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| from sympy.holonomic import (DifferentialOperator, HolonomicFunction, | |
| DifferentialOperators, from_hyper, | |
| from_meijerg, expr_to_holonomic) | |
| from sympy.holonomic.recurrence import RecurrenceOperators, HolonomicSequence | |
| from sympy.core import EulerGamma | |
| from sympy.core.numbers import (I, Rational, pi) | |
| from sympy.core.singleton import S | |
| from sympy.core.symbol import (Symbol, symbols) | |
| from sympy.functions.elementary.exponential import (exp, log) | |
| from sympy.functions.elementary.hyperbolic import (asinh, cosh) | |
| from sympy.functions.elementary.miscellaneous import sqrt | |
| from sympy.functions.elementary.trigonometric import (cos, sin) | |
| from sympy.functions.special.bessel import besselj | |
| from sympy.functions.special.beta_functions import beta | |
| from sympy.functions.special.error_functions import (Ci, Si, erf, erfc) | |
| from sympy.functions.special.gamma_functions import gamma | |
| from sympy.functions.special.hyper import (hyper, meijerg) | |
| from sympy.printing.str import sstr | |
| from sympy.series.order import O | |
| from sympy.simplify.hyperexpand import hyperexpand | |
| from sympy.polys.domains.integerring import ZZ | |
| from sympy.polys.domains.rationalfield import QQ | |
| from sympy.polys.domains.realfield import RR | |
| def test_DifferentialOperator(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') | |
| assert Dx == R.derivative_operator | |
| assert Dx == DifferentialOperator([R.base.zero, R.base.one], R) | |
| assert x * Dx + x**2 * Dx**2 == DifferentialOperator([0, x, x**2], R) | |
| assert (x**2 + 1) + Dx + x * \ | |
| Dx**5 == DifferentialOperator([x**2 + 1, 1, 0, 0, 0, x], R) | |
| assert (x * Dx + x**2 + 1 - Dx * (x**3 + x))**3 == (-48 * x**6) + \ | |
| (-57 * x**7) * Dx + (-15 * x**8) * Dx**2 + (-x**9) * Dx**3 | |
| p = (x * Dx**2 + (x**2 + 3) * Dx**5) * (Dx + x**2) | |
| q = (2 * x) + (4 * x**2) * Dx + (x**3) * Dx**2 + \ | |
| (20 * x**2 + x + 60) * Dx**3 + (10 * x**3 + 30 * x) * Dx**4 + \ | |
| (x**4 + 3 * x**2) * Dx**5 + (x**2 + 3) * Dx**6 | |
| assert p == q | |
| def test_HolonomicFunction_addition(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') | |
| p = HolonomicFunction(Dx**2 * x, x) | |
| q = HolonomicFunction((2) * Dx + (x) * Dx**2, x) | |
| assert p == q | |
| p = HolonomicFunction(x * Dx + 1, x) | |
| q = HolonomicFunction(Dx + 1, x) | |
| r = HolonomicFunction((x - 2) + (x**2 - 2) * Dx + (x**2 - x) * Dx**2, x) | |
| assert p + q == r | |
| p = HolonomicFunction(x * Dx + Dx**2 * (x**2 + 2), x) | |
| q = HolonomicFunction(Dx - 3, x) | |
| r = HolonomicFunction((-54 * x**2 - 126 * x - 150) + (-135 * x**3 - 252 * x**2 - 270 * x + 140) * Dx +\ | |
| (-27 * x**4 - 24 * x**2 + 14 * x - 150) * Dx**2 + \ | |
| (9 * x**4 + 15 * x**3 + 38 * x**2 + 30 * x +40) * Dx**3, x) | |
| assert p + q == r | |
| p = HolonomicFunction(Dx**5 - 1, x) | |
| q = HolonomicFunction(x**3 + Dx, x) | |
| r = HolonomicFunction((-x**18 + 45*x**14 - 525*x**10 + 1575*x**6 - x**3 - 630*x**2) + \ | |
| (-x**15 + 30*x**11 - 195*x**7 + 210*x**3 - 1)*Dx + (x**18 - 45*x**14 + 525*x**10 - \ | |
| 1575*x**6 + x**3 + 630*x**2)*Dx**5 + (x**15 - 30*x**11 + 195*x**7 - 210*x**3 + \ | |
| 1)*Dx**6, x) | |
| assert p+q == r | |
| p = x**2 + 3*x + 8 | |
| q = x**3 - 7*x + 5 | |
| p = p*Dx - p.diff() | |
| q = q*Dx - q.diff() | |
| r = HolonomicFunction(p, x) + HolonomicFunction(q, x) | |
| s = HolonomicFunction((6*x**2 + 18*x + 14) + (-4*x**3 - 18*x**2 - 62*x + 10)*Dx +\ | |
| (x**4 + 6*x**3 + 31*x**2 - 10*x - 71)*Dx**2, x) | |
| assert r == s | |
| def test_HolonomicFunction_multiplication(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') | |
| p = HolonomicFunction(Dx+x+x*Dx**2, x) | |
| q = HolonomicFunction(x*Dx+Dx*x+Dx**2, x) | |
| r = HolonomicFunction((8*x**6 + 4*x**4 + 6*x**2 + 3) + (24*x**5 - 4*x**3 + 24*x)*Dx + \ | |
| (8*x**6 + 20*x**4 + 12*x**2 + 2)*Dx**2 + (8*x**5 + 4*x**3 + 4*x)*Dx**3 + \ | |
| (2*x**4 + x**2)*Dx**4, x) | |
| assert p*q == r | |
| p = HolonomicFunction(Dx**2+1, x) | |
| q = HolonomicFunction(Dx-1, x) | |
| r = HolonomicFunction((2) + (-2)*Dx + (1)*Dx**2, x) | |
| assert p*q == r | |
| p = HolonomicFunction(Dx**2+1+x+Dx, x) | |
| q = HolonomicFunction((Dx*x-1)**2, x) | |
| r = HolonomicFunction((4*x**7 + 11*x**6 + 16*x**5 + 4*x**4 - 6*x**3 - 7*x**2 - 8*x - 2) + \ | |
| (8*x**6 + 26*x**5 + 24*x**4 - 3*x**3 - 11*x**2 - 6*x - 2)*Dx + \ | |
| (8*x**6 + 18*x**5 + 15*x**4 - 3*x**3 - 6*x**2 - 6*x - 2)*Dx**2 + (8*x**5 + \ | |
