, True), ((s + 1)**(-2), 0, True)],
[((s + 1)**(-2), 0, True), (1/(s - 1), 1, True)]
])
def test_issue_8368_7173():
LT = laplace_transform
# hyperbolic
assert LT(sinh(x), x, s) == (1/(s**2 - 1), 1, True)
assert LT(cosh(x), x, s) == (s/(s**2 - 1), 1, True)
assert LT(sinh(x + 3), x, s) == (
(-s + (s + 1)*exp(6) + 1)*exp(-3)/(s - 1)/(s + 1)/2, 1, True)
assert LT(sinh(x)*cosh(x), x, s) == (
1/(s**2 - 4), 2, Ne(s/2, 1))
# trig (make sure they are not being rewritten in terms of exp)
assert LT(cos(x + 3), x, s) == ((s*cos(3) - sin(3))/(s**2 + 1), 0, True)
def test_inverse_laplace_transform():
from sympy import sinh, cosh, besselj, besseli, simplify, factor_terms
ILT = inverse_laplace_transform
a, b, c, = symbols('a b c', positive=True, finite=True)
t = symbols('t')
def simp_hyp(expr):
return factor_terms(expand_mul(expr)).rewrite(sin)
# just test inverses of all of the above
assert ILT(1/s, s, t) == Heaviside(t)
assert ILT(1/s**2, s, t) == t*Heaviside(t)
assert ILT(1/s**5, s, t) == t**4*Heaviside(t)/24
assert ILT(exp(-a*s)/s, s, t) == Heaviside(t - a)
assert ILT(exp(-a*s)/(s + b), s, t) == exp(b*(a - t))*Heaviside(-a + t)
assert ILT(a/(s**2 + a**2), s, t) == sin(a*t)*Heaviside(t)
assert ILT(s/(s**2 + a**2), s, t) == cos(a*t)*Heaviside(t)
# TODO is there a way around simp_hyp?
assert simp_hyp(ILT(a/(s**2 - a**2), s, t)) == sinh(a*t)*Heaviside(t)
assert simp_hyp(ILT(s/(s**2 - a**2), s, t)) == cosh(a*t)*Heaviside(t)
assert ILT(a/((s + b)**2 + a**2), s, t) == exp(-b*t)*sin(a*t)*Heaviside(t)
assert ILT(
(s + b)/((s + b)**2 + a**2), s, t) == exp(-b*t)*cos(a*t)*Heaviside(t)
# TODO sinh/cosh shifted come out a mess. also delayed trig is a mess
# TODO should this simplify further?
assert ILT(exp(-a*s)/s**b, s, t) == \
(t - a)**(b - 1)*Heaviside(t - a)/gamma(b)
assert ILT(exp(-a*s)/sqrt(1 + s**2), s, t) == \
Heaviside(t - a)*besselj(0, a - t) # note: besselj(0, x) is even
# XXX ILT turns these branch factor into trig functions ...
assert simplify(ILT(a**b*(s + sqrt(s**2 - a**2))**(-b)/sqrt(s**2 - a**2),
s, t).rewrite(exp)) == \
Heaviside(t)*besseli(b, a*t)
assert ILT(a**b*(s + sqrt(s**2 + a**2))**(-b)/sqrt(s**2 + a**2),
s, t).rewrite(exp) == \
Heaviside(t)*besselj(b, a*t)
assert ILT(1/(s*sqrt(s + 1)), s, t) == Heaviside(t)*erf(sqrt(t))
# TODO can we make erf(t) work?
assert ILT(1/(s**2*(s**2 + 1)),s,t) == (t - sin(t))*Heaviside(t)
assert ILT( (s * eye(2) - Matrix([[1, 0], [0, 2]])).inv(), s, t) ==\
Matrix([[exp(t)*Heaviside(t), 0], [0, exp(2*t)*Heaviside(t)]])
def test_inverse_laplace_transform_delta():
from sympy import DiracDelta
ILT = inverse_laplace_transform
t = symbols('t')
assert ILT(2, s, t) == 2*DiracDelta(t)
assert ILT(2*exp(3*s) - 5*exp(-7*s), s, t) == \
2*DiracDelta(t + 3) - 5*DiracDelta(t - 7)
a = cos(sin(7)/2)
assert ILT(a*exp(-3*s), s, t) == a*DiracDelta(t - 3)
assert ILT(exp(2*s), s, t) == DiracDelta(t + 2)
r = Symbol('r', real=True)
assert ILT(exp(r*s), s, t) == DiracDelta(t + r)
def test_inverse_laplace_transform_delta_cond():
from sympy import DiracDelta, Eq, im, Heaviside
ILT = inverse_laplace_transform
t = symbols('t')
r = Symbol('r', real=True)
assert ILT(exp(r*s), s, t, noconds=False) == (DiracDelta(t + r), True)
z = Symbol('z')
assert ILT(exp(z*s), s, t, noconds=False) == \
(DiracDelta(t + z), Eq(im(z), 0))
# inversion does not exist: verify it doesn't evaluate to DiracDelta
for z in (Symbol('z', real=False),
Symbol('z', imaginary=True, zero=False)):
f = ILT(exp(z*s), s, t, noconds=False)
f = f[0] if isinstance(f, tuple) else f
assert f.func != DiracDelta
# issue 15043
assert ILT(1/s + exp(r*s)/s, s, t, noconds=False) == (
Heaviside(t) + Heaviside(r + t), True)
def test_fourier_transform():
from sympy import simplify, expand, expand_complex, factor, expand_trig
FT = fourier_transform
IFT = inverse_fourier_transform
def simp(x):
return simplify(expand_trig(expand_complex(expand(x))))
def sinc(x):
return sin(pi*x)/(pi*x)
k = symbols('k', real=True)
f = Function("f")
# TODO for this to work with real a, need to expand abs(a*x) to abs(a)*abs(x)
a = symbols('a', positive=True)
b = symbols('b', positive=True)
posk = symbols('posk', positive=True)
# Test unevaluated form
assert fourier_transform(f(x), x, k) == FourierTransform(f(x), x, k)
assert inverse_fourier_transform(
f(k), k, x) == InverseFourierTransform(f(k), k, x)
# basic examples from wikipedia
assert simp(FT(Heaviside(1 - abs(2*a*x)), x, k)) == sinc(k/a)/a
# TODO IFT is a *mess*
assert simp(FT(Heaviside(1 - abs(a*x))*(1 - abs(a*x)), x, k)) == sinc(k/a)**2/a
# TODO IFT
assert factor(FT(exp(-a*x)*Heaviside(x), x, k), extension=I) == \
1/(a + 2*pi*I*k)
# NOTE: the ift comes out in pieces
assert IFT(1/(a + 2*pi*I*x), x, posk,
noconds=False) == (exp(-a*posk), True)
assert IFT(1/(a + 2*pi*I*x), x, -posk,
noconds=False) == (0, True)
assert IFT(1/(a + 2*pi*I*x), x, symbols('k', negative=True),
noconds=False) == (0, True)
# TODO IFT without factoring comes out as meijer g
assert factor(FT(x*exp(-a*x)*Heaviside(x), x, k), extension=I) == \
1/(a + 2*pi*I*k)**2
assert FT(exp(-a*x)*sin(b*x)*Heaviside(x), x, k) == \
b/(b**2 + (a + 2*I*pi*k)**2)
assert FT(exp(-a*x**2), x, k) == sqrt(pi)*exp(-pi**2*k**2/a)/sqrt(a)
assert IFT(sqrt(pi/a)*exp(-(pi*k)**2/a), k, x) == exp(-a*x**2)
assert FT(exp(-a*abs(x)), x, k) == 2*a/(a**2 + 4*pi**2*k**2)
# TODO IFT (comes out as meijer G)
# TODO besselj(n, x), n an integer > 0 actually can be done...
# TODO are there other common transforms (no distributions!)?
def test_sine_transform():
from sympy import EulerGamma
t = symbols("t")
w = symbols("w")
a = symbols("a")
f = Function("f")
# Test unevaluated form
assert sine_transform(f(t), t, w) == SineTransform(f(t), t, w)
assert inverse_sine_transform(
f(w), w, t) == InverseSineTransform(f(w), w, t)
assert sine_transform(1/sqrt(t), t, w) == 1/sqrt(w)
assert inverse_sine_transform(1/sqrt(w), w, t) == 1/sqrt(t)
assert sine_transform((1/sqrt(t))**3, t, w) == 2*sqrt(w)
assert sine_transform(t**(-a), t, w) == 2**(
-a + S(1)/2)*w**(a - 1)*gamma(-a/2 + 1)/gamma((a + 1)/2)
assert inverse_sine_transform(2**(-a + S(
1)/2)*w**(a - 1)*gamma(-a/2 + 1)/gamma(a/2 + S(1)/2), w, t) == t**(-a)
assert sine_transform(
exp(-a*t), t, w) == sqrt(2)*w/(sqrt(pi)*(a**2 + w**2))
assert inverse_sine_transform(
sqrt(2)*w/(sqrt(pi)*(a**2 + w**2)), w, t) == exp(-a*t)
assert sine_transform(
log(t)/t, t, w) == -sqrt(2)*sqrt(pi)*(log(w**2) + 2*EulerGamma)/4
assert sine_transform(
t*exp(-a*t**2), t, w) == sqrt(2)*w*exp(-w**2/(4*a))/(4*a**(S(3)/2))
assert inverse_sine_transform(
sqrt(2)*w*exp(-w**2/(4*a))/(4*a**(S(3)/2)), w, t) == t*exp(-a*t**2)
def test_cosine_transform():
from sympy import Si, Ci
t = symbols("t")
w = symbols("w")
a = symbols("a")
f = Function("f")
# Test unevaluated form
assert cosine_transform(f(t), t, w) == CosineTransform(f(t), t, w)
assert inverse_cosine_transform(
f(w), w, t) == InverseCosineTransform(f(w), w, t)
assert cosine_transform(1/sqrt(t), t, w) == 1/sqrt(w)
assert inverse_cosine_transform(1/sqrt(w), w, t) == 1/sqrt(t)
assert cosine_transform(1/(
a**2 + t**2), t, w) == sqrt(2)*sqrt(pi)*exp(-a*w)/(2*a)
assert cosine_transform(t**(
-a), t, w) == 2**(-a + S(1)/2)*w**(a - 1)*gamma((-a + 1)/2)/gamma(a/2)
assert inverse_cosine_transform(2**(-a + S(
1)/2)*w**(a - 1)*gamma(-a/2 + S(1)/2)/gamma(a/2), w, t) == t**(-a)
assert cosine_transform(
exp(-a*t), t, w) == sqrt(2)*a/(sqrt(pi)*(a**2 + w**2))
assert inverse_cosine_transform(
sqrt(2)*a/(sqrt(pi)*(a**2 + w**2)), w, t) == exp(-a*t)
assert cosine_transform(exp(-a*sqrt(t))*cos(a*sqrt(
t)), t, w) == a*exp(-a**2/(2*w))/(2*w**(S(3)/2))
assert cosine_transform(1/(a + t), t, w) == sqrt(2)*(
(-2*Si(a*w) + pi)*sin(a*w)/2 - cos(a*w)*Ci(a*w))/sqrt(pi)
assert inverse_cosine_transform(sqrt(2)*meijerg(((S(1)/2, 0), ()), (
(S(1)/2, 0, 0), (S(1)/2,)), a**2*w**2/4)/(2*pi), w, t) == 1/(a + t)
assert cosine_transform(1/sqrt(a**2 + t**2), t, w) == sqrt(2)*meijerg(
((S(1)/2,), ()), ((0, 0), (S(1)/2,)), a**2*w**2/4)/(2*sqrt(pi))
assert inverse_cosine_transform(sqrt(2)*meijerg(((S(1)/2,), ()), ((0, 0), (S(1)/2,)), a**2*w**2/4)/(2*sqrt(pi)), w, t) == 1/(t*sqrt(a**2/t**2 + 1))
def test_hankel_transform():
from sympy import gamma, sqrt, exp
r = Symbol("r")
k = Symbol("k")
nu = Symbol("nu")
m = Symbol("m")
a = symbols("a")
assert hankel_transform(1/r, r, k, 0) == 1/k
assert inverse_hankel_transform(1/k, k, r, 0) == 1/r
assert hankel_transform(
1/r**m, r, k, 0) == 2**(-m + 1)*k**(m - 2)*gamma(-m/2 + 1)/gamma(m/2)
assert inverse_hankel_transform(
2**(-m + 1)*k**(m - 2)*gamma(-m/2 + 1)/gamma(m/2), k, r, 0) == r**(-m)
assert hankel_transform(1/r**m, r, k, nu) == (
2*2**(-m)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2))
assert inverse_hankel_transform(2**(-m + 1)*k**(
m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2), k, r, nu) == r**(-m)
assert hankel_transform(r**nu*exp(-a*r), r, k, nu) == \
2**(nu + 1)*a*k**(-nu - 3)*(a**2/k**2 + 1)**(-nu - S(
3)/2)*gamma(nu + S(3)/2)/sqrt(pi)
assert inverse_hankel_transform(
2**(nu + 1)*a*k**(-nu - 3)*(a**2/k**2 + 1)**(-nu - S(3)/2)*gamma(
nu + S(3)/2)/sqrt(pi), k, r, nu) == r**nu*exp(-a*r)
def test_issue_7181():
assert mellin_transform(1/(1 - x), x, s) != None
def test_issue_8882():
# This is the original test.
# from sympy import diff, Integral, integrate
# r = Symbol('r')
# psi = 1/r*sin(r)*exp(-(a0*r))
# h = -1/2*diff(psi, r, r) - 1/r*psi
# f = 4*pi*psi*h*r**2
# assert integrate(f, (r, -oo, 3), meijerg=True).has(Integral) == True
# To save time, only the critical part is included.
F = -a**(-s + 1)*(4 + 1/a**2)**(-s/2)*sqrt(1/a**2)*exp(-s*I*pi)* \
sin(s*atan(sqrt(1/a**2)/2))*gamma(s)
raises(IntegralTransformError, lambda:
inverse_mellin_transform(F, s, x, (-1, oo),
**{'as_meijerg': True, 'needeval': True}))
def test_issue_7173():
from sympy import cse
x0, x1, x2, x3 = symbols('x:4')
ans = laplace_transform(sinh(a*x)*cosh(a*x), x, s)
r, e = cse(ans)
assert r == [
(x0, arg(a)),
(x1, Abs(x0)),
(x2, pi/2),
(x3, Abs(x0 + pi))]
assert e == [
a/(-4*a**2 + s**2),
0,
((x1 <= x2) | (x1 < x2)) & ((x3 <= x2) | (x3 < x2))]
def test_issue_8514():
from sympy import simplify
a, b, c, = symbols('a b c', positive=True)
t = symbols('t', positive=True)
ft = simplify(inverse_laplace_transform(1/(a*s**2+b*s+c),s, t))
assert ft == (I*exp(t*cos(atan2(0, -4*a*c + b**2)/2)*sqrt(Abs(4*a*c -
b**2))/a)*sin(t*sin(atan2(0, -4*a*c + b**2)/2)*sqrt(Abs(
4*a*c - b**2))/(2*a)) + exp(t*cos(atan2(0, -4*a*c + b**2)
/2)*sqrt(Abs(4*a*c - b**2))/a)*cos(t*sin(atan2(0, -4*a*c
+ b**2)/2)*sqrt(Abs(4*a*c - b**2))/(2*a)) + I*sin(t*sin(
atan2(0, -4*a*c + b**2)/2)*sqrt(Abs(4*a*c - b**2))/(2*a))
- cos(t*sin(atan2(0, -4*a*c + b**2)/2)*sqrt(Abs(4*a*c -
b**2))/(2*a)))*exp(-t*(b + cos(atan2(0, -4*a*c + b**2)/2)
*sqrt(Abs(4*a*c - b**2)))/(2*a))/sqrt(-4*a*c + b**2)
def test_issue_12591():
x, y = symbols("x y", real=True)
assert fourier_transform(exp(x), x, y) == FourierTransform(exp(x), x, y)
def test_issue_14692():
b = Symbol('b', negative=True)
assert laplace_transform(1/(I*x - b), x, s) == \
(-I*exp(I*b*s)*expint(1, b*s*exp_polar(I*pi/2)), 0, True)
|
e28ffa57bea13d47e04a8ab3e587abeb5b9dd3ba39f59c8b49c9bb1e880b9084
|
'''
Parser for FullForm[Downvalues[]] of Mathematica rules.
This parser is customised to parse the output in MatchPy rules format. Multiple
`Constraints` are divided into individual `Constraints` because it helps the
MatchPy's `ManyToOneReplacer` to backtrack earlier and improve the speed.
Parsed output is formatted into readable format by using `sympify` and print the
expression using `sstr`. This replaces `And`, `Mul`, 'Pow' by their respective
symbols.
Mathematica
===========
To get the full form from Wolfram Mathematica, type:
```
ShowSteps = False
Import["RubiLoader.m"]
Export["output.txt", ToString@FullForm@DownValues@Int]
```
The file ``output.txt`` will then contain the rules in parseable format.
References
==========
[1] http://reference.wolfram.com/language/ref/FullForm.html
[2] http://reference.wolfram.com/language/ref/DownValues.html
[3] https://gist.github.com/Upabjojr/bc07c49262944f9c1eb0
'''
import re
import os
import inspect
from sympy import sympify, Function, Set, Symbol
from sympy.core.compatibility import string_types
from sympy.printing import sstr, StrPrinter
from sympy.utilities.misc import debug
class RubiStrPrinter(StrPrinter):
def _print_Not(self, expr):
return "Not(%s)" % self._print(expr.args[0])
def rubi_printer(expr, **settings):
return RubiStrPrinter(settings).doprint(expr)
replacements = dict( # Mathematica equivalent functions in SymPy
Times="Mul",
Plus="Add",
Power="Pow",
Log='log',
Exp='exp',
Sqrt='sqrt',
Cos='cos',
Sin='sin',
Tan='tan',
Cot='1/tan',
cot='1/tan',
Sec='1/cos',
sec='1/cos',
Csc='1/sin',
csc='1/sin',
ArcSin='asin',
ArcCos='acos',
# ArcTan='atan',
ArcCot='acot',
ArcSec='asec',
ArcCsc='acsc',
Sinh='sinh',
Cosh='cosh',
Tanh='tanh',
Coth='1/tanh',
coth='1/tanh',
Sech='1/cosh',
sech='1/cosh',
Csch='1/sinh',
csch='1/sinh',
ArcSinh='asinh',
ArcCosh='acosh',
ArcTanh='atanh',
ArcCoth='acoth',
ArcSech='asech',
ArcCsch='acsch',
Expand='expand',
Im='im',
Re='re',
Flatten='flatten',
Polylog='polylog',
Cancel='cancel',
#Gamma='gamma',
TrigExpand='expand_trig',
Sign='sign',
Simplify='simplify',
Defer='UnevaluatedExpr',
Identity = 'S',
Sum = 'Sum_doit',
Module = 'With',
Block = 'With',
Null = 'None'
)
temporary_variable_replacement = { # Temporarily rename because it can raise errors while sympifying
'gcd' : "_gcd",
'jn' : "_jn",
}
permanent_variable_replacement = { # Permamenely rename these variables
r"\[ImaginaryI]" : 'ImaginaryI',
"$UseGamma": '_UseGamma',
}
#these functions have different return type in different cases. So better to use a try and except in the constraints, when any of these appear
f_diff_return_type = ['BinomialParts', 'BinomialDegree', 'TrinomialParts', 'GeneralizedBinomialParts', 'GeneralizedTrinomialParts', 'PseudoBinomialParts', 'PerfectPowerTest',
'SquareFreeFactorTest', 'SubstForFractionalPowerOfQuotientOfLinears', 'FractionalPowerOfQuotientOfLinears', 'InverseFunctionOfQuotientOfLinears',
'FractionalPowerOfSquareQ', 'FunctionOfLinear', 'FunctionOfInverseLinear', 'FunctionOfTrig', 'FindTrigFactor', 'FunctionOfLog',
'PowerVariableExpn', 'FunctionOfSquareRootOfQuadratic', 'SubstForFractionalPowerOfLinear', 'FractionalPowerOfLinear', 'InverseFunctionOfLinear',
'Divides', 'DerivativeDivides', 'TrigSquare', 'SplitProduct', 'SubstForFractionalPowerOfQuotientOfLinears', 'InverseFunctionOfQuotientOfLinears',
'FunctionOfHyperbolic', 'SplitSum']
def contains_diff_return_type(a):
'''
This function returns whether an expression contains functions which have different return types in
diiferent cases.
'''
if isinstance(a, list):
for i in a:
if contains_diff_return_type(i):
return True
elif type(a) == Function('With') or type(a) == Function('Module'):
for i in f_diff_return_type:
if a.has(Function(i)):
return True
else:
if a in f_diff_return_type:
return True
return False
def parse_full_form(wmexpr):
'''
Parses FullForm[Downvalues[]] generated by Mathematica
'''
out = []
stack = [out]
generator = re.finditer(r'[\[\],]', wmexpr)
last_pos = 0
for match in generator:
if match is None:
break
position = match.start()
last_expr = wmexpr[last_pos:position].replace(',', '').replace(']', '').replace('[', '').strip()
if match.group() == ',':
if last_expr != '':
stack[-1].append(last_expr)
elif match.group() == ']':
if last_expr != '':
stack[-1].append(last_expr)
stack.pop()
current_pos = stack[-1]
elif match.group() == '[':
stack[-1].append([last_expr])
stack.append(stack[-1][-1])
last_pos = match.end()
return out[0]
def get_default_values(parsed, default_values={}):
'''
Returns Optional variables and their values in the pattern
'''
if not isinstance(parsed, list):
return default_values
if parsed[0] == "Times": # find Default arguments for "Times"
for i in parsed[1:]:
if i[0] == "Optional":
default_values[(i[1][1])] = 1
if parsed[0] == "Plus": # find Default arguments for "Plus"
for i in parsed[1:]:
if i[0] == "Optional":
default_values[(i[1][1])] = 0
if parsed[0] == "Power": # find Default arguments for "Power"
for i in parsed[1:]:
if i[0] == "Optional":
default_values[(i[1][1])] = 1
if len(parsed) == 1:
return default_values
for i in parsed:
default_values = get_default_values(i, default_values)
return default_values
def add_wildcards(string, optional={}):
'''
Replaces `Pattern(variable)` by `variable` in `string`.
Returns the free symbols present in the string.
'''
symbols = [] # stores symbols present in the expression
p = r'(Optional\(Pattern\((\w+), Blank\)\))'
matches = re.findall(p, string)
for i in matches:
string = string.replace(i[0], "WC('{}', S({}))".format(i[1], optional[i[1]]))
symbols.append(i[1])
p = r'(Pattern\((\w+), Blank\))'
matches = re.findall(p, string)
for i in matches:
string = string.replace(i[0], i[1] + '_')
symbols.append(i[1])
p = r'(Pattern\((\w+), Blank\(Symbol\)\))'
matches = re.findall(p, string)
for i in matches:
string = string.replace(i[0], i[1] + '_')
symbols.append(i[1])
return string, symbols
def seperate_freeq(s, variables=[], x=None):
'''
Returns list of symbols in FreeQ.
'''
if s[0] == 'FreeQ':
if len(s[1]) == 1:
variables = [s[1]]
else:
variables = s[1][1:]
x = s[2]
else:
for i in s[1:]:
variables, x = seperate_freeq(i, variables, x)
return variables, x
return variables, x
def parse_freeq(l, x, cons_index, cons_dict, cons_import, symbols=None):
'''
Converts FreeQ constraints into MatchPy constraint
'''
res = []
cons = ''
for i in l:
if isinstance(i, string_types):
r = ' return FreeQ({}, {})'.format(i, x)
# First it checks if a constraint is already present in `cons_dict`, If yes, use it else create a new one.
if r not in cons_dict.values():
cons_index += 1
c = '\n def cons_f{}({}, {}):\n'.format(cons_index, i, x)
c += r
c += '\n\n cons{} = CustomConstraint({})\n'.format(cons_index, 'cons_f{}'.format(cons_index))
cons_name = 'cons{}'.format(cons_index)
cons_dict[cons_name] = r
else:
c = ''
cons_name = next(key for key, value in cons_dict.items() if value == r)
elif isinstance(i, list):
s = list(set(get_free_symbols(i, symbols)))
s = ', '.join(s)
r = ' return FreeQ({}, {})'.format(generate_sympy_from_parsed(i), x)
if r not in cons_dict.values():
cons_index += 1
c = '\n def cons_f{}({}):\n'.format(cons_index, s)
c += r
c += '\n\n cons{} = CustomConstraint({})\n'.format(cons_index, 'cons_f{}'.format(cons_index))
cons_name = 'cons{}'.format(cons_index)
cons_dict[cons_name] = r
else:
c = ''
cons_name = next(key for key, value in cons_dict.items() if value == r)
if cons_name not in cons_import:
cons_import.append(cons_name)
res.append(cons_name)
cons += c
if res != []:
return ', ' + ', '.join(res), cons, cons_index
return '', cons, cons_index
def generate_sympy_from_parsed(parsed, wild=False, symbols=[], replace_Int=False):
'''
Parses list into Python syntax.
Parameters
==========
wild : When set to True, the symbols are replaced as wild symbols.
symbols : Symbols already present in the pattern.
replace_Int: when set to True, `Int` is replaced by `Integral`(used to parse pattern).
'''
out = ""
if not isinstance(parsed, list):
try: #return S(number) if parsed is Number
float(parsed)
return "S({})".format(parsed)
except:
pass
if parsed in symbols:
if wild:
return parsed + '_'
return parsed
if parsed[0] == 'Rational':
return 'S({})/S({})'.format(generate_sympy_from_parsed(parsed[1], wild=wild, symbols=symbols, replace_Int=replace_Int), generate_sympy_from_parsed(parsed[2], wild=wild, symbols=symbols, replace_Int=replace_Int))
if parsed[0] in replacements:
out += replacements[parsed[0]]
elif parsed[0] == 'Int' and replace_Int:
out += 'Integral'
else:
out += parsed[0]
if len(parsed) == 1:
return out
result = [generate_sympy_from_parsed(i, wild=wild, symbols=symbols, replace_Int=replace_Int) for i in parsed[1:]]
if '' in result:
result.remove('')
out += "("
out += ", ".join(result)
out += ")"
return out
def get_free_symbols(s, symbols, free_symbols=[]):
'''
Returns free_symbols present in `s`.
'''
if not isinstance(s, list):
if s in symbols:
free_symbols.append(s)
return free_symbols
for i in s:
free_symbols = get_free_symbols(i, symbols, free_symbols)
return free_symbols
def set_matchq_in_constraint(a, cons_index):
'''
Takes care of the case, when a pattern matching has to be done inside a constraint.
'''
lst = []
res = ''
if isinstance(a, list):
if a[0] == 'MatchQ':
s = a
optional = get_default_values(s, {})
r = generate_sympy_from_parsed(s, replace_Int=True)
r, free_symbols = add_wildcards(r, optional=optional)
free_symbols = list(set(free_symbols)) #remove common symbols
r = sympify(r, locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not")})
pattern = r.args[1].args[0]
cons = r.args[1].args[1]
pattern = rubi_printer(pattern, sympy_integers=True)
pattern = setWC(pattern)
res = ' def _cons_f_{}({}):\n return {}\n'.format(cons_index, ', '.join(free_symbols), cons)
res += ' _cons_{} = CustomConstraint(_cons_f_{})\n'.format(cons_index, cons_index)
res += ' pat = Pattern(UtilityOperator({}, x), _cons_{})\n'.format(pattern, cons_index)
res += ' result_matchq = is_match(UtilityOperator({}, x), pat)'.format(r.args[0])
return "result_matchq", res
else:
for i in a:
if isinstance(i, list):
r = set_matchq_in_constraint(i, cons_index)
lst.append(r[0])
res = r[1]
else:
lst.append(i)
return (lst, res)
def _divide_constriant(s, symbols, cons_index, cons_dict, cons_import):
# Creates a CustomConstraint of the form `CustomConstraint(lambda a, x: FreeQ(a, x))`
lambda_symbols = list(set(get_free_symbols(s, symbols, [])))
r = generate_sympy_from_parsed(s)
r = sympify(r, locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not")})
if r.has(Function('MatchQ')):
match_res = set_matchq_in_constraint(s, cons_index)
res = match_res[1]
res += '\n return {}'.format(rubi_printer(sympify(generate_sympy_from_parsed(match_res[0]), locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not")}), sympy_integers = True))
elif contains_diff_return_type(s):
res = ' try:\n return {}\n except (TypeError, AttributeError):\n return False'.format(rubi_printer(r, sympy_integers=True))
else:
res = ' return {}'.format(rubi_printer(r, sympy_integers=True))
# First it checks if a constraint is already present in `cons_dict`, If yes, use it else create a new one.
if not res in cons_dict.values():
cons_index += 1
cons = '\n def cons_f{}({}):\n'.format(cons_index, ', '.join(lambda_symbols))
if 'x' in lambda_symbols:
cons += ' if isinstance(x, (int, Integer, float, Float)):\n return False\n'
cons += res
cons += '\n\n cons{} = CustomConstraint({})\n'.format(cons_index, 'cons_f{}'.format(cons_index))
cons_name = 'cons{}'.format(cons_index)
cons_dict[cons_name] = res
else:
cons = ''
cons_name = next(key for key, value in cons_dict.items() if value == res)
if cons_name not in cons_import:
cons_import.append(cons_name)
return (cons_name, cons, cons_index)
def divide_constraint(s, symbols, cons_index, cons_dict, cons_import):
'''
Divides multiple constraints into smaller constraints.
Parameters
==========
s : constraint as list
symbols : all the symbols present in the expression
'''
result =[]
cons = ''
if s[0] == 'And':
for i in s[1:]:
if i[0]!= 'FreeQ':
a = _divide_constriant(i, symbols, cons_index, cons_dict, cons_import)
result.append(a[0])
cons += a[1]
cons_index = a[2]
else:
a = _divide_constriant(s, symbols, cons_index, cons_dict, cons_import)
result.append(a[0])
cons += a[1]
cons_index = a[2]
r = ['']
for i in result:
if i != '':
r.append(i)
return ', '.join(r),cons, cons_index
def setWC(string):
'''
Replaces `WC(a, b)` by `WC('a', S(b))`
'''
p = r'(WC\((\w+), S\(([-+]?\d)\)\))'
matches = re.findall(p, string)
for i in matches:
string = string.replace(i[0], "WC('{}', S({}))".format(i[1], i[2]))
return string
def process_return_type(a1, L):
'''
Functions like `Set`, `With` and `CompoundExpression` has to be taken special care.
