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from sympy.core.containers import Tuple
from sympy.core.function import Function
from sympy.core.numbers import oo, Rational
from sympy.core.singleton import S
from sympy.core.symbol import symbols, Symbol
from sympy.functions.combinatorial.numbers import tribonacci, fibonacci
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import cos, sin
from sympy.series import EmptySequence
from sympy.series.sequences import (SeqMul, SeqAdd, SeqPer, SeqFormula,
sequence)
from sympy.sets.sets import Interval
from sympy.tensor.indexed import Indexed, Idx
from sympy.series.sequences import SeqExpr, SeqExprOp, RecursiveSeq
from sympy.testing.pytest import raises, slow
x, y, z = symbols('x y z')
n, m = symbols('n m')
def test_EmptySequence():
assert S.EmptySequence is EmptySequence
assert S.EmptySequence.interval is S.EmptySet
assert S.EmptySequence.length is S.Zero
assert list(S.EmptySequence) == []
def test_SeqExpr():
#SeqExpr is a baseclass and does not take care of
#ensuring all arguments are Basics hence the use of
#Tuple(...) here.
s = SeqExpr(Tuple(1, n, y), Tuple(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(Tuple(1, 2, 3), Tuple(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) \
== ([Rational(19, 2), -20, Rational(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))
def test_RecursiveSeq():
y = Function('y')
n = Symbol('n')
fib = RecursiveSeq(y(n - 1) + y(n - 2), y(n), n, [0, 1])
assert fib.coeff(3) == 2
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