| 10*x**4 + 6*x**3 - 2*x**2 - 4*x)*Dx**3 + (4*x**5 + 3*x**4 - x**2)*Dx**4, x) | |
| assert p*q == r | |
| p = HolonomicFunction(x*Dx**2-1, x) | |
| q = HolonomicFunction(Dx*x-x, x) | |
| r = HolonomicFunction((x - 3) + (-2*x + 2)*Dx + (x)*Dx**2, x) | |
| assert p*q == r | |
| def test_HolonomicFunction_power(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') | |
| p = HolonomicFunction(Dx+x+x*Dx**2, x) | |
| a = HolonomicFunction(Dx, x) | |
| for n in range(10): | |
| assert a == p**n | |
| a *= p | |
| def test_addition_initial_condition(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') | |
| p = HolonomicFunction(Dx-1, x, 0, [3]) | |
| q = HolonomicFunction(Dx**2+1, x, 0, [1, 0]) | |
| r = HolonomicFunction(-1 + Dx - Dx**2 + Dx**3, x, 0, [4, 3, 2]) | |
| assert p + q == r | |
| p = HolonomicFunction(Dx - x + Dx**2, x, 0, [1, 2]) | |
| q = HolonomicFunction(Dx**2 + x, x, 0, [1, 0]) | |
| r = HolonomicFunction((-x**4 - x**3/4 - x**2 + Rational(1, 4)) + (x**3 + x**2/4 + x*Rational(3, 4) + 1)*Dx + \ | |
| (x*Rational(-3, 2) + Rational(7, 4))*Dx**2 + (x**2 - x*Rational(7, 4) + Rational(1, 4))*Dx**3 + (x**2 + x/4 + S.Half)*Dx**4, x, 0, [2, 2, -2, 2]) | |
| assert p + q == r | |
| p = HolonomicFunction(Dx**2 + 4*x*Dx + x**2, x, 0, [3, 4]) | |
| q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1]) | |
| r = HolonomicFunction((x**6 + 2*x**4 - 5*x**2 - 6) + (4*x**5 + 36*x**3 - 32*x)*Dx + \ | |
| (x**6 + 3*x**4 + 5*x**2 - 9)*Dx**2 + (4*x**5 + 36*x**3 - 32*x)*Dx**3 + (x**4 + \ | |
| 10*x**2 - 3)*Dx**4, x, 0, [4, 5, -1, -17]) | |
| assert p + q == r | |
| q = HolonomicFunction(Dx**3 + x, x, 2, [3, 0, 1]) | |
| p = HolonomicFunction(Dx - 1, x, 2, [1]) | |
| r = HolonomicFunction((-x**2 - x + 1) + (x**2 + x)*Dx + (-x - 2)*Dx**3 + \ | |
| (x + 1)*Dx**4, x, 2, [4, 1, 2, -5 ]) | |
| assert p + q == r | |
| p = expr_to_holonomic(sin(x)) | |
| q = expr_to_holonomic(1/x, x0=1) | |
| r = HolonomicFunction((x**2 + 6) + (x**3 + 2*x)*Dx + (x**2 + 6)*Dx**2 + (x**3 + 2*x)*Dx**3, \ | |
| x, 1, [sin(1) + 1, -1 + cos(1), -sin(1) + 2]) | |
| assert p + q == r | |
| C_1 = symbols('C_1') | |
| p = expr_to_holonomic(sqrt(x)) | |
| q = expr_to_holonomic(sqrt(x**2-x)) | |
| r = (p + q).to_expr().subs(C_1, -I/2).expand() | |
| assert r == I*sqrt(x)*sqrt(-x + 1) + sqrt(x) | |
| def test_multiplication_initial_condition(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') | |
| p = HolonomicFunction(Dx**2 + x*Dx - 1, x, 0, [3, 1]) | |
| q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1]) | |
| r = HolonomicFunction((x**4 + 14*x**2 + 60) + 4*x*Dx + (x**4 + 9*x**2 + 20)*Dx**2 + \ | |
| (2*x**3 + 18*x)*Dx**3 + (x**2 + 10)*Dx**4, x, 0, [3, 4, 2, 3]) | |
| assert p * q == r | |
| p = HolonomicFunction(Dx**2 + x, x, 0, [1, 0]) | |
| q = HolonomicFunction(Dx**3 - x**2, x, 0, [3, 3, 3]) | |
| r = HolonomicFunction((x**8 - 37*x**7/27 - 10*x**6/27 - 164*x**5/9 - 184*x**4/9 + \ | |
| 160*x**3/27 + 404*x**2/9 + 8*x + Rational(40, 3)) + (6*x**7 - 128*x**6/9 - 98*x**5/9 - 28*x**4/9 + \ | |
| 8*x**3/9 + 28*x**2 + x*Rational(40, 9) - 40)*Dx + (3*x**6 - 82*x**5/9 + 76*x**4/9 + 4*x**3/3 + \ | |
| 220*x**2/9 - x*Rational(80, 3))*Dx**2 + (-2*x**6 + 128*x**5/27 - 2*x**4/3 -80*x**2/9 + Rational(200, 9))*Dx**3 + \ | |
| (3*x**5 - 64*x**4/9 - 28*x**3/9 + 6*x**2 - x*Rational(20, 9) - Rational(20, 3))*Dx**4 + (-4*x**3 + 64*x**2/9 + \ | |
| x*Rational(8, 3))*Dx**5 + (x**4 - 64*x**3/27 - 4*x**2/3 + Rational(20, 9))*Dx**6, x, 0, [3, 3, 3, -3, -12, -24]) | |
| assert p * q == r | |
| p = HolonomicFunction(Dx - 1, x, 0, [2]) | |
| q = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]) | |
| r = HolonomicFunction(2 -2*Dx + Dx**2, x, 0, [0, 2]) | |
| assert p * q == r | |
| q = HolonomicFunction(x*Dx**2 + 1 + 2*Dx, x, 0,[0, 1]) | |
| r = HolonomicFunction((x - 1) + (-2*x + 2)*Dx + x*Dx**2, x, 0, [0, 2]) | |
| assert p * q == r | |
| p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 3]) | |
| q = HolonomicFunction(Dx**3 + 1, x, 0, [1, 2, 1]) | |
| r = HolonomicFunction(6*Dx + 3*Dx**2 + 2*Dx**3 - 3*Dx**4 + Dx**6, x, 0, [1, 5, 14, 17, 17, 2]) | |
| assert p * q == r | |
| p = expr_to_holonomic(sin(x)) | |
| q = expr_to_holonomic(1/x, x0=1) | |
| r = HolonomicFunction(x + 2*Dx + x*Dx**2, x, 1, [sin(1), -sin(1) + cos(1)]) | |
| assert p * q == r | |
| p = expr_to_holonomic(sqrt(x)) | |
| q = expr_to_holonomic(sqrt(x**2-x)) | |
| r = (p * q).to_expr() | |
| assert r == I*x*sqrt(-x + 1) | |
| def test_HolonomicFunction_composition(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') | |