'''
a = sympify(a1[1])
x =''
processed = False
return_value = ''
if type(a) == Function('With') or type(a) == Function('Module'):
for i in a.args:
for s in i.args:
if isinstance(s, Set) and not s in L:
x += '\n {} = {}'.format(s.args[0], rubi_printer(s.args[1], sympy_integers=True))
if not type(i) in (Function('List'), Function('CompoundExpression')) and not i.has(Function('CompoundExpression')):
return_value = i
processed = True
elif type(i) ==Function('CompoundExpression'):
return_value = i.args[-1]
processed = True
elif type(i.args[0]) == Function('CompoundExpression'):
C = i.args[0]
return_value = '{}({}, {})'.format(i.func, C.args[-1], i.args[1])
processed = True
return x, return_value, processed
def extract_set(s, L):
'''
this function extracts all `Set` functions
'''
lst = []
if isinstance(s, Set) and not s in L:
lst.append(s)
else:
try:
for i in s.args:
lst += extract_set(i, L)
except: #when s has no attribute args (like `bool`)
pass
return lst
def replaceWith(s, symbols, index):
'''
Replaces `With` and `Module by python functions`
'''
return_type = None
with_value = ''
if type(s) == Function('With') or type(s) == Function('Module'):
constraints = ' '
result = ' def With{}({}):'.format(index, ', '.join(symbols))
if type(s.args[0]) == Function('List'): # get all local variables of With and Module
L = list(s.args[0].args)
else:
L = [s.args[0]]
lst = []
for i in s.args[1:]:
lst+=extract_set(i, L)
L+=lst
for i in L: # define local variables
if isinstance(i, Set):
with_value += '\n {} = {}'.format(i.args[0], rubi_printer(i.args[1], sympy_integers=True))
elif isinstance(i, Symbol):
with_value += "\n {} = Symbol('{}')".format(i, i)
#result += with_value
if type(s.args[1]) == Function('CompoundExpression'): # Expand CompoundExpression
C = s.args[1]
result += with_value
if isinstance(C.args[0], Set):
result += '\n {} = {}'.format(C.args[0].args[0], C.args[0].args[1])
result += '\n rubi.append({})\n return {}'.format(index, rubi_printer(C.args[1], sympy_integers=True))
return result, constraints, return_type
elif type(s.args[1]) == Function('Condition'):
C = s.args[1]
if len(C.args) == 2:
if all(j in symbols for j in [str(i) for i in C.free_symbols]):
result += with_value
#constraints += 'CustomConstraint(lambda {}: {})'.format(', '.join([str(i) for i in C.free_symbols]), sstr(C.args[1], sympy_integers=True))
result += '\n rubi.append({})\n return {}'.format(index, rubi_printer(C.args[0], sympy_integers=True))
else:
if 'x' in symbols:
result += '\n if isinstance(x, (int, Integer, float, Float)):\n return False'
if contains_diff_return_type(s):
n_with_value = with_value.replace('\n', '\n ')
result += '\n try:{}\n res = {}'.format(n_with_value, rubi_printer(C.args[1], sympy_integers=True))
result += '\n except (TypeError, AttributeError):\n return False'
result += '\n if res:'
else:
result+=with_value
result += '\n if {}:'.format(rubi_printer(C.args[1], sympy_integers=True))
return_type = (with_value, rubi_printer(C.args[0], sympy_integers=True))
return_type1 = process_return_type(return_type, L)
if return_type1[2]:
return_type = ( with_value+return_type1[0], rubi_printer(return_type1[1]))
result += '\n return True'
result += '\n return False'
constraints = ', CustomConstraint(With{})'.format(index)
return result, constraints, return_type
elif type(s.args[1]) == Function('Module') or type(s.args[1]) == Function('With'):
C = s.args[1]
result += with_value
return_type = (with_value, rubi_printer(C, sympy_integers=True))
return_type1 = process_return_type(return_type, L)
if return_type1[2]:
return_type = ( with_value+return_type1[0], rubi_printer(return_type1[1]))
result+=return_type1[0]
result+='\n rubi.append({})\n return {}'.format(index, rubi_printer(return_type1[1]))
return result, constraints, None
elif s.args[1].has(Function("CompoundExpression")):
C = s.args[1].args[0]
result += with_value
if isinstance(C.args[0], Set):
result += '\n {} = {}'.format(C.args[0].args[0], C.args[0].args[1])
result += '\n return {}({}, {})'.format(s.args[1].func, C.args[-1], s.args[1].args[1])
return result, constraints, None
result += with_value
result += '\n rubi.append({})\n return {}'.format(index, rubi_printer(s.args[1], sympy_integers=True))
return result, constraints, return_type
else:
return rubi_printer(s, sympy_integers=True), '', return_type
def downvalues_rules(r, header, cons_dict, cons_index, index):
'''
Function which generates parsed rules by substituting all possible
combinations of default values.
'''
rules = '['
parsed = '\n\n'
cons = ''
cons_import = [] # it contains name of constraints that need to be imported for rules.
for i in r:
debug('parsing rule {}'.format(r.index(i) + 1))
# Parse Pattern
if i[1][1][0] == 'Condition':
p = i[1][1][1].copy()
else:
p = i[1][1].copy()
optional = get_default_values(p, {})
pattern = generate_sympy_from_parsed(p.copy(), replace_Int=True)
pattern, free_symbols = add_wildcards(pattern, optional=optional)
free_symbols = list(set(free_symbols)) #remove common symbols
# Parse Transformed Expression and Constraints
if i[2][0] == 'Condition': # parse rules without constraints separately
constriant, constraint_def, cons_index = divide_constraint(i[2][2], free_symbols, cons_index, cons_dict, cons_import) # separate And constraints into individual constraints
FreeQ_vars, FreeQ_x = seperate_freeq(i[2][2].copy()) # separate FreeQ into individual constraints
transformed = generate_sympy_from_parsed(i[2][1].copy(), symbols=free_symbols)
else:
constriant = ''
constraint_def = ''
FreeQ_vars, FreeQ_x = [], []
transformed = generate_sympy_from_parsed(i[2].copy(), symbols=free_symbols)
FreeQ_constraint, free_cons_def, cons_index = parse_freeq(FreeQ_vars, FreeQ_x, cons_index, cons_dict, cons_import, free_symbols)
pattern = sympify(pattern, locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not") })
pattern = rubi_printer(pattern, sympy_integers=True)
pattern = setWC(pattern)
transformed = sympify(transformed, locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not") })
constraint_def = constraint_def + free_cons_def
cons+=constraint_def
index += 1
# below are certain if - else condition depending on various situation that may be encountered
if type(transformed) == Function('With') or type(transformed) == Function('Module'): # define separate function when With appears
transformed, With_constraints, return_type = replaceWith(transformed, free_symbols, index)
if return_type is None:
parsed += '{}'.format(transformed)
parsed += '\n pattern' + str(index) +' = Pattern(' + pattern + '' + FreeQ_constraint + '' + constriant + ')'
parsed += '\n ' + 'rule' + str(index) +' = ReplacementRule(' + 'pattern' + rubi_printer(index, sympy_integers=True) + ', With{}'.format(index) + ')\n'
else:
parsed += '{}'.format(transformed)
parsed += '\n pattern' + str(index) +' = Pattern(' + pattern + '' + FreeQ_constraint + '' + constriant + With_constraints + ')'
parsed += '\n def replacement{}({}):\n'.format(index, ', '.join(free_symbols)) + return_type[0] + '\n rubi.append({})\n return '.format(index) + return_type[1]
parsed += '\n ' + 'rule' + str(index) +' = ReplacementRule(' + 'pattern' + rubi_printer(index, sympy_integers=True) + ', replacement{}'.format(index) + ')\n'
else:
transformed = rubi_printer(transformed, sympy_integers=True)
parsed += ' pattern' + str(index) +' = Pattern(' + pattern + '' + FreeQ_constraint + '' + constriant + ')'
parsed += '\n def replacement{}({}):\n rubi.append({})\n return '.format(index, ', '.join(free_symbols), index) + transformed
parsed += '\n ' + 'rule' + str(index) +' = ReplacementRule(' + 'pattern' + rubi_printer(index, sympy_integers=True) + ', replacement{}'.format(index) + ')\n'
rules += 'rule{}, '.format(index)
rules += ']'
parsed += ' return ' + rules +'\n'
header += ' from sympy.integrals.rubi.constraints import ' + ', '.join(word for word in cons_import)
parsed = header + parsed
return parsed, cons_index, cons, index
def rubi_rule_parser(fullform, header=None, module_name='rubi_object'):
'''
Parses rules in MatchPy format.
Parameters
==========
fullform : FullForm of the rule as string.
header : Header imports for the file. Uses default imports if None.
module_name : name of RUBI module
References
==========
[1] http://reference.wolfram.com/language/ref/FullForm.html
[2] http://reference.wolfram.com/language/ref/DownValues.html
[3] https://gist.github.com/Upabjojr/bc07c49262944f9c1eb0
'''
if header is None: # use default header values
path_header = os.path.dirname(os.path.abspath(inspect.getfile(inspect.currentframe())))
header = open(os.path.join(path_header, "header.py.txt"), "r").read()
header = header.format(module_name)
cons_dict = {} # dict keeps track of constraints that has been encountered, thus avoids repetition of constraints.
cons_index =0 # for index of a constraint
index = 0 # indicates the number of a rule.
cons = ''
# Temporarily rename these variables because it
# can raise errors while sympifying
for i in temporary_variable_replacement:
fullform = fullform.replace(i, temporary_variable_replacement[i])
# Permanently rename these variables
for i in permanent_variable_replacement:
fullform = fullform.replace(i, permanent_variable_replacement[i])
rules = []
for i in parse_full_form(fullform): # separate all rules
if i[0] == 'RuleDelayed':
rules.append(i)
parsed = downvalues_rules(rules, header, cons_dict, cons_index, index)
result = parsed[0].strip() + '\n'
cons_index = parsed[1]
cons += parsed[2]
index = parsed[3]
# Replace temporary variables by actual values
for i in temporary_variable_replacement:
cons = cons.replace(temporary_variable_replacement[i], i)
result = result.replace(temporary_variable_replacement[i], i)
cons = "\n".join(header.split("\n")[:-2])+ '\n' + cons
return result, cons
|
519a2c5942c7f114a178ad667d3e832bbc27cd098d9c1515d81fbdf1f08495b1
|
import sys
from sympy.external import import_module
matchpy = import_module("matchpy")
if not matchpy:
#bin/test will not execute any tests now
disabled = True
if sys.version_info[:2] < (3, 6):
disabled = True
from sympy.integrals.rubi.utility_function import (Int, Set, With, Module, Scan, MapAnd, FalseQ, ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ, PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ, ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ, NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart, FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest, SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient, Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart, IntegerPart, AppellF1, EllipticPi, PolynomialQuotient,
EllipticE, EllipticF, ArcTan, ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec, ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less, Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ, PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ, ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ, Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ, SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator, NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,
InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ, EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree, PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts, TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ, NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll, ExpandLinearProduct, GCD, ContentFactor, NumericFactor, NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst, ExpandExpression, Apart, SmartApart, MatchQ, PolynomialQuotientRemainder, FreeFactors, NonfreeFactors, RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms, ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup, AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor, RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon, MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ, GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList, PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ, RationalFunctionFactors, NonrationalFunctionFactors, Reverse, RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand, SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree, CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree, GeneralizedBinomialParts, GeneralizedTrinomialDegree, GeneralizedTrinomialParts, MonomialQ, MonomialSumQ, MinimumMonomialExponent, MonomialExponent, LinearMatchQ, PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ, TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ, QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms, NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial, PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD, AlgebraicFunctionFactors, NonalgebraicFunctionFactors, QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ, Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors, NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop, CombineExponents, FactorInteger, FactorAbsurdNumber, SubstForInverseFunction, SubstForFractionalPower, SubstForFractionalPowerOfQuotientOfLinears, FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ, SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ, FractionalPowerSubexpressionQ, Apply, FactorNumericGcd, MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ, TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest, OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors, PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn, PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree, FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify, FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand, NormalizeIntegrandAux, NormalizeIntegrandFactor, NormalizeIntegrandFactorBase, NormalizeTogether, NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors, SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm, TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum, UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear, PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ, IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor, FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ, FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator, SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand, SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM, SubstForFractionalPowerOfLinear, FractionalPowerOfLinear, InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig, FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ, PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ, KnownTangentIntegrandQ, KnownCotangentIntegrandQ, KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst, AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand, ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp, ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ, FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ, PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ, FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ, FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ, FunctionOfLog, PowerVariableExpn, PowerVariableDegree, PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic, SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ, Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ, SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2, ConstantFactor, SameQ, ReplacePart, CommonFactors, MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential, FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux, FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev, rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent, RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct, SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma, FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ, _SimpFixFactor,
DerivativeDivides, SimpFixFactor, _FixSimplify, FixSimplify, _SimplifyAntiderivativeSum, SimplifyAntiderivativeSum, PureFunctionOfCothQ, _SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux, TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, IntegralFreeQ, Sum_doit, log,
PolynomialRemainder, CoprimeQ, Distribute, ProductLog, Floor, PolyGamma, process_trig, replace_pow_exp)
from sympy.core.symbol import symbols, S
from sympy.functions.elementary.trigonometric import atan, acsc, asin, acot, acos, asec, atan2
from sympy.functions.elementary.hyperbolic import acosh, asinh, atanh, acsch, cosh, sinh, tanh, coth, sech, csch, acoth
from sympy.functions import (sin, cos, tan, cot, sec, csc, sqrt, log as sym_log)
from sympy import (I, E, pi, hyper, Add, Wild, simplify, Symbol, exp, UnevaluatedExpr, Pow, li, Ei, expint,
Si, Ci, Shi, Chi, loggamma, zeta, zoo, gamma, polylog, oo, polygamma)
from sympy import Integral, nsimplify, Min
A, B, a, b, c, d, e, f, g, h, y, z, m, n, p, q, u, v, w, F = symbols('A B a b c d e f g h y z m n p q u v w F', real=True, imaginary=False)
x = Symbol('x')
def test_ZeroQ():
e = b*(n*p + n + 1)
d = a
assert ZeroQ(a*e - b*d*(n*(p + S(1)) + S(1)))
assert ZeroQ(S(0))
assert not ZeroQ(S(10))
assert not ZeroQ(S(-2))
assert ZeroQ(0, 2-2)
assert ZeroQ([S(2), (4), S(0), S(8)]) == [False, False, True, False]
assert ZeroQ([S(2), S(4), S(8)]) == [False, False, False]
def test_NonzeroQ():
assert NonzeroQ(S(1)) == True
def test_FreeQ():
l = [a*b, x, a + b]
assert FreeQ(l, x) == False
l = [a*b, a + b]
assert FreeQ(l, x) == True
def test_List():
assert List(a, b, c) == [a, b, c]
def test_Log():
assert Log(a) == log(a)
def test_PositiveIntegerQ():
assert PositiveIntegerQ(S(1))
assert not PositiveIntegerQ(S(-3))
assert not PositiveIntegerQ(S(0))
def test_NegativeIntegerQ():
assert not NegativeIntegerQ(S(1))
assert NegativeIntegerQ(S(-3))
assert not NegativeIntegerQ(S(0))
def test_PositiveQ():
assert PositiveQ(S(1))
assert not PositiveQ(S(-3))
assert not PositiveQ(S(0))
assert not PositiveQ(zoo)
assert not PositiveQ(I)
assert PositiveQ(b/(b*(b*c/(-a*d + b*c)) - a*(b*d/(-a*d + b*c))))
def test_IntegerQ():
assert IntegerQ(S(1))
assert not IntegerQ(S(-1.9))
assert not IntegerQ(S(0.0))
assert IntegerQ(S(-1))
def test_FracPart():
assert FracPart(S(10)) == 0
assert FracPart(S(10)+0.5) == 10.5
def test_IntPart():
assert IntPart(m*n) == 0
assert IntPart(S(10)) == 10
assert IntPart(1 + m) == 1
def test_NegQ():
assert NegQ(-S(3))
assert not NegQ(S(0))
assert not NegQ(S(0))
def test_RationalQ():
assert RationalQ(S(5)/6)
assert RationalQ(S(5)/6, S(4)/5)
assert not RationalQ(Sqrt(1.6))
assert not RationalQ(Sqrt(1.6), S(5)/6)
assert not RationalQ(log(2))
def test_ArcCosh():
assert ArcCosh(x) == acosh(x)
def test_LinearQ():
assert not LinearQ(a, x)
assert LinearQ(3*x + y**2, x)
assert not LinearQ(3*x + y**2, y)
assert not LinearQ(S(3), x)
def test_Sqrt():
assert Sqrt(x) == sqrt(x)
assert Sqrt(25) == 5
def test_Util_Coefficient():
from sympy.integrals.rubi.utility_function import Util_Coefficient
assert Util_Coefficient(a + b*x + c*x**3, x, a) == Util_Coefficient(a + b*x + c*x**3, x, a)
assert Util_Coefficient(a + b*x + c*x**3, x, 4).doit() == 0
def test_Coefficient():
assert Coefficient(7 + 2*x + 4*x**3, x, 1) == 2
assert Coefficient(a + b*x + c*x**3, x, 0) == a
assert Coefficient(a + b*x + c*x**3, x, 4) == 0
assert Coefficient(b*x + c*x**3, x, 3) == c
assert Coefficient(x, x, -1) == 0
def test_Denominator():
assert Denominator((-S(1)/S(2) + I/3)) == 6
assert Denominator((-a/b)**3) == (b)**(3)
assert Denominator(S(3)/2) == 2
assert Denominator(x/y) == y
assert Denominator(S(4)/5) == 5
def test_Hypergeometric2F1():
assert Hypergeometric2F1(1, 2, 3, x) == hyper((1, 2), (3,), x)
def test_ArcTan():
assert ArcTan(x) == atan(x)
assert ArcTan(x, y) == atan2(x, y)
def test_Not():
a = 10
assert Not(a == 2)
def test_FractionalPart():
assert FractionalPart(S(3.0)) == 0.0
def test_IntegerPart():
assert IntegerPart(3.6) == 3
assert IntegerPart(-3.6) == -4
def test_AppellF1():
assert AppellF1(1,0,0.5,1,0.5,0.25).evalf() == 1.154700538379251529018298
assert AppellF1(a, b, c, d, e, f) == AppellF1(a, b, c, d, e, f)
def test_Simplify():
assert Simplify(sin(x)**2 + cos(x)**2) == 1
assert Simplify((x**3 + x**2 - x - 1)/(x**2 + 2*x + 1)) == x - 1
def test_ArcTanh():
assert ArcTanh(a) == atanh(a)
def test_ArcSin():
assert ArcSin(a) == asin(a)
def test_ArcSinh():
assert ArcSinh(a) == asinh(a)
def test_ArcCos():
assert ArcCos(a) == acos(a)
def test_ArcCsc():
assert ArcCsc(a) == acsc(a)
def test_ArcCsch():
assert ArcCsch(a) == acsch(a)
def test_Equal():
assert Equal(a, a)
assert not Equal(a, b)
def test_LessEqual():
assert LessEqual(1, 2, 3)
assert LessEqual(1, 1)
assert not LessEqual(3, 2, 1)
def test_With():
assert With(Set(x, 3), x + y) == 3 + y
assert With(List(Set(x, 3), Set(y, c)), x + y) == 3 + c
def test_Less():
assert Less(1, 2, 3)
assert not Less(1, 1, 3)
def test_Greater():
assert Greater(3, 2, 1)
assert not Greater(3, 2, 2)
def test_GreaterEqual():
assert GreaterEqual(3, 2, 1)
assert GreaterEqual(3, 2, 2)
assert not GreaterEqual(2, 3)
def test_Unequal():
assert Unequal(1, 2)
assert not Unequal(1, 1)
def test_FractionQ():
assert not FractionQ(S('3'))
assert FractionQ(S('3')/S('2'))
def test_Expand():
assert Expand((1 + x)**10) == x**10 + 10*x**9 + 45*x**8 + 120*x**7 + 210*x**6 + 252*x**5 + 210*x**4 + 120*x**3 + 45*x**2 + 10*x + 1
def test_Scan():
assert list(Scan(sin, [a, b])) == [sin(a), sin(b)]
def test_MapAnd():
assert MapAnd(PositiveQ, [S(1), S(2), S(3), S(0)]) == False
assert MapAnd(PositiveQ, [S(1), S(2), S(3)]) == True
def test_FalseQ():
assert FalseQ(True) == False
assert FalseQ(False) == True
def test_ComplexNumberQ():
assert ComplexNumberQ(1 + I*2, I) == True
assert ComplexNumberQ(a + b, I) == False
def test_Re():
assert Re(1 + I) == 1
def test_Im():
assert Im(1 + 2*I) == 2
assert Im(a*I) == a
def test_PositiveOrZeroQ():
assert PositiveOrZeroQ(S(0)) == True
assert PositiveOrZeroQ(S(1)) == True
assert PositiveOrZeroQ(-S(1)) == False
def test_RealNumericQ():
assert RealNumericQ(S(1)) == True
assert RealNumericQ(-S(1)) == True
def test_NegativeOrZeroQ():
assert NegativeOrZeroQ(S(0)) == True
assert NegativeOrZeroQ(-S(1)) == True
assert NegativeOrZeroQ(S(1)) == False
def test_FractionOrNegativeQ():
assert FractionOrNegativeQ(S(1)/2) == True
assert FractionOrNegativeQ(-S(1)) == True
def test_ProductQ():
assert ProductQ(a*b) == True
assert ProductQ(a + b) == False
def test_SumQ():
assert SumQ(a*b) == False
assert SumQ(a + b) == True
def test_NonsumQ():
assert NonsumQ(a*b) == True
assert NonsumQ(a + b) == False
def test_SqrtNumberQ():
assert SqrtNumberQ(sqrt(2)) == True
def test_IntLinearcQ():
assert IntLinearcQ(1, 2, 3, 4, 5, 6, x) == True
assert IntLinearcQ(S(1)/100, S(2)/100, S(3)/100, S(4)/100, S(5)/100, S(6)/100, x) == False
def test_IndependentQ():
assert IndependentQ(a + b*x, x) == False
assert IndependentQ(a + b, x) == True
def test_PowerQ():
assert PowerQ(a**b) == True
assert PowerQ(a + b) == False
def test_IntegerPowerQ():
assert IntegerPowerQ(a**2) == True
assert IntegerPowerQ(a**0.5) == False
def test_PositiveIntegerPowerQ():
assert PositiveIntegerPowerQ(a**3) == True
assert PositiveIntegerPowerQ(a**(-2)) == False
def test_FractionalPowerQ():
assert FractionalPowerQ(a**(S(2)/S(3)))
assert FractionalPowerQ(a**sqrt(2)) == False
def test_AtomQ():
assert AtomQ(x)
assert not AtomQ(x+1)
assert not AtomQ([a, b])
def test_ExpQ():
assert ExpQ(E**2)
assert not ExpQ(2**E)
