Esempio n. 1
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def test_integrate_Abs_sign():
    assert integrate(Abs(x), (x, -2, 1)) == S(5)/2
    assert integrate(Abs(x), (x, 0, 1)) == S(1)/2
    assert integrate(Abs(x + 1), (x, 0, 1)) == S(3)/2
    assert integrate(Abs(x**2 - 1), (x, -2, 2)) == 4
    assert integrate(Abs(x**2 - 3*x), (x, -15, 15)) == 2259
    assert integrate(sign(x), (x, -1, 2)) == 1
    assert integrate(sign(x)*sin(x), (x, -pi, pi)) == 4
    assert integrate(sign(x - 2) * x**2, (x, 0, 3)) == S(11)/3

    t, s = symbols('t s', real=True)
    assert integrate(Abs(t), t) == Piecewise(
        (-t**2/2, t <= 0), (t**2/2, True))
    assert integrate(Abs(2*t - 6), t) == Piecewise(
        (-t**2 + 6*t, t <= 3), (t**2 - 6*t + 18, True))
    assert (integrate(abs(t - s**2), (t, 0, 2)) ==
        2*s**2*Min(2, s**2) - 2*s**2 - Min(2, s**2)**2 + 2)
    assert integrate(exp(-Abs(t)), t) == Piecewise(
        (exp(t), t <= 0), (2 - exp(-t), True))
    assert integrate(sign(2*t - 6), t) == Piecewise(
        (-t, t < 3), (t - 6, True))
    assert integrate(2*t*sign(t**2 - 1), t) == Piecewise(
        (t**2, t < -1), (-t**2 + 2, t < 1), (t**2, True))
    assert integrate(sign(t), (t, s + 1)) == Piecewise(
        (s + 1, s + 1 > 0), (-s - 1, s + 1 < 0), (0, True))
Esempio n. 2
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def test_issue_1572_1364_1368():
    assert solve((sqrt(x**2 - 1) - 2)) in ([sqrt(5), -sqrt(5)],
                                           [-sqrt(5), sqrt(5)])
    assert set(solve((2**exp(y**2/x) + 2)/(x**2 + 15), y)) == set([
        -sqrt(x)*sqrt(-log(log(2)) + log(log(2) + I*pi)),
        sqrt(x)*sqrt(-log(log(2)) + log(log(2) + I*pi))])

    C1, C2 = symbols('C1 C2')
    f = Function('f')
    assert solve(C1 + C2/x**2 - exp(-f(x)), f(x)) == [log(x**2/(C1*x**2 + C2))]
    a = Symbol('a')
    E = S.Exp1
    assert solve(1 - log(a + 4*x**2), x) in (
        [-sqrt(-a + E)/2, sqrt(-a + E)/2],
        [sqrt(-a + E)/2, -sqrt(-a + E)/2]
    )
    assert solve(log(a**(-3) - x**2)/a, x) in (
        [-sqrt(-1 + a**(-3)), sqrt(-1 + a**(-3))],
        [sqrt(-1 + a**(-3)), -sqrt(-1 + a**(-3))],)
    assert solve(1 - log(a + 4*x**2), x) in (
        [-sqrt(-a + E)/2, sqrt(-a + E)/2],
        [sqrt(-a + E)/2, -sqrt(-a + E)/2],)
    assert set(solve((
        a**2 + 1) * (sin(a*x) + cos(a*x)), x)) == set([-pi/(4*a), 3*pi/(4*a)])
    assert solve(3 - (sinh(a*x) + cosh(a*x)), x) == [2*atanh(S.Half)/a]
    assert set(solve(3 - (sinh(a*x) + cosh(a*x)**2), x)) == \
        set([
            2*atanh(-1 + sqrt(2))/a,
            2*atanh(S(1)/2 + sqrt(5)/2)/a,
            2*atanh(-sqrt(2) - 1)/a,
            2*atanh(-sqrt(5)/2 + S(1)/2)/a
             ])
    assert solve(atan(x) - 1) == [tan(1)]
Esempio n. 3
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def test_tan_rewrite():
    x = Symbol('x')
    neg_exp, pos_exp = exp(-x*I), exp(x*I)
    assert tan(x).rewrite(exp) == I*(neg_exp-pos_exp)/(neg_exp+pos_exp)
    assert tan(x).rewrite(sin) == 2*sin(x)**2/sin(2*x)
    assert tan(x).rewrite(cos) == -cos(x + S.Pi/2)/cos(x)
    assert tan(x).rewrite(cot) == 1/cot(x)
Esempio n. 4
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def test_derivative_by_array():
    from sympy.abc import a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z

    bexpr = x*y**2*exp(z)*log(t)
    sexpr = sin(bexpr)
    cexpr = cos(bexpr)

    a = Array([sexpr])

    assert derive_by_array(sexpr, t) == x*y**2*exp(z)*cos(x*y**2*exp(z)*log(t))/t
    assert derive_by_array(sexpr, [x, y, z]) == Array([bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr, bexpr*cexpr])
    assert derive_by_array(a, [x, y, z]) == Array([[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr], [bexpr*cexpr]])

    assert derive_by_array(sexpr, [[x, y], [z, t]]) == Array([[bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr], [bexpr*cexpr, bexpr/log(t)/t*cexpr]])
    assert derive_by_array(a, [[x, y], [z, t]]) == Array([[[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr]], [[bexpr*cexpr], [bexpr/log(t)/t*cexpr]]])
    assert derive_by_array([[x, y], [z, t]], [x, y]) == Array([[[1, 0], [0, 0]], [[0, 1], [0, 0]]])
    assert derive_by_array([[x, y], [z, t]], [[x, y], [z, t]]) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]],
                                                                         [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])

    assert diff(sexpr, t) == x*y**2*exp(z)*cos(x*y**2*exp(z)*log(t))/t
    assert diff(sexpr, Array([x, y, z])) == Array([bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr, bexpr*cexpr])
    assert diff(a, Array([x, y, z])) == Array([[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr], [bexpr*cexpr]])

    assert diff(sexpr, Array([[x, y], [z, t]])) == Array([[bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr], [bexpr*cexpr, bexpr/log(t)/t*cexpr]])
    assert diff(a, Array([[x, y], [z, t]])) == Array([[[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr]], [[bexpr*cexpr], [bexpr/log(t)/t*cexpr]]])
    assert diff(Array([[x, y], [z, t]]), Array([x, y])) == Array([[[1, 0], [0, 0]], [[0, 1], [0, 0]]])
    assert diff(Array([[x, y], [z, t]]), Array([[x, y], [z, t]])) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]],
                                                                         [[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
Esempio n. 5
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def test_cot_rewrite():
    x = Symbol('x')
    neg_exp, pos_exp = exp(-x*I), exp(x*I)
    assert cot(x).rewrite(exp) == I*(pos_exp+neg_exp)/(pos_exp-neg_exp)
    assert cot(x).rewrite(sin) == 2*sin(2*x)/sin(x)**2
    assert cot(x).rewrite(cos) == -cos(x)/cos(x + S.Pi/2)
    assert cot(x).rewrite(tan) == 1/tan(x)
Esempio n. 6
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    def __init__(self, difficulty):


        a = not_named_yet.randint_no_zero(-3, 2)
        k = not_named_yet.randint_no_zero(-2, 2)
        c = not_named_yet.randint_no_zero(-5, 5)

        if a == 1:
            a += 1

        if difficulty == 1:
            self.equation = sympy.exp(k * x) + c
        elif difficulty == 2:
            self.equation = a * sympy.exp(k * x) + c
        elif difficulty == 3:
            inner_function = request_linear(difficulty=3).equation
            self.equation = a * sympy.exp(inner_function) + c
        else:
            raise ValueError('You have given an invalid difficulty level! Please use difficulty levels 1-3')

        self.domain = sympy.Interval(-sympy.oo, sympy.oo, True, True)
        if a < 0:
            self.range = sympy.Interval(-sympy.oo, c, True, True)
        else:
            self.range = sympy.Interval(c, sympy.oo, True, True)
Esempio n. 7
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def test_simple_3():
    assert Order(x) + x == Order(x)
    assert Order(x) + 2 == 2 + Order(x)
    assert Order(x) + x ** 2 == Order(x)
    assert Order(x) + 1 / x == 1 / x + Order(x)
    assert Order(1 / x) + 1 / x ** 2 == 1 / x ** 2 + Order(1 / x)
    assert Order(x) + exp(1 / x) == Order(x) + exp(1 / x)
Esempio n. 8
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def test_evalf_ramanujan():
    assert NS(exp(pi*sqrt(163)) - 640320**3 - 744, 10) == '-7.499274028e-13'
    # A related identity
    A = 262537412640768744*exp(-pi*sqrt(163))
    B = 196884*exp(-2*pi*sqrt(163))
    C = 103378831900730205293632*exp(-3*pi*sqrt(163))
    assert NS(1 - A - B + C, 10) == '1.613679005e-59'
Esempio n. 9
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    def eq_sympy(input_dim, output_dim, ARD=False):
        """
        Latent force model covariance, exponentiated quadratic with multiple outputs. Derived from a diffusion equation with the initial spatial condition layed down by a Gaussian process with lengthscale given by shared_lengthscale.

        See IEEE Trans Pattern Anal Mach Intell. 2013 Nov;35(11):2693-705. doi: 10.1109/TPAMI.2013.86. Linear latent force models using Gaussian processes. Alvarez MA, Luengo D, Lawrence ND.

        :param input_dim: Dimensionality of the kernel
        :type input_dim: int
        :param output_dim: number of outputs in the covariance function.
        :type output_dim: int
        :param ARD: whether or not to user ARD (default False).
        :type ARD: bool

