Python randtest.test_numerically函数代码示例

def test_roots_quartic():
    assert roots_quartic(Poly(x**4, x)) == [0, 0, 0, 0]
    assert roots_quartic(Poly(x**4 + x**3, x)) in [
        [-1, 0, 0, 0],
        [0, -1, 0, 0],
        [0, 0, -1, 0],
        [0, 0, 0, -1]
    ]
    assert roots_quartic(Poly(x**4 - x**3, x)) in [
        [1, 0, 0, 0],
        [0, 1, 0, 0],
        [0, 0, 1, 0],
        [0, 0, 0, 1]
    ]

    lhs = roots_quartic(Poly(x**4 + x, x))
    rhs = [S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2, S.Zero, -S.One]

    assert sorted(lhs, key=hash) == sorted(rhs, key=hash)

    # test of all branches of roots quartic
    for i, (a, b, c, d) in enumerate([(1, 2, 3, 0),
                                      (3, -7, -9, 9),
                                      (1, 2, 3, 4),
                                      (1, 2, 3, 4),
                                      (-7, -3, 3, -6),
                                      (-3, 5, -6, -4),
                                      (6, -5, -10, -3)]):
        if i == 2:
            c = -a*(a**2/S(8) - b/S(2))
        elif i == 3:
            d = a*(a*(3*a**2/S(256) - b/S(16)) + c/S(4))
        eq = x**4 + a*x**3 + b*x**2 + c*x + d
        ans = roots_quartic(Poly(eq, x))
        assert all(eq.subs(x, ai).n(chop=True) == 0 for ai in ans)

    # not all symbolic quartics are unresolvable
    eq = Poly(q*x + q/4 + x**4 + x**3 + 2*x**2 - Rational(1, 3), x)
    sol = roots_quartic(eq)
    assert all(test_numerically(eq.subs(x, i), 0) for i in sol)
    z = symbols('z', negative=True)
    eq = x**4 + 2*x**3 + 3*x**2 + x*(z + 11) + 5
    zans = roots_quartic(Poly(eq, x))
    assert all([test_numerically(eq.subs(((x, i), (z, -1))), 0) for i in zans])
    # but some are (see also issue 4989)
    # it's ok if the solution is not Piecewise, but the tests below should pass
    eq = Poly(y*x**4 + x**3 - x + z, x)
    ans = roots_quartic(eq)
    assert all(type(i) == Piecewise for i in ans)
    reps = (
        dict(y=-Rational(1, 3), z=-Rational(1, 4)),  # 4 real
        dict(y=-Rational(1, 3), z=-Rational(1, 2)),  # 2 real
        dict(y=-Rational(1, 3), z=-2))  # 0 real
    for rep in reps:
        sol = roots_quartic(Poly(eq.subs(rep), x))
        assert all([test_numerically(w.subs(rep) - s, 0) for w, s in zip(ans, sol)])

 def t(a, b, arg, n):
     from sympy import Mul
     m1 = meijerg(a, b, arg)
     m2 = Mul(*_inflate_g(m1, n))
     # NOTE: (the random number)**9 must still be on the principal sheet.
     # Thus make b&d small to create random numbers of small imaginary part.
     return test_numerically(m1.subs(subs), m2.subs(subs), x, b=0.1, d=-0.1)

def test_roots_quartic():
    assert roots_quartic(Poly(x ** 4, x)) == [0, 0, 0, 0]
    assert roots_quartic(Poly(x ** 4 + x ** 3, x)) in [[-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, -1]]
    assert roots_quartic(Poly(x ** 4 - x ** 3, x)) in [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]

    lhs = roots_quartic(Poly(x ** 4 + x, x))
    rhs = [S.Half + I * sqrt(3) / 2, S.Half - I * sqrt(3) / 2, S.Zero, -S.One]

    assert sorted(lhs, key=hash) == sorted(rhs, key=hash)