def test_LogQ():
assert LogQ(log(x))
assert not LogQ(sin(x) + log(x))
def test_Head():
assert Head(sin(x)) == sin
assert Head(log(x**3 + 3)) in (sym_log, log)
def test_MemberQ():
assert MemberQ([a, b, c], b)
assert MemberQ([sin, cos, log, tan], Head(sin(x)))
assert MemberQ([[sin, cos], [tan, cot]], [sin, cos])
assert not MemberQ([[sin, cos], [tan, cot]], [sin, tan])
def test_TrigQ():
assert TrigQ(sin(x))
assert TrigQ(tan(x**2 + 2))
assert not TrigQ(sin(x) + tan(x))
def test_SinQ():
assert SinQ(sin(x))
assert not SinQ(tan(x))
def test_CosQ():
assert CosQ(cos(x))
assert not CosQ(csc(x))
def test_TanQ():
assert TanQ(tan(x))
assert not TanQ(cot(x))
def test_CotQ():
assert not CotQ(tan(x))
assert CotQ(cot(x))
def test_SecQ():
assert SecQ(sec(x))
assert not SecQ(csc(x))
def test_CscQ():
assert not CscQ(sec(x))
assert CscQ(csc(x))
def test_HyperbolicQ():
assert HyperbolicQ(sinh(x))
assert HyperbolicQ(cosh(x))
assert HyperbolicQ(tanh(x))
assert not HyperbolicQ(sinh(x) + cosh(x) + tanh(x))
def test_SinhQ():
assert SinhQ(sinh(x))
assert not SinhQ(cosh(x))
def test_CoshQ():
assert not CoshQ(sinh(x))
assert CoshQ(cosh(x))
def test_TanhQ():
assert TanhQ(tanh(x))
assert not TanhQ(coth(x))
def test_CothQ():
assert not CothQ(tanh(x))
assert CothQ(coth(x))
def test_SechQ():
assert SechQ(sech(x))
assert not SechQ(csch(x))
def test_CschQ():
assert not CschQ(sech(x))
assert CschQ(csch(x))
def test_InverseTrigQ():
assert InverseTrigQ(acot(x))
assert InverseTrigQ(asec(x))
assert not InverseTrigQ(acsc(x) + asec(x))
def test_SinCosQ():
assert SinCosQ(sin(x))
assert SinCosQ(cos(x))
assert SinCosQ(sec(x))
assert not SinCosQ(acsc(x))
def test_SinhCoshQ():
assert not SinhCoshQ(sin(x))
assert SinhCoshQ(cosh(x))
assert SinhCoshQ(sech(x))
assert SinhCoshQ(csch(x))
def test_LeafCount():
assert LeafCount(1 + a + x**2) == 6
def test_Numerator():
assert Numerator((-S(1)/S(2) + I/3)) == -3 + 2*I
assert Numerator((-a/b)**3) == (-a)**(3)
assert Numerator(S(3)/2) == 3
assert Numerator(x/y) == x
def test_Length():
assert Length(a + b) == 2
assert Length(sin(a)*cos(a)) == 2
def test_ListQ():
assert ListQ([1, 2])
assert not ListQ(a)
def test_InverseHyperbolicQ():
assert InverseHyperbolicQ(acosh(a))
def test_InverseFunctionQ():
assert InverseFunctionQ(log(a))
assert InverseFunctionQ(acos(a))
assert not InverseFunctionQ(a)
assert InverseFunctionQ(acosh(a))
assert InverseFunctionQ(polylog(a, b))
def test_EqQ():
assert EqQ(a, a)
assert not EqQ(a, b)
def test_FactorSquareFree():
assert FactorSquareFree(x**5 - x**3 - x**2 + 1) == (x**3 + 2*x**2 + 2*x + 1)*(x - 1)**2
def test_FactorSquareFreeList():
assert FactorSquareFreeList(x**5-x**3-x**2 + 1) == [[1, 1], [x**3 + 2*x**2 + 2*x + 1, 1], [x - 1, 2]]
assert FactorSquareFreeList(x**4 - 2*x**2 + 1) == [[1, 1], [x**2 - 1, 2]]
def test_PerfectPowerTest():
assert not PerfectPowerTest(sqrt(x), x)
assert not PerfectPowerTest(x**5-x**3-x**2 + 1, x)
assert PerfectPowerTest(x**4 - 2*x**2 + 1, x) == (x**2 - 1)**2
def test_SquareFreeFactorTest():
assert not SquareFreeFactorTest(sqrt(x), x)
assert SquareFreeFactorTest(x**5 - x**3 - x**2 + 1, x) == (x**3 + 2*x**2 + 2*x + 1)*(x - 1)**2
def test_Rest():
assert Rest([2, 3, 5, 7]) == [3, 5, 7]
assert Rest(a + b + c) == b + c
assert Rest(a*b*c) == b*c
assert Rest(1/b) == -1
def test_First():
assert First([2, 3, 5, 7]) == 2
assert First(y**S(2)) == y
assert First(a + b + c) == a
assert First(a*b*c) == a
def test_ComplexFreeQ():
assert ComplexFreeQ(a)
assert not ComplexFreeQ(a + 2*I)
def test_FractionalPowerFreeQ():
assert not FractionalPowerFreeQ(x**(S(2)/3))
assert FractionalPowerFreeQ(x)
def test_Exponent():
assert Exponent(x**2 + x + 1 + 5, x, Min) == 0
assert Exponent(x**2 + x + 1 + 5, x, List) == [0, 1, 2]
assert Exponent(x**2 + x + 1, x, List) == [0, 1, 2]
assert Exponent(x**2 + 2*x + 1, x, List) == [0, 1, 2]
assert Exponent(x**3 + x + 1, x) == 3
assert Exponent(x**2 + 2*x + 1, x) == 2
assert Exponent(x**3, x, List) == [3]
assert Exponent(S(1), x) == 0
assert Exponent(x**(-3), x) == 0
def test_Expon():
assert Expon(x**2+2*x+1, x) == 2
assert Expon(x**3, x, List) == [3]
def test_QuadraticQ():
assert not QuadraticQ([x**2+x+1, 5*x**2], x)
assert QuadraticQ([x**2+x+1, 5*x**2+3*x+6], x)
assert not QuadraticQ(x**2+1+x**3, x)
assert QuadraticQ(x**2+1+x, x)
assert not QuadraticQ(x**2, x)
def test_BinomialQ():
assert BinomialQ(x**9, x)
assert not BinomialQ((1 + x)**3, x)
def test_BinomialParts():
assert BinomialParts(2 + x*(9*x), x) == [2, 9, 2]
assert BinomialParts(x**9, x) == [0, 1, 9]
assert BinomialParts(2*x**3, x) == [0, 2, 3]
assert BinomialParts(2 + x, x) == [2, 1, 1]
def test_BinomialDegree():
assert BinomialDegree(b + 2*c*x**n, x) == n
assert BinomialDegree(2 + x*(9*x), x) == 2
assert BinomialDegree(x**9, x) == 9
def test_PolynomialQ():
assert not PolynomialQ(x*(-1 + x**2), (1 + x)**(S(1)/2))
assert not PolynomialQ((16*x + 1)/((x + 5)**2*(x**2 + x + 1)), 2*x)
C = Symbol('C')
assert not PolynomialQ(A + b*x + c*x**2, x**2)
assert PolynomialQ(A + B*x + C*x**2)
assert PolynomialQ(A + B*x**4 + C*x**2, x**2)
assert PolynomialQ(x**3, x)
assert not PolynomialQ(sqrt(x), x)
def test_PolyQ():
assert PolyQ(-2*a*d**3*e**2 + x**6*(a*e**5 - b*d*e**4 + c*d**2*e**3)\
+ x**4*(-2*a*d*e**4 + 2*b*d**2*e**3 - 2*c*d**3*e**2) + x**2*(2*a*d**2*e**3 - 2*b*d**3*e**2), x)
assert not PolyQ(1/sqrt(a + b*x**2 - c*x**4), x**2)
assert PolyQ(x, x, 1)
assert PolyQ(x**2, x, 2)
assert not PolyQ(x**3, x, 2)
def test_EvenQ():
assert EvenQ(S(2))
assert not EvenQ(S(1))
def test_OddQ():
assert OddQ(S(1))
assert not OddQ(S(2))
def test_PerfectSquareQ():
assert PerfectSquareQ(S(4))
assert PerfectSquareQ(a**S(2)*b**S(4))
assert not PerfectSquareQ(S(1)/3)
def test_NiceSqrtQ():
assert NiceSqrtQ(S(1)/3)
assert not NiceSqrtQ(-S(1))
assert NiceSqrtQ(pi**2)
assert NiceSqrtQ(pi**2*sin(4)**4)
assert not NiceSqrtQ(pi**2*sin(4)**3)
def test_Together():
assert Together(1/a + b/2) == (a*b + 2)/(2*a)
def test_PosQ():
#assert not PosQ((b*e - c*d)/(c*e))
assert not PosQ(S(0))
assert PosQ(S(1))
assert PosQ(pi)
assert PosQ(pi**3)
assert PosQ((-pi)**4)
assert PosQ(sin(1)**2*pi**4)
def test_NumericQ():
assert NumericQ(sin(cos(2)))
def test_NumberQ():
assert NumberQ(pi)
def test_CoefficientList():
assert CoefficientList(1 + a*x, x) == [1, a]
assert CoefficientList(1 + a*x**3, x) == [1, 0, 0, a]
assert CoefficientList(sqrt(x), x) == []
def test_ReplaceAll():
assert ReplaceAll(x, {x: a}) == a
assert ReplaceAll(a*x, {x: a + b}) == a*(a + b)
assert ReplaceAll(a*x, {a: b, x: a + b}) == b*(a + b)
def test_ExpandLinearProduct():
assert ExpandLinearProduct(log(x), x**2, a, b, x) == a**2*log(x)/b**2 - 2*a*(a + b*x)*log(x)/b**2 + (a + b*x)**2*log(x)/b**2
assert ExpandLinearProduct((a + b*x)**n, x**3, a, b, x) == -a**3*(a + b*x)**n/b**3 + 3*a**2*(a + b*x)**(n + 1)/b**3 - 3*a*(a + b*x)**(n + 2)/b**3 + (a + b*x)**(n + 3)/b**3
def test_PolynomialDivide():
assert PolynomialDivide((a*c - b*c*x)**2, (a + b*x)**2, x) == -4*a*b*c**2*x/(a + b*x)**2 + c**2
assert PolynomialDivide(x + x**2, x, x) == x + 1
assert PolynomialDivide((1 + x)**3, (1 + x)**2, x) == x + 1
assert PolynomialDivide((a + b*x)**3, x**3, x) == a*(a**2 + 3*a*b*x + 3*b**2*x**2)/x**3 + b**3
assert PolynomialDivide(x**3*(a + b*x), S(1), x) == b*x**4 + a*x**3
assert PolynomialDivide(x**6, (a + b*x)**2, x) == -a**5*(5*a + 6*b*x)/(b**6*(a + b*x)**2) + 5*a**4/b**6 - 4*a**3*x/b**5 + 3*a**2*x**2/b**4 - 2*a*x**3/b**3 + x**4/b**2
def test_MatchQ():
a_ = Wild('a', exclude=[x])
b_ = Wild('b', exclude=[x])
c_ = Wild('c', exclude=[x])
assert MatchQ(a*b + c, a_*b_ + c_, a_, b_, c_) == (a, b, c)
def test_PolynomialQuotientRemainder():
assert PolynomialQuotientRemainder(x**2, x+a, x) == [-a + x, a**2]
def test_FreeFactors():
assert FreeFactors(a, x) == a
assert FreeFactors(x + a, x) == 1
assert FreeFactors(a*b*x, x) == a*b
def test_NonfreeFactors():
assert NonfreeFactors(a, x) == 1
assert NonfreeFactors(x + a, x) == x + a
assert NonfreeFactors(a*b*x, x) == x
def test_FreeTerms():
assert FreeTerms(a, x) == a
assert FreeTerms(x*a, x) == 0
assert FreeTerms(a*x + b, x) == b
def test_NonfreeTerms():
assert NonfreeTerms(a, x) == 0
assert NonfreeTerms(a*x, x) == a*x
assert NonfreeTerms(a*x + b, x) == a*x
def test_RemoveContent():
assert RemoveContent(a + b*x, x) == a + b*x
def test_ExpandAlgebraicFunction():
assert ExpandAlgebraicFunction((a + b)*x, x) == a*x + b*x
assert ExpandAlgebraicFunction((a + b)**2*x, x)== a**2*x + 2*a*b*x + b**2*x
assert ExpandAlgebraicFunction((a + b)**2*x**2, x) == a**2*x**2 + 2*a*b*x**2 + b**2*x**2
def test_CollectReciprocals():
assert CollectReciprocals(-1/(1 + 1*x) - 1/(1 - 1*x), x) == -2/(-x**2 + 1)
assert CollectReciprocals(1/(1 + 1*x) - 1/(1 - 1*x), x) == -2*x/(-x**2 + 1)
def test_ExpandCleanup():
assert ExpandCleanup(a + b, x) == a*(1 + b/a)
assert ExpandCleanup(b**2/(a**2*(a + b*x)**2) + 1/(a**2*x**2) + 2*b**2/(a**3*(a + b*x)) - 2*b/(a**3*x), x) == b**2/(a**2*(a + b*x)**2) + 1/(a**2*x**2) + 2*b**2/(a**3*(a + b*x)) - 2*b/(a**3*x)
def test_AlgebraicFunctionQ():
assert not AlgebraicFunctionQ(1/(a + c*x**(2*n)), x)
assert AlgebraicFunctionQ(a, x) == True
assert AlgebraicFunctionQ(a*b, x) == True
assert AlgebraicFunctionQ(x**2, x) == True
assert AlgebraicFunctionQ(x**2*a, x) == True
assert AlgebraicFunctionQ(x**2 + a, x) == True
assert AlgebraicFunctionQ(sin(x), x) == False
assert AlgebraicFunctionQ([], x) == True
assert AlgebraicFunctionQ([a, a*b], x) == True
assert AlgebraicFunctionQ([sin(x)], x) == False
def test_MonomialQ():
assert not MonomialQ(2*x**7 + 6, x)
assert MonomialQ(2*x**7, x)
assert not MonomialQ(2*x**7 + 5*x**3, x)
assert not MonomialQ([2*x**7 + 6, 2*x**7], x)
assert MonomialQ([2*x**7, 5*x**3], x)
def test_MonomialSumQ():
assert MonomialSumQ(2*x**7 + 6, x) == True
assert MonomialSumQ(x**2 + x**3 + 5*x, x) == True
def test_MinimumMonomialExponent():
assert MinimumMonomialExponent(x**2 + 5*x**2 + 3*x**5, x) == 2
assert MinimumMonomialExponent(x**2 + 5*x**2 + 1, x) == 0
def test_MonomialExponent():
assert MonomialExponent(3*x**7, x) == 7
assert not MonomialExponent(3+x**3, x)
def test_LinearMatchQ():
assert LinearMatchQ(2 + 3*x, x)
assert LinearMatchQ(3*x, x)
assert not LinearMatchQ(3*x**2, x)
def test_SimplerQ():
a1, b1 = symbols('a1 b1')
assert SimplerQ(a1, b1)
assert SimplerQ(2*a, a + 2)
assert SimplerQ(2, x)
assert not SimplerQ(x**2, x)
assert SimplerQ(2*x, x + 2 + 6*x**3)
def test_GeneralizedTrinomialParts():
assert not GeneralizedTrinomialParts((7 + 2*x**6 + 3*x**12), x)
assert GeneralizedTrinomialParts(x**2 + x**3 + x**4, x) == [1, 1, 1, 3, 2]
assert not GeneralizedTrinomialParts(2*x + 3*x + 4*x, x)
def test_TrinomialQ():
assert TrinomialQ((7 + 2*x**6 + 3*x**12), x)
assert not TrinomialQ(x**2, x)
def test_GeneralizedTrinomialDegree():
assert not GeneralizedTrinomialDegree((7 + 2*x**6 + 3*x**12), x)
assert GeneralizedTrinomialDegree(x**2 + x**3 + x**4, x) == 1
def test_GeneralizedBinomialParts():
assert GeneralizedBinomialParts(3*x*(3 + x**6), x) == [9, 3, 7, 1]
assert GeneralizedBinomialParts((3*x + x**7), x) == [3, 1, 7, 1]
def test_GeneralizedBinomialDegree():
assert GeneralizedBinomialDegree(3*x*(3 + x**6), x) == 6
assert GeneralizedBinomialDegree((3*x + x**7), x) == 6
def test_PowerOfLinearQ():
assert PowerOfLinearQ((6*x), x)
assert not PowerOfLinearQ((3 + 6*x**3), x)
assert PowerOfLinearQ((3 + 6*x)**3, x)
def test_LinearPairQ():
assert not LinearPairQ(6*x**2 + 4, 3*x**2 + 2, x)
assert LinearPairQ(6*x + 4, 3*x + 2, x)
assert not LinearPairQ(6*x, 3*x + 2, x)
assert LinearPairQ(6*x, 3*x, x)
def test_LeadTerm():
assert LeadTerm(a*b*c) == a*b*c
assert LeadTerm(a + b + c) == a
def test_RemainingTerms():
assert RemainingTerms(a*b*c) == a*b*c
assert RemainingTerms(a + b + c) == b + c
def test_LeadFactor():
assert LeadFactor(a*b*c) == a
assert LeadFactor(a + b + c) == a + b + c
assert LeadFactor(b*I) == I
assert LeadFactor(c*a**b) == a**b
assert LeadFactor(S(2)) == S(2)
def test_RemainingFactors():
assert RemainingFactors(a*b*c) == b*c
assert RemainingFactors(a + b + c) == 1
assert RemainingFactors(a*I) == a
def test_LeadBase():
assert LeadBase(a**b) == a
assert LeadBase(a**b*c) == a
def test_LeadDegree():
assert LeadDegree(a**b) == b
assert LeadDegree(a**b*c) == b
def test_Numer():
assert Numer(a/b) == a
assert Numer(a**(-2)) == 1
assert Numer(a**(-2)*a/b) == 1
def test_Denom():
assert Denom(a/b) == b
assert Denom(a**(-2)) == a**2
assert Denom(a**(-2)*a/b) == a*b
def test_Coeff():
assert Coeff(7 + 2*x + 4*x**3, x, 1) == 2
assert Coeff(a + b*x + c*x**3, x, 0) == a
assert Coeff(a + b*x + c*x**3, x, 4) == 0
assert Coeff(b*x + c*x**3, x, 3) == c
def test_MergeMonomials():
assert MergeMonomials(x**2*(1 + 1*x)**3*(1 + 1*x)**n, x) == x**2*(x + 1)**(n + 3)
assert MergeMonomials(x**2*(1 + 1*x)**2*(1*(1 + 1*x)**1)**2, x) == x**2*(x + 1)**4
assert MergeMonomials(b**2/a**3, x) == b**2/a**3
def test_RationalFunctionQ():
assert RationalFunctionQ(a, x)
assert RationalFunctionQ(x**2, x)
assert RationalFunctionQ(x**3 + x**4, x)
assert RationalFunctionQ(x**3*S(2), x)
assert not RationalFunctionQ(x**3 + x**(0.5), x)
assert not RationalFunctionQ(x**(S(2)/3)*(a + b*x)**2, x)
def test_Apart():
assert Apart(1/(x**2*(a + b*x)**2), x) == b**2/(a**2*(a + b*x)**2) + 1/(a**2*x**2) + 2*b**2/(a**3*(a + b*x)) - 2*b/(a**3*x)
assert Apart(x**(S(2)/3)*(a + b*x)**2, x) == x**(S(2)/3)*(a + b*x)**2
def test_RationalFunctionFactors():
assert RationalFunctionFactors(a, x) == a
assert RationalFunctionFactors(sqrt(x), x) == 1
assert RationalFunctionFactors(x*x**3, x) == x*x**3
assert RationalFunctionFactors(x*sqrt(x), x) == 1
def test_NonrationalFunctionFactors():
assert NonrationalFunctionFactors(x, x) == 1
assert NonrationalFunctionFactors(sqrt(x), x) == sqrt(x)
assert NonrationalFunctionFactors(sqrt(x)*log(x), x) == sqrt(x)*log(x)
def test_Reverse():
assert Reverse([1, 2, 3]) == [3, 2, 1]
assert Reverse(a**b) == b**a
def test_RationalFunctionExponents():
assert RationalFunctionExponents(sqrt(x), x) == [0, 0]
assert RationalFunctionExponents(a, x) == [0, 0]
assert RationalFunctionExponents(x, x) == [1, 0]
assert RationalFunctionExponents(x**(-1), x)== [0, 1]
assert RationalFunctionExponents(x**(-1)*a, x) == [0, 1]
assert RationalFunctionExponents(x**(-1) + a, x) == [1, 1]
def test_PolynomialGCD():
assert PolynomialGCD(x**2 - 1, x**2 - 3*x + 2) == x - 1
def test_PolyGCD():
assert PolyGCD(x**2 - 1, x**2 - 3*x + 2, x) == x - 1
def test_AlgebraicFunctionFactors():
assert AlgebraicFunctionFactors(sin(x)*x, x) == x
assert AlgebraicFunctionFactors(sin(x), x) == 1
assert AlgebraicFunctionFactors(x, x) == x
def test_NonalgebraicFunctionFactors():
assert NonalgebraicFunctionFactors(sin(x)*x, x) == sin(x)
assert NonalgebraicFunctionFactors(sin(x), x) == sin(x)
assert NonalgebraicFunctionFactors(x, x) == 1
def test_QuotientOfLinearsP():
assert QuotientOfLinearsP((a + b*x)/(x), x)
assert QuotientOfLinearsP(x*a, x)
assert not QuotientOfLinearsP(x**2*a, x)
assert not QuotientOfLinearsP(x**2 + a, x)
assert QuotientOfLinearsP(x + a, x)
assert QuotientOfLinearsP(x, x)
assert QuotientOfLinearsP(1 + x, x)
def test_QuotientOfLinearsParts():
assert QuotientOfLinearsParts((b*x)/(c), x) == [0, b/c, 1, 0]
assert QuotientOfLinearsParts((b*x)/(c + x), x) == [0, b, c, 1]
assert QuotientOfLinearsParts((b*x)/(c + d*x), x) == [0, b, c, d]
assert QuotientOfLinearsParts((a + b*x)/(c + d*x), x) == [a, b, c, d]
assert QuotientOfLinearsParts(x**2 + a, x) == [a + x**2, 0, 1, 0]
assert QuotientOfLinearsParts(a/x, x) == [a, 0, 0, 1]
assert QuotientOfLinearsParts(1/x, x) == [1, 0, 0, 1]
assert QuotientOfLinearsParts(a*x + 1, x) == [1, a, 1, 0]
assert QuotientOfLinearsParts(x, x) == [0, 1, 1, 0]
assert QuotientOfLinearsParts(a, x) == [a, 0, 1, 0]
def test_QuotientOfLinearsQ():
assert not QuotientOfLinearsQ((a + x), x)
assert QuotientOfLinearsQ((a + x)/(x), x)
assert QuotientOfLinearsQ((a + b*x)/(x), x)
def test_Flatten():
assert Flatten([a, b, [c, [d, e]]]) == [a, b, c, d, e]
def test_Sort():
assert Sort([b, a, c]) == [a, b, c]
assert Sort([b, a, c], True) == [c, b, a]
def test_AbsurdNumberQ():
assert AbsurdNumberQ(S(1))
assert not AbsurdNumberQ(a*x)
assert not AbsurdNumberQ(a**(S(1)/2))
assert AbsurdNumberQ((S(1)/3)**(S(1)/3))
def test_AbsurdNumberFactors():
assert AbsurdNumberFactors(S(1)) == S(1)
assert AbsurdNumberFactors((S(1)/3)**(S(1)/3)) == S(3)**(S(2)/3)/S(3)
assert AbsurdNumberFactors(a) == S(1)
def test_NonabsurdNumberFactors():
assert NonabsurdNumberFactors(a) == a
assert NonabsurdNumberFactors(S(1)) == S(1)
assert NonabsurdNumberFactors(a*S(2)) == a
def test_NumericFactor():
assert NumericFactor(S(1)) == S(1)
assert NumericFactor(1*I) == S(1)
assert NumericFactor(S(1) + I) == S(1)
assert NumericFactor(a**(S(1)/3)) == S(1)
assert NumericFactor(a*S(3)) == S(3)
assert NumericFactor(a + b) == S(1)
def test_NonnumericFactors():
assert NonnumericFactors(S(3)) == S(1)
assert NonnumericFactors(I) == I
assert NonnumericFactors(S(3) + I) == S(3) + I
assert NonnumericFactors((S(1)/3)**(S(1)/3)) == S(1)
assert NonnumericFactors(log(a)) == log(a)
def test_Prepend():
assert Prepend([1, 2, 3], [4, 5]) == [4, 5, 1, 2, 3]
def test_SumSimplerQ():
assert not SumSimplerQ(S(4 + x),S(3 + x**3))
assert SumSimplerQ(S(4 + x), S(3 - x))
def test_SumSimplerAuxQ():
assert SumSimplerAuxQ(S(4 + x), S(3 - x))
assert not SumSimplerAuxQ(S(4), S(3))
def test_SimplerSqrtQ():
assert SimplerSqrtQ(S(2), S(16*x**3))
assert not SimplerSqrtQ(S(x*2), S(16))
assert not SimplerSqrtQ(S(-4), S(16))
assert SimplerSqrtQ(S(4), S(16))
assert not SimplerSqrtQ(S(4), S(0))
def test_TrinomialParts():
assert TrinomialParts((1 + 5*x**3)**2, x) == [1, 10, 25, 3]
assert TrinomialParts(1 + 5*x**3 + 2*x**6, x) == [1, 5, 2, 3]
assert TrinomialParts(((1 + 5*x**3)**2) + 6, x) == [7, 10, 25, 3]
assert not TrinomialParts(1 + 5*x**3 + 2*x**5, x)
def test_TrinomialDegree():
assert TrinomialDegree((7 + 2*x**6)**2, x) == 6
assert TrinomialDegree(1 + 5*x**3 + 2*x**6, x) == 3
assert not TrinomialDegree(1 + 5*x**3 + 2*x**5, x)
def test_CubicMatchQ():
assert not CubicMatchQ(S(3 + x**6), x)
assert CubicMatchQ(S(x**3), x)
assert not CubicMatchQ(S(3), x)
assert CubicMatchQ(S(3 + x**3), x)
assert CubicMatchQ(S(3 + x**3 + 2*x), x)
def test_BinomialMatchQ():
assert BinomialMatchQ(x, x)
assert BinomialMatchQ(2 + 3*x**5, x)
assert BinomialMatchQ(3*x**5, x)
assert BinomialMatchQ(3*x, x)
assert not BinomialMatchQ(x + x**2 + x**3, x)
def test_TrinomialMatchQ():
assert not TrinomialMatchQ((5 + 2*x**6)**2, x)
assert not TrinomialMatchQ((7 + 8*x**6), x)
assert TrinomialMatchQ((7 + 2*x**6 + 3*x**3), x)
assert TrinomialMatchQ(b*x**2 + c*x**4, x)
def test_GeneralizedBinomialMatchQ():
assert not GeneralizedBinomialMatchQ((1 + x**4), x)
assert GeneralizedBinomialMatchQ((3*x + x**7), x)
def test_QuadraticMatchQ():
assert not QuadraticMatchQ((a + b*x)*(c + d*x), x)
assert QuadraticMatchQ(x**2 + x, x)
assert QuadraticMatchQ(x**2+1+x, x)
assert QuadraticMatchQ(x**2, x)
def test_PowerOfLinearMatchQ():
assert PowerOfLinearMatchQ(x, x)
assert not PowerOfLinearMatchQ(S(6)**3, x)
assert not PowerOfLinearMatchQ(S(6 + 3*x**2)**3, x)
assert PowerOfLinearMatchQ(S(6 + 3*x)**3, x)
def test_GeneralizedTrinomialMatchQ():
assert not GeneralizedTrinomialMatchQ(7 + 2*x**6 + 3*x**12, x)
assert not GeneralizedTrinomialMatchQ(7 + 2*x**6 + 3*x**3, x)
assert not GeneralizedTrinomialMatchQ(7 + 2*x**6 + 3*x**5, x)
assert GeneralizedTrinomialMatchQ(x**2 + x**3 + x**4, x)
def test_QuotientOfLinearsMatchQ():
assert QuotientOfLinearsMatchQ((1 + x)*(3 + 4*x**2)/(2 + 4*x), x)
assert not QuotientOfLinearsMatchQ(x*(3 + 4*x**2)/(2 + 4*x**3), x)
assert QuotientOfLinearsMatchQ(x*(3 + 4*x)/(2 + 4*x), x)
assert QuotientOfLinearsMatchQ(2*(3 + 4*x)/(2 + 4*x), x)
def test_PolynomialTermQ():
assert not PolynomialTermQ(S(3), x)
assert PolynomialTermQ(3*x**6, x)
assert not PolynomialTermQ(3*x**6+5*x, x)
def test_PolynomialTerms():
assert PolynomialTerms(x + 6*x**3 + log(x), x) == 6*x**3 + x
assert PolynomialTerms(x + 6*x**3 + 6*x, x) == 6*x**3 + 7*x
assert PolynomialTerms(x + 6*x**3 + 6, x) == 6*x**3 + x
def test_NonpolynomialTerms():
assert NonpolynomialTerms(x + 6*x**3 + log(x), x) == log(x)
assert NonpolynomialTerms(x + 6*x**3 + 6*x, x) == 0
assert NonpolynomialTerms(x + 6*x**3 + 6, x) == 6
def test_PseudoBinomialQ():
assert PseudoBinomialQ(3 + 5*(x)**6, x)
assert PseudoBinomialQ(3 + 5*(2 + 5*x)**6, x)
def test_PseudoBinomialParts():
assert PseudoBinomialParts(3 + 7*(1 + x)**6, x) == [3, 1, 7**(S(1)/S(6)), 7**(S(1)/S(6)), 6]
assert PseudoBinomialParts(3 + 7*(1 + x)**3, x) == [3, 1, 7**(S(1)/S(3)), 7**(S(1)/S(3)), 3]
assert not PseudoBinomialParts(3 + 7*(1 + x)**2, x)
assert PseudoBinomialParts(3 + 7*(x)**5, x) == [3, 1, 0, 7**(S(1)/S(5)), 5]
def test_PseudoBinomialPairQ():
assert not PseudoBinomialPairQ(3 + 5*(x)**6,3 + (x)**6, x)
assert not PseudoBinomialPairQ(3 + 5*(1 + x)**6,3 + (1 + x)**6, x)
def test_NormalizePseudoBinomial():
assert NormalizePseudoBinomial(3 + 5*(1 + x)**6, x) == 3+(5**(S(1)/S(6))+5**(S(1)/S(6))*x)**S(6)
assert NormalizePseudoBinomial(3 + 5*(x)**6, x) == 3+5*x**6
def test_CancelCommonFactors():
assert CancelCommonFactors(S(x*y*S(6))**S(6), S(x*y*S(6))) == [46656*x**6*y**6, 6*x*y]
assert CancelCommonFactors(S(y*6)**S(6), S(x*y*S(6))) == [46656*y**6, 6*x*y]
assert CancelCommonFactors(S(6), S(3)) == [6, 3]
def test_SimplerIntegrandQ():
assert SimplerIntegrandQ(S(5), 4*x, x)
assert not SimplerIntegrandQ(S(x + 5*x**3), S(x**2 + 3*x), x)
assert SimplerIntegrandQ(S(x + 8), S(x**2 + 3*x), x)
def test_Drop():
assert Drop([1, 2, 3, 4, 5, 6], [2, 4]) == [1, 5, 6]
assert Drop([1, 2, 3, 4, 5, 6], -3) == [1, 2, 3]
assert Drop([1, 2, 3, 4, 5, 6], 2) == [3, 4, 5, 6]
assert Drop(a*b*c, 1) == b*c
def test_SubstForInverseFunction():
assert SubstForInverseFunction(x, a, b, x) == b
assert SubstForInverseFunction(a, a, b, x) == a
assert SubstForInverseFunction(x**a, x**a, b, x) == x
assert SubstForInverseFunction(a*x**a, a, b, x) == a*b**a
def test_SubstForFractionalPower():
assert SubstForFractionalPower(a, b, n, c, x) == a
assert SubstForFractionalPower(x, b, n, c, x) == c
assert SubstForFractionalPower(a**(S(1)/2), a, n, b, x) == x**(n/2)
def test_CombineExponents():
assert True
def test_FractionalPowerOfSquareQ():
assert not FractionalPowerOfSquareQ(x)
assert not FractionalPowerOfSquareQ((a + b)**(S(2)/S(3)))
assert not FractionalPowerOfSquareQ((a + b)**(S(2)/S(3))*c)
assert FractionalPowerOfSquareQ(((a + b*x)**(S(2)))**(S(1)/3)) == (a + b*x)**S(2)
def test_FractionalPowerSubexpressionQ():
assert not FractionalPowerSubexpressionQ(x, a, x)
assert FractionalPowerSubexpressionQ(x**(S(2)/S(3)), a, x)
assert not FractionalPowerSubexpressionQ(b*a, a, x)
def test_FactorNumericGcd():
assert FactorNumericGcd(5*a**2*e**4 + 2*a*b*d*e**3 + 2*a*c*d**2*e**2 + b**2*d**2*e**2 - 6*b*c*d**3*e + 21*c**2*d**4) ==\
5*a**2*e**4 + 2*a*b*d*e**3 + 2*a*c*d**2*e**2 + b**2*d**2*e**2 - 6*b*c*d**3*e + 21*c**2*d**4
assert FactorNumericGcd(x**(S(2))) == x**S(2)
assert FactorNumericGcd(log(x)) == log(x)
assert FactorNumericGcd(log(x)*x) == x*log(x)
assert FactorNumericGcd(log(x) + x**S(2)) == log(x) + x**S(2)
def test_Apply():
assert Apply(List, [a, b, c]) == [a, b, c]
def test_TrigSimplify():
assert TrigSimplify(a*sin(x)**2 + a*cos(x)**2 + v) == a + v
assert TrigSimplify(a*sec(x)**2 - a*tan(x)**2 + v) == a + v
assert TrigSimplify(a*csc(x)**2 - a*cot(x)**2 + v) == a + v
