363 lines
10 KiB
Python
363 lines
10 KiB
Python
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"""Test sparse rational functions. """
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from sympy.polys.fields import field, sfield, FracField, FracElement
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from sympy.polys.rings import ring
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from sympy.polys.domains import ZZ, QQ
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from sympy.polys.orderings import lex
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from sympy.testing.pytest import raises, XFAIL
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from sympy.core import symbols, E
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from sympy.core.numbers import Rational
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from sympy.functions.elementary.exponential import (exp, log)
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from sympy.functions.elementary.miscellaneous import sqrt
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def test_FracField___init__():
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F1 = FracField("x,y", ZZ, lex)
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F2 = FracField("x,y", ZZ, lex)
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F3 = FracField("x,y,z", ZZ, lex)
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assert F1.x == F1.gens[0]
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assert F1.y == F1.gens[1]
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assert F1.x == F2.x
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assert F1.y == F2.y
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assert F1.x != F3.x
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assert F1.y != F3.y
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def test_FracField___hash__():
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F, x, y, z = field("x,y,z", QQ)
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assert hash(F)
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def test_FracField___eq__():
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assert field("x,y,z", QQ)[0] == field("x,y,z", QQ)[0]
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assert field("x,y,z", QQ)[0] is field("x,y,z", QQ)[0]
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assert field("x,y,z", QQ)[0] != field("x,y,z", ZZ)[0]
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assert field("x,y,z", QQ)[0] is not field("x,y,z", ZZ)[0]
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assert field("x,y,z", ZZ)[0] != field("x,y,z", QQ)[0]
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assert field("x,y,z", ZZ)[0] is not field("x,y,z", QQ)[0]
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assert field("x,y,z", QQ)[0] != field("x,y", QQ)[0]
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assert field("x,y,z", QQ)[0] is not field("x,y", QQ)[0]
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assert field("x,y", QQ)[0] != field("x,y,z", QQ)[0]
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assert field("x,y", QQ)[0] is not field("x,y,z", QQ)[0]
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def test_sfield():
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x = symbols("x")
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F = FracField((E, exp(exp(x)), exp(x)), ZZ, lex)
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e, exex, ex = F.gens
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assert sfield(exp(x)*exp(exp(x) + 1 + log(exp(x) + 3)/2)**2/(exp(x) + 3)) \
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== (F, e**2*exex**2*ex)
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F = FracField((x, exp(1/x), log(x), x**QQ(1, 3)), ZZ, lex)
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_, ex, lg, x3 = F.gens
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assert sfield(((x-3)*log(x)+4*x**2)*exp(1/x+log(x)/3)/x**2) == \
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(F, (4*F.x**2*ex + F.x*ex*lg - 3*ex*lg)/x3**5)
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F = FracField((x, log(x), sqrt(x + log(x))), ZZ, lex)
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_, lg, srt = F.gens
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assert sfield((x + 1) / (x * (x + log(x))**QQ(3, 2)) - 1/(x * log(x)**2)) \
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== (F, (F.x*lg**2 - F.x*srt + lg**2 - lg*srt)/
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(F.x**2*lg**2*srt + F.x*lg**3*srt))
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def test_FracElement___hash__():
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F, x, y, z = field("x,y,z", QQ)
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assert hash(x*y/z)
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def test_FracElement_copy():
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F, x, y, z = field("x,y,z", ZZ)
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f = x*y/3*z
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g = f.copy()
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assert f == g
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g.numer[(1, 1, 1)] = 7
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assert f != g