        """
        real_input_dim = input_dim
        if output_dim>1:
            real_input_dim -= 1
        X = sp.symbols('x_:' + str(real_input_dim))
        Z = sp.symbols('z_:' + str(real_input_dim))
        scale = sp.var('scale_i scale_j',positive=True)
        if ARD:
            lengthscales = [sp.var('lengthscale%i_i lengthscale%i_j' % i, positive=True) for i in range(real_input_dim)]
            shared_lengthscales = [sp.var('shared_lengthscale%i' % i, positive=True) for i in range(real_input_dim)]
            dist_string = ' + '.join(['(x_%i-z_%i)**2/(shared_lengthscale%i**2 + lengthscale%i_i**2 + lengthscale%i_j**2)' % (i, i, i) for i in range(real_input_dim)])
            dist = parse_expr(dist_string)
            f =  variance*sp.exp(-dist/2.)
        else:
            lengthscales = sp.var('lengthscale_i lengthscale_j',positive=True)
            shared_lengthscale = sp.var('shared_lengthscale',positive=True)
            dist_string = ' + '.join(['(x_%i-z_%i)**2' % (i, i) for i in range(real_input_dim)])
            dist = parse_expr(dist_string)
            f =  scale_i*scale_j*sp.exp(-dist/(2*(lengthscale_i**2 + lengthscale_j**2 + shared_lengthscale**2)))
        return kern(input_dim, [spkern(input_dim, f, output_dim=output_dim, name='eq_sympy')])
Esempio n. 10
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def test_issue_1364():
    assert solve(-a*x + 2*x*log(x), x) == [exp(a/2)]
    assert solve(a/x + exp(x/2), x) == [2*LambertW(-a/2)]
    assert solve(x**x) == []
    assert solve(x**x - 2) == [exp(LambertW(log(2)))]
    assert solve(((x - 3)*(x - 2))**((x - 3)*(x - 4))) == [2]
    assert solve((a/x + exp(x/2)).diff(x), x) == [4*LambertW(sqrt(2)*sqrt(a)/4)]
Esempio n. 11
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def test_solve_args():
    #implicit symbol to solve for
    assert set(int(tmp) for tmp in solve(x**2-4)) == set([2,-2])
    assert solve([x+y-3,x-y-5]) == {x: 4, y: -1}
    #no symbol to solve for
    assert solve(42) == []
    assert solve([1, 2]) is None
    #multiple symbols: take the first linear solution
    assert solve(x + y - 3, [x, y]) == [{x: 3 - y}]
    # unless it is an undetermined coefficients system
    assert solve(a + b*x - 2, [a, b]) == {a: 2, b: 0}
    # failing undetermined system
    assert solve(a*x + b**2/(x + 4) - 3*x - 4/x, a, b) == \
        [{a: (-b**2*x + 3*x**3 + 12*x**2 + 4*x + 16)/(x**2*(x + 4))}]
    # failed single equation
    assert solve(1/(1/x - y + exp(y))) ==  []
    raises(NotImplementedError, 'solve(exp(x) + sin(x) + exp(y) + sin(y))')
    # failed system
    # --  when no symbols given, 1 fails
    assert solve([y, exp(x) + x]) == [{x: -LambertW(1), y: 0}]
    #     both fail
    assert solve((exp(x) - x, exp(y) - y)) == [{x: -LambertW(-1), y: -LambertW(-1)}]
    # --  when symbols given
    solve([y, exp(x) + x], x, y) == [(-LambertW(1), 0)]
    #symbol is not a symbol or function
    raises(TypeError, "solve(x**2-pi, pi)")
    # no equations
    assert solve([], [x]) == []
    # overdetermined system
    # - nonlinear
    assert solve([(x + y)**2 - 4, x + y - 2]) == [{x: -y + 2}]
    # - linear
    assert solve((x + y - 2, 2*x + 2*y - 4)) == {x: -y + 2}
Esempio n. 12
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def test_branching():
    from sympy import exp_polar, polar_lift, Symbol, I, exp
    assert besselj(polar_lift(k), x) == besselj(k, x)
    assert besseli(polar_lift(k), x) == besseli(k, x)

    n = Symbol('n', integer=True)
    assert besselj(n, exp_polar(2*pi*I)*x) == besselj(n, x)
    assert besselj(n, polar_lift(x)) == besselj(n, x)
    assert besseli(n, exp_polar(2*pi*I)*x) == besseli(n, x)
    assert besseli(n, polar_lift(x)) == besseli(n, x)

    def tn(func, s):
        from random import uniform
        c = uniform(1, 5)
        expr = func(s, c*exp_polar(I*pi)) - func(s, c*exp_polar(-I*pi))
        eps = 1e-15
        expr2 = func(s + eps, -c + eps*I) - func(s + eps, -c - eps*I)
        return abs(expr.n() - expr2.n()).n() < 1e-10

    nu = Symbol('nu')
    assert besselj(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besselj(nu, x)
    assert besseli(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besseli(nu, x)
    assert tn(besselj, 2)
    assert tn(besselj, pi)
    assert tn(besselj, I)
    assert tn(besseli, 2)
    assert tn(besseli, pi)
    assert tn(besseli, I)
Esempio n. 13
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def test_polarify():
    from sympy import polar_lift, polarify
    x = Symbol('x')
    z = Symbol('z', polar=True)
    f = Function('f')
    ES = {}

    assert polarify(-1) == (polar_lift(-1), ES)
    assert polarify(1 + I) == (polar_lift(1 + I), ES)

    assert polarify(exp(x), subs=False) == exp(x)
    assert polarify(1 + x, subs=False) == 1 + x
    assert polarify(f(I) + x, subs=False) == f(polar_lift(I)) + x

    assert polarify(x, lift=True) == polar_lift(x)
    assert polarify(z, lift=True) == z
    assert polarify(f(x), lift=True) == f(polar_lift(x))
    assert polarify(1 + x, lift=True) == polar_lift(1 + x)
    assert polarify(1 + f(x), lift=True) == polar_lift(1 + f(polar_lift(x)))

    newex, subs = polarify(f(x) + z)
    assert newex.subs(subs) == f(x) + z

    mu = Symbol("mu")
    sigma = Symbol("sigma", positive=True)

    # Make sure polarify(lift=True) doesn't try to lift the integration
    # variable
    assert polarify(
        Integral(sqrt(2)*x*exp(-(-mu + x)**2/(2*sigma**2))/(2*sqrt(pi)*sigma),
        (x, -oo, oo)), lift=True) == Integral(sqrt(2)*(sigma*exp_polar(0))**exp_polar(I*pi)*
        exp((sigma*exp_polar(0))**(2*exp_polar(I*pi))*exp_polar(I*pi)*polar_lift(-mu + x)**
        (2*exp_polar(0))/2)*exp_polar(0)*polar_lift(x)/(2*sqrt(pi)), (x, -oo, oo))
Esempio n. 14
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def test_issue_6920():
    e = [cos(x) + I*sin(x), cos(x) - I*sin(x),
        cosh(x) - sinh(x), cosh(x) + sinh(x)]
    ok = [exp(I*x), exp(-I*x), exp(-x), exp(x)]
    # wrap in f to show that the change happens wherever ei occurs
    f = Function('f')
    assert [simplify(f(ei)).args[0] for ei in e] == ok
Esempio n. 15
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def test_evalf_bugs():
    assert NS(sin(1)+exp(-10**10),10) == NS(sin(1),10)
    assert NS(exp(10**10)+sin(1),10) == NS(exp(10**10),10)
    assert NS('log(1+1/10**50)',20) == '1.0000000000000000000e-50'
    assert NS('log(10**100,10)',10) == '100.0000000'
    assert NS('log(2)',10) == '0.6931471806'
    assert NS('(sin(x)-x)/x**3', 15, subs={x:'1/10**50'}) == '-0.166666666666667'
    assert NS(sin(1)+Rational(1,10**100)*I,15) == '0.841470984807897 + 1.00000000000000e-100*I'
    assert x.evalf() == x
    assert NS((1+I)**2*I, 6) == '-2.00000'
    d={n: (-1)**Rational(6,7), y: (-1)**Rational(4,7), x: (-1)**Rational(2,7)}
    assert NS((x*(1+y*(1 + n))).subs(d).evalf(),6) == '0.346011 + 0.433884*I'
    assert NS(((-I-sqrt(2)*I)**2).evalf()) == '-5.82842712474619'
    assert NS((1+I)**2*I,15) == '-2.00000000000000'
    #1659 (1/2):
    assert NS(pi.evalf(69) - pi) == '-4.43863937855894e-71'
    #1659 (2/2): With the bug present, this still only fails if the
    # terms are in the order given here. This is not generally the case,
    # because the order depends on the hashes of the terms.
    assert NS(20 - 5008329267844*n**25 - 477638700*n**37 - 19*n,
              subs={n:.01}) == '19.8100000000000'
    assert NS(((x - 1)*((1 - x))**1000).n()) == '(-x + 1.00000000000000)**1000*(x - 1.00000000000000)'
    assert NS((-x).n()) == '-x'
    assert NS((-2*x).n()) == '-2.00000000000000*x'
    assert NS((-2*x*y).n()) == '-2.00000000000000*x*y'
Esempio n. 16
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def test_case():
    ob = FCodePrinter()
    x,x_,x__,y,X,X_,Y = symbols('x,x_,x__,y,X,X_,Y')
    assert fcode(exp(x_) + sin(x*y) + cos(X*Y)) == \
                        '      exp(x_) + sin(x*y) + cos(X__*Y_)'
    assert fcode(exp(x__) + 2*x*Y*X_**Rational(7, 2)) == \
                        '      2*X_**(7.0d0/2.0d0)*Y*x + exp(x__)'
    assert fcode(exp(x_) + sin(x*y) + cos(X*Y), name_mangling=False) == \
                        '      exp(x_) + sin(x*y) + cos(X*Y)'
    assert fcode(x - cos(X), name_mangling=False) == '      x - cos(X)'
    assert ob.doprint(X*sin(x) + x_, assign_to='me') == '      me = X*sin(x_) + x__'
    assert ob.doprint(X*sin(x), assign_to='mu') == '      mu = X*sin(x_)'
    assert ob.doprint(x_, assign_to='ad') == '      ad = x__'
    n, m = symbols('n,m', integer=True)
    A = IndexedBase('A')
    x = IndexedBase('x')
    y = IndexedBase('y')
    i = Idx('i', m)
    I = Idx('I', n)
    assert fcode(A[i, I]*x[I], assign_to=y[i], source_format='free') == (
                                            "do i = 1, m\n"
                                            "   y(i) = 0\n"
                                            "end do\n"
                                            "do i = 1, m\n"
                                            "   do I_ = 1, n\n"
                                            "      y(i) = A(i, I_)*x(I_) + y(i)\n"
                                            "   end do\n"
                                            "end do" )
Esempio n. 17
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def test_pde_separate_add():
    x, y, z, t = symbols("x,y,z,t")
    F, T, X, Y, Z, u = map(Function, 'FTXYZu')

    eq = Eq(D(u(x, t), x), D(u(x, t), t)*exp(u(x, t)))
    res = pde_separate_add(eq, u(x, t), [X(x), T(t)])
    assert res == [D(X(x), x)*exp(-X(x)), D(T(t), t)*exp(T(t))]
Esempio n. 18
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def test_manualintegrate_exponentials():
    assert manualintegrate(exp(2*x), x) == exp(2*x) / 2
    assert manualintegrate(2**x, x) == (2 ** x) / log(2)

    assert manualintegrate(1 / x, x) == log(x)
    assert manualintegrate(1 / (2*x + 3), x) == log(2*x + 3) / 2
    assert manualintegrate(log(x)**2 / x, x) == log(x)**3 / 3
Esempio n. 19
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def test_manual_subs():
    x, y = symbols('x y')
    expr = log(x) + exp(x)
    # if log(x) is y, then exp(y) is x
    assert manual_subs(expr, log(x), y) == y + exp(exp(y))
    # if exp(x) is y, then log(y) need not be x
    assert manual_subs(expr, exp(x), y) == log(x) + y
Esempio n. 20
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def test_fisher_z():
    d1 = Symbol("d1", positive=True)
    d2 = Symbol("d2", positive=True)

    X = FisherZ("x", d1, d2)
    assert density(X)(x) == (2*d1**(d1/2)*d2**(d2/2)*(d1*exp(2*x) + d2)
                             **(-d1/2 - d2/2)*exp(d1*x)/beta(d1/2, d2/2))
Esempio n. 21
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def test_lognormal():
    mean = Symbol('mu', real=True, finite=True)
    std = Symbol('sigma', positive=True, real=True, finite=True)
    X = LogNormal('x', mean, std)
    # The sympy integrator can't do this too well
    #assert E(X) == exp(mean+std**2/2)
    #assert variance(X) == (exp(std**2)-1) * exp(2*mean + std**2)

    # Right now, only density function and sampling works
    # Test sampling: Only e^mean in sample std of 0
    for i in range(3):
        X = LogNormal('x', i, 0)
        assert S(sample(X)) == N(exp(i))
    # The sympy integrator can't do this too well
    #assert E(X) ==

    mu = Symbol("mu", real=True)
    sigma = Symbol("sigma", positive=True)