assert TrigSimplify(S(1) - sin(x)**2) == cos(x)**2
assert TrigSimplify(1 + tan(x)**2) == sec(x)**2
assert TrigSimplify(1 + cot(x)**2) == csc(x)**2
assert TrigSimplify(-S(1) + sec(x)**2) == tan(x)**2
assert TrigSimplify(-1 + csc(x)**2) == cot(x)**2
def test_MergeFactors():
assert simplify(MergeFactors(b/(a - c)**3 , 8*c**3*(b*x + c)**(3/2)/(3*b**4) - 24*c**2*(b*x + c)**(5/2)/(5*b**4) + \
24*c*(b*x + c)**(7/2)/(7*b**4) - 8*(b*x + c)**(9/2)/(9*b**4)) - (8*c**3*(b*x + c)**1.5/(3*b**3) - 24*c**2*(b*x + c)**2.5/(5*b**3) + \
24*c*(b*x + c)**3.5/(7*b**3) - 8*(b*x + c)**4.5/(9*b**3))/(a - c)**3) == 0
assert MergeFactors(x, x) == x**2
assert MergeFactors(x*y, x) == x**2*y
def test_FactorInteger():
assert FactorInteger(2434500) == [(2, 2), (3, 2), (5, 3), (541, 1)]
def test_ContentFactor():
assert ContentFactor(a*b + a*c) == a*(b + c)
def test_Order():
assert Order(a, b) == 1
assert Order(b, a) == -1
assert Order(a, a) == 0
def test_FactorOrder():
assert FactorOrder(1, 1) == 0
assert FactorOrder(1, 2) == -1
assert FactorOrder(2, 1) == 1
assert FactorOrder(a, b) == 1
def test_Smallest():
assert Smallest([2, 1, 3, 4]) == 1
assert Smallest(1, 2) == 1
assert Smallest(-1, -2) == -2
def test_MostMainFactorPosition():
assert MostMainFactorPosition([S(1), S(2), S(3)]) == 1
assert MostMainFactorPosition([S(1), S(7), S(3), S(4), S(5)]) == 2
def test_OrderedQ():
assert OrderedQ([a, b])
assert not OrderedQ([b, a])
def test_MinimumDegree():
assert MinimumDegree(S(1), S(2)) == 1
assert MinimumDegree(S(1), sqrt(2)) == 1
assert MinimumDegree(sqrt(2), S(1)) == 1
assert MinimumDegree(sqrt(3), sqrt(2)) == sqrt(2)
assert MinimumDegree(sqrt(2), sqrt(2)) == sqrt(2)
def test_PositiveFactors():
assert PositiveFactors(S(0)) == 1
assert PositiveFactors(-S(1)) == S(1)
assert PositiveFactors(sqrt(2)) == sqrt(2)
assert PositiveFactors(-log(2)) == log(2)
assert PositiveFactors(sqrt(2)*S(-1)) == sqrt(2)
def test_NonpositiveFactors():
assert NonpositiveFactors(S(0)) == 0
assert NonpositiveFactors(-S(1)) == -1
assert NonpositiveFactors(sqrt(2)) == 1
assert NonpositiveFactors(-log(2)) == -1
def test_Sign():
assert Sign(S(0)) == 0
assert Sign(S(1)) == 1
assert Sign(-S(1)) == -1
def test_PolynomialInQ():
v = log(x)
assert PolynomialInQ(S(1), v, x)
assert PolynomialInQ(v, v, x)
assert PolynomialInQ(1 + v**2, v, x)
assert PolynomialInQ(1 + a*v**2, v, x)
assert not PolynomialInQ(sqrt(v), v, x)
def test_ExponentIn():
v = log(x)
assert ExponentIn(S(1), log(x), x) == 0
assert ExponentIn(S(1) + v, log(x), x) == 1
assert ExponentIn(S(1) + v + v**3, log(x), x) == 3
assert ExponentIn(S(2)*sqrt(v)*v**3, log(x), x) == 3.5
def test_PolynomialInSubst():
v = log(x)
assert PolynomialInSubst(S(1) + log(x)**3, log(x), x) == 1 + x**3
assert PolynomialInSubst(S(1) + log(x), log(x), x) == x + 1
def test_Distrib():
assert Distrib(x, a) == x*a
assert Distrib(x, a + b) == a*x + b*x
def test_DistributeDegree():
assert DistributeDegree(x, m) == x**m
assert DistributeDegree(x**a, m) == x**(a*m)
assert DistributeDegree(a*b, m) == a**m * b**m
def test_FunctionOfPower():
assert FunctionOfPower(a, x) == None
assert FunctionOfPower(x, x) == 1
assert FunctionOfPower(x**3, x) == 3
assert FunctionOfPower(x**3*cos(x**6), x) == 3
def test_DivideDegreesOfFactors():
assert DivideDegreesOfFactors(a**b, S(3)) == a**(b/3)
assert DivideDegreesOfFactors(a**b*c, S(3)) == a**(b/3)*c**(c/3)
def test_MonomialFactor():
assert MonomialFactor(a, x) == [0, a]
assert MonomialFactor(x, x) == [1, 1]
assert MonomialFactor(x + y, x) == [0, x + y]
assert MonomialFactor(log(x), x) == [0, log(x)]
assert MonomialFactor(log(x)*x, x) == [1, log(x)]
def test_NormalizeIntegrand():
assert NormalizeIntegrand((x**2 + 8), x) == x**2 + 8
assert NormalizeIntegrand((x**2 + 3*x)**2, x) == x**2*(x + 3)**2
assert NormalizeIntegrand(a**2*(a + b*x)**2, x) == a**2*(a + b*x)**2
assert NormalizeIntegrand(b**2/(a**2*(a + b*x)**2), x) == b**2/(a**2*(a + b*x)**2)
def test_NormalizeIntegrandAux():
v = (6*A*a*c - 2*A*b**2 + B*a*b)/(a*x**2) - (6*A*a**2*c**2 - 10*A*a*b**2*c - 8*A*a*b*c**2*x + 2*A*b**4 + 2*A*b**3*c*x + 5*B*a**2*b*c + 4*B*a**2*c**2*x - B*a*b**3 - B*a*b**2*c*x)/(a**2*(a + b*x + c*x**2)) + (-2*A*b + B*a)*(4*a*c - b**2)/(a**2*x)
assert NormalizeIntegrandAux(v, x) == (6*A*a*c - 2*A*b**2 + B*a*b)/(a*x**2) - (6*A*a**2*c**2 - 10*A*a*b**2*c + 2*A*b**4 + 5*B*a**2*b*c - B*a*b**3 + x*(-8*A*a*b*c**2 + 2*A*b**3*c + 4*B*a**2*c**2 - B*a*b**2*c))/(a**2*(a + b*x + c*x**2)) + (-2*A*b + B*a)*(4*a*c - b**2)/(a**2*x)
assert NormalizeIntegrandAux((x**2 + 3*x)**2, x) == x**2*(x + 3)**2
assert NormalizeIntegrandAux((x**2 + 8), x) == x**2 + 8
def test_NormalizeIntegrandFactor():
assert NormalizeIntegrandFactor((3*x + x**3)**2, x) == x**2*(x**2 + 3)**2
assert NormalizeIntegrandFactor((x**2 + 8), x) == x**2 + 8
def test_NormalizeIntegrandFactorBase():
assert NormalizeIntegrandFactorBase((x**2 + 8)**3, x) == (x**2 + 8)**3
assert NormalizeIntegrandFactorBase((x**2 + 8), x) == x**2 + 8
assert NormalizeIntegrandFactorBase(a**2*(a + b*x)**2, x) == a**2*(a + b*x)**2
def test_AbsorbMinusSign():
assert AbsorbMinusSign((x + 2)**5*(x + 3)**5) == (-x - 3)**5*(x + 2)**5
assert AbsorbMinusSign((x + 2)**5*(x + 3)**2) == -(x + 2)**5*(x + 3)**2
def test_NormalizeLeadTermSigns():
assert NormalizeLeadTermSigns((-x + 3)*(x**2 + 3)) == (-x + 3)*(x**2 + 3)
assert NormalizeLeadTermSigns(x + 3) == x + 3
def test_SignOfFactor():
assert SignOfFactor(S(-x + 3)) == [1, -x + 3]
assert SignOfFactor(S(-x)) == [-1, x]
def test_NormalizePowerOfLinear():
assert NormalizePowerOfLinear((x + 3)**5, x) == (x + 3)**5
assert NormalizePowerOfLinear(((x + 3)**2) + 3, x) == x**2 + 6*x + 12
def test_SimplifyIntegrand():
assert SimplifyIntegrand((x**2 + 3)**2, x) == (x**2 + 3)**2
assert SimplifyIntegrand(x**2 + 3 + (x**6) + 6, x) == x**6 + x**2 + 9
def test_SimplifyTerm():
assert SimplifyTerm(a**2/b**2, x) == a**2/b**2
assert SimplifyTerm(-6*x/5 + (5*x + 3)**2/25 - 9/25, x) == x**2
def test_togetherSimplify():
assert TogetherSimplify(-6*x/5 + (5*x + 3)**2/25 - 9/25) == x**2
def test_ExpandToSum():
qq = 6
Pqq = e**3
Pq = (d+e*x**2)**3
aa = 2
nn = 2
cc = 1
pp = -1/2
bb = 3
assert nsimplify(ExpandToSum(Pq - Pqq*x**qq - Pqq*(aa*x**(-2*nn + qq)*(-2*nn + qq + 1) + bb*x**(-nn + qq)*(nn*(pp - 1) + qq + 1))/(cc*(2*nn*pp + qq + 1)), x) - \
(d**3 + x**4*(3*d*e**2 - 2.4*e**3) + x**2*(3*d**2*e - 1.2*e**3))) == 0
assert ExpandToSum(x**2 + 3*x + 3, x**3 + 3, x) == x**3*(x**2 + 3*x + 3) + 3*x**2 + 9*x + 9
assert ExpandToSum(x**3 + 6, x) == x**3 + 6
assert ExpandToSum(S(x**2 + 3*x + 3)*3, x) == 3*x**2 + 9*x + 9
assert ExpandToSum((a + b*x), x) == a + b*x
def test_UnifySum():
assert UnifySum((3 + x + 6*x**3 + sin(x)), x) == 6*x**3 + x + sin(x) + 3
assert UnifySum((3 + x + 6*x**3)*3, x) == 18*x**3 + 3*x + 9
def test_FunctionOfInverseLinear():
assert FunctionOfInverseLinear((x)/(a + b*x), x) == [a, b]
assert FunctionOfInverseLinear((c + d*x)/(a + b*x), x) == [a, b]
assert not FunctionOfInverseLinear(1/(a + b*x), x)
def test_PureFunctionOfSinhQ():
v = log(x)
f = sinh(v)
assert PureFunctionOfSinhQ(f, v, x)
assert not PureFunctionOfSinhQ(cosh(v), v, x)
assert PureFunctionOfSinhQ(f**2, v, x)
def test_PureFunctionOfTanhQ():
v = log(x)
f = tanh(v)
assert PureFunctionOfTanhQ(f, v, x)
assert not PureFunctionOfTanhQ(cosh(v), v, x)
assert PureFunctionOfTanhQ(f**2, v, x)
def test_PureFunctionOfCoshQ():
v = log(x)
f = cosh(v)
assert PureFunctionOfCoshQ(f, v, x)
assert not PureFunctionOfCoshQ(sinh(v), v, x)
assert PureFunctionOfCoshQ(f**2, v, x)
def test_IntegerQuotientQ():
u = S(2)*sin(x)
v = sin(x)
assert IntegerQuotientQ(u, v)
assert IntegerQuotientQ(u, u)
assert not IntegerQuotientQ(S(1), S(2))
def test_OddQuotientQ():
u = S(3)*sin(x)
v = sin(x)
assert OddQuotientQ(u, v)
assert OddQuotientQ(u, u)
assert not OddQuotientQ(S(1), S(2))
def test_EvenQuotientQ():
u = S(2)*sin(x)
v = sin(x)
assert EvenQuotientQ(u, v)
assert not EvenQuotientQ(u, u)
assert not EvenQuotientQ(S(1), S(2))
def test_FunctionOfSinhQ():
v = log(x)
assert FunctionOfSinhQ(cos(sinh(v)), v, x)
assert FunctionOfSinhQ(sinh(v), v, x)
assert FunctionOfSinhQ(sinh(v)*cos(sinh(v)), v, x)
def test_FunctionOfCoshQ():
v = log(x)
assert FunctionOfCoshQ(cos(cosh(v)), v, x)
assert FunctionOfCoshQ(cosh(v), v, x)
assert FunctionOfCoshQ(cosh(v)*cos(cosh(v)), v, x)
def test_FunctionOfTanhQ():
v = log(x)
t = Tanh(v)
c = Coth(v)
assert FunctionOfTanhQ(t, v, x)
assert FunctionOfTanhQ(c, v, x)
assert FunctionOfTanhQ(t + c, v, x)
assert FunctionOfTanhQ(t*c, v, x)
assert not FunctionOfTanhQ(sin(x), v, x)
def test_FunctionOfTanhWeight():
v = log(x)
t = Tanh(v)
c = Coth(v)
assert FunctionOfTanhWeight(x, v, x) == 0
assert FunctionOfTanhWeight(sinh(v), v, x) == 0
assert FunctionOfTanhWeight(tanh(v), v, x) == 1
assert FunctionOfTanhWeight(coth(v), v, x) == -1
assert FunctionOfTanhWeight(t**2, v, x) == 1
assert FunctionOfTanhWeight(sinh(v)**2, v, x) == -1
assert FunctionOfTanhWeight(coth(v)*sinh(v)**2, v, x) == -2
def test_FunctionOfHyperbolicQ():
v = log(x)
s = Sinh(v)
t = Tanh(v)
assert not FunctionOfHyperbolicQ(x, v, x)
assert FunctionOfHyperbolicQ(s + t, v, x)
assert FunctionOfHyperbolicQ(sinh(t), v, x)
def test_SmartNumerator():
assert SmartNumerator(x**(-2)) == 1
assert SmartNumerator(x**(2)*a) == x**2*a
def test_SmartDenominator():
assert SmartDenominator(x**(-2)) == x**2
assert SmartDenominator(x**(-2)*1/S(3)) == x**2*3
def test_SubstForAux():
v = log(x)
assert SubstForAux(v, v, x) == x
assert SubstForAux(v**2, v, x) == x**2
assert SubstForAux(x, v, x) == x
assert SubstForAux(v**2, v**4, x) == sqrt(x)
assert SubstForAux(v**2*v, v, x) == x**3
def test_SubstForTrig():
v = log(x)
s, c, t = sin(v), cos(v), tan(v)
assert SubstForTrig(cos(a/2 + b*x/2), x/sqrt(x**2 + 1), 1/sqrt(x**2 + 1), a/2 + b*x/2, x) == 1/sqrt(x**2 + 1)
assert SubstForTrig(s, sin, cos, v, x) == sin
assert SubstForTrig(t, sin(v), cos(v), v, x) == sin(log(x))/cos(log(x))
assert SubstForTrig(sin(2*v), sin(x), cos(x), v, x) == 2*sin(x)*cos(x)
assert SubstForTrig(s*t, sin(x), cos(x), v, x) == sin(x)**2/cos(x)
def test_SubstForHyperbolic():
v = log(x)
s, c, t = sinh(v), cosh(v), tanh(v)
assert SubstForHyperbolic(s, sinh(x), cosh(x), v, x) == sinh(x)
assert SubstForHyperbolic(t, sinh(x), cosh(x), v, x) == sinh(x)/cosh(x)
assert SubstForHyperbolic(sinh(2*v), sinh(x), cosh(x), v, x) == 2*sinh(x)*cosh(x)
assert SubstForHyperbolic(s*t, sinh(x), cosh(x), v, x) == sinh(x)**2/cosh(x)
def test_SubstForFractionalPowerOfLinear():
u = a + b*x
assert not SubstForFractionalPowerOfLinear(u, x)
assert not SubstForFractionalPowerOfLinear(u**(S(2)), x)
assert SubstForFractionalPowerOfLinear(u**(S(1)/2), x) == [x**2, 2, a + b*x, 1/b]
def test_InverseFunctionOfLinear():
u = a + b*x
assert InverseFunctionOfLinear(log(u)*sin(x), x) == log(u)
assert InverseFunctionOfLinear(log(u), x) == log(u)
def test_InertTrigQ():
s = sin(x)
c = cos(x)
assert not InertTrigQ(sin(x), csc(x), cos(h))
assert InertTrigQ(sin(x), csc(x))
assert not InertTrigQ(s, c)
assert InertTrigQ(c)
def test_PowerOfInertTrigSumQ():
func = sin
assert PowerOfInertTrigSumQ((1 + S(2)*(S(3)*func(x**2))**S(5))**3, func, x)
assert PowerOfInertTrigSumQ((1 + 2*(S(3)*func(x**2))**3 + 4*(S(5)*func(x**2))**S(3))**2, func, x)
def test_PiecewiseLinearQ():
assert PiecewiseLinearQ(a + b*x, x)
assert not PiecewiseLinearQ(Log(c*sin(a)**S(3)), x)
assert not PiecewiseLinearQ(x**3, x)
assert PiecewiseLinearQ(atanh(tanh(a + b*x)), x)
assert PiecewiseLinearQ(tanh(atanh(a + b*x)), x)
assert not PiecewiseLinearQ(coth(atanh(a + b*x)), x)
def test_KnownTrigIntegrandQ():
func = sin(a + b*x)
assert KnownTrigIntegrandQ([sin], S(1), x)
assert KnownTrigIntegrandQ([sin], (a + b*func)**m, x)
assert KnownTrigIntegrandQ([sin], (a + b*func)**m*(1 + 2*func), x)
assert KnownTrigIntegrandQ([sin], a + c*func**2, x)
assert KnownTrigIntegrandQ([sin], a + b*func + c*func**2, x)
assert KnownTrigIntegrandQ([sin], (a + b*func)**m*(c + d*func**2), x)
assert KnownTrigIntegrandQ([sin], (a + b*func)**m*(c + d*func + e*func**2), x)
assert not KnownTrigIntegrandQ([cos], (a + b*func)**m, x)
def test_KnownSineIntegrandQ():
assert KnownSineIntegrandQ((a + b*sin(a + b*x))**m, x)
def test_KnownTangentIntegrandQ():
assert KnownTangentIntegrandQ((a + b*tan(a + b*x))**m, x)
def test_KnownCotangentIntegrandQ():
assert KnownCotangentIntegrandQ((a + b*cot(a + b*x))**m, x)
def test_KnownSecantIntegrandQ():
assert KnownSecantIntegrandQ((a + b*sec(a + b*x))**m, x)
def test_TryPureTanSubst():
assert TryPureTanSubst(atan(c*(a + b*tan(a + b*x))), x)
assert TryPureTanSubst(atanh(c*(a + b*cot(a + b*x))), x)
assert not TryPureTanSubst(tan(c*(a + b*cot(a + b*x))), x)
def test_TryPureTanhSubst():
assert not TryPureTanhSubst(log(x), x)
assert TryPureTanhSubst(sin(x), x)
assert not TryPureTanhSubst(atanh(a*tanh(x)), x)
assert not TryPureTanhSubst((a + b*x)**S(2), x)
def test_TryTanhSubst():
assert not TryTanhSubst(log(x), x)
assert not TryTanhSubst(a*(b + c)**3, x)
assert not TryTanhSubst(1/(a + b*sinh(x)**S(3)), x)
assert not TryTanhSubst(sinh(S(3)*x)*cosh(S(4)*x), x)
assert not TryTanhSubst(a*(b*sech(x)**3)**c, x)
def test_GeneralizedBinomialQ():
assert GeneralizedBinomialQ(a*x**q + b*x**n, x)
assert not GeneralizedBinomialQ(a*x**q, x)
def test_GeneralizedTrinomialQ():
assert not GeneralizedTrinomialQ(7 + 2*x**6 + 3*x**12, x)
assert not GeneralizedTrinomialQ(a*x**q + c*x**(2*n-q), x)
def test_SubstForFractionalPowerOfQuotientOfLinears():
assert SubstForFractionalPowerOfQuotientOfLinears(((a + b*x)/(c + d*x))**(S(3)/2), x) == [x**4/(b - d*x**2)**2, 2, (a + b*x)/(c + d*x), -a*d + b*c]
def test_SubstForFractionalPowerQ():
assert SubstForFractionalPowerQ(x, sin(x), x)
assert SubstForFractionalPowerQ(x**2, sin(x), x)
assert not SubstForFractionalPowerQ(x**(S(3)/2), sin(x), x)
assert SubstForFractionalPowerQ(sin(x)**(S(3)/2), sin(x), x)
def test_AbsurdNumberGCD():
assert AbsurdNumberGCD(S(4)) == 4
assert AbsurdNumberGCD(S(4), S(8), S(12)) == 4
assert AbsurdNumberGCD(S(2), S(3), S(12)) == 1
def test_TrigReduce():
assert TrigReduce(cos(x)**2) == cos(2*x)/2 + 1/2
assert TrigReduce(cos(x)**2*sin(x)) == sin(x)/4 + sin(3*x)/4
assert TrigReduce(cos(x)**2+sin(x)) == sin(x) + cos(2*x)/2 + 1/2
assert TrigReduce(cos(x)**2*sin(x)**5) == 5*sin(x)/64 + sin(3*x)/64 - 3*sin(5*x)/64 + sin(7*x)/64
assert TrigReduce(2*sin(x)*cos(x) + 2*cos(x)**2) == sin(2*x) + cos(2*x) + 1
assert TrigReduce(sinh(a + b*x)**2) == cosh(2*a + 2*b*x)/2 - 1/2
assert TrigReduce(sinh(a + b*x)*cosh(a + b*x)) == sinh(2*a + 2*b*x)/2
def test_FunctionOfDensePolynomialsQ():
assert FunctionOfDensePolynomialsQ(x**2 + 3, x)
assert not FunctionOfDensePolynomialsQ(x**2, x)
assert not FunctionOfDensePolynomialsQ(x, x)
assert FunctionOfDensePolynomialsQ(S(2), x)
def test_PureFunctionOfSinQ():
v = log(x)
f = sin(v)
assert PureFunctionOfSinQ(f, v, x)
assert not PureFunctionOfSinQ(cos(v), v, x)
assert PureFunctionOfSinQ(f**2, v, x)
def test_PureFunctionOfTanQ():
v = log(x)
f = tan(v)
assert PureFunctionOfTanQ(f, v, x)
assert not PureFunctionOfTanQ(cos(v), v, x)
assert PureFunctionOfTanQ(f**2, v, x)
def test_PowerVariableSubst():
assert PowerVariableSubst((2*x)**3, 2, x) == 8*x**(3/2)
assert PowerVariableSubst((2*x)**3, 2, x) == 8*x**(3/2)
assert PowerVariableSubst((2*x), 2, x) == 2*x
assert PowerVariableSubst((2*x)**3, 2, x) == 8*x**(3/2)
assert PowerVariableSubst((2*x)**7, 2, x) == 128*x**(7/2)
assert PowerVariableSubst((6+2*x)**7, 2, x) == (2*x + 6)**7
assert PowerVariableSubst((2*x)**7+3, 2, x) == 128*x**(7/2) + 3
def test_PowerVariableDegree():
assert PowerVariableDegree(S(2), 0, 2*x, x) == [0, 2*x]
assert PowerVariableDegree((2*x)**2, 0, 2*x, x) == [2, 1]
assert PowerVariableDegree(x**2, 0, 2*x, x) == [2, 1]
assert PowerVariableDegree(S(4), 0, 2*x, x) == [0, 2*x]
def test_PowerVariableExpn():
assert not PowerVariableExpn((x)**3, 2, x)
assert not PowerVariableExpn((2*x)**3, 2, x)
assert PowerVariableExpn((2*x)**2, 4, x) == [4*x**3, 2, 1]
def test_FunctionOfQ():
assert FunctionOfQ(x**2, sqrt(-exp(2*x**2) + 1)*exp(x**2),x)
assert not FunctionOfQ(S(x**3), x*2, x)
assert FunctionOfQ(S(a), x*2, x)
assert FunctionOfQ(S(3*x), x*2, x)
def test_ExpandTrigExpand():
assert ExpandTrigExpand(1, cos(x), x**2, 2, 2, x) == 4*cos(x**2)**4 - 4*cos(x**2)**2 + 1
assert ExpandTrigExpand(1, cos(x) + sin(x), x**2, 2, 2, x) == 4*sin(x**2)**2*cos(x**2)**2 + 8*sin(x**2)*cos(x**2)**3 - 4*sin(x**2)*cos(x**2) + 4*cos(x**2)**4 - 4*cos(x**2)**2 + 1
def test_TrigToExp():
from sympy.integrals.rubi.utility_function import exp
assert TrigToExp(sin(x)) == -I*(exp(I*x) - exp(-I*x))/2
assert TrigToExp(cos(x)) == exp(I*x)/2 + exp(-I*x)/2
assert TrigToExp(cos(x)*tan(x**2)) == I*(exp(I*x)/2 + exp(-I*x)/2)*(-exp(I*x**2) + exp(-I*x**2))/(exp(I*x**2) + exp(-I*x**2))
assert TrigToExp(cos(x) + sin(x)**2) == -(exp(I*x) - exp(-I*x))**2/4 + exp(I*x)/2 + exp(-I*x)/2
assert Simplify(TrigToExp(cos(x)*tan(x**S(2))*sin(x)**S(2))-(-I*(exp(I*x)/S(2) + exp(-I*x)/S(2))*(exp(I*x) - exp(-I*x))**S(2)*(-exp(I*x**S(2)) + exp(-I*x**S(2)))/(S(4)*(exp(I*x**S(2)) + exp(-I*x**S(2)))))) == 0
def test_ExpandTrigReduce():
assert ExpandTrigReduce(2*cos(3 + x)**3, x) == 3*cos(x + 3)/2 + cos(3*x + 9)/2
assert ExpandTrigReduce(2*sin(x)**3+cos(2 + x), x) == 3*sin(x)/2 - sin(3*x)/2 + cos(x + 2)
assert ExpandTrigReduce(cos(x + 3)**2, x) == cos(2*x + 6)/2 + 1/2
def test_NormalizeTrig():
assert NormalizeTrig(S(2*sin(2 + x)), x) == 2*sin(x + 2)
assert NormalizeTrig(S(2*sin(2 + x)**3), x) == 2*sin(x + 2)**3
assert NormalizeTrig(S(2*sin((2 + x)**2)**3), x) == 2*sin(x**2 + 4*x + 4)**3
def test_FunctionOfTrigQ():
v = log(x)
s = sin(v)
t = tan(v)
assert not FunctionOfTrigQ(x, v, x)
assert FunctionOfTrigQ(s + t, v, x)
assert FunctionOfTrigQ(sin(t), v, x)
def test_RationalFunctionExpand():
assert RationalFunctionExpand(x**S(5)*(e + f*x)**n/(a + b*x**S(3)), x) == -a*x**2*(e + f*x)**n/(b*(a + b*x**3)) +\
e**2*(e + f*x)**n/(b*f**2) - 2*e*(e + f*x)**(n + 1)/(b*f**2) + (e + f*x)**(n + 2)/(b*f**2)
assert RationalFunctionExpand(x**S(3)*(S(2)*x + 2)**S(2)/(2*x**2 + 1), x) == 2*x**3 + 4*x**2 + x + (- x + 2)/(2*x**2 + 1) - 2
assert RationalFunctionExpand((a + b*x + c*x**4)*log(x)**3, x) == a*log(x)**3 + b*x*log(x)**3 + c*x**4*log(x)**3
assert RationalFunctionExpand(a + b*x + c*x**4, x) == a + b*x + c*x**4
def test_SameQ():
assert SameQ(1, 1, 1)
assert not SameQ(1, 1, 2)
def test_Map2():
assert Map2(Add, [a, b, c], [x, y, z]) == [a + x, b + y, c + z]
def test_ConstantFactor():
assert ConstantFactor(a + a*x**3, x) == [a, x**3 + 1]
assert ConstantFactor(a, x) == [a, 1]
assert ConstantFactor(x, x) == [1, x]
assert ConstantFactor(x**S(3), x) == [1, x**3]
assert ConstantFactor(x**(S(3)/2), x) == [1, x**(3/2)]
assert ConstantFactor(a*x**3, x) == [a, x**3]
assert ConstantFactor(a + x**3, x) == [1, a + x**3]
def test_CommonFactors():
assert CommonFactors([a, a, a]) == [a, 1, 1, 1]
assert CommonFactors([x*S(2), x**S(3)*S(2), sin(x)*x*S(2)]) == [2, x, x**3, x*sin(x)]
assert CommonFactors([x, x**S(3), sin(x)*x]) == [1, x, x**3, x*sin(x)]
assert CommonFactors([S(2), S(4), S(6)]) == [2, 1, 2, 3]
def test_FunctionOfLinear():
f = sin(a + b*x)
assert FunctionOfLinear(f, x) == [sin(x), a, b]
assert FunctionOfLinear(a + b*x, x) == [x, a, b]
assert not FunctionOfLinear(a, x)
def test_FunctionOfExponentialQ():
assert FunctionOfExponentialQ(exp(x + exp(x) + exp(exp(x))), x)
assert FunctionOfExponentialQ(a**(a + b*x), x)
assert FunctionOfExponentialQ(a**(b*x), x)
assert not FunctionOfExponentialQ(a**sin(a + b*x), x)
def test_FunctionOfExponential():
assert FunctionOfExponential(a**(a + b*x), x)
def test_FunctionOfExponentialFunction():
assert FunctionOfExponentialFunction(a**(a + b*x), x) == x
assert FunctionOfExponentialFunction(S(2)*a**(a + b*x), x) == 2*x
def test_FunctionOfTrig():
assert FunctionOfTrig(sin(x + 1), x + 1, x) == x + 1
assert FunctionOfTrig(sin(x), x) == x
assert not FunctionOfTrig(cos(x**2 + 1), x)
assert FunctionOfTrig(sin(a+b*x)**3, x) == a+b*x
def test_AlgebraicTrigFunctionQ():
assert AlgebraicTrigFunctionQ(sin(x + 3), x)
assert AlgebraicTrigFunctionQ(x, x)
assert AlgebraicTrigFunctionQ(x + 1, x)
assert AlgebraicTrigFunctionQ(sinh(x + 1), x)
assert AlgebraicTrigFunctionQ(sinh(x + 1)**2, x)
assert not AlgebraicTrigFunctionQ(sinh(x**2 + 1)**2, x)
def test_FunctionOfHyperbolic():
assert FunctionOfTrig(sin(x + 1), x + 1, x) == x + 1
assert FunctionOfTrig(sin(x), x) == x
assert not FunctionOfTrig(cos(x**2 + 1), x)
def test_FunctionOfExpnQ():
assert FunctionOfExpnQ(x, x, x) == 1
assert FunctionOfExpnQ(x**2, x, x) == 2
assert FunctionOfExpnQ(x**2.1, x, x) == 1
assert not FunctionOfExpnQ(x, x**2, x)
assert not FunctionOfExpnQ(x + 1, (x + 5)**2, x)
assert not FunctionOfExpnQ(x + 1, (x + 1)**2, x)
def test_PureFunctionOfCosQ():
v = log(x)
f = cos(v)
assert PureFunctionOfCosQ(f, v, x)
assert not PureFunctionOfCosQ(sin(v), v, x)
assert PureFunctionOfCosQ(f**2, v, x)
def test_PureFunctionOfCotQ():
v = log(x)
f = cot(v)
assert PureFunctionOfCotQ(f, v, x)
assert not PureFunctionOfCotQ(sin(v), v, x)
assert PureFunctionOfCotQ(f**2, v, x)
def test_FunctionOfSinQ():
v = log(x)
assert FunctionOfSinQ(cos(sin(v)), v, x)
assert FunctionOfSinQ(sin(v), v, x)
assert FunctionOfSinQ(sin(v)*cos(sin(v)), v, x)
def test_FunctionOfCosQ():
v = log(x)
assert FunctionOfCosQ(cos(cos(v)), v, x)
assert FunctionOfCosQ(cos(v), v, x)
assert FunctionOfCosQ(cos(v)*cos(cos(v)), v, x)
def test_FunctionOfTanQ():
v = log(x)
t = tan(v)
c = cot(v)
assert FunctionOfTanQ(t, v, x)
assert FunctionOfTanQ(c, v, x)
assert FunctionOfTanQ(t + c, v, x)
assert FunctionOfTanQ(t*c, v, x)
assert not FunctionOfTanQ(sin(x), v, x)
def test_FunctionOfTanWeight():
v = log(x)
t = tan(v)
c = cot(v)
assert FunctionOfTanWeight(x, v, x) == 0
assert FunctionOfTanWeight(sin(v), v, x) == 0
assert FunctionOfTanWeight(tan(v), v, x) == 1
assert FunctionOfTanWeight(cot(v), v, x) == -1
assert FunctionOfTanWeight(t**2, v, x) == 1
assert FunctionOfTanWeight(sin(v)**2, v, x) == -1
assert FunctionOfTanWeight(cot(v)*sin(v)**2, v, x) == -2
def test_OddTrigPowerQ():
assert not OddTrigPowerQ(sin(x)**3, 1, x)
assert OddTrigPowerQ(sin(3),1,x)
assert OddTrigPowerQ(sin(3*x),x,x)
assert OddTrigPowerQ(sin(3*x)**3,x,x)
def test_FunctionOfLog():
assert not FunctionOfLog(x**2*(a + b*x)**3*exp(-a - b*x) ,False, False, x)
assert FunctionOfLog(log(2*x**8)*2 + log(2*x**8) + 1, x) == [3*x + 1, 2*x**8, 8]
assert FunctionOfLog(log(2*x)**2,x) == [x**2, 2*x, 1]
assert FunctionOfLog(log(3*x**3)**2 + 1,x) == [x**2 + 1, 3*x**3, 3]
assert FunctionOfLog(log(2*x**8)*2,x) == [2*x, 2*x**8, 8]
assert not FunctionOfLog(2*sin(x)*2,x)
def test_EulerIntegrandQ():
assert EulerIntegrandQ((2*x + 3*((x + 1)**3)**1.5)**(-3), x)
assert not EulerIntegrandQ((2*x + (2*x**2)**2)**3, x)
assert not EulerIntegrandQ(3*x**2 + 5*x + 1, x)
def test_Divides():
assert not Divides(x, a*x**2, x)
assert Divides(x, a*x, x) == a
def test_EasyDQ():
assert EasyDQ(3*x**2, x)
assert EasyDQ(3*x**3 - 6, x)
assert EasyDQ(x**3, x)
assert EasyDQ(sin(x**log(3)), x)
def test_ProductOfLinearPowersQ():
assert ProductOfLinearPowersQ(S(1), x)
assert ProductOfLinearPowersQ((x + 1)**3, x)
assert not ProductOfLinearPowersQ((x**2 + 1)**3, x)
assert ProductOfLinearPowersQ(x + 1, x)
def test_Rt():
b = symbols('b')
assert Rt(-b**2, 4) == (-b**2)**(S(1)/S(4))
assert Rt(x**2, 2) == x
assert Rt(S(2 + 3*I), S(8)) == (2 + 3*I)**(1/8)
assert Rt(x**2 + 4 + 4*x, 2) == x + 2
assert Rt(S(8), S(3)) == 2