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def test_FracElement_as_expr():
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F, x, y, z = field("x,y,z", ZZ)
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f = (3*x**2*y - x*y*z)/(7*z**3 + 1)
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X, Y, Z = F.symbols
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g = (3*X**2*Y - X*Y*Z)/(7*Z**3 + 1)
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assert f != g
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assert f.as_expr() == g
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X, Y, Z = symbols("x,y,z")
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g = (3*X**2*Y - X*Y*Z)/(7*Z**3 + 1)
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assert f != g
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assert f.as_expr(X, Y, Z) == g
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raises(ValueError, lambda: f.as_expr(X))
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def test_FracElement_from_expr():
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x, y, z = symbols("x,y,z")
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F, X, Y, Z = field((x, y, z), ZZ)
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f = F.from_expr(1)
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assert f == 1 and isinstance(f, F.dtype)
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f = F.from_expr(Rational(3, 7))
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assert f == F(3)/7 and isinstance(f, F.dtype)
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f = F.from_expr(x)
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assert f == X and isinstance(f, F.dtype)
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f = F.from_expr(Rational(3,7)*x)
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assert f == X*Rational(3, 7) and isinstance(f, F.dtype)
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f = F.from_expr(1/x)
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assert f == 1/X and isinstance(f, F.dtype)
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f = F.from_expr(x*y*z)
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assert f == X*Y*Z and isinstance(f, F.dtype)
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f = F.from_expr(x*y/z)
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assert f == X*Y/Z and isinstance(f, F.dtype)
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f = F.from_expr(x*y*z + x*y + x)
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assert f == X*Y*Z + X*Y + X and isinstance(f, F.dtype)
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f = F.from_expr((x*y*z + x*y + x)/(x*y + 7))
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assert f == (X*Y*Z + X*Y + X)/(X*Y + 7) and isinstance(f, F.dtype)
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f = F.from_expr(x**3*y*z + x**2*y**7 + 1)
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assert f == X**3*Y*Z + X**2*Y**7 + 1 and isinstance(f, F.dtype)
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raises(ValueError, lambda: F.from_expr(2**x))
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raises(ValueError, lambda: F.from_expr(7*x + sqrt(2)))
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assert isinstance(ZZ[2**x].get_field().convert(2**(-x)),
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FracElement)
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assert isinstance(ZZ[x**2].get_field().convert(x**(-6)),
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FracElement)
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assert isinstance(ZZ[exp(Rational(1, 3))].get_field().convert(E),
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FracElement)
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def test_FracField_nested():
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a, b, x = symbols('a b x')
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F1 = ZZ.frac_field(a, b)
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F2 = F1.frac_field(x)
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frac = F2(a + b)
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assert frac.numer == F1.poly_ring(x)(a + b)
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assert frac.numer.coeffs() == [F1(a + b)]
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assert frac.denom == F1.poly_ring(x)(1)
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F3 = ZZ.poly_ring(a, b)
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F4 = F3.frac_field(x)
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frac = F4(a + b)
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assert frac.numer == F3.poly_ring(x)(a + b)
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assert frac.numer.coeffs() == [F3(a + b)]
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assert frac.denom == F3.poly_ring(x)(1)
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frac = F2(F3(a + b))
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assert frac.numer == F1.poly_ring(x)(a + b)
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assert frac.numer.coeffs() == [F1(a + b)]
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assert frac.denom == F1.poly_ring(x)(1)
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frac = F4(F1(a + b))