    X = LogNormal('x', mu, sigma)
    assert density(X)(x) == (sqrt(2)*exp(-(-mu + log(x))**2
                                    /(2*sigma**2))/(2*x*sqrt(pi)*sigma))

    X = LogNormal('x', 0, 1)  # Mean 0, standard deviation 1
    assert density(X)(x) == sqrt(2)*exp(-log(x)**2/2)/(2*x*sqrt(pi))
Esempio n. 22
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def test_hankel_transform():
    from sympy import sinh, cosh, gamma, sqrt, exp

    r = Symbol("r")
    k = Symbol("k")
    nu = Symbol("nu")
    m = Symbol("m")
    a = symbols("a")

    assert hankel_transform(1/r, r, k, 0) == 1/k
    assert inverse_hankel_transform(1/k, k, r, 0) == 1/r

    assert hankel_transform(
        1/r**m, r, k, 0) == 2**(-m + 1)*k**(m - 2)*gamma(-m/2 + 1)/gamma(m/2)
    assert inverse_hankel_transform(
        2**(-m + 1)*k**(m - 2)*gamma(-m/2 + 1)/gamma(m/2), k, r, 0) == r**(-m)

    assert hankel_transform(1/r**m, r, k, nu) == (
        2*2**(-m)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2))
    assert inverse_hankel_transform(2**(-m + 1)*k**(
        m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2), k, r, nu) == r**(-m)

    assert hankel_transform(r**nu*exp(-a*r), r, k, nu) == \
        2**(nu + 1)*a*k**(-nu - 3)*(a**2/k**2 + 1)**(-nu - S(
                                                     3)/2)*gamma(nu + S(3)/2)/sqrt(pi)
    assert inverse_hankel_transform(
        2**(nu + 1)*a*k**(-nu - 3)*(a**2/k**2 + 1)**(-nu - S(3)/2)*gamma(
        nu + S(3)/2)/sqrt(pi), k, r, nu) == r**nu*exp(-a*r)
Esempio n. 23
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def test_deriv_wrt_function():
    t = Symbol('t')
    xfunc = Function('x')
    yfunc = Function('y')
    x = xfunc(t)
    xd = diff(x, t)
    xdd = diff(xd, t)
    y = yfunc(t)
    yd = diff(y, t)
    ydd = diff(yd, t)

    assert diff(x, t) == xd
    assert diff(2 * x + 4, t) == 2 * xd
    assert diff(2 * x + 4 + y, t) == 2 * xd + yd
    assert diff(2 * x + 4 + y * x, t) == 2 * xd + x * yd + xd * y
    assert diff(2 * x + 4 + y * x, x) == 2 + y
    assert (diff(4 * x**2 + 3 * x + x * y, t) == 3 * xd + x * yd + xd * y +
            8 * x * xd)
    assert (diff(4 * x**2 + 3 * xd + x * y, t) ==  3 * xdd + x * yd + xd * y +
            8 * x * xd)
    assert diff(4 * x**2 + 3 * xd + x * y, xd) == 3
    assert diff(4 * x**2 + 3 * xd + x * y, xdd) == 0
    assert diff(sin(x), t) == xd * cos(x)
    assert diff(exp(x), t) == xd * exp(x)
    assert diff(sqrt(x), t) == xd / (2 * sqrt(x))
Esempio n. 24
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def test_integrate_linearterm_pow():
    # check integrate((a*x+b)^c, x)  --  issue 3499
    y = Symbol('y', positive=True)
    # TODO: Remove conds='none' below, let the assumption take care of it.
    assert integrate(x**y, x, conds='none') == x**(y + 1)/(y + 1)
    assert integrate((exp(y)*x + 1/y)**(1 + sin(y)), x, conds='none') == \
        exp(-y)*(exp(y)*x + 1/y)**(2 + sin(y)) / (2 + sin(y))
Esempio n. 25
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def test_transform():
    a = Integral(x**2 + 1, (x, -1, 2))
    fx = x
    fy = 3*y + 1
    assert a.doit() == a.transform(fx, fy).doit()
    assert a.transform(fx, fy).transform(fy, fx) == a
    fx = 3*x + 1
    fy = y
    assert a.transform(fx, fy).transform(fy, fx) == a
    a = Integral(sin(1/x), (x, 0, 1))
    assert a.transform(x, 1/y) == Integral(sin(y)/y**2, (y, 1, oo))
    assert a.transform(x, 1/y).transform(y, 1/x) == a
    a = Integral(exp(-x**2), (x, -oo, oo))
    assert a.transform(x, 2*y) == Integral(2*exp(-4*y**2), (y, -oo, oo))
    # < 3 arg limit handled properly
    assert Integral(x, x).transform(x, a*y).doit() == \
        Integral(y*a**2, y).doit()
    _3 = S(3)
    assert Integral(x, (x, 0, -_3)).transform(x, 1/y).doit() == \
        Integral(-1/x**3, (x, -oo, -1/_3)).doit()
    assert Integral(x, (x, 0, _3)).transform(x, 1/y) == \
        Integral(y**(-3), (y, 1/_3, oo))
    # issue 8400
    i = Integral(x + y, (x, 1, 2), (y, 1, 2))
    assert i.transform(x, (x + 2*y, x)).doit() == \
        i.transform(x, (x + 2*z, x)).doit() == 3
Esempio n. 26
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def test_tan_rewrite():
    neg_exp, pos_exp = exp(-x*I), exp(x*I)
    assert tan(x).rewrite(exp) == I*(neg_exp - pos_exp)/(neg_exp + pos_exp)
    assert tan(x).rewrite(sin) == 2*sin(x)**2/sin(2*x)
    assert tan(x).rewrite(cos) == -cos(x + S.Pi/2)/cos(x)
    assert tan(x).rewrite(cot) == 1/cot(x)
    assert tan(sinh(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sinh(3)).n()
    assert tan(cosh(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cosh(3)).n()
    assert tan(tanh(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tanh(3)).n()
    assert tan(coth(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, coth(3)).n()
    assert tan(sin(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sin(3)).n()
    assert tan(cos(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cos(3)).n()
    assert tan(tan(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tan(3)).n()
    assert tan(cot(x)).rewrite(
        exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cot(3)).n()
    assert tan(log(x)).rewrite(Pow) == I*(x**-I - x**I)/(x**-I + x**I)
    assert 0 == (cos(pi/15)*tan(pi/15) - sin(pi/15)).rewrite(pow)
    assert tan(pi/19).rewrite(pow) == tan(pi/19)
    assert tan(8*pi/19).rewrite(sqrt) == tan(8*pi/19)
Esempio n. 27
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def test_issue_8368():
    assert integrate(exp(-s*x)*cosh(x), (x, 0, oo)) == \
        Piecewise(
            (   pi*Piecewise(
                    (   -s/(pi*(-s**2 + 1)),
                        Abs(s**2) < 1),
                    (   1/(pi*s*(1 - 1/s**2)),
                        Abs(s**(-2)) < 1),
                    (   meijerg(
                            ((S(1)/2,), (0, 0)),
                            ((0, S(1)/2), (0,)),
                            polar_lift(s)**2),
                        True)
                ),
                And(
                    Abs(periodic_argument(polar_lift(s)**2, oo)) < pi,
                    cos(Abs(periodic_argument(polar_lift(s)**2, oo))/2)*sqrt(Abs(s**2)) - 1 > 0,
                    Ne(s**2, 1))
            ),
            (
                Integral(exp(-s*x)*cosh(x), (x, 0, oo)),
                True))
    assert integrate(exp(-s*x)*sinh(x), (x, 0, oo)) == \
        Piecewise(
            (   -1/(s + 1)/2 - 1/(-s + 1)/2,
                And(
                    Ne(1/s, 1),
                    Abs(periodic_argument(s, oo)) < pi/2,
                    Abs(periodic_argument(s, oo)) <= pi/2,
                    cos(Abs(periodic_argument(s, oo)))*Abs(s) - 1 > 0)),
            (   Integral(exp(-s*x)*sinh(x), (x, 0, oo)),
                True))
Esempio n. 28
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def test_ode_solutions():
    # only a few examples here, the rest will be tested in the actual dsolve tests
    assert ode_renumber(constantsimp(C1*exp(2*x)+exp(x)*(C2+C3), x, 3), 'C', 1, 3) == \
        ode_renumber(C1*exp(x)+C2*exp(2*x), 'C', 1, 2)
    assert ode_renumber(constantsimp(Eq(f(x),I*C1*sinh(x/3) + C2*cosh(x/3)), x, 2),
        'C', 1, 2) == ode_renumber(Eq(f(x), C1*sinh(x/3) + C2*cosh(x/3)), 'C', 1, 2)
    assert ode_renumber(constantsimp(Eq(f(x),acos((-C1)/cos(x))), x, 1), 'C', 1, 1) == \
        Eq(f(x),acos(C1/cos(x)))
    assert ode_renumber(constantsimp(Eq(log(f(x)/C1) + 2*exp(x/f(x)), 0), x, 1),
        'C', 1, 1) ==  Eq(log(C1*f(x)) + 2*exp(x/f(x)), 0)
    assert ode_renumber(constantsimp(Eq(log(x*2**Rational(1,2)*(1/x)**Rational(1,2)*f(x)\
        **Rational(1,2)/C1) + x**2/(2*f(x)**2), 0), x, 1), 'C', 1, 1) == \
        Eq(log(C1*x*(1/x)**Rational(1,2)*f(x)**Rational(1,2)) + x**2/(2*f(x)**2), 0)
    assert ode_renumber(constantsimp(Eq(-exp(-f(x)/x)*sin(f(x)/x)/2 + log(x/C1) - \
        cos(f(x)/x)*exp(-f(x)/x)/2, 0), x, 1), 'C', 1, 1) == \
        Eq(-exp(-f(x)/x)*sin(f(x)/x)/2 + log(C1*x) - cos(f(x)/x)*exp(-f(x)/x)/2, 0)
    u2 = Symbol('u2')
    _a = Symbol('_a')
    assert ode_renumber(constantsimp(Eq(-Integral(-1/((1 - u2**2)**Rational(1,2)*u2), \
        (u2, _a, x/f(x))) + log(f(x)/C1), 0), x, 1), 'C', 1, 1) == \
        Eq(-Integral(-1/(u2*(1 - u2**2)**Rational(1,2)), (u2, _a, x/f(x))) + \
        log(C1*f(x)), 0)
    assert map(lambda i: ode_renumber(constantsimp(i, x, 1), 'C', 1, 1),
        [Eq(f(x), (-C1*x + x**2)**Rational(1,2)), Eq(f(x), -(-C1*x +
        x**2)**Rational(1,2))]) == [Eq(f(x), (C1*x + x**2)**Rational(1,2)),
        Eq(f(x), -(C1*x + x**2)**Rational(1,2))]
Esempio n. 29
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def test_convolution_fft():
    assert all(convolution_fft([], x, dps=y) == [] for x in ([], [1]) for y in (None, 3))
    assert convolution_fft([1, 2, 3], [4, 5, 6]) == [4, 13, 28, 27, 18]
    assert convolution_fft([1], [5, 6, 7]) == [5, 6, 7]
    assert convolution_fft([1, 3], [5, 6, 7]) == [5, 21, 25, 21]