| p = HolonomicFunction(Dx-1, x).composition(x**2+x) | |
| r = HolonomicFunction((-2*x - 1) + Dx, x) | |
| assert p == r | |
| p = HolonomicFunction(Dx**2+1, x).composition(x**5+x**2+1) | |
| r = HolonomicFunction((125*x**12 + 150*x**9 + 60*x**6 + 8*x**3) + (-20*x**3 - 2)*Dx + \ | |
| (5*x**4 + 2*x)*Dx**2, x) | |
| assert p == r | |
| p = HolonomicFunction(Dx**2*x+x, x).composition(2*x**3+x**2+1) | |
| r = HolonomicFunction((216*x**9 + 324*x**8 + 180*x**7 + 152*x**6 + 112*x**5 + \ | |
| 36*x**4 + 4*x**3) + (24*x**4 + 16*x**3 + 3*x**2 - 6*x - 1)*Dx + (6*x**5 + 5*x**4 + \ | |
| x**3 + 3*x**2 + x)*Dx**2, x) | |
| assert p == r | |
| p = HolonomicFunction(Dx**2+1, x).composition(1-x**2) | |
| r = HolonomicFunction((4*x**3) - Dx + x*Dx**2, x) | |
| assert p == r | |
| p = HolonomicFunction(Dx**2+1, x).composition(x - 2/(x**2 + 1)) | |
| r = HolonomicFunction((x**12 + 6*x**10 + 12*x**9 + 15*x**8 + 48*x**7 + 68*x**6 + \ | |
| 72*x**5 + 111*x**4 + 112*x**3 + 54*x**2 + 12*x + 1) + (12*x**8 + 32*x**6 + \ | |
| 24*x**4 - 4)*Dx + (x**12 + 6*x**10 + 4*x**9 + 15*x**8 + 16*x**7 + 20*x**6 + 24*x**5+ \ | |
| 15*x**4 + 16*x**3 + 6*x**2 + 4*x + 1)*Dx**2, x) | |
| assert p == r | |
| def test_from_hyper(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') | |
| p = hyper([1, 1], [Rational(3, 2)], x**2/4) | |
| q = HolonomicFunction((4*x) + (5*x**2 - 8)*Dx + (x**3 - 4*x)*Dx**2, x, 1, [2*sqrt(3)*pi/9, -4*sqrt(3)*pi/27 + Rational(4, 3)]) | |
| r = from_hyper(p) | |
| assert r == q | |
| p = from_hyper(hyper([1], [Rational(3, 2)], x**2/4)) | |
| q = HolonomicFunction(-x + (-x**2/2 + 2)*Dx + x*Dx**2, x) | |
| # x0 = 1 | |
| y0 = '[sqrt(pi)*exp(1/4)*erf(1/2), -sqrt(pi)*exp(1/4)*erf(1/2)/2 + 1]' | |
| assert sstr(p.y0) == y0 | |
| assert q.annihilator == p.annihilator | |
| def test_from_meijerg(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') | |
| p = from_meijerg(meijerg(([], [Rational(3, 2)]), ([S.Half], [S.Half, 1]), x)) | |
| q = HolonomicFunction(x/2 - Rational(1, 4) + (-x**2 + x/4)*Dx + x**2*Dx**2 + x**3*Dx**3, x, 1, \ | |
| [1/sqrt(pi), 1/(2*sqrt(pi)), -1/(4*sqrt(pi))]) | |
| assert p == q | |
| p = from_meijerg(meijerg(([], []), ([0], []), x)) | |
| q = HolonomicFunction(1 + Dx, x, 0, [1]) | |
| assert p == q | |
| p = from_meijerg(meijerg(([1], []), ([S.Half], [0]), x)) | |
| q = HolonomicFunction((x + S.Half)*Dx + x*Dx**2, x, 1, [sqrt(pi)*erf(1), exp(-1)]) | |
| assert p == q | |
| p = from_meijerg(meijerg(([0], [1]), ([0], []), 2*x**2)) | |
| q = HolonomicFunction((3*x**2 - 1)*Dx + x**3*Dx**2, x, 1, [-exp(Rational(-1, 2)) + 1, -exp(Rational(-1, 2))]) | |
| assert p == q | |
| def test_to_Sequence(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') | |
| n = symbols('n', integer=True) | |
| _, Sn = RecurrenceOperators(ZZ.old_poly_ring(n), 'Sn') | |
| p = HolonomicFunction(x**2*Dx**4 + x + Dx, x).to_sequence() | |
| q = [(HolonomicSequence(1 + (n + 2)*Sn**2 + (n**4 + 6*n**3 + 11*n**2 + 6*n)*Sn**3), 0, 1)] | |
| assert p == q | |
| p = HolonomicFunction(x**2*Dx**4 + x**3 + Dx**2, x).to_sequence() | |
| q = [(HolonomicSequence(1 + (n**4 + 14*n**3 + 72*n**2 + 163*n + 140)*Sn**5), 0, 0)] | |
| assert p == q | |
| p = HolonomicFunction(x**3*Dx**4 + 1 + Dx**2, x).to_sequence() | |
| q = [(HolonomicSequence(1 + (n**4 - 2*n**3 - n**2 + 2*n)*Sn + (n**2 + 3*n + 2)*Sn**2), 0, 0)] | |
| assert p == q | |
| p = HolonomicFunction(3*x**3*Dx**4 + 2*x*Dx + x*Dx**3, x).to_sequence() | |
| q = [(HolonomicSequence(2*n + (3*n**4 - 6*n**3 - 3*n**2 + 6*n)*Sn + (n**3 + 3*n**2 + 2*n)*Sn**2), 0, 1)] | |
| assert p == q | |
| def test_to_Sequence_Initial_Coniditons(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') | |
| n = symbols('n', integer=True) | |
| _, Sn = RecurrenceOperators(QQ.old_poly_ring(n), 'Sn') | |
| p = HolonomicFunction(Dx - 1, x, 0, [1]).to_sequence() | |
| q = [(HolonomicSequence(-1 + (n + 1)*Sn, 1), 0)] | |
| assert p == q | |
| p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]).to_sequence() | |
| q = [(HolonomicSequence(1 + (n**2 + 3*n + 2)*Sn**2, [0, 1]), 0)] | |
| assert p == q | |
| p = HolonomicFunction(Dx**2 + 1 + x**3*Dx, x, 0, [2, 3]).to_sequence() | |
| q = [(HolonomicSequence(n + Sn**2 + (n**2 + 7*n + 12)*Sn**4, [2, 3, -1, Rational(-1, 2), Rational(1, 12)]), 1)] | |
| assert p == q | |
| p = HolonomicFunction(x**3*Dx**5 + 1 + Dx, x).to_sequence() | |
| q = [(HolonomicSequence(1 + (n + 1)*Sn + (n**5 - 5*n**3 + 4*n)*Sn**2), 0, 3)] | |
| assert p == q | |
| C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3') | |