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assert frac.numer == F3.poly_ring(x)(a + b)
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assert frac.numer.coeffs() == [F3(a + b)]
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assert frac.denom == F3.poly_ring(x)(1)
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def test_FracElement__lt_le_gt_ge__():
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F, x, y = field("x,y", ZZ)
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assert F(1) < 1/x < 1/x**2 < 1/x**3
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assert F(1) <= 1/x <= 1/x**2 <= 1/x**3
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assert -7/x < 1/x < 3/x < y/x < 1/x**2
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assert -7/x <= 1/x <= 3/x <= y/x <= 1/x**2
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assert 1/x**3 > 1/x**2 > 1/x > F(1)
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assert 1/x**3 >= 1/x**2 >= 1/x >= F(1)
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assert 1/x**2 > y/x > 3/x > 1/x > -7/x
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assert 1/x**2 >= y/x >= 3/x >= 1/x >= -7/x
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def test_FracElement___neg__():
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F, x,y = field("x,y", QQ)
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f = (7*x - 9)/y
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g = (-7*x + 9)/y
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assert -f == g
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assert -g == f
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def test_FracElement___add__():
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F, x,y = field("x,y", QQ)
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f, g = 1/x, 1/y
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assert f + g == g + f == (x + y)/(x*y)
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assert x + F.ring.gens[0] == F.ring.gens[0] + x == 2*x
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F, x,y = field("x,y", ZZ)
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assert x + 3 == 3 + x
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assert x + QQ(3,7) == QQ(3,7) + x == (7*x + 3)/7
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Fuv, u,v = field("u,v", ZZ)
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Fxyzt, x,y,z,t = field("x,y,z,t", Fuv)
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f = (u*v + x)/(y + u*v)
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assert dict(f.numer) == {(1, 0, 0, 0): 1, (0, 0, 0, 0): u*v}
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assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): u*v}
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Ruv, u,v = ring("u,v", ZZ)
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Fxyzt, x,y,z,t = field("x,y,z,t", Ruv)
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f = (u*v + x)/(y + u*v)
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assert dict(f.numer) == {(1, 0, 0, 0): 1, (0, 0, 0, 0): u*v}
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assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): u*v}
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def test_FracElement___sub__():
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F, x,y = field("x,y", QQ)
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f, g = 1/x, 1/y
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assert f - g == (-x + y)/(x*y)
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assert x - F.ring.gens[0] == F.ring.gens[0] - x == 0
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F, x,y = field("x,y", ZZ)
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assert x - 3 == -(3 - x)
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assert x - QQ(3,7) == -(QQ(3,7) - x) == (7*x - 3)/7
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Fuv, u,v = field("u,v", ZZ)
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Fxyzt, x,y,z,t = field("x,y,z,t", Fuv)
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f = (u*v - x)/(y - u*v)
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assert dict(f.numer) == {(1, 0, 0, 0):-1, (0, 0, 0, 0): u*v}
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assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0):-u*v}
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Ruv, u,v = ring("u,v", ZZ)
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Fxyzt, x,y,z,t = field("x,y,z,t", Ruv)
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f = (u*v - x)/(y - u*v)
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assert dict(f.numer) == {(1, 0, 0, 0):-1, (0, 0, 0, 0): u*v}
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assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0):-u*v}
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def test_FracElement___mul__():
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F, x,y = field("x,y", QQ)
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f, g = 1/x, 1/y
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assert f*g == g*f == 1/(x*y)
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assert x*F.ring.gens[0] == F.ring.gens[0]*x == x**2