assert Rt(S(16807), S(5)) == 7
def test_NthRoot():
assert NthRoot(S(14580), S(3)) == 9*2**(S(2)/S(3))*5**(S(1)/S(3))
assert NthRoot(9, 2) == 3.0
assert NthRoot(81, 2) == 9.0
assert NthRoot(81, 4) == 3.0
def test_AtomBaseQ():
assert not AtomBaseQ(x**2)
assert AtomBaseQ(x**3)
assert AtomBaseQ(x)
assert AtomBaseQ(S(2)**3)
assert not AtomBaseQ(sin(x))
def test_SumBaseQ():
assert not SumBaseQ((x + 1)**2)
assert SumBaseQ((x + 1)**3)
assert SumBaseQ((3*x+3))
assert not SumBaseQ(x)
def test_NegSumBaseQ():
assert not NegSumBaseQ(-x + 1)
assert NegSumBaseQ(x - 1)
assert not NegSumBaseQ((x - 1)**2)
assert NegSumBaseQ((x - 1)**3)
def test_AllNegTermQ():
x = Symbol('x', negative=True)
assert AllNegTermQ(x)
assert not AllNegTermQ(x + 2)
assert AllNegTermQ(x - 2)
assert AllNegTermQ((x - 2)**3)
assert not AllNegTermQ((x - 2)**2)
def test_TrigSquareQ():
assert TrigSquareQ(sin(x)**2)
assert TrigSquareQ(cos(x)**2)
assert not TrigSquareQ(tan(x)**2)
def test_Inequality():
assert not Inequality(S('0'), Less, m, LessEqual, S('1'))
assert Inequality(S('0'), Less, S('1'))
assert Inequality(S('0'), Less, S('1'), LessEqual, S('5'))
def test_SplitProduct():
assert SplitProduct(OddQ, S(3)*x) == [3, x]
assert not SplitProduct(OddQ, S(2)*x)
def test_SplitSum():
assert SplitSum(FracPart, sin(x)) == [sin(x), 0]
assert SplitSum(FracPart, sin(x) + S(2)) == [sin(x), S(2)]
def test_Complex():
assert Complex(a, b) == a + I*b
def test_SimpFixFactor():
assert SimpFixFactor((a*c + b*c)**S(4), x) == (a*c + b*c)**4
assert SimpFixFactor((a*Complex(0, c) + b*Complex(0, d))**S(3), x) == -I*(a*c + b*d)**3
assert SimpFixFactor((a*Complex(0, d) + b*Complex(0, e) + c*Complex(0, f))**S(2), x) == -(a*d + b*e + c*f)**2
assert SimpFixFactor((a + b*x**(-1/S(2))*x**S(3))**S(3), x) == (a + b*x**(5/2))**3
assert SimpFixFactor((a*c + b*c**S(2)*x**S(2))**S(3), x) == c**3*(a + b*c*x**2)**3
assert SimpFixFactor((a*c**S(2) + b*c**S(1)*x**S(2))**S(3), x) == c**3*(a*c + b*x**2)**3
assert SimpFixFactor(a*cos(x)**2 + a*sin(x)**2 + v, x) == a*cos(x)**2 + a*sin(x)**2 + v
def test_SimplifyAntiderivative():
assert SimplifyAntiderivative(acoth(coth(x)), x) == x
assert SimplifyAntiderivative(a*x, x) == a*x
assert SimplifyAntiderivative(atanh(cot(x)), x) == atanh(2*sin(x)*cos(x))/2
assert SimplifyAntiderivative(a*cos(x)**2 + a*sin(x)**2 + v, x) == a*cos(x)**2 + a*sin(x)**2
def test_FixSimplify():
assert FixSimplify(x*Complex(0, a)*(v*Complex(0, b) + w)**S(3)) == a*x*(b*v - I*w)**3
def test_TrigSimplifyAux():
assert TrigSimplifyAux(a*cos(x)**2 + a*sin(x)**2 + v) == a + v
assert TrigSimplifyAux(x**2) == x**2
def test_SubstFor():
assert SubstFor(x**2 + 1, tanh(x), x) == tanh(x)
assert SubstFor(x**2, sinh(x), x) == sinh(sqrt(x))
def test_FresnelS():
assert FresnelS(oo) == 1/2
assert FresnelS(0) == 0
def test_FresnelC():
assert FresnelC(0) == 0
assert FresnelC(oo) == 1/2
def test_Erfc():
assert Erfc(0) == 1
assert Erfc(oo) == 0
def test_Erfi():
assert Erfi(oo) == oo
assert Erfi(0) == 0
def test_Gamma():
assert Gamma(u) == gamma(u)
def test_ElementaryFunctionQ():
assert ElementaryFunctionQ(x + y)
assert ElementaryFunctionQ(sin(x + y))
assert ElementaryFunctionQ(E**(x*a))
def test_Util_Part():
from sympy.integrals.rubi.utility_function import Util_Part
assert Util_Part(1, a + b).doit() == a
assert Util_Part(c, a + b).doit() == Util_Part(c, a + b)
def test_Part():
assert Part([1, 2, 3], 1) == 1
assert Part(a*b, 1) == a
def test_PolyLog():
assert PolyLog(a, b) == polylog(a, b)
def test_PureFunctionOfCothQ():
v = log(x)
assert PureFunctionOfCothQ(coth(v), v, x)
assert PureFunctionOfCothQ(a + coth(v), v, x)
assert not PureFunctionOfCothQ(sin(v), v, x)
def test_ExpandIntegrand():
assert ExpandIntegrand(sqrt(a + b*x**S(2) + c*x**S(4)), (f*x)**(S(3)/2)*(d + e*x**S(2)), x) == \
d*(f*x)**(3/2)*sqrt(a + b*x**2 + c*x**4) + e*(f*x)**(7/2)*sqrt(a + b*x**2 + c*x**4)/f**2
assert ExpandIntegrand((6*A*a*c - 2*A*b**2 + B*a*b - 2*c*x*(A*b - 2*B*a))/(x**2*(a + b*x + c*x**2)), x) == \
(6*A*a*c - 2*A*b**2 + B*a*b)/(a*x**2) + (-6*A*a**2*c**2 + 10*A*a*b**2*c - 2*A*b**4 - 5*B*a**2*b*c + B*a*b**3 + x*(8*A*a*b*c**2 - 2*A*b**3*c - 4*B*a**2*c**2 + B*a*b**2*c))/(a**2*(a + b*x + c*x**2)) + (-2*A*b + B*a)*(4*a*c - b**2)/(a**2*x)
assert ExpandIntegrand(x**2*(e + f*x)**3*F**(a + b*(c + d*x)**1), x) == F**(a + b*(c + d*x))*e**2*(e + f*x)**3/f**2 - 2*F**(a + b*(c + d*x))*e*(e + f*x)**4/f**2 + F**(a + b*(c + d*x))*(e + f*x)**5/f**2
assert ExpandIntegrand((x)*(a + b*x)**2*f**(e*(c + d*x)**n), x) == a**2*f**(e*(c + d*x)**n)*x + 2*a*b*f**(e*(c + d*x)**n)*x**2 + b**2*f**(e*(c + d*x)**n)*x**3
assert ExpandIntegrand(sin(x)**3*(a + b*(1/sin(x)))**2, x) == a**2*sin(x)**3 + 2*a*b*sin(x)**2 + b**2*sin(x)
assert ExpandIntegrand(x*(a + b*ArcSin(c + d*x))**n, x) == -c*(a + b*asin(c + d*x))**n/d + (a + b*asin(c + d*x))**n*(c + d*x)/d
assert ExpandIntegrand((a + b*x)**S(3)*(A + B*x)/(c + d*x), x) == B*(a + b*x)**3/d + b*(a + b*x)**2*(A*d - B*c)/d**2 + b*(a + b*x)*(A*d - B*c)*(a*d - b*c)/d**3 + b*(A*d - B*c)*(a*d - b*c)**2/d**4 + (A*d - B*c)*(a*d - b*c)**3/(d**4*(c + d*x))
assert ExpandIntegrand((x**2)*(S(3)*x)**(S(1)/2), x) ==sqrt(3)*x**(5/2)
assert ExpandIntegrand((x)*(sin(x))**(S(1)/2), x) == x*sqrt(sin(x))
assert ExpandIntegrand(x*(e + f*x)**2*F**(b*(c + d*x)), x) == -F**(b*(c + d*x))*e*(e + f*x)**2/f + F**(b*(c + d*x))*(e + f*x)**3/f
assert ExpandIntegrand(x**m*(e + f*x)**2*F**(b*(c + d*x)**n), x) == F**(b*(c + d*x)**n)*e**2*x**m + 2*F**(b*(c + d*x)**n)*e*f*x*x**m + F**(b*(c + d*x)**n)*f**2*x**2*x**m
assert simplify(ExpandIntegrand((S(1) - S(1)*x**S(2))**(-S(3)), x) - (-S(3)/(8*(x**2 - 1)) + S(3)/(16*(x + 1)**2) + S(1)/(S(8)*(x + 1)**3) + S(3)/(S(16)*(x - 1)**2) - S(1)/(S(8)*(x - 1)**3))) == 0
assert ExpandIntegrand(-S(1), 1/((-q - x)**3*(q - x)**3), x) == 1/(8*q**3*(q + x)**3) - 1/(8*q**3*(-q + x)**3) - 3/(8*q**4*(-q**2 + x**2)) + 3/(16*q**4*(q + x)**2) + 3/(16*q**4*(-q + x)**2)
assert ExpandIntegrand((1 + 1*x)**(3)/(2 + 1*x), x) == x**2 + x + 1 - 1/(x + 2)
assert ExpandIntegrand((c + d*x**1 + e*x**2)/(1 - x**3), x) == (c - (-1)**(S(1)/3)*d + (-1)**(S(2)/3)*e)/(-3*(-1)**(S(2)/3)*x + 3) + (c + (-1)**(S(2)/3)*d - (-1)**(S(1)/3)*e)/(3*(-1)**(S(1)/3)*x + 3) + (c + d + e)/(-3*x + 3)
assert ExpandIntegrand((c + d*x**1 + e*x**2 + f*x**3)/(1 - x**4), x) == (c + I*d - e - I*f)/(4*I*x + 4) + (c - I*d - e + I*f)/(-4*I*x + 4) + (c - d + e - f)/(4*x + 4) + (c + d + e + f)/(-4*x + 4)
assert ExpandIntegrand((d + e*(f + g*x))/(2 + 3*x + 1*x**2), x) == (-2*d - 2*e*f + 4*e*g)/(2*x + 4) + (2*d + 2*e*f - 2*e*g)/(2*x + 2)
assert ExpandIntegrand(x/(a*x**3 + b*Sqrt(c + d*x**6)), x) == a*x**4/(-b**2*c + x**6*(a**2 - b**2*d)) + b*x*sqrt(c + d*x**6)/(b**2*c + x**6*(-a**2 + b**2*d))
assert simplify(ExpandIntegrand(x**1*(1 - x**4)**(-2), x) - (x/(S(4)*(x**2 + 1)) + x/(S(4)*(x**2 + 1)**2) - x/(S(4)*(x**2 - 1)) + x/(S(4)*(x**2 - 1)**2))) == 0
assert simplify(ExpandIntegrand((-1 + x**S(6))**(-3), x) - (S(3)/(S(8)*(x**6 - 1)) - S(3)/(S(16)*(x**S(3) + S(1))**S(2)) - S(1)/(S(8)*(x**S(3) + S(1))**S(3)) - S(3)/(S(16)*(x**S(3) - S(1))**S(2)) + S(1)/(S(8)*(x**S(3) - S(1))**S(3)))) == 0
assert simplify(ExpandIntegrand(u**1*(a + b*u**2 + c*u**4)**(-1), x)) == simplify(1/(2*b*(u + sqrt(-(a + c*u**4)/b))) - 1/(2*b*(-u + sqrt(-(a + c*u**4)/b))))
assert simplify(ExpandIntegrand((1 + 1*u + 1*u**2)**(-2), x) - (S(1)/(S(2)*(-u - 1)*(-u**2 - u - 1)) + S(1)/(S(4)*(-u - 1)*(u + sqrt(-u - 1))**2) + S(1)/(S(4)*(-u - 1)*(u - sqrt(-u - 1))**2))) == 0
assert ExpandIntegrand(x*(a + b*Log(c*(d*(e + f*x)**p)**q))**n, x) == -e*(a + b*log(c*(d*(e + f*x)**p)**q))**n/f + (a + b*log(c*(d*(e + f*x)**p)**q))**n*(e + f*x)/f
assert ExpandIntegrand(x*f**(e*(c + d*x)*S(1)), x) == f**(e*(c + d*x))*x
assert simplify(ExpandIntegrand((x)*(a + b*x)**m*Log(c*(d + e*x**n)**p), x) - (-a*(a + b*x)**m*log(c*(d + e*x**n)**p)/b + (a + b*x)**(m + S(1))*log(c*(d + e*x**n)**p)/b)) == 0
assert simplify(ExpandIntegrand(u*(a + b*F**v)**S(2)*(c + d*F**v)**S(-3), x) - (b**2*u/(d**2*(F**v*d + c)) + 2*b*u*(a*d - b*c)/(d**2*(F**v*d + c)**2) + u*(a*d - b*c)**2/(d**2*(F**v*d + c)**3))) == 0
assert ExpandIntegrand((S(1) + 1*x)**S(2)*f**(e*(1 + S(1)*x)**n)/(g + h*x), x) == f**(e*(x + 1)**n)*(x + 1)/h + f**(e*(x + 1)**n)*(-g + h)/h**2 + f**(e*(x + 1)**n)*(g - h)**2/(h**2*(g + h*x))
assert ExpandIntegrand((a*c - b*c*x)**2/(a + b*x)**2, x) == 4*a**2*c**2/(a + b*x)**2 - 4*a*c**2/(a + b*x) + c**2
assert simplify(ExpandIntegrand(x**2*(1 - 1*x**2)**(-2), x) - (1/(S(2)*(x**2 - 1)) + 1/(S(4)*(x + 1)**2) + 1/(S(4)*(x - 1)**2))) == 0
assert ExpandIntegrand((a + x)**2, x) == a**2 + 2*a*x + x**2
assert ExpandIntegrand((a + b*x)**S(2)/x**3, x) == a**2/x**3 + 2*a*b/x**2 + b**2/x
assert ExpandIntegrand(1/(x**2*(a + b*x)**2), x) == b**2/(a**2*(a + b*x)**2) + 1/(a**2*x**2) + 2*b**2/(a**3*(a + b*x)) - 2*b/(a**3*x)
assert ExpandIntegrand((1 + x)**3/x, x) == x**2 + 3*x + 3 + 1/x
assert ExpandIntegrand((1 + 2*(3 + 4*x**2))/(2 + 3*x**2 + 1*x**4), x) == 18/(2*x**2 + 4) - 2/(2*x**2 + 2)
assert ExpandIntegrand((c + d*x**2 + e*x**3)/(1 - 1*x**4), x) == (c - d - I*e)/(4*I*x + 4) + (c - d + I*e)/(-4*I*x + 4) + (c + d - e)/(4*x + 4) + (c + d + e)/(-4*x + 4)
assert ExpandIntegrand((a + b*x)**2/(c + d*x), x) == b*(a + b*x)/d + b*(a*d - b*c)/d**2 + (a*d - b*c)**2/(d**2*(c + d*x))
assert ExpandIntegrand(x**2*(a + b*Log(c*(d*(e + f*x)**p)**q))**n, x) == e**2*(a + b*log(c*(d*(e + f*x)**p)**q))**n/f**2 - 2*e*(a + b*log(c*(d*(e + f*x)**p)**q))**n*(e + f*x)/f**2 + (a + b*log(c*(d*(e + f*x)**p)**q))**n*(e + f*x)**2/f**2
assert ExpandIntegrand(x*(1 + 2*x)**3*log(2*(1 + 1*x**2)**1), x) == 8*x**4*log(2*x**2 + 2) + 12*x**3*log(2*x**2 + 2) + 6*x**2*log(2*x**2 + 2) + x*log(2*x**2 + 2)
assert ExpandIntegrand((1 + 1*x)**S(3)*f**(e*(1 + 1*x)**n)/(g + h*x), x) == f**(e*(x + 1)**n)*(x + 1)**2/h + f**(e*(x + 1)**n)*(-g + h)*(x + 1)/h**2 + f**(e*(x + 1)**n)*(-g + h)**2/h**3 - f**(e*(x + 1)**n)*(g - h)**3/(h**3*(g + h*x))
def test_Dist():
assert Dist(x, a + b, x) == a*x + b*x
assert Dist(x, Integral(a + b , x), x) == x*Integral(a + b, x)
assert Dist(3*x,(a+b), x) - Dist(2*x, (a+b), x) == a*x + b*x
assert Dist(3*x,(a+b), x) + Dist(2*x, (a+b), x) == 5*a*x + 5*b*x
assert Dist(x, c*Integral((a + b), x), x) == c*x*Integral(a + b, x)
def test_IntegralFreeQ():
assert not IntegralFreeQ(Integral(a, x))
assert IntegralFreeQ(a + b)
def test_OneQ():
from sympy.integrals.rubi.utility_function import OneQ
assert OneQ(S(1))
assert not OneQ(S(2))
def test_DerivativeDivides():
assert not DerivativeDivides(x, x, x)
assert not DerivativeDivides(a, x + y, b)
assert DerivativeDivides(a + x, a, x) == a
assert DerivativeDivides(a + b, x + y, b) == x + y
def test_LogIntegral():
from sympy.integrals.rubi.utility_function import LogIntegral
assert LogIntegral(a) == li(a)
def test_SinIntegral():
from sympy.integrals.rubi.utility_function import SinIntegral
assert SinIntegral(a) == Si(a)
def test_CosIntegral():
from sympy.integrals.rubi.utility_function import CosIntegral
assert CosIntegral(a) == Ci(a)
def test_SinhIntegral():
from sympy.integrals.rubi.utility_function import SinhIntegral
assert SinhIntegral(a) == Shi(a)
def test_CoshIntegral():
from sympy.integrals.rubi.utility_function import CoshIntegral
assert CoshIntegral(a) == Chi(a)
def test_ExpIntegralEi():
from sympy.integrals.rubi.utility_function import ExpIntegralEi
assert ExpIntegralEi(a) == Ei(a)
def test_ExpIntegralE():
from sympy.integrals.rubi.utility_function import ExpIntegralE
assert ExpIntegralE(a, z) == expint(a, z)
def test_LogGamma():
from sympy.integrals.rubi.utility_function import LogGamma
assert LogGamma(a) == loggamma(a)
def test_Factorial():
from sympy.integrals.rubi.utility_function import Factorial
assert Factorial(S(5)) == 120
def test_Zeta():
from sympy.integrals.rubi.utility_function import Zeta
assert Zeta(a, z) == zeta(a, z)
def test_HypergeometricPFQ():
from sympy.integrals.rubi.utility_function import HypergeometricPFQ
assert HypergeometricPFQ([a, b], [c], z) == hyper([a, b], [c], z)
def test_PolyGamma():
assert PolyGamma(S(2), S(3)) == polygamma(2, 3)
def test_ProductLog():
from sympy import N
assert N(ProductLog(S(5.0)), 5) == N(1.32672466524220, 5)
assert N(ProductLog(S(2), S(3.5)), 5) == N(-1.14064876353898 + 10.8912237027092*I, 5)
def test_PolynomialQuotient():
assert PolynomialQuotient(log((-a*d + b*c)/(b*(c + d*x)))/(c + d*x), a + b*x, e) == log((-a*d + b*c)/(b*(c + d*x)))/((a + b*x)*(c + d*x))
assert PolynomialQuotient(x**2, x + a, x) == -a + x
def test_PolynomialRemainder():
assert PolynomialRemainder(log((-a*d + b*c)/(b*(c + d*x)))/(c + d*x), a + b*x, e) == 0
assert PolynomialRemainder(x**2, x + a, x) == a**2
def test_Floor():
assert Floor(S(7.5)) == 7
assert Floor(S(15.5), S(6)) == 12
def test_Factor():
from sympy.integrals.rubi.utility_function import Factor
assert Factor(a*b + a*c) == a*(b + c)
def test_Rule():
from sympy.integrals.rubi.utility_function import Rule
assert Rule(x, S(5)) == {x: 5}
def test_Distribute():
assert Distribute((a + b)*c + (a + b)*d, Add) == c*(a + b) + d*(a + b)
assert Distribute((a + b)*(c + e), Add) == a*c + a*e + b*c + b*e
def test_CoprimeQ():
assert CoprimeQ(S(7), S(5))
assert not CoprimeQ(S(6), S(3))
def test_Discriminant():
from sympy.integrals.rubi.utility_function import Discriminant
assert Discriminant(a*x**2 + b*x + c, x) == b**2 - 4*a*c
assert Discriminant(1/x, x) == Discriminant(1/x, x)
def test_Sum_doit():
assert Sum_doit(2*x + 2, [x, 0, 1.7]) == 6
def test_DeactivateTrig():
assert DeactivateTrig(sec(a + b*x), x) == sec(a + b*x)
def test_Negative():
from sympy.integrals.rubi.utility_function import Negative
assert Negative(S(-2))
assert not Negative(S(0))
def test_Quotient():
from sympy.integrals.rubi.utility_function import Quotient
assert Quotient(17, 5) == 3
def test_process_trig():
assert process_trig(x*cot(x)) == x/tan(x)
assert process_trig(coth(x)*csc(x)) == S(1)/(tanh(x)*sin(x))
def test_replace_pow_exp():
from sympy.integrals.rubi.utility_function import exp as rubi_exp
assert replace_pow_exp(rubi_exp(S(5))) == exp(S(5))
def test_rubi_unevaluated_expr():
from sympy.integrals.rubi.utility_function import rubi_unevaluated_expr
assert rubi_unevaluated_expr(a)*rubi_unevaluated_expr(b) == rubi_unevaluated_expr(b)*rubi_unevaluated_expr(a)
def test_rubi_exp():
# class name in utility_function is `exp`. To avoid confusion `rubi_exp` has been used here
from sympy.integrals.rubi.utility_function import exp as rubi_exp
assert isinstance(rubi_exp(a), Pow)
def test_rubi_log():
# class name in utility_function is `log`. To avoid confusion `rubi_log` has been used here
from sympy.integrals.rubi.utility_function import exp as rubi_exp, log as rubi_log
assert rubi_log(rubi_exp(S(a))) == a
|
ba0b01844ff526e21c199605bf7bd58f41297470cf1f0e7bce7dede1a89476a1
|
from sympy import (
Symbol, Wild, sin, cos, exp, sqrt, pi, Function, Derivative,
Integer, Eq, symbols, Add, I, Float, log, Rational,
Lambda, atan2, cse, cot, tan, S, Tuple, Basic, Dict,
Piecewise, oo, Mul, factor, nsimplify, zoo, Subs, RootOf,
AccumBounds, Matrix, zeros, ZeroMatrix)
from sympy.core.basic import _aresame
from sympy.utilities.pytest import XFAIL, raises
from sympy.abc import a, x, y, z
def test_subs():
n3 = Rational(3)
e = x
e = e.subs(x, n3)
assert e == Rational(3)
e = 2*x
assert e == 2*x
e = e.subs(x, n3)
assert e == Rational(6)
def test_subs_Matrix():
z = zeros(2)
z1 = ZeroMatrix(2, 2)
assert (x*y).subs({x:z, y:0}) in [z, z1]
assert (x*y).subs({y:z, x:0}) == 0
assert (x*y).subs({y:z, x:0}, simultaneous=True) in [z, z1]
assert (x + y).subs({x: z, y: z}, simultaneous=True) in [z, z1]
assert (x + y).subs({x: z, y: z}) in [z, z1]
# Issue #15528
assert Mul(Matrix([[3]]), x).subs(x, 2.0) == Matrix([[6.0]])
# Does not raise a TypeError, see comment on the MatAdd postprocessor
assert Add(Matrix([[3]]), x).subs(x, 2.0) == Add(Matrix([[3]]), 2.0)
def test_subs_AccumBounds():
e = x
e = e.subs(x, AccumBounds(1, 3))
assert e == AccumBounds(1, 3)
e = 2*x
e = e.subs(x, AccumBounds(1, 3))
assert e == AccumBounds(2, 6)
e = x + x**2
e = e.subs(x, AccumBounds(-1, 1))
assert e == AccumBounds(-1, 2)
def test_trigonometric():
n3 = Rational(3)
e = (sin(x)**2).diff(x)
assert e == 2*sin(x)*cos(x)
e = e.subs(x, n3)
assert e == 2*cos(n3)*sin(n3)
e = (sin(x)**2).diff(x)
assert e == 2*sin(x)*cos(x)
e = e.subs(sin(x), cos(x))
assert e == 2*cos(x)**2
assert exp(pi).subs(exp, sin) == 0
assert cos(exp(pi)).subs(exp, sin) == 1
i = Symbol('i', integer=True)
zoo = S.ComplexInfinity
assert tan(x).subs(x, pi/2) is zoo
assert cot(x).subs(x, pi) is zoo
assert cot(i*x).subs(x, pi) is zoo
assert tan(i*x).subs(x, pi/2) == tan(i*pi/2)
assert tan(i*x).subs(x, pi/2).subs(i, 1) is zoo
o = Symbol('o', odd=True)
assert tan(o*x).subs(x, pi/2) == tan(o*pi/2)
def test_powers():
assert sqrt(1 - sqrt(x)).subs(x, 4) == I
assert (sqrt(1 - x**2)**3).subs(x, 2) == - 3*I*sqrt(3)
assert (x**Rational(1, 3)).subs(x, 27) == 3
assert (x**Rational(1, 3)).subs(x, -27) == 3*(-1)**Rational(1, 3)
assert ((-x)**Rational(1, 3)).subs(x, 27) == 3*(-1)**Rational(1, 3)
n = Symbol('n', negative=True)
assert (x**n).subs(x, 0) is S.ComplexInfinity
assert exp(-1).subs(S.Exp1, 0) is S.ComplexInfinity
assert (x**(4.0*y)).subs(x**(2.0*y), n) == n**2.0
assert (2**(x + 2)).subs(2, 3) == 3**(x + 3)
def test_logexppow(): # no eval()
x = Symbol('x', real=True)
w = Symbol('w')
e = (3**(1 + x) + 2**(1 + x))/(3**x + 2**x)
assert e.subs(2**x, w) != e
assert e.subs(exp(x*log(Rational(2))), w) != e
def test_bug():
x1 = Symbol('x1')
x2 = Symbol('x2')
y = x1*x2
assert y.subs(x1, Float(3.0)) == Float(3.0)*x2
def test_subbug1():
# see that they don't fail
(x**x).subs(x, 1)
(x**x).subs(x, 1.0)
def test_subbug2():
# Ensure this does not cause infinite recursion
assert Float(7.7).epsilon_eq(abs(x).subs(x, -7.7))
def test_dict_set():
a, b, c = map(Wild, 'abc')
f = 3*cos(4*x)
r = f.match(a*cos(b*x))
assert r == {a: 3, b: 4}
e = a/b*sin(b*x)
assert e.subs(r) == r[a]/r[b]*sin(r[b]*x)
assert e.subs(r) == 3*sin(4*x) / 4
s = set(r.items())
assert e.subs(s) == r[a]/r[b]*sin(r[b]*x)
assert e.subs(s) == 3*sin(4*x) / 4
assert e.subs(r) == r[a]/r[b]*sin(r[b]*x)
assert e.subs(r) == 3*sin(4*x) / 4
assert x.subs(Dict((x, 1))) == 1
def test_dict_ambigous(): # see issue 3566
f = x*exp(x)
g = z*exp(z)
df = {x: y, exp(x): y}
dg = {z: y, exp(z): y}
assert f.subs(df) == y**2
assert g.subs(dg) == y**2
# and this is how order can affect the result
assert f.subs(x, y).subs(exp(x), y) == y*exp(y)
assert f.subs(exp(x), y).subs(x, y) == y**2
# length of args and count_ops are the same so
# default_sort_key resolves ordering...if one
# doesn't want this result then an unordered
# sequence should not be used.
e = 1 + x*y
assert e.subs({x: y, y: 2}) == 5
# here, there are no obviously clashing keys or values
# but the results depend on the order
assert exp(x/2 + y).subs({exp(y + 1): 2, x: 2}) == exp(y + 1)
def test_deriv_sub_bug3():
f = Function('f')
pat = Derivative(f(x), x, x)
assert pat.subs(y, y**2) == Derivative(f(x), x, x)
assert pat.subs(y, y**2) != Derivative(f(x), x)
def test_equality_subs1():
f = Function('f')
eq = Eq(f(x)**2, x)
res = Eq(Integer(16), x)
assert eq.subs(f(x), 4) == res
def test_equality_subs2():
f = Function('f')
eq = Eq(f(x)**2, 16)
assert bool(eq.subs(f(x), 3)) is False
assert bool(eq.subs(f(x), 4)) is True
def test_issue_3742():
e = sqrt(x)*exp(y)
assert e.subs(sqrt(x), 1) == exp(y)
def test_subs_dict1():
assert (1 + x*y).subs(x, pi) == 1 + pi*y
assert (1 + x*y).subs({x: pi, y: 2}) == 1 + 2*pi
c2, c3, q1p, q2p, c1, s1, s2, s3 = symbols('c2 c3 q1p q2p c1 s1 s2 s3')
test = (c2**2*q2p*c3 + c1**2*s2**2*q2p*c3 + s1**2*s2**2*q2p*c3
- c1**2*q1p*c2*s3 - s1**2*q1p*c2*s3)
assert (test.subs({c1**2: 1 - s1**2, c2**2: 1 - s2**2, c3**3: 1 - s3**2})
== c3*q2p*(1 - s2**2) + c3*q2p*s2**2*(1 - s1**2)
- c2*q1p*s3*(1 - s1**2) + c3*q2p*s1**2*s2**2 - c2*q1p*s3*s1**2)
def test_mul():
x, y, z, a, b, c = symbols('x y z a b c')
A, B, C = symbols('A B C', commutative=0)
assert (x*y*z).subs(z*x, y) == y**2
assert (z*x).subs(1/x, z) == 1
assert (x*y/z).subs(1/z, a) == a*x*y
assert (x*y/z).subs(x/z, a) == a*y
assert (x*y/z).subs(y/z, a) == a*x
assert (x*y/z).subs(x/z, 1/a) == y/a
assert (x*y/z).subs(x, 1/a) == y/(z*a)
assert (2*x*y).subs(5*x*y, z) != 2*z/5
assert (x*y*A).subs(x*y, a) == a*A
assert (x**2*y**(3*x/2)).subs(x*y**(x/2), 2) == 4*y**(x/2)
assert (x*exp(x*2)).subs(x*exp(x), 2) == 2*exp(x)
assert ((x**(2*y))**3).subs(x**y, 2) == 64
assert (x*A*B).subs(x*A, y) == y*B
assert (x*y*(1 + x)*(1 + x*y)).subs(x*y, 2) == 6*(1 + x)
assert ((1 + A*B)*A*B).subs(A*B, x*A*B)
assert (x*a/z).subs(x/z, A) == a*A
assert (x**3*A).subs(x**2*A, a) == a*x
assert (x**2*A*B).subs(x**2*B, a) == a*A
assert (x**2*A*B).subs(x**2*A, a) == a*B
assert (b*A**3/(a**3*c**3)).subs(a**4*c**3*A**3/b**4, z) == \
b*A**3/(a**3*c**3)
assert (6*x).subs(2*x, y) == 3*y
assert (y*exp(3*x/2)).subs(y*exp(x), 2) == 2*exp(x/2)
assert (y*exp(3*x/2)).subs(y*exp(x), 2) == 2*exp(x/2)
assert (A**2*B*A**2*B*A**2).subs(A*B*A, C) == A*C**2*A
assert (x*A**3).subs(x*A, y) == y*A**2
assert (x**2*A**3).subs(x*A, y) == y**2*A
assert (x*A**3).subs(x*A, B) == B*A**2
assert (x*A*B*A*exp(x*A*B)).subs(x*A, B) == B**2*A*exp(B*B)
assert (x**2*A*B*A*exp(x*A*B)).subs(x*A, B) == B**3*exp(B**2)
assert (x**3*A*exp(x*A*B)*A*exp(x*A*B)).subs(x*A, B) == \
x*B*exp(B**2)*B*exp(B**2)
assert (x*A*B*C*A*B).subs(x*A*B, C) == C**2*A*B
assert (-I*a*b).subs(a*b, 2) == -2*I
# issue 6361
assert (-8*I*a).subs(-2*a, 1) == 4*I
assert (-I*a).subs(-a, 1) == I
# issue 6441
assert (4*x**2).subs(2*x, y) == y**2
assert (2*4*x**2).subs(2*x, y) == 2*y**2
assert (-x**3/9).subs(-x/3, z) == -z**2*x
assert (-x**3/9).subs(x/3, z) == -z**2*x
assert (-2*x**3/9).subs(x/3, z) == -2*x*z**2
assert (-2*x**3/9).subs(-x/3, z) == -2*x*z**2
assert (-2*x**3/9).subs(-2*x, z) == z*x**2/9
assert (-2*x**3/9).subs(2*x, z) == -z*x**2/9
assert (2*(3*x/5/7)**2).subs(3*x/5, z) == 2*(S(1)/7)**2*z**2
assert (4*x).subs(-2*x, z) == 4*x # try keep subs literal
def test_subs_simple():
a = symbols('a', commutative=True)
x = symbols('x', commutative=False)
assert (2*a).subs(1, 3) == 2*a
assert (2*a).subs(2, 3) == 3*a
assert (2*a).subs(a, 3) == 6
assert sin(2).subs(1, 3) == sin(2)
assert sin(2).subs(2, 3) == sin(3)
assert sin(a).subs(a, 3) == sin(3)
assert (2*x).subs(1, 3) == 2*x
assert (2*x).subs(2, 3) == 3*x
assert (2*x).subs(x, 3) == 6
assert sin(x).subs(x, 3) == sin(3)
def test_subs_constants():
a, b = symbols('a b', commutative=True)
x, y = symbols('x y', commutative=False)
assert (a*b).subs(2*a, 1) == a*b
assert (1.5*a*b).subs(a, 1) == 1.5*b
assert (2*a*b).subs(2*a, 1) == b
assert (2*a*b).subs(4*a, 1) == 2*a*b
assert (x*y).subs(2*x, 1) == x*y
assert (1.5*x*y).subs(x, 1) == 1.5*y
assert (2*x*y).subs(2*x, 1) == y
assert (2*x*y).subs(4*x, 1) == 2*x*y
def test_subs_commutative():
a, b, c, d, K = symbols('a b c d K', commutative=True)
assert (a*b).subs(a*b, K) == K
assert (a*b*a*b).subs(a*b, K) == K**2
assert (a*a*b*b).subs(a*b, K) == K**2
assert (a*b*c*d).subs(a*b*c, K) == d*K
assert (a*b**c).subs(a, K) == K*b**c
assert (a*b**c).subs(b, K) == a*K**c
assert (a*b**c).subs(c, K) == a*b**K
assert (a*b*c*b*a).subs(a*b, K) == c*K**2
assert (a**3*b**2*a).subs(a*b, K) == a**2*K**2
def test_subs_noncommutative():
w, x, y, z, L = symbols('w x y z L', commutative=False)
alpha = symbols('alpha', commutative=True)
someint = symbols('someint', commutative=True, integer=True)
