MTtranslateService/Lib/site-packages/sympy/solvers/tests/test_polysys.py

"""Tests for solvers of systems of polynomial equations. """

from sympy.core.numbers import (I, Integer, Rational)
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.polys.domains.rationalfield import QQ
from sympy.polys.polytools import Poly
from sympy.solvers.solvers import solve
from sympy.utilities.iterables import flatten
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.testing.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, y*Rational(3, 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), Rational(-1, 2)), (I*sqrt(S.Half), Rational(-1, 2))]

    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))

    raises(NotImplementedError, lambda: solve_poly_system(
          [x-1,], (x, y)))
    raises(NotImplementedError, lambda: solve_poly_system(
          [y-1,], (x, y)))


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, Rational(3, 2)),
         (1 + sqrt((2*r - 1)*(2*r + 1))/2, Rational(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 - Rational(5, 10))**2/250000) - 1, x], x, y)
    assert roots == [(0, S.Half - 15*sqrt(1111)), (0, S.Half + 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)