    assert convolution_fft([1 + 2*I], [2 + 3*I]) == [-4 + 7*I]
    assert convolution_fft([1 + 2*I, 3 + 4*I, 5 + S(3)/5*I], [S(2)/5 + S(4)/7*I]) == \
            [-S(26)/35 + 48*I/35, -S(38)/35 + 116*I/35, S(58)/35 + 542*I/175]

    assert convolution_fft([S(3)/4, S(5)/6], [S(7)/8, S(1)/3, S(2)/5]) == \
                                    [S(21)/32, S(47)/48, S(26)/45, S(1)/3]

    assert convolution_fft([S(1)/9, S(2)/3, S(3)/5], [S(2)/5, S(3)/7, S(4)/9]) == \
                                [S(2)/45, S(11)/35, S(8152)/14175, S(523)/945, S(4)/15]

    assert convolution_fft([pi, E, sqrt(2)], [sqrt(3), 1/pi, 1/E]) == \
                    [sqrt(3)*pi, 1 + sqrt(3)*E, E/pi + pi*exp(-1) + sqrt(6),
                                            sqrt(2)/pi + 1, sqrt(2)*exp(-1)]

    assert convolution_fft([2321, 33123], [5321, 6321, 71323]) == \
                        [12350041, 190918524, 374911166, 2362431729]

    assert convolution_fft([312313, 31278232], [32139631, 319631]) == \
                        [10037624576503, 1005370659728895, 9997492572392]

    raises(TypeError, lambda: convolution_fft(x, y))
    raises(ValueError, lambda: convolution_fft([x, y], [y, x]))
Esempio n. 30
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def test_ode_solutions():
    # only a few examples here, the rest will be tested in the actual dsolve tests
    assert constant_renumber(constantsimp(C1*exp(2*x)+exp(x)*(C2+C3), x, 3), 'C', 1, 3) == \
        constant_renumber(C1*exp(x)+C2*exp(2*x), 'C', 1, 2)
    assert constant_renumber(constantsimp(Eq(f(x),I*C1*sinh(x/3) + C2*cosh(x/3)), x, 2),
        'C', 1, 2) == constant_renumber(Eq(f(x), C1*sinh(x/3) + C2*cosh(x/3)), 'C', 1, 2)
    assert constant_renumber(constantsimp(Eq(f(x),acos((-C1)/cos(x))), x, 1), 'C', 1, 1) == \
        Eq(f(x),acos(C1/cos(x)))
    assert constant_renumber(constantsimp(Eq(log(f(x)/C1) + 2*exp(x/f(x)), 0), x, 1),
        'C', 1, 1) ==  Eq(log(C1*f(x)) + 2*exp(x/f(x)), 0)
    assert constant_renumber(constantsimp(Eq(log(x*sqrt(2)*sqrt(1/x)*sqrt(f(x))\
        /C1) + x**2/(2*f(x)**2), 0), x, 1), 'C', 1, 1) == \
        Eq(log(C1*x*sqrt(1/x)*sqrt(f(x))) + x**2/(2*f(x)**2), 0)
    assert constant_renumber(constantsimp(Eq(-exp(-f(x)/x)*sin(f(x)/x)/2 + log(x/C1) - \
        cos(f(x)/x)*exp(-f(x)/x)/2, 0), x, 1), 'C', 1, 1) == \
        Eq(-exp(-f(x)/x)*sin(f(x)/x)/2 + log(C1*x) - cos(f(x)/x)*exp(-f(x)/x)/2, 0)
    u2 = Symbol('u2')
    _a = Symbol('_a')
    assert constant_renumber(constantsimp(Eq(-Integral(-1/(sqrt(1 - u2**2)*u2), \
        (u2, _a, x/f(x))) + log(f(x)/C1), 0), x, 1), 'C', 1, 1) == \
        Eq(-Integral(-1/(u2*sqrt(1 - u2**2)), (u2, _a, x/f(x))) + \
        log(C1*f(x)), 0)
    assert [constant_renumber(constantsimp(i, x, 1), 'C', 1, 1) for i in
        [Eq(f(x), sqrt(-C1*x + x**2)), Eq(f(x), -sqrt(-C1*x +
        x**2))]] == [Eq(f(x), sqrt(C1*x + x**2)),
        Eq(f(x), -sqrt(C1*x + x**2))]
Esempio n. 31
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def test_collect_respecting_exponentials():
    # If this test passes, lines 1306-1311 (at the time of this commit)
    # of sympy/solvers/ode.py should be removed.
    sol = 1 + exp(x / 2)
    assert sol == collect(sol, exp(x / 3))
Esempio n. 32
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def test_exp():
    x = Symbol('x', integer=True)
    assert refine(exp(pi * I * 2 * x)) == 1
    assert refine(exp(pi * I * 2 * (x + S.Half))) == -1
    assert refine(exp(pi * I * 2 * (x + Rational(1, 4)))) == I
    assert refine(exp(pi * I * 2 * (x + Rational(3, 4)))) == -I
Esempio n. 33
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# vector containing the system variables
U = [tau, u, e, z]

# system parameters
parameters = [nu, kappa, D, cnu, A, q, k, G]

#
# f_0(U)_t + F_1(U)_x = (B(U)U_x)_x
#

f0 = [tau, u, e + u**2 / 2, z]

f1 = [
    -u - spd * tau, G * e / tau - spd * u,
    G * e * u / tau - spd * (e + u**2 / 2), -spd * z
]

BU = [[0, 0, 0, 0], [0, nu / tau, 0, 0],
      [0, nu * u / tau, kappa / (cnu * tau), 0], [0, 0, 0, D / tau**2]]

G = [
    0, 0, -q * k * exp(-A / (e / cnu - Ti)) * z,
    k * exp(-A / (e / cnu - Ti)) * z
]

# file path to location where code will be written
file_path = '/Users/blake/Dropbox/stablab20/rNS/matlab'

create_code(U, parameters, f0, f1, G, BU, file_path)
def test_ceiling():

    assert ceiling(nan) == nan

    assert ceiling(oo) == oo
    assert ceiling(-oo) == -oo
    assert ceiling(zoo) == zoo

    assert ceiling(0) == 0

    assert ceiling(1) == 1
    assert ceiling(-1) == -1

    assert ceiling(E) == 3
    assert ceiling(-E) == -2

    assert ceiling(2 * E) == 6
    assert ceiling(-2 * E) == -5

    assert ceiling(pi) == 4
    assert ceiling(-pi) == -3

    assert ceiling(Rational(1, 2)) == 1
    assert ceiling(-Rational(1, 2)) == 0

    assert ceiling(Rational(7, 3)) == 3
    assert ceiling(-Rational(7, 3)) == -2

    assert ceiling(Float(17.0)) == 17
    assert ceiling(-Float(17.0)) == -17

    assert ceiling(Float(7.69)) == 8
    assert ceiling(-Float(7.69)) == -7

    assert ceiling(I) == I
    assert ceiling(-I) == -I
    e = ceiling(i)
    assert e.func is ceiling and e.args[0] == i

    assert ceiling(oo * I) == oo * I
    assert ceiling(-oo * I) == -oo * I
    assert ceiling(exp(I * pi / 4) * oo) == exp(I * pi / 4) * oo

    assert ceiling(2 * I) == 2 * I
    assert ceiling(-2 * I) == -2 * I

    assert ceiling(I / 2) == I
    assert ceiling(-I / 2) == 0

    assert ceiling(E + 17) == 20
    assert ceiling(pi + 2) == 6

    assert ceiling(E + pi) == ceiling(E + pi)
    assert ceiling(I + pi) == ceiling(I + pi)

    assert ceiling(ceiling(pi)) == 4
    assert ceiling(ceiling(y)) == ceiling(y)
    assert ceiling(ceiling(x)) == ceiling(ceiling(x))

    assert ceiling(x) == ceiling(x)
    assert ceiling(2 * x) == ceiling(2 * x)
    assert ceiling(k * x) == ceiling(k * x)

    assert ceiling(k) == k
    assert ceiling(2 * k) == 2 * k
    assert ceiling(k * n) == k * n

    assert ceiling(k / 2) == ceiling(k / 2)

    assert ceiling(x + y) == ceiling(x + y)

    assert ceiling(x + 3) == ceiling(x + 3)
    assert ceiling(x + k) == ceiling(x + k)

    assert ceiling(y + 3) == ceiling(y) + 3
    assert ceiling(y + k) == ceiling(y) + k

    assert ceiling(3 + pi + y * I) == 7 + ceiling(y) * I

    assert ceiling(k + n) == k + n

    assert ceiling(x * I) == ceiling(x * I)
    assert ceiling(k * I) == k * I

    assert ceiling(Rational(23, 10) - E * I) == 3 - 2 * I

    assert ceiling(sin(1)) == 1
    assert ceiling(sin(-1)) == 0

    assert ceiling(exp(2)) == 8

    assert ceiling(-log(8) / log(2)) != -2
    assert int(ceiling(-log(8) / log(2)).evalf(chop=True)) == -3

    assert ceiling(factorial(50)/exp(1)) == \
        11188719610782480504630258070757734324011354208865721592720336801

    assert (ceiling(y) >= y) == True
    assert (ceiling(y) < y) == False
    assert (ceiling(x) >= x).is_Relational  # x could be non-real
    assert (ceiling(x) < x).is_Relational
    assert (ceiling(x) >= y).is_Relational  # arg is not same as rhs
    assert (ceiling(x) < y).is_Relational

    assert ceiling(y).rewrite(floor) == -floor(-y)
    assert ceiling(y).rewrite(frac) == y + frac(-y)
    assert ceiling(y).rewrite(floor).subs(y, -pi) == -floor(pi)
    assert ceiling(y).rewrite(floor).subs(y, E) == -floor(-E)
    assert ceiling(y).rewrite(frac).subs(y, pi) == ceiling(pi)
    assert ceiling(y).rewrite(frac).subs(y, -E) == ceiling(-E)

    assert Eq(ceiling(y), y + frac(-y))
    assert Eq(ceiling(y), -floor(-y))
def test_floor():

    assert floor(nan) == nan

    assert floor(oo) == oo
    assert floor(-oo) == -oo
    assert floor(zoo) == zoo

    assert floor(0) == 0

    assert floor(1) == 1
    assert floor(-1) == -1

    assert floor(E) == 2
    assert floor(-E) == -3

    assert floor(2 * E) == 5
    assert floor(-2 * E) == -6

    assert floor(pi) == 3
    assert floor(-pi) == -4

    assert floor(Rational(1, 2)) == 0
    assert floor(-Rational(1, 2)) == -1

    assert floor(Rational(7, 3)) == 2
    assert floor(-Rational(7, 3)) == -3

    assert floor(Float(17.0)) == 17
    assert floor(-Float(17.0)) == -17

    assert floor(Float(7.69)) == 7
    assert floor(-Float(7.69)) == -8

    assert floor(I) == I
    assert floor(-I) == -I
    e = floor(i)
    assert e.func is floor and e.args[0] == i

    assert floor(oo * I) == oo * I
    assert floor(-oo * I) == -oo * I
    assert floor(exp(I * pi / 4) * oo) == exp(I * pi / 4) * oo

    assert floor(2 * I) == 2 * I
    assert floor(-2 * I) == -2 * I

    assert floor(I / 2) == 0
    assert floor(-I / 2) == -I

    assert floor(E + 17) == 19
    assert floor(pi + 2) == 5

    assert floor(E + pi) == floor(E + pi)
    assert floor(I + pi) == floor(I + pi)