assert (x*y).subs(x*y, L) == L
assert (w*y*x).subs(x*y, L) == w*y*x
assert (w*x*y*z).subs(x*y, L) == w*L*z
assert (x*y*x*y).subs(x*y, L) == L**2
assert (x*x*y).subs(x*y, L) == x*L
assert (x*x*y*y).subs(x*y, L) == x*L*y
assert (w*x*y).subs(x*y*z, L) == w*x*y
assert (x*y**z).subs(x, L) == L*y**z
assert (x*y**z).subs(y, L) == x*L**z
assert (x*y**z).subs(z, L) == x*y**L
assert (w*x*y*z*x*y).subs(x*y*z, L) == w*L*x*y
assert (w*x*y*y*w*x*x*y*x*y*y*x*y).subs(x*y, L) == w*L*y*w*x*L**2*y*L
# Check fractional power substitutions. It should not do
# substitutions that choose a value for noncommutative log,
# or inverses that don't already appear in the expressions.
assert (x*x*x).subs(x*x, L) == L*x
assert (x*x*x*y*x*x*x*x).subs(x*x, L) == L*x*y*L**2
for p in range(1, 5):
for k in range(10):
assert (y * x**k).subs(x**p, L) == y * L**(k//p) * x**(k % p)
assert (x**(S(3)/2)).subs(x**(S(1)/2), L) == x**(S(3)/2)
assert (x**(S(1)/2)).subs(x**(S(1)/2), L) == L
assert (x**(-S(1)/2)).subs(x**(S(1)/2), L) == x**(-S(1)/2)
assert (x**(-S(1)/2)).subs(x**(-S(1)/2), L) == L
assert (x**(2*someint)).subs(x**someint, L) == L**2
assert (x**(2*someint + 3)).subs(x**someint, L) == L**2*x**3
assert (x**(3*someint + 3)).subs(x**someint, L) == L**3*x**3
assert (x**(3*someint)).subs(x**(2*someint), L) == L * x**someint
assert (x**(4*someint)).subs(x**(2*someint), L) == L**2
assert (x**(4*someint + 1)).subs(x**(2*someint), L) == L**2 * x
assert (x**(4*someint)).subs(x**(3*someint), L) == L * x**someint
assert (x**(4*someint + 1)).subs(x**(3*someint), L) == L * x**(someint + 1)
assert (x**(2*alpha)).subs(x**alpha, L) == x**(2*alpha)
assert (x**(2*alpha + 2)).subs(x**2, L) == x**(2*alpha + 2)
assert ((2*z)**alpha).subs(z**alpha, y) == (2*z)**alpha
assert (x**(2*someint*alpha)).subs(x**someint, L) == x**(2*someint*alpha)
assert (x**(2*someint + alpha)).subs(x**someint, L) == x**(2*someint + alpha)
# This could in principle be substituted, but is not currently
# because it requires recognizing that someint**2 is divisible by
# someint.
assert (x**(someint**2 + 3)).subs(x**someint, L) == x**(someint**2 + 3)
# alpha**z := exp(log(alpha) z) is usually well-defined
assert (4**z).subs(2**z, y) == y**2
# Negative powers
assert (x**(-1)).subs(x**3, L) == x**(-1)
assert (x**(-2)).subs(x**3, L) == x**(-2)
assert (x**(-3)).subs(x**3, L) == L**(-1)
assert (x**(-4)).subs(x**3, L) == L**(-1) * x**(-1)
assert (x**(-5)).subs(x**3, L) == L**(-1) * x**(-2)
assert (x**(-1)).subs(x**(-3), L) == x**(-1)
assert (x**(-2)).subs(x**(-3), L) == x**(-2)
assert (x**(-3)).subs(x**(-3), L) == L
assert (x**(-4)).subs(x**(-3), L) == L * x**(-1)
assert (x**(-5)).subs(x**(-3), L) == L * x**(-2)
assert (x**1).subs(x**(-3), L) == x
assert (x**2).subs(x**(-3), L) == x**2
assert (x**3).subs(x**(-3), L) == L**(-1)
assert (x**4).subs(x**(-3), L) == L**(-1) * x
assert (x**5).subs(x**(-3), L) == L**(-1) * x**2
def test_subs_basic_funcs():
a, b, c, d, K = symbols('a b c d K', commutative=True)
w, x, y, z, L = symbols('w x y z L', commutative=False)
assert (x + y).subs(x + y, L) == L
assert (x - y).subs(x - y, L) == L
assert (x/y).subs(x, L) == L/y
assert (x**y).subs(x, L) == L**y
assert (x**y).subs(y, L) == x**L
assert ((a - c)/b).subs(b, K) == (a - c)/K
assert (exp(x*y - z)).subs(x*y, L) == exp(L - z)
assert (a*exp(x*y - w*z) + b*exp(x*y + w*z)).subs(z, 0) == \
a*exp(x*y) + b*exp(x*y)
assert ((a - b)/(c*d - a*b)).subs(c*d - a*b, K) == (a - b)/K
assert (w*exp(a*b - c)*x*y/4).subs(x*y, L) == w*exp(a*b - c)*L/4
def test_subs_wild():
R, S, T, U = symbols('R S T U', cls=Wild)
assert (R*S).subs(R*S, T) == T
assert (S*R).subs(R*S, T) == T
assert (R + S).subs(R + S, T) == T
assert (R**S).subs(R, T) == T**S
assert (R**S).subs(S, T) == R**T
assert (R*S**T).subs(R, U) == U*S**T
assert (R*S**T).subs(S, U) == R*U**T
assert (R*S**T).subs(T, U) == R*S**U
def test_subs_mixed():
a, b, c, d, K = symbols('a b c d K', commutative=True)
w, x, y, z, L = symbols('w x y z L', commutative=False)
R, S, T, U = symbols('R S T U', cls=Wild)
assert (a*x*y).subs(x*y, L) == a*L
assert (a*b*x*y*x).subs(x*y, L) == a*b*L*x
assert (R*x*y*exp(x*y)).subs(x*y, L) == R*L*exp(L)
assert (a*x*y*y*x - x*y*z*exp(a*b)).subs(x*y, L) == a*L*y*x - L*z*exp(a*b)
e = c*y*x*y*x**(R*S - a*b) - T*(a*R*b*S)
assert e.subs(x*y, L).subs(a*b, K).subs(R*S, U) == \
c*y*L*x**(U - K) - T*(U*K)
def test_division():
a, b, c = symbols('a b c', commutative=True)
x, y, z = symbols('x y z', commutative=True)
assert (1/a).subs(a, c) == 1/c
assert (1/a**2).subs(a, c) == 1/c**2
assert (1/a**2).subs(a, -2) == Rational(1, 4)
assert (-(1/a**2)).subs(a, -2) == -Rational(1, 4)
assert (1/x).subs(x, z) == 1/z
assert (1/x**2).subs(x, z) == 1/z**2
assert (1/x**2).subs(x, -2) == Rational(1, 4)
assert (-(1/x**2)).subs(x, -2) == -Rational(1, 4)
#issue 5360
assert (1/x).subs(x, 0) == 1/S(0)
def test_add():
a, b, c, d, x, y, t = symbols('a b c d x y t')
assert (a**2 - b - c).subs(a**2 - b, d) in [d - c, a**2 - b - c]
assert (a**2 - c).subs(a**2 - c, d) == d
assert (a**2 - b - c).subs(a**2 - c, d) in [d - b, a**2 - b - c]
assert (a**2 - x - c).subs(a**2 - c, d) in [d - x, a**2 - x - c]
assert (a**2 - b - sqrt(a)).subs(a**2 - sqrt(a), c) == c - b
assert (a + b + exp(a + b)).subs(a + b, c) == c + exp(c)
assert (c + b + exp(c + b)).subs(c + b, a) == a + exp(a)
assert (a + b + c + d).subs(b + c, x) == a + d + x
assert (a + b + c + d).subs(-b - c, x) == a + d - x
assert ((x + 1)*y).subs(x + 1, t) == t*y
assert ((-x - 1)*y).subs(x + 1, t) == -t*y
assert ((x - 1)*y).subs(x + 1, t) == y*(t - 2)
assert ((-x + 1)*y).subs(x + 1, t) == y*(-t + 2)
# this should work every time:
e = a**2 - b - c
assert e.subs(Add(*e.args[:2]), d) == d + e.args[2]
assert e.subs(a**2 - c, d) == d - b
# the fallback should recognize when a change has
# been made; while .1 == Rational(1, 10) they are not the same
# and the change should be made
assert (0.1 + a).subs(0.1, Rational(1, 10)) == Rational(1, 10) + a
e = (-x*(-y + 1) - y*(y - 1))
ans = (-x*(x) - y*(-x)).expand()
assert e.subs(-y + 1, x) == ans
def test_subs_issue_4009():
assert (I*Symbol('a')).subs(1, 2) == I*Symbol('a')
def test_functions_subs():
f, g = symbols('f g', cls=Function)
l = Lambda((x, y), sin(x) + y)
assert (g(y, x) + cos(x)).subs(g, l) == sin(y) + x + cos(x)
assert (f(x)**2).subs(f, sin) == sin(x)**2
assert (f(x, y)).subs(f, log) == log(x, y)
assert (f(x, y)).subs(f, sin) == f(x, y)
assert (sin(x) + atan2(x, y)).subs([[atan2, f], [sin, g]]) == \
f(x, y) + g(x)
assert (g(f(x + y, x))).subs([[f, l], [g, exp]]) == exp(x + sin(x + y))
def test_derivative_subs():
f = Function('f')
g = Function('g')
assert Derivative(f(x), x).subs(f(x), y) != 0
# need xreplace to put the function back, see #13803
assert Derivative(f(x), x).subs(f(x), y).xreplace({y: f(x)}) == \
Derivative(f(x), x)
# issues 5085, 5037
assert cse(Derivative(f(x), x) + f(x))[1][0].has(Derivative)
assert cse(Derivative(f(x, y), x) +
Derivative(f(x, y), y))[1][0].has(Derivative)
eq = Derivative(g(x), g(x))
assert eq.subs(g, f) == Derivative(f(x), f(x))
assert eq.subs(g(x), f(x)) == Derivative(f(x), f(x))
assert eq.subs(g, cos) == Subs(Derivative(y, y), y, cos(x))
def test_derivative_subs2():
f_func, g_func = symbols('f g', cls=Function)
f, g = f_func(x, y, z), g_func(x, y, z)
assert Derivative(f, x, y).subs(Derivative(f, x, y), g) == g
assert Derivative(f, y, x).subs(Derivative(f, x, y), g) == g
assert Derivative(f, x, y).subs(Derivative(f, x), g) == Derivative(g, y)
assert Derivative(f, x, y).subs(Derivative(f, y), g) == Derivative(g, x)
assert (Derivative(f, x, y, z).subs(
Derivative(f, x, z), g) == Derivative(g, y))
assert (Derivative(f, x, y, z).subs(
Derivative(f, z, y), g) == Derivative(g, x))
assert (Derivative(f, x, y, z).subs(
Derivative(f, z, y, x), g) == g)
# Issue 9135
assert (Derivative(f, x, x, y).subs(
Derivative(f, y, y), g) == Derivative(f, x, x, y))
assert (Derivative(f, x, y, y, z).subs(
Derivative(f, x, y, y, y), g) == Derivative(f, x, y, y, z))
assert Derivative(f, x, y).subs(Derivative(f_func(x), x, y), g) == Derivative(f, x, y)
def test_derivative_subs3():
dex = Derivative(exp(x), x)
assert Derivative(dex, x).subs(dex, exp(x)) == dex
assert dex.subs(exp(x), dex) == Derivative(exp(x), x, x)
def test_issue_5284():
A, B = symbols('A B', commutative=False)
assert (x*A).subs(x**2*A, B) == x*A
assert (A**2).subs(A**3, B) == A**2
assert (A**6).subs(A**3, B) == B**2
def test_subs_iter():
assert x.subs(reversed([[x, y]])) == y
it = iter([[x, y]])
assert x.subs(it) == y
assert x.subs(Tuple((x, y))) == y
def test_subs_dict():
a, b, c, d, e = symbols('a b c d e')
assert (2*x + y + z).subs(dict(x=1, y=2)) == 4 + z
l = [(sin(x), 2), (x, 1)]
assert (sin(x)).subs(l) == \
(sin(x)).subs(dict(l)) == 2
assert sin(x).subs(reversed(l)) == sin(1)
expr = sin(2*x) + sqrt(sin(2*x))*cos(2*x)*sin(exp(x)*x)
reps = dict([
(sin(2*x), c),
(sqrt(sin(2*x)), a),
(cos(2*x), b),
(exp(x), e),
(x, d),
])
assert expr.subs(reps) == c + a*b*sin(d*e)
l = [(x, 3), (y, x**2)]
assert (x + y).subs(l) == 3 + x**2
assert (x + y).subs(reversed(l)) == 12
# If changes are made to convert lists into dictionaries and do
# a dictionary-lookup replacement, these tests will help to catch
# some logical errors that might occur
l = [(y, z + 2), (1 + z, 5), (z, 2)]
assert (y - 1 + 3*x).subs(l) == 5 + 3*x
l = [(y, z + 2), (z, 3)]
assert (y - 2).subs(l) == 3
def test_no_arith_subs_on_floats():
assert (x + 3).subs(x + 3, a) == a
assert (x + 3).subs(x + 2, a) == a + 1
assert (x + y + 3).subs(x + 3, a) == a + y
assert (x + y + 3).subs(x + 2, a) == a + y + 1
assert (x + 3.0).subs(x + 3.0, a) == a
assert (x + 3.0).subs(x + 2.0, a) == x + 3.0
assert (x + y + 3.0).subs(x + 3.0, a) == a + y
assert (x + y + 3.0).subs(x + 2.0, a) == x + y + 3.0
def test_issue_5651():
a, b, c, K = symbols('a b c K', commutative=True)
assert (a/(b*c)).subs(b*c, K) == a/K
assert (a/(b**2*c**3)).subs(b*c, K) == a/(c*K**2)
assert (1/(x*y)).subs(x*y, 2) == S.Half
assert ((1 + x*y)/(x*y)).subs(x*y, 1) == 2
assert (x*y*z).subs(x*y, 2) == 2*z
assert ((1 + x*y)/(x*y)/z).subs(x*y, 1) == 2/z
def test_issue_6075():
assert Tuple(1, True).subs(1, 2) == Tuple(2, True)
def test_issue_6079():
# since x + 2.0 == x + 2 we can't do a simple equality test
assert _aresame((x + 2.0).subs(2, 3), x + 2.0)
assert _aresame((x + 2.0).subs(2.0, 3), x + 3)
assert not _aresame(x + 2, x + 2.0)
assert not _aresame(Basic(cos, 1), Basic(cos, 1.))
assert _aresame(cos, cos)
assert not _aresame(1, S(1))
assert not _aresame(x, symbols('x', positive=True))
def test_issue_4680():
N = Symbol('N')
assert N.subs(dict(N=3)) == 3
def test_issue_6158():
assert (x - 1).subs(1, y) == x - y
assert (x - 1).subs(-1, y) == x + y
assert (x - oo).subs(oo, y) == x - y
assert (x - oo).subs(-oo, y) == x + y
def test_Function_subs():
f, g, h, i = symbols('f g h i', cls=Function)
p = Piecewise((g(f(x, y)), x < -1), (g(x), x <= 1))
assert p.subs(g, h) == Piecewise((h(f(x, y)), x < -1), (h(x), x <= 1))
assert (f(y) + g(x)).subs({f: h, g: i}) == i(x) + h(y)
def test_simultaneous_subs():
reps = {x: 0, y: 0}
assert (x/y).subs(reps) != (y/x).subs(reps)
assert (x/y).subs(reps, simultaneous=True) == \
(y/x).subs(reps, simultaneous=True)
reps = reps.items()
assert (x/y).subs(reps) != (y/x).subs(reps)
assert (x/y).subs(reps, simultaneous=True) == \
(y/x).subs(reps, simultaneous=True)
assert Derivative(x, y, z).subs(reps, simultaneous=True) == \
Subs(Derivative(0, y, z), y, 0)
def test_issue_6419_6421():
assert (1/(1 + x/y)).subs(x/y, x) == 1/(1 + x)
assert (-2*I).subs(2*I, x) == -x
assert (-I*x).subs(I*x, x) == -x
assert (-3*I*y**4).subs(3*I*y**2, x) == -x*y**2
def test_issue_6559():
assert (-12*x + y).subs(-x, 1) == 12 + y
# though this involves cse it generated a failure in Mul._eval_subs
x0, x1 = symbols('x0 x1')
e = -log(-12*sqrt(2) + 17)/24 - log(-2*sqrt(2) + 3)/12 + sqrt(2)/3
# XXX modify cse so x1 is eliminated and x0 = -sqrt(2)?
assert cse(e) == (
[(x0, sqrt(2))], [x0/3 - log(-12*x0 + 17)/24 - log(-2*x0 + 3)/12])
def test_issue_5261():
x = symbols('x', real=True)
e = I*x
assert exp(e).subs(exp(x), y) == y**I
assert (2**e).subs(2**x, y) == y**I
eq = (-2)**e
assert eq.subs((-2)**x, y) == eq
def test_issue_6923():
assert (-2*x*sqrt(2)).subs(2*x, y) == -sqrt(2)*y
def test_2arg_hack():
N = Symbol('N', commutative=False)
ans = Mul(2, y + 1, evaluate=False)
assert (2*x*(y + 1)).subs(x, 1, hack2=True) == ans
assert (2*(y + 1 + N)).subs(N, 0, hack2=True) == ans
@XFAIL
def test_mul2():
"""When this fails, remove things labelled "2-arg hack"
1) remove special handling in the fallback of subs that
was added in the same commit as this test
2) remove the special handling in Mul.flatten
"""
assert (2*(x + 1)).is_Mul
def test_noncommutative_subs():
x,y = symbols('x,y', commutative=False)
assert (x*y*x).subs([(x, x*y), (y, x)], simultaneous=True) == (x*y*x**2*y)
def test_issue_2877():
f = Float(2.0)
assert (x + f).subs({f: 2}) == x + 2
def r(a, b, c):
return factor(a*x**2 + b*x + c)
e = r(5.0/6, 10, 5)
assert nsimplify(e) == 5*x**2/6 + 10*x + 5
def test_issue_5910():
t = Symbol('t')
assert (1/(1 - t)).subs(t, 1) == zoo
n = t
d = t - 1
assert (n/d).subs(t, 1) == zoo
assert (-n/-d).subs(t, 1) == zoo
def test_issue_5217():
s = Symbol('s')
z = (1 - 2*x*x)
w = (1 + 2*x*x)
q = 2*x*x*2*y*y
sub = {2*x*x: s}
assert w.subs(sub) == 1 + s
assert z.subs(sub) == 1 - s
assert q == 4*x**2*y**2
assert q.subs(sub) == 2*y**2*s
def test_issue_10829():
assert (4**x).subs(2**x, y) == y**2
assert (9**x).subs(3**x, y) == y**2
def test_pow_eval_subs_no_cache():
# Tests pull request 9376 is working
from sympy.core.cache import clear_cache
s = 1/sqrt(x**2)
# This bug only appeared when the cache was turned off.
# We need to approximate running this test without the cache.
# This creates approximately the same situation.
clear_cache()
# This used to fail with a wrong result.
# It incorrectly returned 1/sqrt(x**2) before this pull request.
result = s.subs(sqrt(x**2), y)
assert result == 1/y
def test_RootOf_issue_10092():
x = Symbol('x', real=True)
eq = x**3 - 17*x**2 + 81*x - 118
r = RootOf(eq, 0)
assert (x < r).subs(x, r) is S.false
def test_issue_8886():
from sympy.physics.mechanics import ReferenceFrame as R
# if something can't be sympified we assume that it
# doesn't play well with SymPy and disallow the
# substitution
v = R('A').x
assert x.subs(x, v) == x
assert v.subs(v, x) == v
assert v.__eq__(x) is False
def test_issue_12657():
# treat -oo like the atom that it is
reps = [(-oo, 1), (oo, 2)]
assert (x < -oo).subs(reps) == (x < 1)
assert (x < -oo).subs(list(reversed(reps))) == (x < 1)
reps = [(-oo, 2), (oo, 1)]
assert (x < oo).subs(reps) == (x < 1)
assert (x < oo).subs(list(reversed(reps))) == (x < 1)
def test_recurse_Application_args():
F = Lambda((x, y), exp(2*x + 3*y))
f = Function('f')
A = f(x, f(x, x))
C = F(x, F(x, x))
assert A.subs(f, F) == A.replace(f, F) == C
def test_Subs_subs():
assert Subs(x*y, x, x).subs(x, y) == Subs(x*y, x, y)
assert Subs(x*y, x, x + 1).subs(x, y) == \
Subs(x*y, x, y + 1)
assert Subs(x*y, y, x + 1).subs(x, y) == \
Subs(y**2, y, y + 1)
a = Subs(x*y*z, (y, x, z), (x + 1, x + z, x))
b = Subs(x*y*z, (y, x, z), (x + 1, y + z, y))
assert a.subs(x, y) == b and \
a.doit().subs(x, y) == a.subs(x, y).doit()
f = Function('f')
g = Function('g')
assert Subs(2*f(x, y) + g(x), f(x, y), 1).subs(y, 2) == Subs(
2*f(x, y) + g(x), (f(x, y), y), (1, 2))
def test_issue_13333():
eq = 1/x
assert eq.subs(dict(x='1/2')) == 2
assert eq.subs(dict(x='(1/2)')) == 2
def test_issue_15234():
x, y = symbols('x y', real=True)
p = 6*x**5 + x**4 - 4*x**3 + 4*x**2 - 2*x + 3
p_subbed = 6*x**5 - 4*x**3 - 2*x + y**4 + 4*y**2 + 3
assert p.subs([(x**i, y**i) for i in [2, 4]]) == p_subbed
x, y = symbols('x y', complex=True)
p = 6*x**5 + x**4 - 4*x**3 + 4*x**2 - 2*x + 3
p_subbed = 6*x**5 - 4*x**3 - 2*x + y**4 + 4*y**2 + 3
assert p.subs([(x**i, y**i) for i in [2, 4]]) == p_subbed
def test_issue_6976():
x, y = symbols('x y')
assert (sqrt(x)**3 + sqrt(x) + x + x**2).subs(sqrt(x), y) == \
y**4 + y**3 + y**2 + y
assert (x**4 + x**3 + x**2 + x + sqrt(x)).subs(x**2, y) == \
sqrt(x) + x**3 + x + y**2 + y
assert x.subs(x**3, y) == x
assert x.subs(x**(S(1)/3), y) == y**3
# More substitutions are possible with nonnegative symbols
x, y = symbols('x y', nonnegative=True)
assert (x**4 + x**3 + x**2 + x + sqrt(x)).subs(x**2, y) == \
y**(S(1)/4) + y**(S(3)/2) + sqrt(y) + y**2 + y
assert x.subs(x**3, y) == y**(S(1)/3)
|
0ee3957c3943566fea7d9049579c3cc8a387f81943c58f0b2ac84164bb950022
|
from sympy import (Add, Basic, Expr, S, Symbol, Wild, Float, Integer, Rational, I,
sin, cos, tan, exp, log, nan, oo, sqrt, symbols, Integral, sympify,
WildFunction, Poly, Function, Derivative, Number, pi, NumberSymbol, zoo,
Piecewise, Mul, Pow, nsimplify, ratsimp, trigsimp, radsimp, powsimp,
simplify, together, collect, factorial, apart, combsimp, factor, refine,
cancel, Tuple, default_sort_key, DiracDelta, gamma, Dummy, Sum, E,
exp_polar, expand, diff, O, Heaviside, Si, Max, UnevaluatedExpr,
integrate, gammasimp)
from sympy.core.function import AppliedUndef
from sympy.core.compatibility import range
from sympy.physics.secondquant import FockState
from sympy.physics.units import meter
from sympy.utilities.pytest import raises, XFAIL
from sympy.abc import a, b, c, n, t, u, x, y, z
class DummyNumber(object):
"""
Minimal implementation of a number that works with SymPy.
If one has a Number class (e.g. Sage Integer, or some other custom class)
that one wants to work well with SymPy, one has to implement at least the
methods of this class DummyNumber, resp. its subclasses I5 and F1_1.
Basically, one just needs to implement either __int__() or __float__() and
then one needs to make sure that the class works with Python integers and
with itself.