| p = expr_to_holonomic(log(1+x**2)) | |
| q = [(HolonomicSequence(n**2 + (n**2 + 2*n)*Sn**2, [0, 0, C_2]), 0, 1)] | |
| assert p.to_sequence() == q | |
| p = p.diff() | |
| q = [(HolonomicSequence((n + 2) + (n + 2)*Sn**2, [C_0, 0]), 1, 0)] | |
| assert p.to_sequence() == q | |
| p = expr_to_holonomic(erf(x) + x).to_sequence() | |
| q = [(HolonomicSequence((2*n**2 - 2*n) + (n**3 + 2*n**2 - n - 2)*Sn**2, [0, 1 + 2/sqrt(pi), 0, C_3]), 0, 2)] | |
| assert p == q | |
| def test_series(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') | |
| p = HolonomicFunction(Dx**2 + 2*x*Dx, x, 0, [0, 1]).series(n=10) | |
| q = x - x**3/3 + x**5/10 - x**7/42 + x**9/216 + O(x**10) | |
| assert p == q | |
| p = HolonomicFunction(Dx - 1, x).composition(x**2, 0, [1]) # e^(x**2) | |
| q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]) # cos(x) | |
| r = (p * q).series(n=10) # expansion of cos(x) * exp(x**2) | |
| s = 1 + x**2/2 + x**4/24 - 31*x**6/720 - 179*x**8/8064 + O(x**10) | |
| assert r == s | |
| t = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1]) # log(1 + x) | |
| r = (p * t + q).series(n=10) | |
| s = 1 + x - x**2 + 4*x**3/3 - 17*x**4/24 + 31*x**5/30 - 481*x**6/720 +\ | |
| 71*x**7/105 - 20159*x**8/40320 + 379*x**9/840 + O(x**10) | |
| assert r == s | |
| p = HolonomicFunction((6+6*x-3*x**2) - (10*x-3*x**2-3*x**3)*Dx + \ | |
| (4-6*x**3+2*x**4)*Dx**2, x, 0, [0, 1]).series(n=7) | |
| q = x + x**3/6 - 3*x**4/16 + x**5/20 - 23*x**6/960 + O(x**7) | |
| assert p == q | |
| p = HolonomicFunction((6+6*x-3*x**2) - (10*x-3*x**2-3*x**3)*Dx + \ | |
| (4-6*x**3+2*x**4)*Dx**2, x, 0, [1, 0]).series(n=7) | |
| q = 1 - 3*x**2/4 - x**3/4 - 5*x**4/32 - 3*x**5/40 - 17*x**6/384 + O(x**7) | |
| assert p == q | |
| p = expr_to_holonomic(erf(x) + x).series(n=10) | |
| C_3 = symbols('C_3') | |
| q = (erf(x) + x).series(n=10) | |
| assert p.subs(C_3, -2/(3*sqrt(pi))) == q | |
| assert expr_to_holonomic(sqrt(x**3 + x)).series(n=10) == sqrt(x**3 + x).series(n=10) | |
| assert expr_to_holonomic((2*x - 3*x**2)**Rational(1, 3)).series() == ((2*x - 3*x**2)**Rational(1, 3)).series() | |
| assert expr_to_holonomic(sqrt(x**2-x)).series() == (sqrt(x**2-x)).series() | |
| assert expr_to_holonomic(cos(x)**2/x**2, y0={-2: [1, 0, -1]}).series(n=10) == (cos(x)**2/x**2).series(n=10) | |
| assert expr_to_holonomic(cos(x)**2/x**2, x0=1).series(n=10).together() == (cos(x)**2/x**2).series(n=10, x0=1).together() | |
| assert expr_to_holonomic(cos(x-1)**2/(x-1)**2, x0=1, y0={-2: [1, 0, -1]}).series(n=10) \ | |
| == (cos(x-1)**2/(x-1)**2).series(x0=1, n=10) | |
| def test_evalf_euler(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') | |
| # log(1+x) | |
| p = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1]) | |
| # path taken is a straight line from 0 to 1, on the real axis | |
| r = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1] | |
| s = '0.699525841805253' # approx. equal to log(2) i.e. 0.693147180559945 | |
| assert sstr(p.evalf(r, method='Euler')[-1]) == s | |
| # path taken is a triangle 0-->1+i-->2 | |
| r = [0.1 + 0.1*I] | |
| for i in range(9): | |
| r.append(r[-1]+0.1+0.1*I) | |
| for i in range(10): | |
| r.append(r[-1]+0.1-0.1*I) | |
| # close to the exact solution 1.09861228866811 | |
| # imaginary part also close to zero | |
| s = '1.07530466271334 - 0.0251200594793912*I' | |
| assert sstr(p.evalf(r, method='Euler')[-1]) == s | |
| # sin(x) | |
| p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]) | |
| s = '0.905546532085401 - 6.93889390390723e-18*I' | |
| assert sstr(p.evalf(r, method='Euler')[-1]) == s | |
| # computing sin(pi/2) using this method | |
| # using a linear path from 0 to pi/2 | |
| r = [0.1] | |
| for i in range(14): | |
| r.append(r[-1] + 0.1) | |
| r.append(pi/2) | |
| s = '1.08016557252834' # close to 1.0 (exact solution) | |
| assert sstr(p.evalf(r, method='Euler')[-1]) == s | |
| # trying different path, a rectangle (0-->i-->pi/2 + i-->pi/2) | |
| # computing the same value sin(pi/2) using different path | |
| r = [0.1*I] | |
| for i in range(9): | |
| r.append(r[-1]+0.1*I) | |
| for i in range(15): | |
| r.append(r[-1]+0.1) | |
| r.append(pi/2+I) | |
| for i in range(10): | |
| r.append(r[-1]-0.1*I) | |
| # close to 1.0 | |
| s = '0.976882381836257 - 1.65557671738537e-16*I' | |
| assert sstr(p.evalf(r, method='Euler')[-1]) == s | |
| # cos(x) | |
| p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]) | |
| # compute cos(pi) along 0-->pi | |
| r = [0.05] | |
| for i in range(61): | |
| r.append(r[-1]+0.05) | |
| r.append(pi) | |