    assert floor(floor(pi)) == 3
    assert floor(floor(y)) == floor(y)
    assert floor(floor(x)) == floor(floor(x))

    assert floor(x) == floor(x)
    assert floor(2 * x) == floor(2 * x)
    assert floor(k * x) == floor(k * x)

    assert floor(k) == k
    assert floor(2 * k) == 2 * k
    assert floor(k * n) == k * n

    assert floor(k / 2) == floor(k / 2)

    assert floor(x + y) == floor(x + y)

    assert floor(x + 3) == floor(x + 3)
    assert floor(x + k) == floor(x + k)

    assert floor(y + 3) == floor(y) + 3
    assert floor(y + k) == floor(y) + k

    assert floor(3 + I * y + pi) == 6 + floor(y) * I

    assert floor(k + n) == k + n

    assert floor(x * I) == floor(x * I)
    assert floor(k * I) == k * I

    assert floor(Rational(23, 10) - E * I) == 2 - 3 * I

    assert floor(sin(1)) == 0
    assert floor(sin(-1)) == -1

    assert floor(exp(2)) == 7

    assert floor(log(8) / log(2)) != 2
    assert int(floor(log(8) / log(2)).evalf(chop=True)) == 3

    assert floor(factorial(50)/exp(1)) == \
        11188719610782480504630258070757734324011354208865721592720336800

    assert (floor(y) <= y) == True
    assert (floor(y) > y) == False
    assert (floor(x) <= x).is_Relational  # x could be non-real
    assert (floor(x) > x).is_Relational
    assert (floor(x) <= y).is_Relational  # arg is not same as rhs
    assert (floor(x) > y).is_Relational

    assert floor(y).rewrite(frac) == y - frac(y)
    assert floor(y).rewrite(ceiling) == -ceiling(-y)
    assert floor(y).rewrite(frac).subs(y, -pi) == floor(-pi)
    assert floor(y).rewrite(frac).subs(y, E) == floor(E)
    assert floor(y).rewrite(ceiling).subs(y, E) == -ceiling(-E)
    assert floor(y).rewrite(ceiling).subs(y, -pi) == -ceiling(pi)

    assert Eq(floor(y), y - frac(y))
    assert Eq(floor(y), -ceiling(-y))
Esempio n. 36
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def test_constantsimp_take_problem():
    c = exp(C1) + 2
    assert len(Poly(constantsimp(exp(C1) + c + c * x, [C1])).gens) == 2
Esempio n. 37
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def test_undetermined_coefficients_match():
    assert _undetermined_coefficients_match(g(x), x) == {'test': False}
    assert _undetermined_coefficients_match(sin(2*x + sqrt(5)), x) == \
        {'test': True, 'trialset':
            {cos(2*x + sqrt(5)), sin(2*x + sqrt(5))}}
    assert _undetermined_coefficients_match(sin(x)*cos(x), x) == \
        {'test': False}
    s = {cos(x), x * cos(x), x**2 * cos(x), x**2 * sin(x), x * sin(x), sin(x)}
    assert _undetermined_coefficients_match(sin(x)*(x**2 + x + 1), x) == \
        {'test': True, 'trialset': s}
    assert _undetermined_coefficients_match(
        sin(x) * x**2 + sin(x) * x + sin(x), x) == {
            'test': True,
            'trialset': s
        }
    assert _undetermined_coefficients_match(
        exp(2 * x) * sin(x) * (x**2 + x + 1), x) == {
            'test': True,
            'trialset': {
                exp(2 * x) * sin(x), x**2 * exp(2 * x) * sin(x),
                cos(x) * exp(2 * x), x**2 * cos(x) * exp(2 * x),
                x * cos(x) * exp(2 * x), x * exp(2 * x) * sin(x)
            }
        }
    assert _undetermined_coefficients_match(1 / sin(x), x) == {'test': False}
    assert _undetermined_coefficients_match(log(x), x) == {'test': False}
    assert _undetermined_coefficients_match(2**(x)*(x**2 + x + 1), x) == \
        {'test': True, 'trialset': {2**x, x*2**x, x**2*2**x}}
    assert _undetermined_coefficients_match(x**y, x) == {'test': False}
    assert _undetermined_coefficients_match(exp(x)*exp(2*x + 1), x) == \
        {'test': True, 'trialset': {exp(1 + 3*x)}}
    assert _undetermined_coefficients_match(sin(x)*(x**2 + x + 1), x) == \
        {'test': True, 'trialset': {x*cos(x), x*sin(x), x**2*cos(x),
        x**2*sin(x), cos(x), sin(x)}}
    assert _undetermined_coefficients_match(sin(x)*(x + sin(x)), x) == \
        {'test': False}
    assert _undetermined_coefficients_match(sin(x)*(x + sin(2*x)), x) == \
        {'test': False}
    assert _undetermined_coefficients_match(sin(x)*tan(x), x) == \
        {'test': False}
    assert _undetermined_coefficients_match(
        x**2 * sin(x) * exp(x) + x * sin(x) + x, x) == {
            'test': True,
            'trialset': {
                x**2 * cos(x) * exp(x), x,
                cos(x), S.One,
                exp(x) * sin(x),
                sin(x), x * exp(x) * sin(x), x * cos(x), x * cos(x) * exp(x),
                x * sin(x),
                cos(x) * exp(x), x**2 * exp(x) * sin(x)
            }
        }
    assert _undetermined_coefficients_match(4 * x * sin(x - 2), x) == {
        'trialset': {x * cos(x - 2), x * sin(x - 2),
                     cos(x - 2),
                     sin(x - 2)},
        'test': True,
    }
    assert _undetermined_coefficients_match(2**x*x, x) == \
        {'test': True, 'trialset': {2**x, x*2**x}}
    assert _undetermined_coefficients_match(2**x*exp(2*x), x) == \
        {'test': True, 'trialset': {2**x*exp(2*x)}}
    assert _undetermined_coefficients_match(exp(-x)/x, x) == \
        {'test': False}
    # Below are from Ordinary Differential Equations,
    #                Tenenbaum and Pollard, pg. 231
    assert _undetermined_coefficients_match(S(4), x) == \
        {'test': True, 'trialset': {S.One}}
    assert _undetermined_coefficients_match(12*exp(x), x) == \
        {'test': True, 'trialset': {exp(x)}}
    assert _undetermined_coefficients_match(exp(I*x), x) == \
        {'test': True, 'trialset': {exp(I*x)}}
    assert _undetermined_coefficients_match(sin(x), x) == \
        {'test': True, 'trialset': {cos(x), sin(x)}}
    assert _undetermined_coefficients_match(cos(x), x) == \
        {'test': True, 'trialset': {cos(x), sin(x)}}
    assert _undetermined_coefficients_match(8 + 6*exp(x) + 2*sin(x), x) == \
        {'test': True, 'trialset': {S.One, cos(x), sin(x), exp(x)}}
    assert _undetermined_coefficients_match(x**2, x) == \
        {'test': True, 'trialset': {S.One, x, x**2}}
    assert _undetermined_coefficients_match(9*x*exp(x) + exp(-x), x) == \
        {'test': True, 'trialset': {x*exp(x), exp(x), exp(-x)}}
    assert _undetermined_coefficients_match(2*exp(2*x)*sin(x), x) == \
        {'test': True, 'trialset': {exp(2*x)*sin(x), cos(x)*exp(2*x)}}
    assert _undetermined_coefficients_match(x - sin(x), x) == \
        {'test': True, 'trialset': {S.One, x, cos(x), sin(x)}}
    assert _undetermined_coefficients_match(x**2 + 2*x, x) == \
        {'test': True, 'trialset': {S.One, x, x**2}}
    assert _undetermined_coefficients_match(4*x*sin(x), x) == \
        {'test': True, 'trialset': {x*cos(x), x*sin(x), cos(x), sin(x)}}
    assert _undetermined_coefficients_match(x*sin(2*x), x) == \
        {'test': True, 'trialset':
            {x*cos(2*x), x*sin(2*x), cos(2*x), sin(2*x)}}
    assert _undetermined_coefficients_match(x**2*exp(-x), x) == \
        {'test': True, 'trialset': {x*exp(-x), x**2*exp(-x), exp(-x)}}
    assert _undetermined_coefficients_match(2*exp(-x) - x**2*exp(-x), x) == \
        {'test': True, 'trialset': {x*exp(-x), x**2*exp(-x), exp(-x)}}
    assert _undetermined_coefficients_match(exp(-2*x) + x**2, x) == \
        {'test': True, 'trialset': {S.One, x, x**2, exp(-2*x)}}
    assert _undetermined_coefficients_match(x*exp(-x), x) == \
        {'test': True, 'trialset': {x*exp(-x), exp(-x)}}
    assert _undetermined_coefficients_match(x + exp(2*x), x) == \
        {'test': True, 'trialset': {S.One, x, exp(2*x)}}
    assert _undetermined_coefficients_match(sin(x) + exp(-x), x) == \
        {'test': True, 'trialset': {cos(x), sin(x), exp(-x)}}
    assert _undetermined_coefficients_match(exp(x), x) == \
        {'test': True, 'trialset': {exp(x)}}
    # converted from sin(x)**2
    assert _undetermined_coefficients_match(S.Half - cos(2*x)/2, x) == \
        {'test': True, 'trialset': {S.One, cos(2*x), sin(2*x)}}
    # converted from exp(2*x)*sin(x)**2
    assert _undetermined_coefficients_match(
        exp(2 * x) * (S.Half + cos(2 * x) / 2), x) == {
            'test': True,
            'trialset':
            {exp(2 * x) * sin(2 * x),
             cos(2 * x) * exp(2 * x),
             exp(2 * x)}
        }
    assert _undetermined_coefficients_match(2*x + sin(x) + cos(x), x) == \
        {'test': True, 'trialset': {S.One, x, cos(x), sin(x)}}
    # converted from sin(2*x)*sin(x)
    assert _undetermined_coefficients_match(cos(x)/2 - cos(3*x)/2, x) == \
        {'test': True, 'trialset': {cos(x), cos(3*x), sin(x), sin(3*x)}}
    assert _undetermined_coefficients_match(cos(x**2), x) == {'test': False}
    assert _undetermined_coefficients_match(2**(x**2), x) == {'test': False}
Esempio n. 38
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def test_classify_ode():
    assert classify_ode(f(x).diff(x, 2), f(x)) == \
        (
        'nth_algebraic',
        'nth_linear_constant_coeff_homogeneous',
        'nth_linear_euler_eq_homogeneous',
        'Liouville',
        '2nd_power_series_ordinary',
        'nth_algebraic_Integral',
        'Liouville_Integral',
        )
    assert classify_ode(f(x),
                        f(x)) == ('nth_algebraic', 'nth_algebraic_Integral')
    assert classify_ode(
        Eq(f(x).diff(x), 0),
        f(x)) == ('nth_algebraic', 'separable', '1st_exact', '1st_linear',
                  'Bernoulli', '1st_homogeneous_coeff_best',
                  '1st_homogeneous_coeff_subs_indep_div_dep',
                  '1st_homogeneous_coeff_subs_dep_div_indep',
                  '1st_power_series', 'lie_group',
                  'nth_linear_constant_coeff_homogeneous',
                  'nth_linear_euler_eq_homogeneous', 'nth_algebraic_Integral',
                  'separable_Integral', '1st_exact_Integral',
                  '1st_linear_Integral', 'Bernoulli_Integral',
                  '1st_homogeneous_coeff_subs_indep_div_dep_Integral',
                  '1st_homogeneous_coeff_subs_dep_div_indep_Integral')
    assert classify_ode(
        f(x).diff(x)**2,
        f(x)) == ('factorable', 'nth_algebraic', 'separable', '1st_exact',
                  '1st_linear', 'Bernoulli', '1st_homogeneous_coeff_best',
                  '1st_homogeneous_coeff_subs_indep_div_dep',
                  '1st_homogeneous_coeff_subs_dep_div_indep',
                  '1st_power_series', 'lie_group',
                  'nth_linear_euler_eq_homogeneous', 'nth_algebraic_Integral',
                  'separable_Integral', '1st_exact_Integral',
                  '1st_linear_Integral', 'Bernoulli_Integral',
                  '1st_homogeneous_coeff_subs_indep_div_dep_Integral',
                  '1st_homogeneous_coeff_subs_dep_div_indep_Integral')
    # issue 4749: f(x) should be cleared from highest derivative before classifying
    a = classify_ode(Eq(f(x).diff(x) + f(x), x), f(x))
    b = classify_ode(f(x).diff(x) * f(x) + f(x) * f(x) - x * f(x), f(x))
    c = classify_ode(f(x).diff(x) / f(x) + f(x) / f(x) - x / f(x), f(x))
    assert a == ('1st_exact', '1st_linear', 'Bernoulli', 'almost_linear',
                 '1st_power_series', "lie_group",
                 'nth_linear_constant_coeff_undetermined_coefficients',
                 'nth_linear_constant_coeff_variation_of_parameters',
                 '1st_exact_Integral', '1st_linear_Integral',
                 'Bernoulli_Integral', 'almost_linear_Integral',
                 'nth_linear_constant_coeff_variation_of_parameters_Integral')
    assert b == ('factorable', '1st_linear', 'Bernoulli', '1st_power_series',
                 'lie_group',
                 'nth_linear_constant_coeff_undetermined_coefficients',
                 'nth_linear_constant_coeff_variation_of_parameters',
                 '1st_linear_Integral', 'Bernoulli_Integral',
                 'nth_linear_constant_coeff_variation_of_parameters_Integral')
    assert c == ('1st_linear', 'Bernoulli', '1st_power_series', 'lie_group',
                 'nth_linear_constant_coeff_undetermined_coefficients',
                 'nth_linear_constant_coeff_variation_of_parameters',
                 '1st_linear_Integral', 'Bernoulli_Integral',
                 'nth_linear_constant_coeff_variation_of_parameters_Integral')