    # test of all branches of roots quartic
    for i, (a, b, c, d) in enumerate(
        [(1, 2, 3, 0), (3, -7, -9, 9), (1, 2, 3, 4), (1, 2, 3, 4), (-7, -3, 3, -6), (-3, 5, -6, -4), (6, -5, -10, -3)]
    ):
        if i == 2:
            c = -a * (a ** 2 / S(8) - b / S(2))
        elif i == 3:
            d = a * (a * (3 * a ** 2 / S(256) - b / S(16)) + c / S(4))
        eq = x ** 4 + a * x ** 3 + b * x ** 2 + c * x + d
        ans = roots_quartic(Poly(eq, x))
        assert all(eq.subs(x, ai).n(chop=True) == 0 for ai in ans)

    # not all symbolic quartics are unresolvable
    eq = Poly(q * x + q / 4 + x ** 4 + x ** 3 + 2 * x ** 2 - Rational(1, 3), x)
    sol = roots_quartic(eq)
    assert all(test_numerically(eq.subs(x, i), 0) for i in sol)
    # but some are (see also iss 1890)
    raises(PolynomialError, lambda: roots_quartic(Poly(y * x ** 4 + x + z, x)))

def test_TR9():
    a = S(1)/2
    b = 3*a
    assert TR9(a) == a
    assert TR9(cos(1) + cos(2)) == 2*cos(a)*cos(b)
    assert TR9(cos(1) - cos(2)) == 2*sin(a)*sin(b)
    assert TR9(sin(1) - sin(2)) == -2*sin(a)*cos(b)
    assert TR9(sin(1) + sin(2)) == 2*sin(b)*cos(a)
    assert TR9(cos(1) + 2*sin(1) + 2*sin(2)) == cos(1) + 4*sin(b)*cos(a)
    assert TR9(cos(4) + cos(2) + 2*cos(1)*cos(3)) == 4*cos(1)*cos(3)
    assert TR9((cos(4) + cos(2))/cos(3)/2 + cos(3)) == 2*cos(1)*cos(2)
    assert TR9(cos(3) + cos(4) + cos(5) + cos(6)) == \
        4*cos(S(1)/2)*cos(1)*cos(S(9)/2)
    assert TR9(cos(3) + cos(3)*cos(2)) == cos(3) + cos(2)*cos(3)
    assert TR9(-cos(y) + cos(x*y)) == -2*sin(x*y/2 - y/2)*sin(x*y/2 + y/2)
    assert TR9(-sin(y) + sin(x*y)) == 2*sin(x*y/2 - y/2)*cos(x*y/2 + y/2)
    c = cos(x)
    s = sin(x)
    for si in ((1, 1), (1, -1), (-1, 1), (-1, -1)):
        for a in ((c, s), (s, c), (cos(x), cos(x*y)), (sin(x), sin(x*y))):
            args = zip(si, a)
            ex = Add(*[Mul(*ai) for ai in args])
            t = TR9(ex)
            assert not (a[0].func == a[1].func and (
                not test_numerically(ex, t.expand(trig=True)) or t.is_Add)
                or a[1].func != a[0].func and ex != t)

def test_reflect():
    b = Symbol('b')
    m = Symbol('m')
    l = Line((0, b), slope=m)
    p = Point(x, y)
    r = p.reflect(l)
    dp = l.perpendicular_segment(p).length
    dr = l.perpendicular_segment(r).length
    assert test_numerically(dp, dr)
    t = Triangle((0, 0), (1, 0), (2, 3))
    assert t.area == -t.reflect(l).area
    e = Ellipse((1, 0), 1, 2)
    assert e.area == -e.reflect(Line((1, 0), slope=0)).area
    assert e.area == -e.reflect(Line((1, 0), slope=oo)).area
    raises(NotImplementedError, lambda: e.reflect(Line((1,0), slope=m)))
    # test entity overrides
    c = Circle((x, y), 3)
    cr = c.reflect(l)
    assert cr == Circle(r, -3)
    assert c.area == -cr.area
    pent = RegularPolygon((1, 2), 1, 5)
    l = Line((0, pi), slope=sqrt(2))
    rpent = pent.reflect(l)
    poly_pent = Polygon(*pent.vertices)
    assert rpent.center == pent.center.reflect(l)
    assert str([w.n(3) for w in rpent.vertices]) == (
        '[Point(-0.586, 4.27), Point(-1.69, 4.66), '
        'Point(-2.41, 3.73), Point(-1.74, 2.76), '
        'Point(-0.616, 3.10)]')
    assert pent.area.equals(-rpent.area)