| # close to -1 (exact answer) | |
| s = '-1.08140824719196' | |
| assert sstr(p.evalf(r, method='Euler')[-1]) == s | |
| # a rectangular path (0 -> i -> 2+i -> 2) | |
| r = [0.1*I] | |
| for i in range(9): | |
| r.append(r[-1]+0.1*I) | |
| for i in range(20): | |
| r.append(r[-1]+0.1) | |
| for i in range(10): | |
| r.append(r[-1]-0.1*I) | |
| p = HolonomicFunction(Dx**2 + 1, x, 0, [1,1]).evalf(r, method='Euler') | |
| s = '0.501421652861245 - 3.88578058618805e-16*I' | |
| assert sstr(p[-1]) == s | |
| def test_evalf_rk4(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') | |
| # log(1+x) | |
| p = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1]) | |
| # path taken is a straight line from 0 to 1, on the real axis | |
| r = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1] | |
| s = '0.693146363174626' # approx. equal to log(2) i.e. 0.693147180559945 | |
| assert sstr(p.evalf(r)[-1]) == s | |
| # path taken is a triangle 0-->1+i-->2 | |
| r = [0.1 + 0.1*I] | |
| for i in range(9): | |
| r.append(r[-1]+0.1+0.1*I) | |
| for i in range(10): | |
| r.append(r[-1]+0.1-0.1*I) | |
| # close to the exact solution 1.09861228866811 | |
| # imaginary part also close to zero | |
| s = '1.098616 + 1.36083e-7*I' | |
| assert sstr(p.evalf(r)[-1].n(7)) == s | |
| # sin(x) | |
| p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]) | |
| s = '0.90929463522785 + 1.52655665885959e-16*I' | |
| assert sstr(p.evalf(r)[-1]) == s | |
| # computing sin(pi/2) using this method | |
| # using a linear path from 0 to pi/2 | |
| r = [0.1] | |
| for i in range(14): | |
| r.append(r[-1] + 0.1) | |
| r.append(pi/2) | |
| s = '0.999999895088917' # close to 1.0 (exact solution) | |
| assert sstr(p.evalf(r)[-1]) == s | |
| # trying different path, a rectangle (0-->i-->pi/2 + i-->pi/2) | |
| # computing the same value sin(pi/2) using different path | |
| r = [0.1*I] | |
| for i in range(9): | |
| r.append(r[-1]+0.1*I) | |
| for i in range(15): | |
| r.append(r[-1]+0.1) | |
| r.append(pi/2+I) | |
| for i in range(10): | |
| r.append(r[-1]-0.1*I) | |
| # close to 1.0 | |
| s = '1.00000003415141 + 6.11940487991086e-16*I' | |
| assert sstr(p.evalf(r)[-1]) == s | |
| # cos(x) | |
| p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]) | |
| # compute cos(pi) along 0-->pi | |
| r = [0.05] | |
| for i in range(61): | |
| r.append(r[-1]+0.05) | |
| r.append(pi) | |
| # close to -1 (exact answer) | |
| s = '-0.999999993238714' | |
| assert sstr(p.evalf(r)[-1]) == s | |
| # a rectangular path (0 -> i -> 2+i -> 2) | |
| r = [0.1*I] | |
| for i in range(9): | |
| r.append(r[-1]+0.1*I) | |
| for i in range(20): | |
| r.append(r[-1]+0.1) | |
| for i in range(10): | |
| r.append(r[-1]-0.1*I) | |
| p = HolonomicFunction(Dx**2 + 1, x, 0, [1,1]).evalf(r) | |
| s = '0.493152791638442 - 1.41553435639707e-15*I' | |
| assert sstr(p[-1]) == s | |
| def test_expr_to_holonomic(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') | |
| p = expr_to_holonomic((sin(x)/x)**2) | |
| q = HolonomicFunction(8*x + (4*x**2 + 6)*Dx + 6*x*Dx**2 + x**2*Dx**3, x, 0, \ | |
| [1, 0, Rational(-2, 3)]) | |
| assert p == q | |
| p = expr_to_holonomic(1/(1+x**2)**2) | |
| q = HolonomicFunction(4*x + (x**2 + 1)*Dx, x, 0, [1]) | |
| assert p == q | |
| p = expr_to_holonomic(exp(x)*sin(x)+x*log(1+x)) | |
| q = HolonomicFunction((2*x**3 + 10*x**2 + 20*x + 18) + (-2*x**4 - 10*x**3 - 20*x**2 \ | |
| - 18*x)*Dx + (2*x**5 + 6*x**4 + 7*x**3 + 8*x**2 + 10*x - 4)*Dx**2 + \ | |
| (-2*x**5 - 5*x**4 - 2*x**3 + 2*x**2 - x + 4)*Dx**3 + (x**5 + 2*x**4 - x**3 - \ | |
| 7*x**2/2 + x + Rational(5, 2))*Dx**4, x, 0, [0, 1, 4, -1]) | |
| assert p == q | |
| p = expr_to_holonomic(x*exp(x)+cos(x)+1) | |
| q = HolonomicFunction((-x - 3)*Dx + (x + 2)*Dx**2 + (-x - 3)*Dx**3 + (x + 2)*Dx**4, x, \ | |
| 0, [2, 1, 1, 3]) | |
| assert p == q | |
| assert (x*exp(x)+cos(x)+1).series(n=10) == p.series(n=10) | |
| p = expr_to_holonomic(log(1 + x)**2 + 1) | |
| q = HolonomicFunction(Dx + (3*x + 3)*Dx**2 + (x**2 + 2*x + 1)*Dx**3, x, 0, [1, 0, 2]) | |
| assert p == q | |
| p = expr_to_holonomic(erf(x)**2 + x) | |
| q = HolonomicFunction((8*x**4 - 2*x**2 + 2)*Dx**2 + (6*x**3 - x/2)*Dx**3 + \ | |
| (x**2+ Rational(1, 4))*Dx**4, x, 0, [0, 1, 8/pi, 0]) | |
| assert p == q | |
| p = expr_to_holonomic(cosh(x)*x) | |
| q = HolonomicFunction((-x**2 + 2) -2*x*Dx + x**2*Dx**2, x, 0, [0, 1]) | |
| assert p == q | |
| p = expr_to_holonomic(besselj(2, x)) | |
| q = HolonomicFunction((x**2 - 4) + x*Dx + x**2*Dx**2, x, 0, [0, 0]) | |
| assert p == q | |