    assert classify_ode(
        2 * x * f(x) * f(x).diff(x) + (1 + x) * f(x)**2 - exp(x),
        f(x)) == ('1st_exact', 'Bernoulli', 'almost_linear', 'lie_group',
                  '1st_exact_Integral', 'Bernoulli_Integral',
                  'almost_linear_Integral')
    assert 'Riccati_special_minus2' in \
        classify_ode(2*f(x).diff(x) + f(x)**2 - f(x)/x + 3*x**(-2), f(x))
    raises(ValueError,
           lambda: classify_ode(x + f(x, y).diff(x).diff(y), f(x, y)))
    # issue 5176
    k = Symbol('k')
    assert classify_ode(f(x).diff(x)/(k*f(x) + k*x*f(x)) + 2*f(x)/(k*f(x) +
        k*x*f(x)) + x*f(x).diff(x)/(k*f(x) + k*x*f(x)) + z, f(x)) == \
        ('separable', '1st_exact', '1st_linear', 'Bernoulli',
        '1st_power_series', 'lie_group', 'separable_Integral', '1st_exact_Integral',
        '1st_linear_Integral', 'Bernoulli_Integral')
    # preprocessing
    ans = (
        'nth_algebraic', 'separable', '1st_exact', '1st_linear', 'Bernoulli',
        '1st_homogeneous_coeff_best',
        '1st_homogeneous_coeff_subs_indep_div_dep',
        '1st_homogeneous_coeff_subs_dep_div_indep', '1st_power_series',
        'lie_group', 'nth_linear_constant_coeff_undetermined_coefficients',
        'nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients',
        'nth_linear_constant_coeff_variation_of_parameters',
        'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters',
        'nth_algebraic_Integral', 'separable_Integral', '1st_exact_Integral',
        '1st_linear_Integral', 'Bernoulli_Integral',
        '1st_homogeneous_coeff_subs_indep_div_dep_Integral',
        '1st_homogeneous_coeff_subs_dep_div_indep_Integral',
        'nth_linear_constant_coeff_variation_of_parameters_Integral',
        'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters_Integral')
    #     w/o f(x) given
    assert classify_ode(diff(f(x) + x, x) + diff(f(x), x)) == ans
    #     w/ f(x) and prep=True
    assert classify_ode(diff(f(x) + x, x) + diff(f(x), x), f(x),
                        prep=True) == ans

    assert classify_ode(Eq(2*x**3*f(x).diff(x), 0), f(x)) == \
        ('factorable', 'nth_algebraic', 'separable', '1st_exact',
         '1st_linear', 'Bernoulli', '1st_power_series',
         'lie_group', 'nth_linear_euler_eq_homogeneous',
         'nth_algebraic_Integral', 'separable_Integral', '1st_exact_Integral',
         '1st_linear_Integral', 'Bernoulli_Integral')


    assert classify_ode(Eq(2*f(x)**3*f(x).diff(x), 0), f(x)) == \
        ('factorable', 'nth_algebraic', 'separable', '1st_exact', '1st_linear',
         'Bernoulli', '1st_power_series', 'lie_group', 'nth_algebraic_Integral',
         'separable_Integral', '1st_exact_Integral', '1st_linear_Integral',
         'Bernoulli_Integral')
    # test issue 13864
    assert classify_ode(Eq(diff(f(x), x) - f(x)**x, 0), f(x)) == \
        ('1st_power_series', 'lie_group')
    assert isinstance(classify_ode(Eq(f(x), 5), f(x), dict=True), dict)

    #This is for new behavior of classify_ode when called internally with default, It should
    # return the first hint which matches therefore, 'ordered_hints' key will not be there.
    assert sorted(classify_ode(Eq(f(x).diff(x), 0), f(x), dict=True).keys()) == \
        ['default', 'nth_linear_constant_coeff_homogeneous', 'order']
    a = classify_ode(2 * x * f(x) * f(x).diff(x) + (1 + x) * f(x)**2 - exp(x),
                     f(x),
                     dict=True,
                     hint='Bernoulli')
    assert sorted(a.keys()) == [
        'Bernoulli', 'Bernoulli_Integral', 'default', 'order', 'ordered_hints'
    ]
Esempio n. 39
0
    def get_sympy_expr(self, params):
        """Generate the required variables for creating a Dfun to be used in the fitting procedure.

		Parameters
		----------
		params : dict
			a dictionary of the model parameters, possibly created by :py:method:`guess`.

		Returns
		-------
			dNdt : sympy.expr.Expr
				expression for the derivate of the model function
			t : sympy.symbol.symbol
			ma	time symbol
			args : tuple of sympy.symbol.symbol
				tuple of symbols of model parameters

		See also
		--------
		curveball.models.make_Dfun
		"""
        t, y0, K, r, nu, q0, v = sympy.symbols('t y0 K r nu q0 v')
        args = [y0, K, r, nu, q0, v]
        # remove fixed params and replace their symbols by their values
        # for params that don't exist (in inherited models), remove them and replace with a default value
        if not params['y0'].vary:
            args.remove(y0)
            y0 = params['y0'].value
        if 'K' not in self.param_names:
            args.remove(K)
            K = np.inf
        elif not params['K'].vary:
            args.remove(K)
            K = params['K'].value
        if not params['r'].vary:
            args.remove(r)
            r = params['r'].value
        if 'nu' not in self.param_names:
            args.remove(nu)
            nu = 1.0
        elif not params['nu'].vary:
            args.remove(nu)
            nu = params['nu'].value
        if 'q0' not in self.param_names:
            args.remove(q0)
            q0 = np.inf
        elif not params['q0'].vary:
            args.remove(q0)
            q0 = params['q0'].value
        if 'v' not in self.param_names:
            args.remove(v)
            v = r
        elif not params['v'].vary:
            args.remove(v)
            v = params['v'].value
        if (isinstance(q0, float) and np.isinf(q0)) or (isinstance(v, float)
                                                        and np.isinf(v)):
            At = t
        else:
            At = t + 1.0 / v * sympy.log((sympy.exp(-v * t) + q0) / (1 + q0))
        dNdt = K / (1.0 - (1.0 -
                           (K / y0)**nu) * sympy.exp(-r * nu * At))**(1.0 / nu)
        return dNdt, t, tuple(args)
Esempio n. 40
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def test_solve_ics():
    # Basic tests that things work from dsolve.
    assert dsolve(f(x).diff(x) - 1/f(x), f(x), ics={f(1): 2}) == \
        Eq(f(x), sqrt(2 * x + 2))
    assert dsolve(f(x).diff(x) - f(x), f(x), ics={f(0): 1}) == Eq(f(x), exp(x))
    assert dsolve(f(x).diff(x) - f(x), f(x), ics={f(x).diff(x).subs(x, 0):
                                                  1}) == Eq(f(x), exp(x))
    assert dsolve(f(x).diff(x, x) + f(x),
                  f(x),
                  ics={
                      f(0): 1,
                      f(x).diff(x).subs(x, 0): 1
                  }) == Eq(f(x),
                           sin(x) + cos(x))
    assert dsolve([f(x).diff(x) - f(x) + g(x),
                   g(x).diff(x) - g(x) - f(x)], [f(x), g(x)],
                  ics={
                      f(0): 1,
                      g(0): 0
                  }) == [Eq(f(x),
                            exp(x) * cos(x)),
                         Eq(g(x),
                            exp(x) * sin(x))]

    # Test cases where dsolve returns two solutions.
    eq = (x**2 * f(x)**2 - x).diff(x)
    assert dsolve(eq, f(x), ics={f(1): 0}) == [
                                     Eq(f(x), -sqrt(x - 1) / x),
                                     Eq(f(x),
                                        sqrt(x - 1) / x)
                                 ]
    assert dsolve(eq, f(x), ics={f(x).diff(x).subs(x, 1): 0}) == [
                                     Eq(f(x), -sqrt(x - S.Half) / x),
                                     Eq(f(x),
                                        sqrt(x - S.Half) / x)
                                 ]

    eq = cos(f(x)) - (x * sin(f(x)) - f(x)**2) * f(x).diff(x)
    assert dsolve(eq, f(x), ics={f(0): 1}, hint='1st_exact',
                  simplify=False) == Eq(x * cos(f(x)) + f(x)**3 / 3,
                                        Rational(1, 3))
    assert dsolve(eq, f(x), ics={f(0): 1}, hint='1st_exact',
                  simplify=True) == Eq(x * cos(f(x)) + f(x)**3 / 3,
                                       Rational(1, 3))

    assert solve_ics([Eq(f(x), C1 * exp(x))], [f(x)], [C1], {f(0): 1}) == {
                                                                 C1: 1
                                                             }
    assert solve_ics([Eq(f(x),
                         C1 * sin(x) + C2 * cos(x))], [f(x)], [C1, C2], {
                             f(0): 1,
                             f(pi / 2): 1
                         }) == {
                             C1: 1,
                             C2: 1
                         }

    assert solve_ics([Eq(f(x),
                         C1 * sin(x) + C2 * cos(x))], [f(x)], [C1, C2], {
                             f(0): 1,
                             f(x).diff(x).subs(x, 0): 1
                         }) == {
                             C1: 1,
                             C2: 1
                         }

    assert solve_ics([Eq(f(x), C1*sin(x) + C2*cos(x))], [f(x)], [C1, C2], {f(0): 1}) == \
        {C2: 1}

