Vehicle-cpp/hkpc/software/sympy/solvers/tests/test_numeric.py

from sympy.core.function import nfloat
from sympy.core.numbers import (Float, I, Rational, pi)
from sympy.core.relational import Eq
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import sin
from sympy.integrals.integrals import Integral
from sympy.matrices.dense import Matrix
from mpmath import mnorm, mpf
from sympy.solvers import nsolve
from sympy.utilities.lambdify import lambdify
from sympy.testing.pytest import raises, XFAIL
from sympy.utilities.decorator import conserve_mpmath_dps

@XFAIL
def test_nsolve_fail():
    x = symbols('x')
    # Sometimes it is better to use the numerator (issue 4829)
    # but sometimes it is not (issue 11768) so leave this to
    # the discretion of the user
    ans = nsolve(x**2/(1 - x)/(1 - 2*x)**2 - 100, x, 0)
    assert ans > 0.46 and ans < 0.47


def test_nsolve_denominator():
    x = symbols('x')
    # Test that nsolve uses the full expression (numerator and denominator).
    ans = nsolve((x**2 + 3*x + 2)/(x + 2), -2.1)
    # The root -2 was divided out, so make sure we don't find it.
    assert ans == -1.0

def test_nsolve():
    # onedimensional
    x = Symbol('x')
    assert nsolve(sin(x), 2) - pi.evalf() < 1e-15
    assert nsolve(Eq(2*x, 2), x, -10) == nsolve(2*x - 2, -10)
    # Testing checks on number of inputs
    raises(TypeError, lambda: nsolve(Eq(2*x, 2)))
    raises(TypeError, lambda: nsolve(Eq(2*x, 2), x, 1, 2))
    # multidimensional
    x1 = Symbol('x1')
    x2 = Symbol('x2')
    f1 = 3 * x1**2 - 2 * x2**2 - 1
    f2 = x1**2 - 2 * x1 + x2**2 + 2 * x2 - 8
    f = Matrix((f1, f2)).T
    F = lambdify((x1, x2), f.T, modules='mpmath')
    for x0 in [(-1, 1), (1, -2), (4, 4), (-4, -4)]:
        x = nsolve(f, (x1, x2), x0, tol=1.e-8)
        assert mnorm(F(*x), 1) <= 1.e-10
    # The Chinese mathematician Zhu Shijie was the very first to solve this
    # nonlinear system 700 years ago (z was added to make it 3-dimensional)
    x = Symbol('x')
    y = Symbol('y')
    z = Symbol('z')
    f1 = -x + 2*y
    f2 = (x**2 + x*(y**2 - 2) - 4*y) / (x + 4)
    f3 = sqrt(x**2 + y**2)*z
    f = Matrix((f1, f2, f3)).T
    F = lambdify((x, y, z), f.T, modules='mpmath')

    def getroot(x0):
        root = nsolve(f, (x, y, z), x0)
        assert mnorm(F(*root), 1) <= 1.e-8
        return root
    assert list(map(round, getroot((1, 1, 1)))) == [2, 1, 0]
    assert nsolve([Eq(
        f1, 0), Eq(f2, 0), Eq(f3, 0)], [x, y, z], (1, 1, 1))  # just see that it works
    a = Symbol('a')
    assert abs(nsolve(1/(0.001 + a)**3 - 6/(0.9 - a)**3, a, 0.3) -
        mpf('0.31883011387318591')) < 1e-15


def test_issue_6408():
    x = Symbol('x')
    assert nsolve(Piecewise((x, x < 1), (x**2, True)), x, 2) == 0.0


def test_issue_6408_integral():
    x, y = symbols('x y')
    assert nsolve(Integral(x*y, (x, 0, 5)), y, 2) == 0.0


@conserve_mpmath_dps
def test_increased_dps():
    # Issue 8564
    import mpmath
    mpmath.mp.dps = 128
    x = Symbol('x')
    e1 = x**2 - pi
    q = nsolve(e1, x, 3.0)

    assert abs(sqrt(pi).evalf(128) - q) < 1e-128

def test_nsolve_precision():
    x, y = symbols('x y')
    sol = nsolve(x**2 - pi, x, 3, prec=128)
    assert abs(sqrt(pi).evalf(128) - sol) < 1e-128
    assert isinstance(sol, Float)

    sols = nsolve((y**2 - x, x**2 - pi), (x, y), (3, 3), prec=128)
    assert isinstance(sols, Matrix)
    assert sols.shape == (2, 1)
    assert abs(sqrt(pi).evalf(128) - sols[0]) < 1e-128
    assert abs(sqrt(sqrt(pi)).evalf(128) - sols[1]) < 1e-128
    assert all(isinstance(i, Float) for i in sols)

def test_nsolve_complex():
    x, y = symbols('x y')

    assert nsolve(x**2 + 2, 1j) == sqrt(2.)*I
    assert nsolve(x**2 + 2, I) == sqrt(2.)*I

    assert nsolve([x**2 + 2, y**2 + 2], [x, y], [I, I]) == Matrix([sqrt(2.)*I, sqrt(2.)*I])
    assert nsolve([x**2 + 2, y**2 + 2], [x, y], [I, I]) == Matrix([sqrt(2.)*I, sqrt(2.)*I])

def test_nsolve_dict_kwarg():
    x, y = symbols('x y')
    # one variable
    assert nsolve(x**2 - 2, 1, dict = True) == \
        [{x: sqrt(2.)}]
    # one variable with complex solution
    assert nsolve(x**2 + 2, I, dict = True) == \
        [{x: sqrt(2.)*I}]
    # two variables
    assert nsolve([x**2 + y**2 - 5, x**2 - y**2 + 1], [x, y], [1, 1], dict = True) == \
        [{x: sqrt(2.), y: sqrt(3.)}]

def test_nsolve_rational():
    x = symbols('x')
    assert nsolve(x - Rational(1, 3), 0, prec=100) == Rational(1, 3).evalf(100)


def test_issue_14950():
    x = Matrix(symbols('t s'))
    x0 = Matrix([17, 23])
    eqn = x + x0
    assert nsolve(eqn, x, x0) == nfloat(-x0)
    assert nsolve(eqn.T, x.T, x0.T) == nfloat(-x0)