| p = expr_to_holonomic(besselj(0, x) + exp(x)) | |
| q = HolonomicFunction((-x**2 - x/2 + S.Half) + (x**2 - x/2 - Rational(3, 2))*Dx + (-x**2 + x/2 + 1)*Dx**2 +\ | |
| (x**2 + x/2)*Dx**3, x, 0, [2, 1, S.Half]) | |
| assert p == q | |
| p = expr_to_holonomic(sin(x)**2/x) | |
| q = HolonomicFunction(4 + 4*x*Dx + 3*Dx**2 + x*Dx**3, x, 0, [0, 1, 0]) | |
| assert p == q | |
| p = expr_to_holonomic(sin(x)**2/x, x0=2) | |
| q = HolonomicFunction((4) + (4*x)*Dx + (3)*Dx**2 + (x)*Dx**3, x, 2, [sin(2)**2/2, | |
| sin(2)*cos(2) - sin(2)**2/4, -3*sin(2)**2/4 + cos(2)**2 - sin(2)*cos(2)]) | |
| assert p == q | |
| p = expr_to_holonomic(log(x)/2 - Ci(2*x)/2 + Ci(2)/2) | |
| q = HolonomicFunction(4*Dx + 4*x*Dx**2 + 3*Dx**3 + x*Dx**4, x, 0, \ | |
| [-log(2)/2 - EulerGamma/2 + Ci(2)/2, 0, 1, 0]) | |
| assert p == q | |
| p = p.to_expr() | |
| q = log(x)/2 - Ci(2*x)/2 + Ci(2)/2 | |
| assert p == q | |
| p = expr_to_holonomic(x**S.Half, x0=1) | |
| q = HolonomicFunction(x*Dx - S.Half, x, 1, [1]) | |
| assert p == q | |
| p = expr_to_holonomic(sqrt(1 + x**2)) | |
| q = HolonomicFunction((-x) + (x**2 + 1)*Dx, x, 0, [1]) | |
| assert p == q | |
| assert (expr_to_holonomic(sqrt(x) + sqrt(2*x)).to_expr()-\ | |
| (sqrt(x) + sqrt(2*x))).simplify() == 0 | |
| assert expr_to_holonomic(3*x+2*sqrt(x)).to_expr() == 3*x+2*sqrt(x) | |
| p = expr_to_holonomic((x**4+x**3+5*x**2+3*x+2)/x**2, lenics=3) | |
| q = HolonomicFunction((-2*x**4 - x**3 + 3*x + 4) + (x**5 + x**4 + 5*x**3 + 3*x**2 + \ | |
| 2*x)*Dx, x, 0, {-2: [2, 3, 5]}) | |
| assert p == q | |
| p = expr_to_holonomic(1/(x-1)**2, lenics=3, x0=1) | |
| q = HolonomicFunction((2) + (x - 1)*Dx, x, 1, {-2: [1, 0, 0]}) | |
| assert p == q | |
| a = symbols("a") | |
| p = expr_to_holonomic(sqrt(a*x), x=x) | |
| assert p.to_expr() == sqrt(a)*sqrt(x) | |
| def test_to_hyper(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') | |
| p = HolonomicFunction(Dx - 2, x, 0, [3]).to_hyper() | |
| q = 3 * hyper([], [], 2*x) | |
| assert p == q | |
| p = hyperexpand(HolonomicFunction((1 + x) * Dx - 3, x, 0, [2]).to_hyper()).expand() | |
| q = 2*x**3 + 6*x**2 + 6*x + 2 | |
| assert p == q | |
| p = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1]).to_hyper() | |
| q = -x**2*hyper((2, 2, 1), (3, 2), -x)/2 + x | |
| assert p == q | |
| p = HolonomicFunction(2*x*Dx + Dx**2, x, 0, [0, 2/sqrt(pi)]).to_hyper() | |
| q = 2*x*hyper((S.Half,), (Rational(3, 2),), -x**2)/sqrt(pi) | |
| assert p == q | |
| p = hyperexpand(HolonomicFunction(2*x*Dx + Dx**2, x, 0, [1, -2/sqrt(pi)]).to_hyper()) | |
| q = erfc(x) | |
| assert p.rewrite(erfc) == q | |
| p = hyperexpand(HolonomicFunction((x**2 - 1) + x*Dx + x**2*Dx**2, | |
| x, 0, [0, S.Half]).to_hyper()) | |
| q = besselj(1, x) | |
| assert p == q | |
| p = hyperexpand(HolonomicFunction(x*Dx**2 + Dx + x, x, 0, [1, 0]).to_hyper()) | |
| q = besselj(0, x) | |
| assert p == q | |
| def test_to_expr(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') | |
| p = HolonomicFunction(Dx - 1, x, 0, [1]).to_expr() | |
| q = exp(x) | |
| assert p == q | |
| p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).to_expr() | |
| q = cos(x) | |
| assert p == q | |
| p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]).to_expr() | |
| q = cosh(x) | |
| assert p == q | |
| p = HolonomicFunction(2 + (4*x - 1)*Dx + \ | |
| (x**2 - x)*Dx**2, x, 0, [1, 2]).to_expr().expand() | |
| q = 1/(x**2 - 2*x + 1) | |
| assert p == q | |
| p = expr_to_holonomic(sin(x)**2/x).integrate((x, 0, x)).to_expr() | |
| q = (sin(x)**2/x).integrate((x, 0, x)) | |
| assert p == q | |
| C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3') | |
| p = expr_to_holonomic(log(1+x**2)).to_expr() | |
| q = C_2*log(x**2 + 1) | |
| assert p == q | |
| p = expr_to_holonomic(log(1+x**2)).diff().to_expr() | |
| q = C_0*x/(x**2 + 1) | |
| assert p == q | |
| p = expr_to_holonomic(erf(x) + x).to_expr() | |
| q = 3*C_3*x - 3*sqrt(pi)*C_3*erf(x)/2 + x + 2*x/sqrt(pi) | |
| assert p == q | |
| p = expr_to_holonomic(sqrt(x), x0=1).to_expr() | |
| assert p == sqrt(x) | |
| assert expr_to_holonomic(sqrt(x)).to_expr() == sqrt(x) | |
| p = expr_to_holonomic(sqrt(1 + x**2)).to_expr() | |
| assert p == sqrt(1+x**2) | |
| p = expr_to_holonomic((2*x**2 + 1)**Rational(2, 3)).to_expr() | |
| assert p == (2*x**2 + 1)**Rational(2, 3) | |
| p = expr_to_holonomic(sqrt(-x**2+2*x)).to_expr() | |
| assert p == sqrt(x)*sqrt(-x + 2) | |
| p = expr_to_holonomic((-2*x**3+7*x)**Rational(2, 3)).to_expr() | |
| q = x**Rational(2, 3)*(-2*x**2 + 7)**Rational(2, 3) | |
| assert p == q | |