"""
def __radd__(self, a):
if isinstance(a, (int, float)):
return a + self.number
return NotImplemented
def __truediv__(a, b):
return a.__div__(b)
def __rtruediv__(a, b):
return a.__rdiv__(b)
def __add__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number + a
return NotImplemented
def __rsub__(self, a):
if isinstance(a, (int, float)):
return a - self.number
return NotImplemented
def __sub__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number - a
return NotImplemented
def __rmul__(self, a):
if isinstance(a, (int, float)):
return a * self.number
return NotImplemented
def __mul__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number * a
return NotImplemented
def __rdiv__(self, a):
if isinstance(a, (int, float)):
return a / self.number
return NotImplemented
def __div__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number / a
return NotImplemented
def __rpow__(self, a):
if isinstance(a, (int, float)):
return a ** self.number
return NotImplemented
def __pow__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number ** a
return NotImplemented
def __pos__(self):
return self.number
def __neg__(self):
return - self.number
class I5(DummyNumber):
number = 5
def __int__(self):
return self.number
class F1_1(DummyNumber):
number = 1.1
def __float__(self):
return self.number
i5 = I5()
f1_1 = F1_1()
# basic sympy objects
basic_objs = [
Rational(2),
Float("1.3"),
x,
y,
pow(x, y)*y,
]
# all supported objects
all_objs = basic_objs + [
5,
5.5,
i5,
f1_1
]
def dotest(s):
for x in all_objs:
for y in all_objs:
s(x, y)
return True
def test_basic():
def j(a, b):
x = a
x = +a
x = -a
x = a + b
x = a - b
x = a*b
x = a/b
x = a**b
assert dotest(j)
def test_ibasic():
def s(a, b):
x = a
x += b
x = a
x -= b
x = a
x *= b
x = a
x /= b
assert dotest(s)
def test_relational():
from sympy import Lt
assert (pi < 3) is S.false
assert (pi <= 3) is S.false
assert (pi > 3) is S.true
assert (pi >= 3) is S.true
assert (-pi < 3) is S.true
assert (-pi <= 3) is S.true
assert (-pi > 3) is S.false
assert (-pi >= 3) is S.false
r = Symbol('r', real=True)
assert (r - 2 < r - 3) is S.false
assert Lt(x + I, x + I + 2).func == Lt # issue 8288
def test_relational_assumptions():
from sympy import Lt, Gt, Le, Ge
m1 = Symbol("m1", nonnegative=False)
m2 = Symbol("m2", positive=False)
m3 = Symbol("m3", nonpositive=False)
m4 = Symbol("m4", negative=False)
assert (m1 < 0) == Lt(m1, 0)
assert (m2 <= 0) == Le(m2, 0)
assert (m3 > 0) == Gt(m3, 0)
assert (m4 >= 0) == Ge(m4, 0)
m1 = Symbol("m1", nonnegative=False, real=True)
m2 = Symbol("m2", positive=False, real=True)
m3 = Symbol("m3", nonpositive=False, real=True)
m4 = Symbol("m4", negative=False, real=True)
assert (m1 < 0) is S.true
assert (m2 <= 0) is S.true
assert (m3 > 0) is S.true
assert (m4 >= 0) is S.true
m1 = Symbol("m1", negative=True)
m2 = Symbol("m2", nonpositive=True)
m3 = Symbol("m3", positive=True)
m4 = Symbol("m4", nonnegative=True)
assert (m1 < 0) is S.true
assert (m2 <= 0) is S.true
assert (m3 > 0) is S.true
assert (m4 >= 0) is S.true
m1 = Symbol("m1", negative=False, real=True)
m2 = Symbol("m2", nonpositive=False, real=True)
m3 = Symbol("m3", positive=False, real=True)
m4 = Symbol("m4", nonnegative=False, real=True)
assert (m1 < 0) is S.false
assert (m2 <= 0) is S.false
assert (m3 > 0) is S.false
assert (m4 >= 0) is S.false
def test_relational_noncommutative():
from sympy import Lt, Gt, Le, Ge
A, B = symbols('A,B', commutative=False)
assert (A < B) == Lt(A, B)
assert (A <= B) == Le(A, B)
assert (A > B) == Gt(A, B)
assert (A >= B) == Ge(A, B)
def test_basic_nostr():
for obj in basic_objs:
raises(TypeError, lambda: obj + '1')
raises(TypeError, lambda: obj - '1')
if obj == 2:
assert obj * '1' == '11'
else:
raises(TypeError, lambda: obj * '1')
raises(TypeError, lambda: obj / '1')
raises(TypeError, lambda: obj ** '1')
def test_series_expansion_for_uniform_order():
assert (1/x + y + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + y + x).series(x, 0, 1) == 1/x + y + O(x)
assert (1/x + 1 + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + 1 + x).series(x, 0, 1) == 1/x + 1 + O(x)
assert (1/x + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + y + y*x + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + y + y*x + x).series(x, 0, 1) == 1/x + y + O(x)
def test_leadterm():
assert (3 + 2*x**(log(3)/log(2) - 1)).leadterm(x) == (3, 0)
assert (1/x**2 + 1 + x + x**2).leadterm(x)[1] == -2
assert (1/x + 1 + x + x**2).leadterm(x)[1] == -1
assert (x**2 + 1/x).leadterm(x)[1] == -1
assert (1 + x**2).leadterm(x)[1] == 0
assert (x + 1).leadterm(x)[1] == 0
assert (x + x**2).leadterm(x)[1] == 1
assert (x**2).leadterm(x)[1] == 2
def test_as_leading_term():
assert (3 + 2*x**(log(3)/log(2) - 1)).as_leading_term(x) == 3
assert (1/x**2 + 1 + x + x**2).as_leading_term(x) == 1/x**2
assert (1/x + 1 + x + x**2).as_leading_term(x) == 1/x
assert (x**2 + 1/x).as_leading_term(x) == 1/x
assert (1 + x**2).as_leading_term(x) == 1
assert (x + 1).as_leading_term(x) == 1
assert (x + x**2).as_leading_term(x) == x
assert (x**2).as_leading_term(x) == x**2
assert (x + oo).as_leading_term(x) == oo
raises(ValueError, lambda: (x + 1).as_leading_term(1))
def test_leadterm2():
assert (x*cos(1)*cos(1 + sin(1)) + sin(1 + sin(1))).leadterm(x) == \
(sin(1 + sin(1)), 0)
def test_leadterm3():
assert (y + z + x).leadterm(x) == (y + z, 0)
def test_as_leading_term2():
assert (x*cos(1)*cos(1 + sin(1)) + sin(1 + sin(1))).as_leading_term(x) == \
sin(1 + sin(1))
def test_as_leading_term3():
assert (2 + pi + x).as_leading_term(x) == 2 + pi
assert (2*x + pi*x + x**2).as_leading_term(x) == (2 + pi)*x
def test_as_leading_term4():
# see issue 6843
n = Symbol('n', integer=True, positive=True)
r = -n**3/(2*n**2 + 4*n + 2) - n**2/(n**2 + 2*n + 1) + \
n**2/(n + 1) - n/(2*n**2 + 4*n + 2) + n/(n*x + x) + 2*n/(n + 1) - \
1 + 1/(n*x + x) + 1/(n + 1) - 1/x
assert r.as_leading_term(x).cancel() == n/2
def test_as_leading_term_stub():
class foo(Function):
pass
assert foo(1/x).as_leading_term(x) == foo(1/x)
assert foo(1).as_leading_term(x) == foo(1)
raises(NotImplementedError, lambda: foo(x).as_leading_term(x))
def test_as_leading_term_deriv_integral():
# related to issue 11313
assert Derivative(x ** 3, x).as_leading_term(x) == 3*x**2
assert Derivative(x ** 3, y).as_leading_term(x) == 0
assert Integral(x ** 3, x).as_leading_term(x) == x**4/4
assert Integral(x ** 3, y).as_leading_term(x) == y*x**3
assert Derivative(exp(x), x).as_leading_term(x) == 1
assert Derivative(log(x), x).as_leading_term(x) == (1/x).as_leading_term(x)
def test_atoms():
assert x.atoms() == {x}
assert (1 + x).atoms() == {x, S(1)}
assert (1 + 2*cos(x)).atoms(Symbol) == {x}
assert (1 + 2*cos(x)).atoms(Symbol, Number) == {S(1), S(2), x}
assert (2*(x**(y**x))).atoms() == {S(2), x, y}
assert Rational(1, 2).atoms() == {S.Half}
assert Rational(1, 2).atoms(Symbol) == set([])
assert sin(oo).atoms(oo) == set()
assert Poly(0, x).atoms() == {S.Zero}
assert Poly(1, x).atoms() == {S.One}
assert Poly(x, x).atoms() == {x}
assert Poly(x, x, y).atoms() == {x}
assert Poly(x + y, x, y).atoms() == {x, y}
assert Poly(x + y, x, y, z).atoms() == {x, y}
assert Poly(x + y*t, x, y, z).atoms() == {t, x, y}
assert (I*pi).atoms(NumberSymbol) == {pi}
assert (I*pi).atoms(NumberSymbol, I) == \
(I*pi).atoms(I, NumberSymbol) == {pi, I}
assert exp(exp(x)).atoms(exp) == {exp(exp(x)), exp(x)}
assert (1 + x*(2 + y) + exp(3 + z)).atoms(Add) == \
{1 + x*(2 + y) + exp(3 + z), 2 + y, 3 + z}
# issue 6132
f = Function('f')
e = (f(x) + sin(x) + 2)
assert e.atoms(AppliedUndef) == \
{f(x)}
assert e.atoms(AppliedUndef, Function) == \
{f(x), sin(x)}
assert e.atoms(Function) == \
{f(x), sin(x)}
assert e.atoms(AppliedUndef, Number) == \
{f(x), S(2)}
assert e.atoms(Function, Number) == \
{S(2), sin(x), f(x)}
def test_is_polynomial():
k = Symbol('k', nonnegative=True, integer=True)
assert Rational(2).is_polynomial(x, y, z) is True
assert (S.Pi).is_polynomial(x, y, z) is True
assert x.is_polynomial(x) is True
assert x.is_polynomial(y) is True
assert (x**2).is_polynomial(x) is True
assert (x**2).is_polynomial(y) is True
assert (x**(-2)).is_polynomial(x) is False
assert (x**(-2)).is_polynomial(y) is True
assert (2**x).is_polynomial(x) is False
assert (2**x).is_polynomial(y) is True
assert (x**k).is_polynomial(x) is False
assert (x**k).is_polynomial(k) is False
assert (x**x).is_polynomial(x) is False
assert (k**k).is_polynomial(k) is False
assert (k**x).is_polynomial(k) is False
assert (x**(-k)).is_polynomial(x) is False
assert ((2*x)**k).is_polynomial(x) is False
assert (x**2 + 3*x - 8).is_polynomial(x) is True
assert (x**2 + 3*x - 8).is_polynomial(y) is True
assert (x**2 + 3*x - 8).is_polynomial() is True
assert sqrt(x).is_polynomial(x) is False
assert (sqrt(x)**3).is_polynomial(x) is False
assert (x**2 + 3*x*sqrt(y) - 8).is_polynomial(x) is True
assert (x**2 + 3*x*sqrt(y) - 8).is_polynomial(y) is False
assert ((x**2)*(y**2) + x*(y**2) + y*x + exp(2)).is_polynomial() is True
assert ((x**2)*(y**2) + x*(y**2) + y*x + exp(x)).is_polynomial() is False
assert (
(x**2)*(y**2) + x*(y**2) + y*x + exp(2)).is_polynomial(x, y) is True
assert (
(x**2)*(y**2) + x*(y**2) + y*x + exp(x)).is_polynomial(x, y) is False
def test_is_rational_function():
assert Integer(1).is_rational_function() is True
assert Integer(1).is_rational_function(x) is True
assert Rational(17, 54).is_rational_function() is True
assert Rational(17, 54).is_rational_function(x) is True
assert (12/x).is_rational_function() is True
assert (12/x).is_rational_function(x) is True
assert (x/y).is_rational_function() is True
assert (x/y).is_rational_function(x) is True
assert (x/y).is_rational_function(x, y) is True
assert (x**2 + 1/x/y).is_rational_function() is True
assert (x**2 + 1/x/y).is_rational_function(x) is True
assert (x**2 + 1/x/y).is_rational_function(x, y) is True
assert (sin(y)/x).is_rational_function() is False
assert (sin(y)/x).is_rational_function(y) is False
assert (sin(y)/x).is_rational_function(x) is True
assert (sin(y)/x).is_rational_function(x, y) is False
assert (S.NaN).is_rational_function() is False
assert (S.Infinity).is_rational_function() is False
assert (-S.Infinity).is_rational_function() is False
assert (S.ComplexInfinity).is_rational_function() is False
def test_is_algebraic_expr():
assert sqrt(3).is_algebraic_expr(x) is True
assert sqrt(3).is_algebraic_expr() is True
eq = ((1 + x**2)/(1 - y**2))**(S(1)/3)
assert eq.is_algebraic_expr(x) is True
assert eq.is_algebraic_expr(y) is True
assert (sqrt(x) + y**(S(2)/3)).is_algebraic_expr(x) is True
assert (sqrt(x) + y**(S(2)/3)).is_algebraic_expr(y) is True
assert (sqrt(x) + y**(S(2)/3)).is_algebraic_expr() is True
assert (cos(y)/sqrt(x)).is_algebraic_expr() is False
assert (cos(y)/sqrt(x)).is_algebraic_expr(x) is True
assert (cos(y)/sqrt(x)).is_algebraic_expr(y) is False
assert (cos(y)/sqrt(x)).is_algebraic_expr(x, y) is False
def test_SAGE1():
#see https://github.com/sympy/sympy/issues/3346
class MyInt:
def _sympy_(self):
return Integer(5)
m = MyInt()
e = Rational(2)*m
assert e == 10
raises(TypeError, lambda: Rational(2)*MyInt)
def test_SAGE2():
class MyInt(object):
def __int__(self):
return 5
assert sympify(MyInt()) == 5
e = Rational(2)*MyInt()
assert e == 10
raises(TypeError, lambda: Rational(2)*MyInt)
def test_SAGE3():
class MySymbol:
def __rmul__(self, other):
return ('mys', other, self)
o = MySymbol()
e = x*o
assert e == ('mys', x, o)
def test_len():
e = x*y
assert len(e.args) == 2
e = x + y + z
assert len(e.args) == 3
def test_doit():
a = Integral(x**2, x)
assert isinstance(a.doit(), Integral) is False
assert isinstance(a.doit(integrals=True), Integral) is False
assert isinstance(a.doit(integrals=False), Integral) is True
assert (2*Integral(x, x)).doit() == x**2
def test_attribute_error():
raises(AttributeError, lambda: x.cos())
raises(AttributeError, lambda: x.sin())
raises(AttributeError, lambda: x.exp())
def test_args():
assert (x*y).args in ((x, y), (y, x))
assert (x + y).args in ((x, y), (y, x))
assert (x*y + 1).args in ((x*y, 1), (1, x*y))
assert sin(x*y).args == (x*y,)
assert sin(x*y).args[0] == x*y
assert (x**y).args == (x, y)
assert (x**y).args[0] == x
assert (x**y).args[1] == y
def test_noncommutative_expand_issue_3757():
A, B, C = symbols('A,B,C', commutative=False)
assert A*B - B*A != 0
assert (A*(A + B)*B).expand() == A**2*B + A*B**2
assert (A*(A + B + C)*B).expand() == A**2*B + A*B**2 + A*C*B
def test_as_numer_denom():
a, b, c = symbols('a, b, c')
assert nan.as_numer_denom() == (nan, 1)
assert oo.as_numer_denom() == (oo, 1)
assert (-oo).as_numer_denom() == (-oo, 1)
assert zoo.as_numer_denom() == (zoo, 1)
assert (-zoo).as_numer_denom() == (zoo, 1)
assert x.as_numer_denom() == (x, 1)
assert (1/x).as_numer_denom() == (1, x)
assert (x/y).as_numer_denom() == (x, y)
assert (x/2).as_numer_denom() == (x, 2)
assert (x*y/z).as_numer_denom() == (x*y, z)
assert (x/(y*z)).as_numer_denom() == (x, y*z)
assert Rational(1, 2).as_numer_denom() == (1, 2)
assert (1/y**2).as_numer_denom() == (1, y**2)
assert (x/y**2).as_numer_denom() == (x, y**2)
assert ((x**2 + 1)/y).as_numer_denom() == (x**2 + 1, y)
assert (x*(y + 1)/y**7).as_numer_denom() == (x*(y + 1), y**7)
assert (x**-2).as_numer_denom() == (1, x**2)
assert (a/x + b/2/x + c/3/x).as_numer_denom() == \
(6*a + 3*b + 2*c, 6*x)
assert (a/x + b/2/x + c/3/y).as_numer_denom() == \
(2*c*x + y*(6*a + 3*b), 6*x*y)
assert (a/x + b/2/x + c/.5/x).as_numer_denom() == \
(2*a + b + 4.0*c, 2*x)
# this should take no more than a few seconds
assert int(log(Add(*[Dummy()/i/x for i in range(1, 705)]
).as_numer_denom()[1]/x).n(4)) == 705
for i in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
assert (i + x/3).as_numer_denom() == \
(x + i, 3)
assert (S.Infinity + x/3 + y/4).as_numer_denom() == \
(4*x + 3*y + S.Infinity, 12)
assert (oo*x + zoo*y).as_numer_denom() == \
(zoo*y + oo*x, 1)
A, B, C = symbols('A,B,C', commutative=False)
assert (A*B*C**-1).as_numer_denom() == (A*B*C**-1, 1)
assert (A*B*C**-1/x).as_numer_denom() == (A*B*C**-1, x)
assert (C**-1*A*B).as_numer_denom() == (C**-1*A*B, 1)
assert (C**-1*A*B/x).as_numer_denom() == (C**-1*A*B, x)
assert ((A*B*C)**-1).as_numer_denom() == ((A*B*C)**-1, 1)
assert ((A*B*C)**-1/x).as_numer_denom() == ((A*B*C)**-1, x)
def test_trunc():
import math
x, y = symbols('x y')
assert math.trunc(2) == 2
assert math.trunc(4.57) == 4
assert math.trunc(-5.79) == -5
assert math.trunc(pi) == 3
assert math.trunc(log(7)) == 1
assert math.trunc(exp(5)) == 148
assert math.trunc(cos(pi)) == -1
assert math.trunc(sin(5)) == 0
raises(TypeError, lambda: math.trunc(x))
raises(TypeError, lambda: math.trunc(x + y**2))
raises(TypeError, lambda: math.trunc(oo))
def test_as_independent():
assert S.Zero.as_independent(x, as_Add=True) == (0, 0)
assert S.Zero.as_independent(x, as_Add=False) == (0, 0)
assert (2*x*sin(x) + y + x).as_independent(x) == (y, x + 2*x*sin(x))
assert (2*x*sin(x) + y + x).as_independent(y) == (x + 2*x*sin(x), y)
assert (2*x*sin(x) + y + x).as_independent(x, y) == (0, y + x + 2*x*sin(x))
assert (x*sin(x)*cos(y)).as_independent(x) == (cos(y), x*sin(x))
assert (x*sin(x)*cos(y)).as_independent(y) == (x*sin(x), cos(y))
assert (x*sin(x)*cos(y)).as_independent(x, y) == (1, x*sin(x)*cos(y))
assert (sin(x)).as_independent(x) == (1, sin(x))
assert (sin(x)).as_independent(y) == (sin(x), 1)
assert (2*sin(x)).as_independent(x) == (2, sin(x))
assert (2*sin(x)).as_independent(y) == (2*sin(x), 1)
# issue 4903 = 1766b
n1, n2, n3 = symbols('n1 n2 n3', commutative=False)
assert (n1 + n1*n2).as_independent(n2) == (n1, n1*n2)
assert (n2*n1 + n1*n2).as_independent(n2) == (0, n1*n2 + n2*n1)
assert (n1*n2*n1).as_independent(n2) == (n1, n2*n1)
assert (n1*n2*n1).as_independent(n1) == (1, n1*n2*n1)
assert (3*x).as_independent(x, as_Add=True) == (0, 3*x)
assert (3*x).as_independent(x, as_Add=False) == (3, x)
assert (3 + x).as_independent(x, as_Add=True) == (3, x)
assert (3 + x).as_independent(x, as_Add=False) == (1, 3 + x)
# issue 5479
assert (3*x).as_independent(Symbol) == (3, x)
# issue 5648
assert (n1*x*y).as_independent(x) == (n1*y, x)
assert ((x + n1)*(x - y)).as_independent(x) == (1, (x + n1)*(x - y))
assert ((x + n1)*(x - y)).as_independent(y) == (x + n1, x - y)
assert (DiracDelta(x - n1)*DiracDelta(x - y)).as_independent(x) \
== (1, DiracDelta(x - n1)*DiracDelta(x - y))
assert (x*y*n1*n2*n3).as_independent(n2) == (x*y*n1, n2*n3)
assert (x*y*n1*n2*n3).as_independent(n1) == (x*y, n1*n2*n3)
assert (x*y*n1*n2*n3).as_independent(n3) == (x*y*n1*n2, n3)
assert (DiracDelta(x - n1)*DiracDelta(y - n1)*DiracDelta(x - n2)).as_independent(y) == \
(DiracDelta(x - n1)*DiracDelta(x - n2), DiracDelta(y - n1))
# issue 5784
assert (x + Integral(x, (x, 1, 2))).as_independent(x, strict=True) == \
(Integral(x, (x, 1, 2)), x)
eq = Add(x, -x, 2, -3, evaluate=False)
assert eq.as_independent(x) == (-1, Add(x, -x, evaluate=False))
eq = Mul(x, 1/x, 2, -3, evaluate=False)
eq.as_independent(x) == (-6, Mul(x, 1/x, evaluate=False))
assert (x*y).as_independent(z, as_Add=True) == (x*y, 0)
@XFAIL
def test_call_2():
# TODO UndefinedFunction does not subclass Expr
f = Function('f')
assert (2*f)(x) == 2*f(x)
def test_replace():
f = log(sin(x)) + tan(sin(x**2))
assert f.replace(sin, cos) == log(cos(x)) + tan(cos(x**2))
assert f.replace(
sin, lambda a: sin(2*a)) == log(sin(2*x)) + tan(sin(2*x**2))
a = Wild('a')
b = Wild('b')
assert f.replace(sin(a), cos(a)) == log(cos(x)) + tan(cos(x**2))
assert f.replace(
sin(a), lambda a: sin(2*a)) == log(sin(2*x)) + tan(sin(2*x**2))
# test exact
assert (2*x).replace(a*x + b, b - a, exact=True) == 2*x
assert (2*x).replace(a*x + b, b - a) == 2*x
assert (2*x).replace(a*x + b, b - a, exact=False) == 2/x
assert (2*x).replace(a*x + b, lambda a, b: b - a, exact=True) == 2*x
assert (2*x).replace(a*x + b, lambda a, b: b - a) == 2*x
assert (2*x).replace(a*x + b, lambda a, b: b - a, exact=False) == 2/x
g = 2*sin(x**3)
assert g.replace(
lambda expr: expr.is_Number, lambda expr: expr**2) == 4*sin(x**9)
assert cos(x).replace(cos, sin, map=True) == (sin(x), {cos(x): sin(x)})
assert sin(x).replace(cos, sin) == sin(x)
cond, func = lambda x: x.is_Mul, lambda x: 2*x
assert (x*y).replace(cond, func, map=True) == (2*x*y, {x*y: 2*x*y})
assert (x*(1 + x*y)).replace(cond, func, map=True) == \
(2*x*(2*x*y + 1), {x*(2*x*y + 1): 2*x*(2*x*y + 1), x*y: 2*x*y})
assert (y*sin(x)).replace(sin, lambda expr: sin(expr)/y, map=True) == \
(sin(x), {sin(x): sin(x)/y})
# if not simultaneous then y*sin(x) -> y*sin(x)/y = sin(x) -> sin(x)/y
assert (y*sin(x)).replace(sin, lambda expr: sin(expr)/y,
simultaneous=False) == sin(x)/y
assert (x**2 + O(x**3)).replace(Pow, lambda b, e: b**e/e) == O(1, x)
assert (x**2 + O(x**3)).replace(Pow, lambda b, e: b**e/e,
simultaneous=False) == x**2/2 + O(x**3)
assert (x*(x*y + 3)).replace(lambda x: x.is_Mul, lambda x: 2 + x) == \
x*(x*y + 5) + 2
e = (x*y + 1)*(2*x*y + 1) + 1
assert e.replace(cond, func, map=True) == (
2*((2*x*y + 1)*(4*x*y + 1)) + 1,
{2*x*y: 4*x*y, x*y: 2*x*y, (2*x*y + 1)*(4*x*y + 1):
2*((2*x*y + 1)*(4*x*y + 1))})
assert x.replace(x, y) == y
assert (x + 1).replace(1, 2) == x + 2
# https://groups.google.com/forum/#!topic/sympy/8wCgeC95tz0
n1, n2, n3 = symbols('n1:4', commutative=False)
f = Function('f')
assert (n1*f(n2)).replace(f, lambda x: x) == n1*n2
assert (n3*f(n2)).replace(f, lambda x: x) == n3*n2
def test_find():
expr = (x + y + 2 + sin(3*x))
assert expr.find(lambda u: u.is_Integer) == {S(2), S(3)}
assert expr.find(lambda u: u.is_Symbol) == {x, y}
assert expr.find(lambda u: u.is_Integer, group=True) == {S(2): 1, S(3): 1}
assert expr.find(lambda u: u.is_Symbol, group=True) == {x: 2, y: 1}
assert expr.find(Integer) == {S(2), S(3)}
assert expr.find(Symbol) == {x, y}
assert expr.find(Integer, group=True) == {S(2): 1, S(3): 1}
assert expr.find(Symbol, group=True) == {x: 2, y: 1}
a = Wild('a')
expr = sin(sin(x)) + sin(x) + cos(x) + x
assert expr.find(lambda u: type(u) is sin) == {sin(x), sin(sin(x))}
assert expr.find(
lambda u: type(u) is sin, group=True) == {sin(x): 2, sin(sin(x)): 1}
assert expr.find(sin(a)) == {sin(x), sin(sin(x))}
assert expr.find(sin(a), group=True) == {sin(x): 2, sin(sin(x)): 1}
assert expr.find(sin) == {sin(x), sin(sin(x))}
assert expr.find(sin, group=True) == {sin(x): 2, sin(sin(x)): 1}
def test_count():
expr = (x + y + 2 + sin(3*x))
assert expr.count(lambda u: u.is_Integer) == 2
assert expr.count(lambda u: u.is_Symbol) == 3
assert expr.count(Integer) == 2
assert expr.count(Symbol) == 3
assert expr.count(2) == 1
a = Wild('a')
assert expr.count(sin) == 1
assert expr.count(sin(a)) == 1
assert expr.count(lambda u: type(u) is sin) == 1
f = Function('f')
assert f(x).count(f(x)) == 1
assert f(x).diff(x).count(f(x)) == 1
assert f(x).diff(x).count(x) == 2
def test_has_basics():
f = Function('f')
g = Function('g')
p = Wild('p')
assert sin(x).has(x)
assert sin(x).has(sin)
assert not sin(x).has(y)
assert not sin(x).has(cos)
assert f(x).has(x)
assert f(x).has(f)
assert not f(x).has(y)
assert not f(x).has(g)
assert f(x).diff(x).has(x)
assert f(x).diff(x).has(f)
assert f(x).diff(x).has(Derivative)
assert not f(x).diff(x).has(y)
assert not f(x).diff(x).has(g)
assert not f(x).diff(x).has(sin)
assert (x**2).has(Symbol)
assert not (x**2).has(Wild)
assert (2*p).has(Wild)
assert not x.has()
def test_has_multiple():
f = x**2*y + sin(2**t + log(z))
assert f.has(x)
assert f.has(y)
assert f.has(z)
assert f.has(t)
assert not f.has(u)
assert f.has(x, y, z, t)
assert f.has(x, y, z, t, u)
i = Integer(4400)
assert not i.has(x)
assert (i*x**i).has(x)
assert not (i*y**i).has(x)
assert (i*y**i).has(x, y)
assert not (i*y**i).has(x, z)
def test_has_piecewise():
f = (x*y + 3/y)**(3 + 2)
g = Function('g')
h = Function('h')
p = Piecewise((g(x), x < -1), (1, x <= 1), (f, True))
assert p.has(x)
assert p.has(y)
assert not p.has(z)
assert p.has(1)
assert p.has(3)
assert not p.has(4)
assert p.has(f)
assert p.has(g)
assert not p.has(h)
def test_has_iterative():
A, B, C = symbols('A,B,C', commutative=False)
f = x*gamma(x)*sin(x)*exp(x*y)*A*B*C*cos(x*A*B)
assert f.has(x)
assert f.has(x*y)
assert f.has(x*sin(x))
assert not f.has(x*sin(y))
assert f.has(x*A)
assert f.has(x*A*B)
assert not f.has(x*A*C)
assert f.has(x*A*B*C)
assert not f.has(x*A*C*B)
assert f.has(x*sin(x)*A*B*C)
assert not f.has(x*sin(x)*A*C*B)
assert not f.has(x*sin(y)*A*B*C)
assert f.has(x*gamma(x))
assert not f.has(x + sin(x))
assert (x & y & z).has(x & z)
def test_has_integrals():
f = Integral(x**2 + sin(x*y*z), (x, 0, x + y + z))
assert f.has(x + y)
assert f.has(x + z)
assert f.has(y + z)
assert f.has(x*y)
assert f.has(x*z)
assert f.has(y*z)
assert not f.has(2*x + y)
assert not f.has(2*x*y)
def test_has_tuple():
f = Function('f')
g = Function('g')
h = Function('h')
assert Tuple(x, y).has(x)
assert not Tuple(x, y).has(z)
assert Tuple(f(x), g(x)).has(x)
assert not Tuple(f(x), g(x)).has(y)
assert Tuple(f(x), g(x)).has(f)
assert Tuple(f(x), g(x)).has(f(x))
assert not Tuple(f, g).has(x)
assert Tuple(f, g).has(f)
assert not Tuple(f, g).has(h)
assert Tuple(True).has(True) is True # .has(1) will also be True
def test_has_units():
from sympy.physics.units import m, s
assert (x*m/s).has(x)
assert (x*m/s).has(y, z) is False
def test_has_polys():
poly = Poly(x**2 + x*y*sin(z), x, y, t)
assert poly.has(x)
assert poly.has(x, y, z)
assert poly.has(x, y, z, t)
def test_has_physics():
assert FockState((x, y)).has(x)
def test_as_poly_as_expr():
f = x**2 + 2*x*y
assert f.as_poly().as_expr() == f
assert f.as_poly(x, y).as_expr() == f
assert (f + sin(x)).as_poly(x, y) is None
p = Poly(f, x, y)
assert p.as_poly() == p
def test_nonzero():
assert bool(S.Zero) is False
assert bool(S.One) is True
assert bool(x) is True
assert bool(x + y) is True
assert bool(x - x) is False
assert bool(x*y) is True
assert bool(x*1) is True
assert bool(x*0) is False
def test_is_number():
assert Float(3.14).is_number is True
assert Integer(737).is_number is True
assert Rational(3, 2).is_number is True
assert Rational(8).is_number is True
assert x.is_number is False
assert (2*x).is_number is False
assert (x + y).is_number is False
assert log(2).is_number is True
assert log(x).is_number is False
assert (2 + log(2)).is_number is True
assert (8 + log(2)).is_number is True
assert (2 + log(x)).is_number is False
assert (8 + log(2) + x).is_number is False
assert (1 + x**2/x - x).is_number is True
assert Tuple(Integer(1)).is_number is False
assert Add(2, x).is_number is False
assert Mul(3, 4).is_number is True
assert Pow(log(2), 2).is_number is True
assert oo.is_number is True
g = WildFunction('g')
assert g.is_number is False
assert (2*g).is_number is False
assert (x**2).subs(x, 3).is_number is True
# test extensibility of .is_number
# on subinstances of Basic
class A(Basic):
pass
a = A()
assert a.is_number is False
def test_as_coeff_add():
assert S(2).as_coeff_add() == (2, ())
assert S(3.0).as_coeff_add() == (0, (S(3.0),))
assert S(-3.0).as_coeff_add() == (0, (S(-3.0),))
assert x.as_coeff_add() == (0, (x,))
assert (x - 1).as_coeff_add() == (-1, (x,))
assert (x + 1).as_coeff_add() == (1, (x,))
assert (x + 2).as_coeff_add() == (2, (x,))
assert (x + y).as_coeff_add(y) == (x, (y,))
assert (3*x).as_coeff_add(y) == (3*x, ())
# don't do expansion
e = (x + y)**2
assert e.as_coeff_add(y) == (0, (e,))
def test_as_coeff_mul():
assert S(2).as_coeff_mul() == (2, ())
assert S(3.0).as_coeff_mul() == (1, (S(3.0),))
assert S(-3.0).as_coeff_mul() == (-1, (S(3.0),))
assert S(-3.0).as_coeff_mul(rational=False) == (-S(3.0), ())
assert x.as_coeff_mul() == (1, (x,))
assert (-x).as_coeff_mul() == (-1, (x,))
assert (2*x).as_coeff_mul() == (2, (x,))
assert (x*y).as_coeff_mul(y) == (x, (y,))
assert (3 + x).as_coeff_mul() == (1, (3 + x,))
assert (3 + x).as_coeff_mul(y) == (3 + x, ())
# don't do expansion
e = exp(x + y)
assert e.as_coeff_mul(y) == (1, (e,))
e = 2**(x + y)
assert e.as_coeff_mul(y) == (1, (e,))
assert (1.1*x).as_coeff_mul(rational=False) == (1.1, (x,))
assert (1.1*x).as_coeff_mul() == (1, (1.1, x))
assert (-oo*x).as_coeff_mul(rational=True) == (-1, (oo, x))
def test_as_coeff_exponent():
assert (3*x**4).as_coeff_exponent(x) == (3, 4)
assert (2*x**3).as_coeff_exponent(x) == (2, 3)
assert (4*x**2).as_coeff_exponent(x) == (4, 2)
assert (6*x**1).as_coeff_exponent(x) == (6, 1)
assert (3*x**0).as_coeff_exponent(x) == (3, 0)
assert (2*x**0).as_coeff_exponent(x) == (2, 0)
assert (1*x**0).as_coeff_exponent(x) == (1, 0)
assert (0*x**0).as_coeff_exponent(x) == (0, 0)
assert (-1*x**0).as_coeff_exponent(x) == (-1, 0)
assert (-2*x**0).as_coeff_exponent(x) == (-2, 0)
assert (2*x**3 + pi*x**3).as_coeff_exponent(x) == (2 + pi, 3)
assert (x*log(2)/(2*x + pi*x)).as_coeff_exponent(x) == \
(log(2)/(2 + pi), 0)
# issue 4784
D = Derivative
f = Function('f')
fx = D(f(x), x)
assert fx.as_coeff_exponent(f(x)) == (fx, 0)
def test_extractions():
assert ((x*y)**3).extract_multiplicatively(x**2 * y) == x*y**2
assert ((x*y)**3).extract_multiplicatively(x**4 * y) is None
assert (2*x).extract_multiplicatively(2) == x
assert (2*x).extract_multiplicatively(3) is None
assert (2*x).extract_multiplicatively(-1) is None
assert (Rational(1, 2)*x).extract_multiplicatively(3) == x/6
assert (sqrt(x)).extract_multiplicatively(x) is None
assert (sqrt(x)).extract_multiplicatively(1/x) is None
assert x.extract_multiplicatively(-x) is None
assert (-2 - 4*I).extract_multiplicatively(-2) == 1 + 2*I
assert (-2 - 4*I).extract_multiplicatively(3) is None
assert (-2*x - 4*y - 8).extract_multiplicatively(-2) == x + 2*y + 4
assert (-2*x*y - 4*x**2*y).extract_multiplicatively(-2*y) == 2*x**2 + x
assert (2*x*y + 4*x**2*y).extract_multiplicatively(2*y) == 2*x**2 + x
assert (-4*y**2*x).extract_multiplicatively(-3*y) is None
assert (2*x).extract_multiplicatively(1) == 2*x
assert (-oo).extract_multiplicatively(5) == -oo
assert (oo).extract_multiplicatively(5) == oo
assert ((x*y)**3).extract_additively(1) is None
assert (x + 1).extract_additively(x) == 1
assert (x + 1).extract_additively(2*x) is None
assert (x + 1).extract_additively(-x) is None
assert (-x + 1).extract_additively(2*x) is None
assert (2*x + 3).extract_additively(x) == x + 3
assert (2*x + 3).extract_additively(2) == 2*x + 1
assert (2*x + 3).extract_additively(3) == 2*x
assert (2*x + 3).extract_additively(-2) is None
assert (2*x + 3).extract_additively(3*x) is None
assert (2*x + 3).extract_additively(2*x) == 3
assert x.extract_additively(0) == x
assert S(2).extract_additively(x) is None
assert S(2.).extract_additively(2) == S.Zero
assert S(2*x + 3).extract_additively(x + 1) == x + 2
assert S(2*x + 3).extract_additively(y + 1) is None
assert S(2*x - 3).extract_additively(x + 1) is None
assert S(2*x - 3).extract_additively(y + z) is None
assert ((a + 1)*x*4 + y).extract_additively(x).expand() == \
4*a*x + 3*x + y
assert ((a + 1)*x*4 + 3*y).extract_additively(x + 2*y).expand() == \
4*a*x + 3*x + y
assert (y*(x + 1)).extract_additively(x + 1) is None
assert ((y + 1)*(x + 1) + 3).extract_additively(x + 1) == \
y*(x + 1) + 3
assert ((x + y)*(x + 1) + x + y + 3).extract_additively(x + y) == \
x*(x + y) + 3
assert (x + y + 2*((x + y)*(x + 1)) + 3).extract_additively((x + y)*(x + 1)) == \
x + y + (x + 1)*(x + y) + 3
assert ((y + 1)*(x + 2*y + 1) + 3).extract_additively(y + 1) == \
(x + 2*y)*(y + 1) + 3
n = Symbol("n", integer=True)
assert (Integer(-3)).could_extract_minus_sign() is True
assert (-n*x + x).could_extract_minus_sign() != \
(n*x - x).could_extract_minus_sign()
assert (x - y).could_extract_minus_sign() != \
(-x + y).could_extract_minus_sign()
assert (1 - x - y).could_extract_minus_sign() is True
assert (1 - x + y).could_extract_minus_sign() is False
assert ((-x - x*y)/y).could_extract_minus_sign() is True
assert (-(x + x*y)/y).could_extract_minus_sign() is True
assert ((x + x*y)/(-y)).could_extract_minus_sign() is True
assert ((x + x*y)/y).could_extract_minus_sign() is False
assert (x*(-x - x**3)).could_extract_minus_sign() is True
assert ((-x - y)/(x + y)).could_extract_minus_sign() is True
class sign_invariant(Function, Expr):
nargs = 1
def __neg__(self):
return self
foo = sign_invariant(x)