def mytn(expr1, expr2, expr3, x, d=0):
    from sympy.utilities.randtest import test_numerically, random_complex_number
    subs = {}
    for a in expr1.free_symbols:
        if a != x:
            subs[a] = random_complex_number()
    return expr2 == expr3 and test_numerically(expr1.subs(subs),
                                               expr2.subs(subs), x, d=d)

def test_TR3():
    assert TR3(cos(y - x*(y - x))) == cos(x*(x - y) + y)
    assert cos(pi/2 + x) == -sin(x)
    assert cos(30*pi/2 + x) == -cos(x)

    for f in (cos, sin, tan, cot, csc, sec):
        i = f(3*pi/7)
        j = TR3(i)
        assert test_numerically(i, j) and i.func != j.func

def test_pow_E():
    assert 2 ** (y / log(2)) == S.Exp1 ** y
    assert 2 ** (y / log(2) / 3) == S.Exp1 ** (y / 3)
    assert 3 ** (1 / log(-3)) != S.Exp1
    assert (3 + 2 * I) ** (1 / (log(-3 - 2 * I) + I * pi)) == S.Exp1
    assert (3 + 2 * I) ** (1 / (log(-3 - 2 * I, 3) / 2 + I * pi / log(3) / 2)) == 9
    assert (3 + 2 * I) ** (1 / (log(3 + 2 * I, 3) / 2)) == 9
    # every time tests are run they will affirm with a different random
    # value that this identity holds
    while 1:
        b = x._random()
        r, i = b.as_real_imag()
        if i:
            break
    assert test_numerically(b ** (1 / (log(-b) + sign(i) * I * pi).n()), S.Exp1)

 def tn(a, b):
     from sympy.utilities.randtest import test_numerically
     from sympy import Dummy
     return test_numerically(a, b, Dummy('x'))

def test_Mod():
    assert Mod(x, 1).func is Mod
    assert pi % pi == S.Zero
    assert Mod(5, 3) == 2
    assert Mod(-5, 3) == 1
    assert Mod(5, -3) == -1
    assert Mod(-5, -3) == -2
    assert type(Mod(3.2, 2, evaluate=False)) == Mod
    assert 5 % x == Mod(5, x)
    assert x % 5 == Mod(x, 5)
    assert x % y == Mod(x, y)
    assert (x % y).subs({x: 5, y: 3}) == 2

    # Float handling
    point3 = Float(3.3) % 1
    assert (x - 3.3) % 1 == Mod(1.*x + 1 - point3, 1)
    assert Mod(-3.3, 1) == 1 - point3
    assert Mod(0.7, 1) == Float(0.7)
    e = Mod(1.3, 1)
    point3 = Float._new(Float(.3)._mpf_, 51)
    assert e == point3 and e.is_Float
    e = Mod(1.3, .7)
    point6 = Float._new(Float(.6)._mpf_, 51)
    assert e == point6 and e.is_Float
    e = Mod(1.3, Rational(7, 10))
    assert e == point6 and e.is_Float
    e = Mod(Rational(13, 10), 0.7)
    assert e == point6 and e.is_Float
    e = Mod(Rational(13, 10), Rational(7, 10))
    assert e == .6 and e.is_Rational

    # check that sign is right
    r2 = sqrt(2)
    r3 = sqrt(3)
    for i in [-r3, -r2, r2, r3]:
        for j in [-r3, -r2, r2, r3]:
            assert test_numerically(i % j, i.n() % j.n())
    for _x in range(4):
        for _y in range(9):
            reps = [(x, _x), (y, _y)]
            assert Mod(3*x + y, 9).subs(reps) == (3*_x + _y) % 9

    # denesting
    #   easy case
    assert Mod(Mod(x, y), y) == Mod(x, y)
    #   in case someone attempts more denesting
    for i in [-3, -2, 2, 3]:
        for j in [-3, -2, 2, 3]:
            for k in range(3):
                # print i, j, k
                assert Mod(Mod(k, i), j) == (k % i) % j