| p = from_hyper(hyper((-2, -3), (S.Half, ), x)) | |
| s = hyperexpand(hyper((-2, -3), (S.Half, ), x)) | |
| D_0 = Symbol('D_0') | |
| C_0 = Symbol('C_0') | |
| assert (p.to_expr().subs({C_0:1, D_0:0}) - s).simplify() == 0 | |
| p.y0 = {0: [1], S.Half: [0]} | |
| assert p.to_expr() == s | |
| assert expr_to_holonomic(x**5).to_expr() == x**5 | |
| assert expr_to_holonomic(2*x**3-3*x**2).to_expr().expand() == \ | |
| 2*x**3-3*x**2 | |
| a = symbols("a") | |
| p = (expr_to_holonomic(1.4*x)*expr_to_holonomic(a*x, x)).to_expr() | |
| q = 1.4*a*x**2 | |
| assert p == q | |
| p = (expr_to_holonomic(1.4*x)+expr_to_holonomic(a*x, x)).to_expr() | |
| q = x*(a + 1.4) | |
| assert p == q | |
| p = (expr_to_holonomic(1.4*x)+expr_to_holonomic(x)).to_expr() | |
| assert p == 2.4*x | |
| def test_integrate(): | |
| x = symbols('x') | |
| R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') | |
| p = expr_to_holonomic(sin(x)**2/x, x0=1).integrate((x, 2, 3)) | |
| q = '0.166270406994788' | |
| assert sstr(p) == q | |
| p = expr_to_holonomic(sin(x)).integrate((x, 0, x)).to_expr() | |
| q = 1 - cos(x) | |
| assert p == q | |
| p = expr_to_holonomic(sin(x)).integrate((x, 0, 3)) | |
| q = 1 - cos(3) | |
| assert p == q | |
| p = expr_to_holonomic(sin(x)/x, x0=1).integrate((x, 1, 2)) | |
| q = '0.659329913368450' | |
| assert sstr(p) == q | |
| p = expr_to_holonomic(sin(x)**2/x, x0=1).integrate((x, 1, 0)) | |
| q = '-0.423690480850035' | |
| assert sstr(p) == q | |
| p = expr_to_holonomic(sin(x)/x) | |
| assert p.integrate(x).to_expr() == Si(x) | |
| assert p.integrate((x, 0, 2)) == Si(2) | |
| p = expr_to_holonomic(sin(x)**2/x) | |
| q = p.to_expr() | |
| assert p.integrate(x).to_expr() == q.integrate((x, 0, x)) | |
| assert p.integrate((x, 0, 1)) == q.integrate((x, 0, 1)) | |
| assert expr_to_holonomic(1/x, x0=1).integrate(x).to_expr() == log(x) | |
| p = expr_to_holonomic((x + 1)**3*exp(-x), x0=-1).integrate(x).to_expr() | |
| q = (-x**3 - 6*x**2 - 15*x + 6*exp(x + 1) - 16)*exp(-x) | |
| assert p == q | |
| p = expr_to_holonomic(cos(x)**2/x**2, y0={-2: [1, 0, -1]}).integrate(x).to_expr() | |
| q = -Si(2*x) - cos(x)**2/x | |
| assert p == q | |
| p = expr_to_holonomic(sqrt(x**2+x)).integrate(x).to_expr() | |
| q = (x**Rational(3, 2)*(2*x**2 + 3*x + 1) - x*sqrt(x + 1)*asinh(sqrt(x)))/(4*x*sqrt(x + 1)) | |
| assert p == q | |
| p = expr_to_holonomic(sqrt(x**2+1)).integrate(x).to_expr() | |
| q = (sqrt(x**2+1)).integrate(x) | |
| assert (p-q).simplify() == 0 | |
| p = expr_to_holonomic(1/x**2, y0={-2:[1, 0, 0]}) | |
| r = expr_to_holonomic(1/x**2, lenics=3) | |
| assert p == r | |
| q = expr_to_holonomic(cos(x)**2) | |
| assert (r*q).integrate(x).to_expr() == -Si(2*x) - cos(x)**2/x | |
| def test_diff(): | |
| x, y = symbols('x, y') | |
| R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') | |
| p = HolonomicFunction(x*Dx**2 + 1, x, 0, [0, 1]) | |
| assert p.diff().to_expr() == p.to_expr().diff().simplify() | |
| p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]) | |
| assert p.diff(x, 2).to_expr() == p.to_expr() | |
| p = expr_to_holonomic(Si(x)) | |
| assert p.diff().to_expr() == sin(x)/x | |
| assert p.diff(y) == 0 | |
| C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3') | |
| q = Si(x) | |
| assert p.diff(x).to_expr() == q.diff() | |
| assert p.diff(x, 2).to_expr().subs(C_0, Rational(-1, 3)).cancel() == q.diff(x, 2).cancel() | |
| assert p.diff(x, 3).series().subs({C_3: Rational(-1, 3), C_0: 0}) == q.diff(x, 3).series() | |
| def test_extended_domain_in_expr_to_holonomic(): | |
| x = symbols('x') | |
| p = expr_to_holonomic(1.2*cos(3.1*x)) | |
| assert p.to_expr() == 1.2*cos(3.1*x) | |
| assert sstr(p.integrate(x).to_expr()) == '0.387096774193548*sin(3.1*x)' | |
| _, Dx = DifferentialOperators(RR.old_poly_ring(x), 'Dx') | |
| p = expr_to_holonomic(1.1329138213*x) | |
| q = HolonomicFunction((-1.1329138213) + (1.1329138213*x)*Dx, x, 0, {1: [1.1329138213]}) | |
| assert p == q | |
| assert p.to_expr() == 1.1329138213*x | |
| assert sstr(p.integrate((x, 1, 2))) == sstr((1.1329138213*x).integrate((x, 1, 2))) | |
| y, z = symbols('y, z') | |
| p = expr_to_holonomic(sin(x*y*z), x=x) | |
| assert p.to_expr() == sin(x*y*z) | |
| assert p.integrate(x).to_expr() == (-cos(x*y*z) + 1)/(y*z) | |
| p = expr_to_holonomic(sin(x*y + z), x=x).integrate(x).to_expr() | |
| q = (cos(z) - cos(x*y + z))/y | |
| assert p == q | |
| a = symbols('a') | |
| p = expr_to_holonomic(a*x, x) | |
| assert p.to_expr() == a*x | |
| assert p.integrate(x).to_expr() == a*x**2/2 | |