    # Some more complicated tests Refer to PR #16098

    assert set(dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0, f(x).diff(x).subs(x, 1):0})) == \
        {Eq(f(x), 0), Eq(f(x), x ** 3 / 6 - x / 2)}
    assert set(dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0})) == \
        {Eq(f(x), 0), Eq(f(x), C2*x + x**3/6)}

    K, r, f0 = symbols('K r f0')
    sol = Eq(
        f(x),
        K * f0 * exp(r * x) / ((-K + f0) * (f0 * exp(r * x) / (-K + f0) - 1)))
    assert (dsolve(Eq(f(x).diff(x),
                      r * f(x) * (1 - f(x) / K)),
                   f(x),
                   ics={f(0): f0})) == sol

    #Order dependent issues Refer to PR #16098
    assert set(dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(x).diff(x).subs(x,0):0, f(0):0})) == \
        {Eq(f(x), 0), Eq(f(x), x ** 3 / 6)}
    assert set(dsolve(f(x).diff(x)*(f(x).diff(x, 2)-x), ics={f(0):0, f(x).diff(x).subs(x,0):0})) == \
        {Eq(f(x), 0), Eq(f(x), x ** 3 / 6)}

    # XXX: Ought to be ValueError
    raises(
        ValueError,
        lambda: solve_ics([Eq(f(x),
                              C1 * sin(x) + C2 * cos(x))], [f(x)], [C1, C2], {
                                  f(0): 1,
                                  f(pi): 1
                              }))

    # Degenerate case. f'(0) is identically 0.
    raises(
        ValueError, lambda: solve_ics([Eq(f(x), sqrt(C1 - x**2))], [f(x)],
                                      [C1], {f(x).diff(x).subs(x, 0): 0}))

    EI, q, L = symbols('EI q L')

    # eq = Eq(EI*diff(f(x), x, 4), q)
    sols = [
        Eq(f(x), C1 + C2 * x + C3 * x**2 + C4 * x**3 + q * x**4 / (24 * EI))
    ]
    funcs = [f(x)]
    constants = [C1, C2, C3, C4]
    # Test both cases, Derivative (the default from f(x).diff(x).subs(x, L)),
    # and Subs
    ics1 = {
        f(0): 0,
        f(x).diff(x).subs(x, 0): 0,
        f(L).diff(L, 2): 0,
        f(L).diff(L, 3): 0
    }
    ics2 = {
        f(0): 0,
        f(x).diff(x).subs(x, 0): 0,
        Subs(f(x).diff(x, 2), x, L): 0,
        Subs(f(x).diff(x, 3), x, L): 0
    }

    solved_constants1 = solve_ics(sols, funcs, constants, ics1)
    solved_constants2 = solve_ics(sols, funcs, constants, ics2)
    assert solved_constants1 == solved_constants2 == {
        C1: 0,
        C2: 0,
        C3: L**2 * q / (4 * EI),
        C4: -L * q / (6 * EI)
    }
Esempio n. 41
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def test_shiftedgompertz():
    b = Symbol("b", positive=True)
    eta = Symbol("eta", positive=True)
    X = ShiftedGompertz("x", b, eta)
    assert density(X)(x) == b * (eta * (1 - exp(-b * x)) + 1) * exp(
        -b * x) * exp(-eta * exp(-b * x))
Esempio n. 42
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def test_dsolve_options():
    eq = x * f(x).diff(x) + f(x)
    a = dsolve(eq, hint='all')
    b = dsolve(eq, hint='all', simplify=False)
    c = dsolve(eq, hint='all_Integral')
    keys = [
        '1st_exact', '1st_exact_Integral', '1st_homogeneous_coeff_best',
        '1st_homogeneous_coeff_subs_dep_div_indep',
        '1st_homogeneous_coeff_subs_dep_div_indep_Integral',
        '1st_homogeneous_coeff_subs_indep_div_dep',
        '1st_homogeneous_coeff_subs_indep_div_dep_Integral', '1st_linear',
        '1st_linear_Integral', 'Bernoulli', 'Bernoulli_Integral',
        'almost_linear', 'almost_linear_Integral', 'best', 'best_hint',
        'default', 'lie_group', 'nth_linear_euler_eq_homogeneous', 'order',
        'separable', 'separable_Integral'
    ]
    Integral_keys = [
        '1st_exact_Integral',
        '1st_homogeneous_coeff_subs_dep_div_indep_Integral',
        '1st_homogeneous_coeff_subs_indep_div_dep_Integral',
        '1st_linear_Integral', 'Bernoulli_Integral', 'almost_linear_Integral',
        'best', 'best_hint', 'default', 'nth_linear_euler_eq_homogeneous',
        'order', 'separable_Integral'
    ]
    assert sorted(a.keys()) == keys
    assert a['order'] == ode_order(eq, f(x))
    assert a['best'] == Eq(f(x), C1 / x)
    assert dsolve(eq, hint='best') == Eq(f(x), C1 / x)
    assert a['default'] == 'separable'
    assert a['best_hint'] == 'separable'
    assert not a['1st_exact'].has(Integral)
    assert not a['separable'].has(Integral)
    assert not a['1st_homogeneous_coeff_best'].has(Integral)
    assert not a['1st_homogeneous_coeff_subs_dep_div_indep'].has(Integral)
    assert not a['1st_homogeneous_coeff_subs_indep_div_dep'].has(Integral)
    assert not a['1st_linear'].has(Integral)
    assert a['1st_linear_Integral'].has(Integral)
    assert a['1st_exact_Integral'].has(Integral)
    assert a['1st_homogeneous_coeff_subs_dep_div_indep_Integral'].has(Integral)
    assert a['1st_homogeneous_coeff_subs_indep_div_dep_Integral'].has(Integral)
    assert a['separable_Integral'].has(Integral)
    assert sorted(b.keys()) == keys
    assert b['order'] == ode_order(eq, f(x))
    assert b['best'] == Eq(f(x), C1 / x)
    assert dsolve(eq, hint='best', simplify=False) == Eq(f(x), C1 / x)
    assert b['default'] == 'separable'
    assert b['best_hint'] == '1st_linear'
    assert a['separable'] != b['separable']
    assert a['1st_homogeneous_coeff_subs_dep_div_indep'] != \
        b['1st_homogeneous_coeff_subs_dep_div_indep']
    assert a['1st_homogeneous_coeff_subs_indep_div_dep'] != \
        b['1st_homogeneous_coeff_subs_indep_div_dep']
    assert not b['1st_exact'].has(Integral)
    assert not b['separable'].has(Integral)
    assert not b['1st_homogeneous_coeff_best'].has(Integral)
    assert not b['1st_homogeneous_coeff_subs_dep_div_indep'].has(Integral)
    assert not b['1st_homogeneous_coeff_subs_indep_div_dep'].has(Integral)
    assert not b['1st_linear'].has(Integral)
    assert b['1st_linear_Integral'].has(Integral)
    assert b['1st_exact_Integral'].has(Integral)
    assert b['1st_homogeneous_coeff_subs_dep_div_indep_Integral'].has(Integral)
    assert b['1st_homogeneous_coeff_subs_indep_div_dep_Integral'].has(Integral)
    assert b['separable_Integral'].has(Integral)
    assert sorted(c.keys()) == Integral_keys
    raises(ValueError, lambda: dsolve(eq, hint='notarealhint'))
    raises(ValueError, lambda: dsolve(eq, hint='Liouville'))
    assert dsolve(f(x).diff(x) - 1/f(x)**2, hint='all')['best'] == \
        dsolve(f(x).diff(x) - 1/f(x)**2, hint='best')
    assert dsolve(f(x) + f(x).diff(x) + sin(x).diff(x) + 1, f(x),
                  hint="1st_linear_Integral") == \
        Eq(f(x), (C1 + Integral((-sin(x).diff(x) - 1)*
                exp(Integral(1, x)), x))*exp(-Integral(1, x)))
Esempio n. 43
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from numpy import inf
from numpy import exp

from scipy.io import savemat
#from mpmath import lambertw
from scipy.special import lambertw
from scipy import optimize
from scipy import interpolate

import sympy
I0, Iph, Rs, Rsh, n, I, V, Vth, V_I0, I_I0, V_n, I_n = sympy.symbols(
    'I0 Iph Rs Rsh n I V Vth V_I0 I_I0 V_n I_n')
#this stuff is from http://dx.doi.org/10.1016/j.solmat.2003.11.018
#symbolic representation for solar cell equation:
lhs = I
rhs = Iph - ((V + I * Rs) / Rsh) - I0 * (sympy.exp(
    (V + I * Rs) / (n * Vth)) - 1)
charEqn = sympy.Eq(lhs, rhs)

#isolate current term in solar cell equation
#current = sympy.solve(charEqn,I)

#isolate voltage term in solar cell equation
#voltage = sympy.solve(charEqn,V)

#isolate I0  in solar cell equation
eyeNot = sympy.solve(charEqn, I0)

#import matplotlib.pyplot as plt
#plt.switch_backend("Qt5Agg")

# the data to be fit
Esempio n. 44
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def test_von_mises():
    mu = Symbol("mu")
    k = Symbol("k", positive=True)

    X = VonMises("x", mu, k)
    assert density(X)(x) == exp(k * cos(x - mu)) / (2 * pi * besseli(0, k))
Esempio n. 45
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def test_cse_ignore_issue_15002():
    l = [w * exp(x) * exp(-z), exp(y) * exp(x) * exp(-z)]
    substs, reduced = cse(l, ignore=(x, ))
    rl = [e.subs(reversed(substs)) for e in reduced]
    assert rl == l
Esempio n. 46
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def test_moment_generating_function():
    t = symbols('t', positive=True)