    # known difference
    assert Mod(5*sqrt(2), sqrt(5)) == 5*sqrt(2) - 3*sqrt(5)
    p = symbols('p', positive=True)
    assert Mod(p + 1, p + 3) == p + 1
    n = symbols('n', negative=True)
    assert Mod(n - 3, n - 1) == -2
    assert Mod(n - 2*p, n - p) == -p
    assert Mod(p - 2*n, p - n) == -n

    # handling sums
    assert (x + 3) % 1 == Mod(x, 1)
    assert (x + 3.0) % 1 == Mod(1.*x, 1)
    assert (x - S(33)/10) % 1 == Mod(x + S(7)/10, 1)
    assert str(Mod(.6*x + y, .3*y)) == str(Mod(0.1*y + 0.6*x, 0.3*y))
    assert (x + 1) % x == 1 % x
    assert (x + y) % x == y % x
    assert (x + y + 2) % x == (y + 2) % x
    assert (a + 3*x + 1) % (2*x) == Mod(a + x + 1, 2*x)
    assert (12*x + 18*y) % (3*x) == 3*Mod(6*y, x)

    # gcd extraction
    assert (-3*x) % (-2*y) == -Mod(3*x, 2*y)
    assert (.6*pi) % (.3*x*pi) == 0.3*pi*Mod(2, x)
    assert (.6*pi) % (.31*x*pi) == pi*Mod(0.6, 0.31*x)
    assert (6*pi) % (.3*x*pi) == pi*Mod(6, 0.3*x)
    assert (6*pi) % (.31*x*pi) == pi*Mod(6, 0.31*x)
    assert (6*pi) % (.42*x*pi) == pi*Mod(6, 0.42*x)
    assert (12*x) % (2*y) == 2*Mod(6*x, y)
    assert (12*x) % (3*5*y) == 3*Mod(4*x, 5*y)
    assert (12*x) % (15*x*y) == 3*x*Mod(4, 5*y)
    assert (-2*pi) % (3*pi) == pi
    assert (2*x + 2) % (x + 1) == 0
    assert (x*(x + 1)) % (x + 1) == (x + 1)*Mod(x, 1)
    assert Mod(5.0*x, 0.1*y) == 0.1*Mod(50*x, y)
    i = Symbol('i', integer=True)
    assert (3*i*x) % (2*i*y) == i*Mod(3*x, 2*y)
    assert Mod(4*i, 4) == 0

 def u(expr, x):
     from sympy import Add, exp, exp_polar
     r = _rewrite_single(expr, x)
     e = Add(*[res[0]*res[2] for res in r[0]]).replace(
         exp_polar, exp)  # XXX Hack?
     assert test_numerically(e, expr, x)

def test_fresnel():
    assert fresnels(0) == 0
    assert fresnels(oo) == S.Half
    assert fresnels(-oo) == -S.Half

    assert fresnels(z) == fresnels(z)
    assert fresnels(-z) == -fresnels(z)
    assert fresnels(I*z) == -I*fresnels(z)
    assert fresnels(-I*z) == I*fresnels(z)

    assert conjugate(fresnels(z)) == fresnels(conjugate(z))

    assert fresnels(z).diff(z) == sin(pi*z**2/2)

    assert fresnels(z).rewrite(erf) == (S.One + I)/4 * (
        erf((S.One + I)/2*sqrt(pi)*z) - I*erf((S.One - I)/2*sqrt(pi)*z))

    assert fresnels(z).rewrite(hyper) == \
        pi*z**3/6 * hyper([S(3)/4], [S(3)/2, S(7)/4], -pi**2*z**4/16)

    assert fresnels(z).series(z, n=15) == \
        pi*z**3/6 - pi**3*z**7/336 + pi**5*z**11/42240 + O(z**15)