assert foo == -foo
assert foo.could_extract_minus_sign() is False
# The results of each of these will vary on different machines, e.g.
# the first one might be False and the other (then) is true or vice versa,
# so both are included.
assert ((-x - y)/(x - y)).could_extract_minus_sign() is False or \
((-x - y)/(y - x)).could_extract_minus_sign() is False
assert (x - y).could_extract_minus_sign() is False
assert (-x + y).could_extract_minus_sign() is True
def test_nan_extractions():
for r in (1, 0, I, nan):
assert nan.extract_additively(r) is None
assert nan.extract_multiplicatively(r) is None
def test_coeff():
assert (x + 1).coeff(x + 1) == 1
assert (3*x).coeff(0) == 0
assert (z*(1 + x)*x**2).coeff(1 + x) == z*x**2
assert (1 + 2*x*x**(1 + x)).coeff(x*x**(1 + x)) == 2
assert (1 + 2*x**(y + z)).coeff(x**(y + z)) == 2
assert (3 + 2*x + 4*x**2).coeff(1) == 0
assert (3 + 2*x + 4*x**2).coeff(-1) == 0
assert (3 + 2*x + 4*x**2).coeff(x) == 2
assert (3 + 2*x + 4*x**2).coeff(x**2) == 4
assert (3 + 2*x + 4*x**2).coeff(x**3) == 0
assert (-x/8 + x*y).coeff(x) == -S(1)/8 + y
assert (-x/8 + x*y).coeff(-x) == S(1)/8
assert (4*x).coeff(2*x) == 0
assert (2*x).coeff(2*x) == 1
assert (-oo*x).coeff(x*oo) == -1
assert (10*x).coeff(x, 0) == 0
assert (10*x).coeff(10*x, 0) == 0
n1, n2 = symbols('n1 n2', commutative=False)
assert (n1*n2).coeff(n1) == 1
assert (n1*n2).coeff(n2) == n1
assert (n1*n2 + x*n1).coeff(n1) == 1 # 1*n1*(n2+x)
assert (n2*n1 + x*n1).coeff(n1) == n2 + x
assert (n2*n1 + x*n1**2).coeff(n1) == n2
assert (n1**x).coeff(n1) == 0
assert (n1*n2 + n2*n1).coeff(n1) == 0
assert (2*(n1 + n2)*n2).coeff(n1 + n2, right=1) == n2
assert (2*(n1 + n2)*n2).coeff(n1 + n2, right=0) == 2
f = Function('f')
assert (2*f(x) + 3*f(x).diff(x)).coeff(f(x)) == 2
expr = z*(x + y)**2
expr2 = z*(x + y)**2 + z*(2*x + 2*y)**2
assert expr.coeff(z) == (x + y)**2
assert expr.coeff(x + y) == 0
assert expr2.coeff(z) == (x + y)**2 + (2*x + 2*y)**2
assert (x + y + 3*z).coeff(1) == x + y
assert (-x + 2*y).coeff(-1) == x
assert (x - 2*y).coeff(-1) == 2*y
assert (3 + 2*x + 4*x**2).coeff(1) == 0
assert (-x - 2*y).coeff(2) == -y
assert (x + sqrt(2)*x).coeff(sqrt(2)) == x
assert (3 + 2*x + 4*x**2).coeff(x) == 2
assert (3 + 2*x + 4*x**2).coeff(x**2) == 4
assert (3 + 2*x + 4*x**2).coeff(x**3) == 0
assert (z*(x + y)**2).coeff((x + y)**2) == z
assert (z*(x + y)**2).coeff(x + y) == 0
assert (2 + 2*x + (x + 1)*y).coeff(x + 1) == y
assert (x + 2*y + 3).coeff(1) == x
assert (x + 2*y + 3).coeff(x, 0) == 2*y + 3
assert (x**2 + 2*y + 3*x).coeff(x**2, 0) == 2*y + 3*x
assert x.coeff(0, 0) == 0
assert x.coeff(x, 0) == 0
n, m, o, l = symbols('n m o l', commutative=False)
assert n.coeff(n) == 1
assert y.coeff(n) == 0
assert (3*n).coeff(n) == 3
assert (2 + n).coeff(x*m) == 0
assert (2*x*n*m).coeff(x) == 2*n*m
assert (2 + n).coeff(x*m*n + y) == 0
assert (2*x*n*m).coeff(3*n) == 0
assert (n*m + m*n*m).coeff(n) == 1 + m
assert (n*m + m*n*m).coeff(n, right=True) == m # = (1 + m)*n*m
assert (n*m + m*n).coeff(n) == 0
assert (n*m + o*m*n).coeff(m*n) == o
assert (n*m + o*m*n).coeff(m*n, right=1) == 1
assert (n*m + n*m*n).coeff(n*m, right=1) == 1 + n # = n*m*(n + 1)
assert (x*y).coeff(z, 0) == x*y
def test_coeff2():
r, kappa = symbols('r, kappa')
psi = Function("psi")
g = 1/r**2 * (2*r*psi(r).diff(r, 1) + r**2 * psi(r).diff(r, 2))
g = g.expand()
assert g.coeff((psi(r).diff(r))) == 2/r
def test_coeff2_0():
r, kappa = symbols('r, kappa')
psi = Function("psi")
g = 1/r**2 * (2*r*psi(r).diff(r, 1) + r**2 * psi(r).diff(r, 2))
g = g.expand()
assert g.coeff(psi(r).diff(r, 2)) == 1
def test_coeff_expand():
expr = z*(x + y)**2
expr2 = z*(x + y)**2 + z*(2*x + 2*y)**2
assert expr.coeff(z) == (x + y)**2
assert expr2.coeff(z) == (x + y)**2 + (2*x + 2*y)**2
def test_integrate():
assert x.integrate(x) == x**2/2
assert x.integrate((x, 0, 1)) == S(1)/2
def test_as_base_exp():
assert x.as_base_exp() == (x, S.One)
assert (x*y*z).as_base_exp() == (x*y*z, S.One)
assert (x + y + z).as_base_exp() == (x + y + z, S.One)
assert ((x + y)**z).as_base_exp() == (x + y, z)
def test_issue_4963():
assert hasattr(Mul(x, y), "is_commutative")
assert hasattr(Mul(x, y, evaluate=False), "is_commutative")
assert hasattr(Pow(x, y), "is_commutative")
assert hasattr(Pow(x, y, evaluate=False), "is_commutative")
expr = Mul(Pow(2, 2, evaluate=False), 3, evaluate=False) + 1
assert hasattr(expr, "is_commutative")
def test_action_verbs():
assert nsimplify((1/(exp(3*pi*x/5) + 1))) == \
(1/(exp(3*pi*x/5) + 1)).nsimplify()
assert ratsimp(1/x + 1/y) == (1/x + 1/y).ratsimp()
assert trigsimp(log(x), deep=True) == (log(x)).trigsimp(deep=True)
assert radsimp(1/(2 + sqrt(2))) == (1/(2 + sqrt(2))).radsimp()
assert radsimp(1/(a + b*sqrt(c)), symbolic=False) == \
(1/(a + b*sqrt(c))).radsimp(symbolic=False)
assert powsimp(x**y*x**z*y**z, combine='all') == \
(x**y*x**z*y**z).powsimp(combine='all')
assert (x**t*y**t).powsimp(force=True) == (x*y)**t
assert simplify(x**y*x**z*y**z) == (x**y*x**z*y**z).simplify()
assert together(1/x + 1/y) == (1/x + 1/y).together()
assert collect(a*x**2 + b*x**2 + a*x - b*x + c, x) == \
(a*x**2 + b*x**2 + a*x - b*x + c).collect(x)
assert apart(y/(y + 2)/(y + 1), y) == (y/(y + 2)/(y + 1)).apart(y)
assert combsimp(y/(x + 2)/(x + 1)) == (y/(x + 2)/(x + 1)).combsimp()
assert gammasimp(gamma(x)/gamma(x-5)) == (gamma(x)/gamma(x-5)).gammasimp()
assert factor(x**2 + 5*x + 6) == (x**2 + 5*x + 6).factor()
assert refine(sqrt(x**2)) == sqrt(x**2).refine()
assert cancel((x**2 + 5*x + 6)/(x + 2)) == ((x**2 + 5*x + 6)/(x + 2)).cancel()
def test_as_powers_dict():
assert x.as_powers_dict() == {x: 1}
assert (x**y*z).as_powers_dict() == {x: y, z: 1}
assert Mul(2, 2, evaluate=False).as_powers_dict() == {S(2): S(2)}
assert (x*y).as_powers_dict()[z] == 0
assert (x + y).as_powers_dict()[z] == 0
def test_as_coefficients_dict():
check = [S(1), x, y, x*y, 1]
assert [Add(3*x, 2*x, y, 3).as_coefficients_dict()[i] for i in check] == \
[3, 5, 1, 0, 3]
assert [Add(3*x, 2*x, y, 3, evaluate=False).as_coefficients_dict()[i]
for i in check] == [3, 5, 1, 0, 3]
assert [(3*x*y).as_coefficients_dict()[i] for i in check] == \
[0, 0, 0, 3, 0]
assert [(3.0*x*y).as_coefficients_dict()[i] for i in check] == \
[0, 0, 0, 3.0, 0]
assert (3.0*x*y).as_coefficients_dict()[3.0*x*y] == 0
def test_args_cnc():
A = symbols('A', commutative=False)
assert (x + A).args_cnc() == \
[[], [x + A]]
assert (x + a).args_cnc() == \
[[a + x], []]
assert (x*a).args_cnc() == \
[[a, x], []]
assert (x*y*A*(A + 1)).args_cnc(cset=True) == \
[{x, y}, [A, 1 + A]]
assert Mul(x, x, evaluate=False).args_cnc(cset=True, warn=False) == \
[{x}, []]
assert Mul(x, x**2, evaluate=False).args_cnc(cset=True, warn=False) == \
[{x, x**2}, []]
raises(ValueError, lambda: Mul(x, x, evaluate=False).args_cnc(cset=True))
assert Mul(x, y, x, evaluate=False).args_cnc() == \
[[x, y, x], []]
# always split -1 from leading number
assert (-1.*x).args_cnc() == [[-1, 1.0, x], []]
def test_new_rawargs():
n = Symbol('n', commutative=False)
a = x + n
assert a.is_commutative is False
assert a._new_rawargs(x).is_commutative
assert a._new_rawargs(x, y).is_commutative
assert a._new_rawargs(x, n).is_commutative is False
assert a._new_rawargs(x, y, n).is_commutative is False
m = x*n
assert m.is_commutative is False
assert m._new_rawargs(x).is_commutative
assert m._new_rawargs(n).is_commutative is False
assert m._new_rawargs(x, y).is_commutative
assert m._new_rawargs(x, n).is_commutative is False
assert m._new_rawargs(x, y, n).is_commutative is False
assert m._new_rawargs(x, n, reeval=False).is_commutative is False
assert m._new_rawargs(S.One) is S.One
def test_issue_5226():
assert Add(evaluate=False) == 0
assert Mul(evaluate=False) == 1
assert Mul(x + y, evaluate=False).is_Add
def test_free_symbols():
# free_symbols should return the free symbols of an object
assert S(1).free_symbols == set()
assert (x).free_symbols == {x}
assert Integral(x, (x, 1, y)).free_symbols == {y}
assert (-Integral(x, (x, 1, y))).free_symbols == {y}
assert meter.free_symbols == set()
assert (meter**x).free_symbols == {x}
def test_issue_5300():
x = Symbol('x', commutative=False)
assert x*sqrt(2)/sqrt(6) == x*sqrt(3)/3
def test_floordiv():
from sympy.functions.elementary.integers import floor
assert x // y == floor(x / y)
def test_as_coeff_Mul():
assert S(0).as_coeff_Mul() == (S.One, S.Zero)
assert Integer(3).as_coeff_Mul() == (Integer(3), Integer(1))
assert Rational(3, 4).as_coeff_Mul() == (Rational(3, 4), Integer(1))
assert Float(5.0).as_coeff_Mul() == (Float(5.0), Integer(1))
assert (Integer(3)*x).as_coeff_Mul() == (Integer(3), x)
assert (Rational(3, 4)*x).as_coeff_Mul() == (Rational(3, 4), x)
assert (Float(5.0)*x).as_coeff_Mul() == (Float(5.0), x)
assert (Integer(3)*x*y).as_coeff_Mul() == (Integer(3), x*y)
assert (Rational(3, 4)*x*y).as_coeff_Mul() == (Rational(3, 4), x*y)
assert (Float(5.0)*x*y).as_coeff_Mul() == (Float(5.0), x*y)
assert (x).as_coeff_Mul() == (S.One, x)
assert (x*y).as_coeff_Mul() == (S.One, x*y)
assert (-oo*x).as_coeff_Mul(rational=True) == (-1, oo*x)
def test_as_coeff_Add():
assert Integer(3).as_coeff_Add() == (Integer(3), Integer(0))
assert Rational(3, 4).as_coeff_Add() == (Rational(3, 4), Integer(0))
assert Float(5.0).as_coeff_Add() == (Float(5.0), Integer(0))
assert (Integer(3) + x).as_coeff_Add() == (Integer(3), x)
assert (Rational(3, 4) + x).as_coeff_Add() == (Rational(3, 4), x)
assert (Float(5.0) + x).as_coeff_Add() == (Float(5.0), x)
assert (Float(5.0) + x).as_coeff_Add(rational=True) == (0, Float(5.0) + x)
assert (Integer(3) + x + y).as_coeff_Add() == (Integer(3), x + y)
assert (Rational(3, 4) + x + y).as_coeff_Add() == (Rational(3, 4), x + y)
assert (Float(5.0) + x + y).as_coeff_Add() == (Float(5.0), x + y)
assert (x).as_coeff_Add() == (S.Zero, x)
assert (x*y).as_coeff_Add() == (S.Zero, x*y)
def test_expr_sorting():
f, g = symbols('f,g', cls=Function)
exprs = [1/x**2, 1/x, sqrt(sqrt(x)), sqrt(x), x, sqrt(x)**3, x**2]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [x, 2*x, 2*x**2, 2*x**3, x**n, 2*x**n, sin(x), sin(x)**n,
sin(x**2), cos(x), cos(x**2), tan(x)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [x + 1, x**2 + x + 1, x**3 + x**2 + x + 1]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [S(4), x - 3*I/2, x + 3*I/2, x - 4*I + 1, x + 4*I + 1]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [f(x), g(x), exp(x), sin(x), cos(x), factorial(x)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [Tuple(x, y), Tuple(x, z), Tuple(x, y, z)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[3], [1, 2]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[1, 2], [2, 3]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[1, 2], [1, 2, 3]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [{x: -y}, {x: y}]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [{1}, {1, 2}]
assert sorted(exprs, key=default_sort_key) == exprs
a, b = exprs = [Dummy('x'), Dummy('x')]
assert sorted([b, a], key=default_sort_key) == exprs
def test_as_ordered_factors():
f, g = symbols('f,g', cls=Function)
assert x.as_ordered_factors() == [x]
assert (2*x*x**n*sin(x)*cos(x)).as_ordered_factors() \
== [Integer(2), x, x**n, sin(x), cos(x)]
args = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
expr = Mul(*args)
assert expr.as_ordered_factors() == args
A, B = symbols('A,B', commutative=False)
assert (A*B).as_ordered_factors() == [A, B]
assert (B*A).as_ordered_factors() == [B, A]
def test_as_ordered_terms():
f, g = symbols('f,g', cls=Function)
assert x.as_ordered_terms() == [x]
assert (sin(x)**2*cos(x) + sin(x)*cos(x)**2 + 1).as_ordered_terms() \
== [sin(x)**2*cos(x), sin(x)*cos(x)**2, 1]
args = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
expr = Add(*args)
assert expr.as_ordered_terms() == args
assert (1 + 4*sqrt(3)*pi*x).as_ordered_terms() == [4*pi*x*sqrt(3), 1]
assert ( 2 + 3*I).as_ordered_terms() == [2, 3*I]
assert (-2 + 3*I).as_ordered_terms() == [-2, 3*I]
assert ( 2 - 3*I).as_ordered_terms() == [2, -3*I]
assert (-2 - 3*I).as_ordered_terms() == [-2, -3*I]
assert ( 4 + 3*I).as_ordered_terms() == [4, 3*I]
assert (-4 + 3*I).as_ordered_terms() == [-4, 3*I]
assert ( 4 - 3*I).as_ordered_terms() == [4, -3*I]
assert (-4 - 3*I).as_ordered_terms() == [-4, -3*I]
f = x**2*y**2 + x*y**4 + y + 2
assert f.as_ordered_terms(order="lex") == [x**2*y**2, x*y**4, y, 2]
assert f.as_ordered_terms(order="grlex") == [x*y**4, x**2*y**2, y, 2]
assert f.as_ordered_terms(order="rev-lex") == [2, y, x*y**4, x**2*y**2]
assert f.as_ordered_terms(order="rev-grlex") == [2, y, x**2*y**2, x*y**4]
k = symbols('k')
assert k.as_ordered_terms(data=True) == ([(k, ((1.0, 0.0), (1,), ()))], [k])
def test_sort_key_atomic_expr():
from sympy.physics.units import m, s
assert sorted([-m, s], key=lambda arg: arg.sort_key()) == [-m, s]
def test_eval_interval():
assert exp(x)._eval_interval(*Tuple(x, 0, 1)) == exp(1) - exp(0)
# issue 4199
# first subs and limit gives NaN
a = x/y
assert a._eval_interval(x, S(0), oo)._eval_interval(y, oo, S(0)) is S.NaN
# second subs and limit gives NaN
assert a._eval_interval(x, S(0), oo)._eval_interval(y, S(0), oo) is S.NaN
# difference gives S.NaN
a = x - y
assert a._eval_interval(x, S(1), oo)._eval_interval(y, oo, S(1)) is S.NaN
raises(ValueError, lambda: x._eval_interval(x, None, None))
a = -y*Heaviside(x - y)
assert a._eval_interval(x, -oo, oo) == -y
assert a._eval_interval(x, oo, -oo) == y
def test_eval_interval_zoo():
# Test that limit is used when zoo is returned
assert Si(1/x)._eval_interval(x, S(0), S(1)) == -pi/2 + Si(1)
def test_primitive():
assert (3*(x + 1)**2).primitive() == (3, (x + 1)**2)
assert (6*x + 2).primitive() == (2, 3*x + 1)
assert (x/2 + 3).primitive() == (S(1)/2, x + 6)
eq = (6*x + 2)*(x/2 + 3)
assert eq.primitive()[0] == 1
eq = (2 + 2*x)**2
assert eq.primitive()[0] == 1
assert (4.0*x).primitive() == (1, 4.0*x)
assert (4.0*x + y/2).primitive() == (S.Half, 8.0*x + y)
assert (-2*x).primitive() == (2, -x)
assert Add(5*z/7, 0.5*x, 3*y/2, evaluate=False).primitive() == \
(S(1)/14, 7.0*x + 21*y + 10*z)
for i in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
assert (i + x/3).primitive() == \
(S(1)/3, i + x)
assert (S.Infinity + 2*x/3 + 4*y/7).primitive() == \
(S(1)/21, 14*x + 12*y + oo)
assert S.Zero.primitive() == (S.One, S.Zero)
def test_issue_5843():
a = 1 + x
assert (2*a).extract_multiplicatively(a) == 2
assert (4*a).extract_multiplicatively(2*a) == 2
assert ((3*a)*(2*a)).extract_multiplicatively(a) == 6*a
def test_is_constant():
from sympy.solvers.solvers import checksol
Sum(x, (x, 1, 10)).is_constant() is True
Sum(x, (x, 1, n)).is_constant() is False
Sum(x, (x, 1, n)).is_constant(y) is True
Sum(x, (x, 1, n)).is_constant(n) is False
Sum(x, (x, 1, n)).is_constant(x) is True
eq = a*cos(x)**2 + a*sin(x)**2 - a
eq.is_constant() is True
assert eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0
assert x.is_constant() is False
assert x.is_constant(y) is True
assert checksol(x, x, Sum(x, (x, 1, n))) is False
assert checksol(x, x, Sum(x, (x, 1, n))) is False
f = Function('f')
assert f(1).is_constant
assert checksol(x, x, f(x)) is False
assert Pow(x, S(0), evaluate=False).is_constant() is True # == 1
assert Pow(S(0), x, evaluate=False).is_constant() is False # == 0 or 1
assert (2**x).is_constant() is False
assert Pow(S(2), S(3), evaluate=False).is_constant() is True
z1, z2 = symbols('z1 z2', zero=True)
assert (z1 + 2*z2).is_constant() is True
assert meter.is_constant() is True
assert (3*meter).is_constant() is True
assert (x*meter).is_constant() is False
assert Poly(3,x).is_constant() is True
def test_equals():
assert (-3 - sqrt(5) + (-sqrt(10)/2 - sqrt(2)/2)**2).equals(0)
assert (x**2 - 1).equals((x + 1)*(x - 1))
assert (cos(x)**2 + sin(x)**2).equals(1)
assert (a*cos(x)**2 + a*sin(x)**2).equals(a)
r = sqrt(2)
assert (-1/(r + r*x) + 1/r/(1 + x)).equals(0)
assert factorial(x + 1).equals((x + 1)*factorial(x))
assert sqrt(3).equals(2*sqrt(3)) is False
assert (sqrt(5)*sqrt(3)).equals(sqrt(3)) is False
assert (sqrt(5) + sqrt(3)).equals(0) is False
assert (sqrt(5) + pi).equals(0) is False
assert meter.equals(0) is False
assert (3*meter**2).equals(0) is False
eq = -(-1)**(S(3)/4)*6**(S(1)/4) + (-6)**(S(1)/4)*I
if eq != 0: # if canonicalization makes this zero, skip the test
assert eq.equals(0)
assert sqrt(x).equals(0) is False
# from integrate(x*sqrt(1 + 2*x), x);
# diff is zero only when assumptions allow
i = 2*sqrt(2)*x**(S(5)/2)*(1 + 1/(2*x))**(S(5)/2)/5 + \
2*sqrt(2)*x**(S(3)/2)*(1 + 1/(2*x))**(S(5)/2)/(-6 - 3/x)
ans = sqrt(2*x + 1)*(6*x**2 + x - 1)/15
diff = i - ans
assert diff.equals(0) is False
assert diff.subs(x, -S.Half/2) == 7*sqrt(2)/120
# there are regions for x for which the expression is True, for
# example, when x < -1/2 or x > 0 the expression is zero
p = Symbol('p', positive=True)
assert diff.subs(x, p).equals(0) is True
assert diff.subs(x, -1).equals(0) is True
# prove via minimal_polynomial or self-consistency
eq = sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3)) - sqrt(10 + 6*sqrt(3))
assert eq.equals(0)
q = 3**Rational(1, 3) + 3
p = expand(q**3)**Rational(1, 3)
assert (p - q).equals(0)
# issue 6829
# eq = q*x + q/4 + x**4 + x**3 + 2*x**2 - S(1)/3
# z = eq.subs(x, solve(eq, x)[0])
q = symbols('q')
z = (q*(-sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S(1)/3) -
S(13)/12)/2 - sqrt((2*q - S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S(1)/3) - S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S(1)/3) - S(13)/6)/2 - S(1)/4) + q/4 + (-sqrt(-2*(-(q
- S(7)/8)**S(2)/8 - S(2197)/13824)**(S(1)/3) - S(13)/12)/2 - sqrt((2*q
- S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S(1)/3) -
S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S(1)/3) -
S(13)/6)/2 - S(1)/4)**4 + (-sqrt(-2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S(1)/3) - S(13)/12)/2 - sqrt((2*q -
S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S(1)/3) -
S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S(1)/3) -
S(13)/6)/2 - S(1)/4)**3 + 2*(-sqrt(-2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S(1)/3) - S(13)/12)/2 - sqrt((2*q -
S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S(1)/3) -
S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S(1)/3) -
S(13)/6)/2 - S(1)/4)**2 - S(1)/3)
assert z.equals(0)
def test_random():
from sympy import posify, lucas
assert posify(x)[0]._random() is not None
assert lucas(n)._random(2, -2, 0, -1, 1) is None
# issue 8662
assert Piecewise((Max(x, y), z))._random() is None
def test_round():
from sympy.abc import x
assert Float('0.1249999').round(2) == 0.12
d20 = 12345678901234567890
ans = S(d20).round(2)
assert ans.is_Float and ans == d20
ans = S(d20).round(-2)
assert ans.is_Float and ans == 12345678901234567900
assert S('1/7').round(4) == 0.1429
assert S('.[12345]').round(4) == 0.1235
assert S('.1349').round(2) == 0.13
n = S(12345)
ans = n.round()
assert ans.is_Float
assert ans == n
ans = n.round(1)
assert ans.is_Float
assert ans == n
ans = n.round(4)
assert ans.is_Float
assert ans == n
assert n.round(-1) == 12350
r = n.round(-4)
assert r == 10000
# in fact, it should equal many values since __eq__
# compares at equal precision
assert all(r == i for i in range(9984, 10049))
assert n.round(-5) == 0
assert (pi + sqrt(2)).round(2) == 4.56
assert (10*(pi + sqrt(2))).round(-1) == 50
raises(TypeError, lambda: round(x + 2, 2))
assert S(2.3).round(1) == 2.3
e = S(12.345).round(2)
assert e == round(12.345, 2)
assert type(e) is Float
assert (Float(.3, 3) + 2*pi).round() == 7
assert (Float(.3, 3) + 2*pi*100).round() == 629
assert (Float(.03, 3) + 2*pi/100).round(5) == 0.09283
assert (Float(.03, 3) + 2*pi/100).round(4) == 0.0928
assert (pi + 2*E*I).round() == 3 + 5*I
assert S.Zero.round() == 0
a = (Add(1, Float('1.' + '9'*27, ''), evaluate=0))
assert a.round(10) == Float('3.0000000000', '')
assert a.round(25) == Float('3.0000000000000000000000000', '')
assert a.round(26) == Float('3.00000000000000000000000000', '')
assert a.round(27) == Float('2.999999999999999999999999999', '')
assert a.round(30) == Float('2.999999999999999999999999999', '')
raises(TypeError, lambda: x.round())
f = Function('f')
raises(TypeError, lambda: f(1).round())
# exact magnitude of 10
assert str(S(1).round()) == '1.'
assert str(S(100).round()) == '100.'
# applied to real and imaginary portions
assert (2*pi + E*I).round() == 6 + 3*I
assert (2*pi + I/10).round() == 6
assert (pi/10 + 2*I).round() == 2*I
# the lhs re and im parts are Float with dps of 2
# and those on the right have dps of 15 so they won't compare
# equal unless we use string or compare components (which will
# then coerce the floats to the same precision) or re-create
# the floats
assert str((pi/10 + E*I).round(2)) == '0.31 + 2.72*I'
assert (pi/10 + E*I).round(2).as_real_imag() == (0.31, 2.72)
assert (pi/10 + E*I).round(2) == Float(0.31, 2) + I*Float(2.72, 3)
# issue 6914
assert (I**(I + 3)).round(3) == Float('-0.208', '')*I
# issue 8720
assert S(-123.6).round() == -124.
assert S(-1.5).round() == -2.
assert S(-100.5).round() == -101.
assert S(-1.5 - 10.5*I).round() == -2.0 - 11.0*I
# issue 7961
assert str(S(0.006).round(2)) == '0.01'
assert str(S(0.00106).round(4)) == '0.0011'
# issue 8147
assert S.NaN.round() == S.NaN
assert S.Infinity.round() == S.Infinity
assert S.NegativeInfinity.round() == S.NegativeInfinity
assert S.ComplexInfinity.round() == S.ComplexInfinity
def test_held_expression_UnevaluatedExpr():
x = symbols("x")
he = UnevaluatedExpr(1/x)
e1 = x*he
assert isinstance(e1, Mul)
assert e1.args == (x, he)
assert e1.doit() == 1
assert UnevaluatedExpr(Derivative(x, x)).doit(deep=False
) == Derivative(x, x)
assert UnevaluatedExpr(Derivative(x, x)).doit() == 1
xx = Mul(x, x, evaluate=False)
assert xx != x**2
ue2 = UnevaluatedExpr(xx)
assert isinstance(ue2, UnevaluatedExpr)
assert ue2.args == (xx,)
assert ue2.doit() == x**2
assert ue2.doit(deep=False) == xx
x2 = UnevaluatedExpr(2)*2
assert type(x2) is Mul
assert x2.args == (2, UnevaluatedExpr(2))
def test_round_exception_nostr():
# Don't use the string form of the expression in the round exception, as
# it's too slow
s = Symbol('bad')
try:
s.round()
except TypeError as e:
assert 'bad' not in str(e)
else:
# Did not raise
raise AssertionError("Did not raise")
def test_extract_branch_factor():
assert exp_polar(2.0*I*pi).extract_branch_factor() == (1, 1)
def test_identity_removal():
assert Add.make_args(x + 0) == (x,)
assert Mul.make_args(x*1) == (x,)
def test_float_0():
assert Float(0.0) + 1 == Float(1.0)
@XFAIL
def test_float_0_fail():
assert Float(0.0)*x == Float(0.0)
assert (x + Float(0.0)).is_Add
def test_issue_6325():
ans = (b**2 + z**2 - (b*(a + b*t) + z*(c + t*z))**2/(
(a + b*t)**2 + (c + t*z)**2))/sqrt((a + b*t)**2 + (c + t*z)**2)
e = sqrt((a + b*t)**2 + (c + z*t)**2)
assert diff(e, t, 2) == ans
e.diff(t, 2) == ans
assert diff(e, t, 2, simplify=False) != ans
def test_issue_7426():
f1 = a % c
f2 = x % z
assert f1.equals(f2) is None
def test_issue_1112():
x = Symbol('x', positive=False)
assert (x > 0) is S.false
def test_issue_10161():
x = symbols('x', real=True)
assert x*abs(x)*abs(x) == x**3
def test_issue_10755():
x = symbols('x')
raises(TypeError, lambda: int(log(x)))
raises(TypeError, lambda: log(x).round(2))
def test_issue_11877():
x = symbols('x')
assert integrate(log(S(1)/2 - x), (x, 0, S(1)/2)) == -S(1)/2 -log(2)/2
def test_normal():
x = symbols('x')
e = Mul(S.Half, 1 + x, evaluate=False)
assert e.normal() == e
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