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F, x,y = field("x,y", ZZ)
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assert x*3 == 3*x
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assert x*QQ(3,7) == QQ(3,7)*x == x*Rational(3, 7)
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Fuv, u,v = field("u,v", ZZ)
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Fxyzt, x,y,z,t = field("x,y,z,t", Fuv)
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f = ((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)
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assert dict(f.numer) == {(1, 1, 0, 0): u + 1, (0, 0, 0, 0): 1}
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assert dict(f.denom) == {(0, 0, 1, 0): v - 1, (0, 0, 0, 1): -u*v, (0, 0, 0, 0): -1}
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Ruv, u,v = ring("u,v", ZZ)
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Fxyzt, x,y,z,t = field("x,y,z,t", Ruv)
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f = ((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)
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assert dict(f.numer) == {(1, 1, 0, 0): u + 1, (0, 0, 0, 0): 1}
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assert dict(f.denom) == {(0, 0, 1, 0): v - 1, (0, 0, 0, 1): -u*v, (0, 0, 0, 0): -1}
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def test_FracElement___truediv__():
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F, x,y = field("x,y", QQ)
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f, g = 1/x, 1/y
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assert f/g == y/x
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assert x/F.ring.gens[0] == F.ring.gens[0]/x == 1
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F, x,y = field("x,y", ZZ)
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assert x*3 == 3*x
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assert x/QQ(3,7) == (QQ(3,7)/x)**-1 == x*Rational(7, 3)
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raises(ZeroDivisionError, lambda: x/0)
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raises(ZeroDivisionError, lambda: 1/(x - x))
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raises(ZeroDivisionError, lambda: x/(x - x))
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Fuv, u,v = field("u,v", ZZ)
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Fxyzt, x,y,z,t = field("x,y,z,t", Fuv)
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f = (u*v)/(x*y)
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assert dict(f.numer) == {(0, 0, 0, 0): u*v}
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assert dict(f.denom) == {(1, 1, 0, 0): 1}
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g = (x*y)/(u*v)
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assert dict(g.numer) == {(1, 1, 0, 0): 1}
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assert dict(g.denom) == {(0, 0, 0, 0): u*v}
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Ruv, u,v = ring("u,v", ZZ)
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Fxyzt, x,y,z,t = field("x,y,z,t", Ruv)
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f = (u*v)/(x*y)
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assert dict(f.numer) == {(0, 0, 0, 0): u*v}
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assert dict(f.denom) == {(1, 1, 0, 0): 1}
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g = (x*y)/(u*v)
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assert dict(g.numer) == {(1, 1, 0, 0): 1}
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assert dict(g.denom) == {(0, 0, 0, 0): u*v}
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def test_FracElement___pow__():
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F, x,y = field("x,y", QQ)
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f, g = 1/x, 1/y
|
||
|
|
|
||
|
|
assert f**3 == 1/x**3
|
||
|
|
assert g**3 == 1/y**3
|
||
|
|
|
||
|
|
assert (f*g)**3 == 1/(x**3*y**3)
|
||
|
|
assert (f*g)**-3 == (x*y)**3
|
||
|
|
|
||
|
|
raises(ZeroDivisionError, lambda: (x - x)**-3)
|
||
|
|
|
||
|
|
def test_FracElement_diff():
|
||
|
|
F, x,y,z = field("x,y,z", ZZ)
|
||
|
|
|
||
|
|
assert ((x**2 + y)/(z + 1)).diff(x) == 2*x/(z + 1)
|
||
|
|
|
||
|
|
@XFAIL
|
||
|
|
def test_FracElement___call__():
|
||
|
|
F, x,y,z = field("x,y,z", ZZ)
|
||
|
|
f = (x**2 + 3*y)/z
|
||
|
|
|
||
|
|
r = f(1, 1, 1)
|
||
|
|
assert r == 4 and not isinstance(r, FracElement)
|
||
|
|
raises(ZeroDivisionError, lambda: f(1, 1, 0))
|
||
|
|
|
||
|
|
def test_FracElement_evaluate():
|
||
|
|
F, x,y,z = field("x,y,z", ZZ)
|
||
|
|
Fyz = field("y,z", ZZ)[0]
|
||
|
|
f = (x**2 + 3*y)/z
|
||
|
|
|
||
|
|
assert f.evaluate(x, 0) == 3*Fyz.y/Fyz.z
|
||
|
|
raises(ZeroDivisionError, lambda: f.evaluate(z, 0))
|
||
|
|
|
||
|
|
def test_FracElement_subs():
|
||
|
|
F, x,y,z = field("x,y,z", ZZ)
|
||
|
|
f = (x**2 + 3*y)/z
|
||
|
|
|
||
|
|
assert f.subs(x, 0) == 3*y/z
|
||
|
|
raises(ZeroDivisionError, lambda: f.subs(z, 0))
|
||
|
|
|
||
|
|
def test_FracElement_compose():
|
||
|
|
pass
|
||
|
|
|
||
|
|
def test_FracField_index():
|
||
|
|
a = symbols("a")
|
||
|
|
F, x, y, z = field('x y z', QQ)
|
||
|
|
assert F.index(x) == 0
|
||
|
|
assert F.index(y) == 1
|
||
|
|
|
||
|
|
raises(ValueError, lambda: F.index(1))
|
||
|
|
raises(ValueError, lambda: F.index(a))
|
||
|
|
pass
|