    assert fresnels(w).is_real is True

    assert fresnels(z).as_real_imag() == \
        ((fresnels(re(z) - I*re(z)*Abs(im(z))/Abs(re(z)))/2 +
          fresnels(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))/2,
          I*(fresnels(re(z) - I*re(z)*Abs(im(z))/Abs(re(z))) -
          fresnels(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))) *
          re(z)*Abs(im(z))/(2*im(z)*Abs(re(z)))))

    assert fresnels(2 + 3*I).as_real_imag() == (
        fresnels(2 + 3*I)/2 + fresnels(2 - 3*I)/2,
        I*(fresnels(2 - 3*I) - fresnels(2 + 3*I))/2
    )

    assert expand_func(integrate(fresnels(z), z)) == \
        z*fresnels(z) + cos(pi*z**2/2)/pi

    assert fresnels(z).rewrite(meijerg) == sqrt(2)*pi*z**(S(9)/4) * \
        meijerg(((), (1,)), ((S(3)/4,),
        (S(1)/4, 0)), -pi**2*z**4/16)/(2*(-z)**(S(3)/4)*(z**2)**(S(3)/4))

    assert fresnelc(0) == 0
    assert fresnelc(oo) == S.Half
    assert fresnelc(-oo) == -S.Half

    assert fresnelc(z) == fresnelc(z)
    assert fresnelc(-z) == -fresnelc(z)
    assert fresnelc(I*z) == I*fresnelc(z)
    assert fresnelc(-I*z) == -I*fresnelc(z)

    assert conjugate(fresnelc(z)) == fresnelc(conjugate(z))

    assert fresnelc(z).diff(z) == cos(pi*z**2/2)

    assert fresnelc(z).rewrite(erf) == (S.One - I)/4 * (
        erf((S.One + I)/2*sqrt(pi)*z) + I*erf((S.One - I)/2*sqrt(pi)*z))

    assert fresnelc(z).rewrite(hyper) == \
        z * hyper([S.One/4], [S.One/2, S(5)/4], -pi**2*z**4/16)

    assert fresnelc(z).series(z, n=15) == \
        z - pi**2*z**5/40 + pi**4*z**9/3456 - pi**6*z**13/599040 + O(z**15)

    assert fresnelc(w).is_real is True

    assert fresnelc(z).as_real_imag() == \
        ((fresnelc(re(z) - I*re(z)*Abs(im(z))/Abs(re(z)))/2 +
          fresnelc(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))/2,
          I*(fresnelc(re(z) - I*re(z)*Abs(im(z))/Abs(re(z))) -
          fresnelc(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))) *
          re(z)*Abs(im(z))/(2*im(z)*Abs(re(z)))))

    assert fresnelc(2 + 3*I).as_real_imag() == (
        fresnelc(2 - 3*I)/2 + fresnelc(2 + 3*I)/2,
        I*(fresnelc(2 - 3*I) - fresnelc(2 + 3*I))/2
    )

    assert expand_func(integrate(fresnelc(z), z)) == \
        z*fresnelc(z) - sin(pi*z**2/2)/pi

    assert fresnelc(z).rewrite(meijerg) == sqrt(2)*pi*z**(S(3)/4) * \
        meijerg(((), (1,)), ((S(1)/4,),
        (S(3)/4, 0)), -pi**2*z**4/16)/(2*(-z)**(S(1)/4)*(z**2)**(S(1)/4))

    from sympy.utilities.randtest import test_numerically

    test_numerically(re(fresnels(z)), fresnels(z).as_real_imag()[0], z)
    test_numerically(im(fresnels(z)), fresnels(z).as_real_imag()[1], z)
    test_numerically(fresnels(z), fresnels(z).rewrite(hyper), z)
    test_numerically(fresnels(z), fresnels(z).rewrite(meijerg), z)

    test_numerically(re(fresnelc(z)), fresnelc(z).as_real_imag()[0], z)
    test_numerically(im(fresnelc(z)), fresnelc(z).as_real_imag()[1], z)
    test_numerically(fresnelc(z), fresnelc(z).rewrite(hyper), z)
    test_numerically(fresnelc(z), fresnelc(z).rewrite(meijerg), z)

 def ok(a, b, n):
     e = (a + I*b)**n
     return test_numerically(e, expand_multinomial(e))