| D_2, C_1 = symbols("D_2, C_1") | |
| p = expr_to_holonomic(x) + expr_to_holonomic(1.2*cos(x)) | |
| p = p.to_expr().subs(D_2, 0) | |
| assert p - x - 1.2*cos(1.0*x) == 0 | |
| p = expr_to_holonomic(x) * expr_to_holonomic(1.2*cos(x)) | |
| p = p.to_expr().subs(C_1, 0) | |
| assert p - 1.2*x*cos(1.0*x) == 0 | |
| def test_to_meijerg(): | |
| x = symbols('x') | |
| assert hyperexpand(expr_to_holonomic(sin(x)).to_meijerg()) == sin(x) | |
| assert hyperexpand(expr_to_holonomic(cos(x)).to_meijerg()) == cos(x) | |
| assert hyperexpand(expr_to_holonomic(exp(x)).to_meijerg()) == exp(x) | |
| assert hyperexpand(expr_to_holonomic(log(x)).to_meijerg()).simplify() == log(x) | |
| assert expr_to_holonomic(4*x**2/3 + 7).to_meijerg() == 4*x**2/3 + 7 | |
| assert hyperexpand(expr_to_holonomic(besselj(2, x), lenics=3).to_meijerg()) == besselj(2, x) | |
| p = hyper((Rational(-1, 2), -3), (), x) | |
| assert from_hyper(p).to_meijerg() == hyperexpand(p) | |
| p = hyper((S.One, S(3)), (S(2), ), x) | |
| assert (hyperexpand(from_hyper(p).to_meijerg()) - hyperexpand(p)).expand() == 0 | |
| p = from_hyper(hyper((-2, -3), (S.Half, ), x)) | |
| s = hyperexpand(hyper((-2, -3), (S.Half, ), x)) | |
| C_0 = Symbol('C_0') | |
| C_1 = Symbol('C_1') | |
| D_0 = Symbol('D_0') | |
| assert (hyperexpand(p.to_meijerg()).subs({C_0:1, D_0:0}) - s).simplify() == 0 | |
| p.y0 = {0: [1], S.Half: [0]} | |
| assert (hyperexpand(p.to_meijerg()) - s).simplify() == 0 | |
| p = expr_to_holonomic(besselj(S.Half, x), initcond=False) | |
| assert (p.to_expr() - (D_0*sin(x) + C_0*cos(x) + C_1*sin(x))/sqrt(x)).simplify() == 0 | |
| p = expr_to_holonomic(besselj(S.Half, x), y0={Rational(-1, 2): [sqrt(2)/sqrt(pi), sqrt(2)/sqrt(pi)]}) | |
| assert (p.to_expr() - besselj(S.Half, x) - besselj(Rational(-1, 2), x)).simplify() == 0 | |
| def test_gaussian(): | |
| mu, x = symbols("mu x") | |
| sd = symbols("sd", positive=True) | |
| Q = QQ[mu, sd].get_field() | |
| e = sqrt(2)*exp(-(-mu + x)**2/(2*sd**2))/(2*sqrt(pi)*sd) | |
| h1 = expr_to_holonomic(e, x, domain=Q) | |
| _, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx') | |
| h2 = HolonomicFunction((-mu/sd**2 + x/sd**2) + (1)*Dx, x) | |
| assert h1 == h2 | |
| def test_beta(): | |
| a, b, x = symbols("a b x", positive=True) | |
| e = x**(a - 1)*(-x + 1)**(b - 1)/beta(a, b) | |
| Q = QQ[a, b].get_field() | |
| h1 = expr_to_holonomic(e, x, domain=Q) | |
| _, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx') | |
| h2 = HolonomicFunction((a + x*(-a - b + 2) - 1) + (x**2 - x)*Dx, x) | |
| assert h1 == h2 | |
| def test_gamma(): | |
| a, b, x = symbols("a b x", positive=True) | |
| e = b**(-a)*x**(a - 1)*exp(-x/b)/gamma(a) | |
| Q = QQ[a, b].get_field() | |
| h1 = expr_to_holonomic(e, x, domain=Q) | |
| _, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx') | |
| h2 = HolonomicFunction((-a + 1 + x/b) + (x)*Dx, x) | |
| assert h1 == h2 | |
| def test_symbolic_power(): | |
| x, n = symbols("x n") | |
| Q = QQ[n].get_field() | |
| _, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx') | |
| h1 = HolonomicFunction((-1) + (x)*Dx, x) ** -n | |
| h2 = HolonomicFunction((n) + (x)*Dx, x) | |
| assert h1 == h2 | |
| def test_negative_power(): | |
| x = symbols("x") | |
| _, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') | |
| h1 = HolonomicFunction((-1) + (x)*Dx, x) ** -2 | |
| h2 = HolonomicFunction((2) + (x)*Dx, x) | |
| assert h1 == h2 | |
| def test_expr_in_power(): | |
| x, n = symbols("x n") | |
| Q = QQ[n].get_field() | |
| _, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx') | |
| h1 = HolonomicFunction((-1) + (x)*Dx, x) ** (n - 3) | |
| h2 = HolonomicFunction((-n + 3) + (x)*Dx, x) | |
| assert h1 == h2 | |
| def test_DifferentialOperatorEqPoly(): | |
| x = symbols('x', integer=True) | |
| R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') | |
| do = DifferentialOperator([x**2, R.base.zero, R.base.zero], R) | |
| do2 = DifferentialOperator([x**2, 1, x], R) | |
| assert not do == do2 | |
| # polynomial comparison issue, see https://github.com/sympy/sympy/pull/15799 | |
| # should work once that is solved | |
| # p = do.listofpoly[0] | |
| # assert do == p | |
| p2 = do2.listofpoly[0] | |
| assert not do2 == p2 | |
| def test_DifferentialOperatorPow(): | |
| x = symbols('x', integer=True) | |
| R, _ = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') | |
| do = DifferentialOperator([x**2, R.base.zero, R.base.zero], R) | |
| a = DifferentialOperator([R.base.one], R) | |
| for n in range(10): | |
| assert a == do**n | |
| a *= do | |