    # Symbolic tests
    a, b, c = symbols('a b c')

    mgf = moment_generating_function(Beta('x', a, b))(t)
    assert mgf == hyper((a, ), (a + b, ), t)

    mgf = moment_generating_function(Chi('x', a))(t)
    assert mgf == sqrt(2)*t*gamma(a/2 + S(1)/2)*\
        hyper((a/2 + S(1)/2,), (S(3)/2,), t**2/2)/gamma(a/2) +\
        hyper((a/2,), (S(1)/2,), t**2/2)

    mgf = moment_generating_function(ChiSquared('x', a))(t)
    assert mgf == (1 - 2 * t)**(-a / 2)

    mgf = moment_generating_function(Erlang('x', a, b))(t)
    assert mgf == (1 - t / b)**(-a)

    mgf = moment_generating_function(Exponential('x', a))(t)
    assert mgf == a / (a - t)

    mgf = moment_generating_function(Gamma('x', a, b))(t)
    assert mgf == (-b * t + 1)**(-a)

    mgf = moment_generating_function(Gumbel('x', a, b))(t)
    assert mgf == exp(b * t) * gamma(-a * t + 1)

    mgf = moment_generating_function(Gompertz('x', a, b))(t)
    assert mgf == b * exp(b) * expint(t / a, b)

    mgf = moment_generating_function(Laplace('x', a, b))(t)
    assert mgf == exp(a * t) / (-b**2 * t**2 + 1)

    mgf = moment_generating_function(Logistic('x', a, b))(t)
    assert mgf == exp(a * t) * beta(-b * t + 1, b * t + 1)

    mgf = moment_generating_function(Normal('x', a, b))(t)
    assert mgf == exp(a * t + b**2 * t**2 / 2)

    mgf = moment_generating_function(Pareto('x', a, b))(t)
    assert mgf == b * (-a * t)**b * uppergamma(-b, -a * t)

    mgf = moment_generating_function(QuadraticU('x', a, b))(t)
    assert str(mgf) == (
        "(3*(t*(-4*b + (a + b)**2) + 4)*exp(b*t) - "
        "3*(t*(a**2 + 2*a*(b - 2) + b**2) + 4)*exp(a*t))/(t**2*(a - b)**3)")

    mgf = moment_generating_function(RaisedCosine('x', a, b))(t)
    assert mgf == pi**2 * exp(a * t) * sinh(b * t) / (b * t *
                                                      (b**2 * t**2 + pi**2))

    mgf = moment_generating_function(Rayleigh('x', a))(t)
    assert mgf == sqrt(2)*sqrt(pi)*a*t*(erf(sqrt(2)*a*t/2) + 1)\
        *exp(a**2*t**2/2)/2 + 1

    mgf = moment_generating_function(Triangular('x', a, b, c))(t)
    assert str(mgf) == ("(-2*(-a + b)*exp(c*t) + 2*(-a + c)*exp(b*t) + "
                        "2*(b - c)*exp(a*t))/(t**2*(-a + b)*(-a + c)*(b - c))")

    mgf = moment_generating_function(Uniform('x', a, b))(t)
    assert mgf == (-exp(a * t) + exp(b * t)) / (t * (-a + b))

    mgf = moment_generating_function(UniformSum('x', a))(t)
    assert mgf == ((exp(t) - 1) / t)**a

    mgf = moment_generating_function(WignerSemicircle('x', a))(t)
    assert mgf == 2 * besseli(1, a * t) / (a * t)

    # Numeric tests

    mgf = moment_generating_function(Beta('x', 1, 1))(t)
    assert mgf.diff(t).subs(t, 1) == hyper((2, ), (3, ), 1) / 2

    mgf = moment_generating_function(Chi('x', 1))(t)
    assert mgf.diff(t).subs(t, 1) == sqrt(2) * hyper(
        (1, ), (S(3) / 2, ),
        S(1) / 2) / sqrt(pi) + hyper((S(3) / 2, ), (S(3) / 2, ),
                                     S(1) / 2) + 2 * sqrt(2) * hyper(
                                         (2, ), (S(5) / 2, ),
                                         S(1) / 2) / (3 * sqrt(pi))

    mgf = moment_generating_function(ChiSquared('x', 1))(t)
    assert mgf.diff(t).subs(t, 1) == I

    mgf = moment_generating_function(Erlang('x', 1, 1))(t)
    assert mgf.diff(t).subs(t, 0) == 1

    mgf = moment_generating_function(Exponential('x', 1))(t)
    assert mgf.diff(t).subs(t, 0) == 1

    mgf = moment_generating_function(Gamma('x', 1, 1))(t)
    assert mgf.diff(t).subs(t, 0) == 1

    mgf = moment_generating_function(Gumbel('x', 1, 1))(t)
    assert mgf.diff(t).subs(t, 0) == EulerGamma + 1

    mgf = moment_generating_function(Gompertz('x', 1, 1))(t)
    assert mgf.diff(t).subs(t, 1) == -e * meijerg(((), (1, 1)),
                                                  ((0, 0, 0), ()), 1)

    mgf = moment_generating_function(Laplace('x', 1, 1))(t)
    assert mgf.diff(t).subs(t, 0) == 1

    mgf = moment_generating_function(Logistic('x', 1, 1))(t)
    assert mgf.diff(t).subs(t, 0) == beta(1, 1)

    mgf = moment_generating_function(Normal('x', 0, 1))(t)
    assert mgf.diff(t).subs(t, 1) == exp(S(1) / 2)

    mgf = moment_generating_function(Pareto('x', 1, 1))(t)
    assert mgf.diff(t).subs(t, 0) == expint(1, 0)

    mgf = moment_generating_function(QuadraticU('x', 1, 2))(t)
    assert mgf.diff(t).subs(t, 1) == -12 * e - 3 * exp(2)

    mgf = moment_generating_function(RaisedCosine('x', 1, 1))(t)
    assert mgf.diff(t).subs(t, 1) == -2*e*pi**2*sinh(1)/\
    (1 + pi**2)**2 + e*pi**2*cosh(1)/(1 + pi**2)

    mgf = moment_generating_function(Rayleigh('x', 1))(t)
    assert mgf.diff(t).subs(t, 0) == sqrt(2) * sqrt(pi) / 2

    mgf = moment_generating_function(Triangular('x', 1, 3, 2))(t)
    assert mgf.diff(t).subs(t, 1) == -e + exp(3)

    mgf = moment_generating_function(Uniform('x', 0, 1))(t)
    assert mgf.diff(t).subs(t, 1) == 1

    mgf = moment_generating_function(UniformSum('x', 1))(t)
    assert mgf.diff(t).subs(t, 1) == 1

    mgf = moment_generating_function(WignerSemicircle('x', 1))(t)
    assert mgf.diff(t).subs(t, 1) == -2*besseli(1, 1) + besseli(2, 1) +\
        besseli(0, 1)
Esempio n. 47
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def test_postprocess():
    eq = (x + 1 + exp((x + 1) / (y + 1)) + cos(y + 1))
    assert cse([eq, Eq(x, z + 1), z - 2, (z + 1)*(x + 1)],
        postprocess=cse_main.cse_separate) == \
        [[(x0, y + 1), (x2, z + 1), (x, x2), (x1, x + 1)],
        [x1 + exp(x1/x0) + cos(x0), z - 2, x1*x2]]
Esempio n. 48
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def test_tsolve_1():
    a, b = symbols('ab')
    x, y, z = symbols('xyz')
    assert solve(exp(x) - 3, x) == [log(3)]
    assert solve((a * x + b) * (exp(x) - 3), x) == [-b / a, log(3)]
    assert solve(cos(x) - y, x) == [acos(y)]
    assert solve(2 * cos(x) - y, x) == [acos(y / 2)]
    raises(NotImplementedError, "solve(Eq(cos(x), sin(x)), x)")

    # XXX in the following test, log(2*y + 2*...) should -> log(2) + log(y +...)
    assert solve(exp(x) + exp(-x) - y, x) == [
        -log(4) + log(2 * y + 2 * (-4 + y**2)**Rational(1, 2)),
        -log(4) + log(2 * y - 2 * (-4 + y**2)**Rational(1, 2))
    ]
    assert solve(exp(x) - 3, x) == [log(3)]
    assert solve(Eq(exp(x), 3), x) == [log(3)]
    assert solve(log(x) - 3, x) == [exp(3)]
    assert solve(sqrt(3 * x) - 4, x) == [Rational(16, 3)]
    assert solve(3**(x + 2), x) == [-oo]
    assert solve(3**(2 - x), x) == [oo]
    assert solve(4 * 3**(5 * x + 2) - 7,
                 x) == [(-log(4) - 2 * log(3) + log(7)) / (5 * log(3))]
    assert solve(x + 2**x, x) == [-LambertW(log(2)) / log(2)]
    assert solve(3*x+5+2**(-5*x+3), x) in \
        [[-Rational(5,3) + LambertW(-10240*2**Rational(1,3)*log(2)/3)/(5*log(2))],\
        [(-25*log(2) + 3*LambertW(-10240*2**(Rational(1, 3))*log(2)/3))/(15*log(2))]]
    assert solve(5*x-1+3*exp(2-7*x), x) == \
        [Rational(1,5) + LambertW(-21*exp(Rational(3,5))/5)/7]
    assert solve(2*x+5+log(3*x-2), x) == \
        [Rational(2,3) + LambertW(2*exp(-Rational(19,3))/3)/2]
    assert solve(3 * x + log(4 * x), x) == [LambertW(Rational(3, 4)) / 3]
    assert solve((2 * x + 8) * (8 + exp(x)), x) == [-4, log(8) + pi * I]
    assert solve(2*exp(3*x+4)-3, x) in [ [-Rational(4,3)+log(Rational(3,2))/3],\
                                         [Rational(-4, 3) - log(2)/3 + log(3)/3]]
    assert solve(2 * log(3 * x + 4) - 3, x) == [(exp(Rational(3, 2)) - 4) / 3]
    assert solve(exp(x) + 1, x) == [pi * I]
    assert solve(x**2 - 2**x, x) == [2]
    assert solve(x**3 - 3**x, x) == [-3 / log(3) * LambertW(-log(3) / 3)]
    assert solve(2*(3*x+4)**5 - 6*7**(3*x+9), x) in \
        [[Rational(-4,3) - 5/log(7)/3*LambertW(-7*2**Rational(4,5)*6**Rational(1,5)*log(7)/10)],\
         [(-5*LambertW(-7*2**(Rational(4, 5))*6**(Rational(1, 5))*log(7)/10) - 4*log(7))/(3*log(7))], \
         [-((4*log(7) + 5*LambertW(-7*2**Rational(4,5)*6**Rational(1,5)*log(7)/10))/(3*log(7)))]]

    assert solve(z * cos(x) - y, x) == [acos(y / z)]
    assert solve(z * cos(2 * x) - y, x) == [acos(y / z) / 2]
    assert solve(z * cos(sin(x)) - y, x) == [asin(acos(y / z))]

    assert solve(z * cos(x), x) == [acos(0)]

    assert solve(exp(x) + exp(-x) - y, x) == [
        -log(4) + log(2 * y + 2 * (-4 + y**2)**(Rational(1, 2))),
        -log(4) + log(2 * y - 2 * (-4 + y**2)**(Rational(1, 2)))
    ]
    # issue #1409
    assert solve(y - b * x / (a + x), x) == [a * y / (b - y)]
    assert solve(y - b * exp(a / x), x) == [a / (-log(b) + log(y))]
    # issue #1408
    assert solve(y - b / (1 + a * x), x) == [(b - y) / (a * y)]
    # issue #1407
    assert solve(y - a * x**b, x) == [y**(1 / b) * (1 / a)**(1 / b)]
    # issue #1406
    assert solve(z**x - y, x) == [log(y) / log(z)]
    # issue #1405
    assert solve(2**x - 10, x) == [log(10) / log(2)]
Esempio n. 49
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def test_sample_continuous():
    Z = ContinuousRV(z, exp(-z), set=Interval(0, oo))
    assert sample(Z) in Z.pspace.domain.set
    sym, val = list(Z.pspace.sample().items())[0]
    assert sym == Z and val in Interval(0, oo)
    assert density(Z)(-1) == 0
Esempio n. 50
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def test_ignore_order_terms():
    eq = exp(x).series(x, 0, 3) + sin(y + x**3) - 1
    assert cse(eq) == ([], [sin(x**3 + y) + x + x**2 / 2 + O(x**3)])
Esempio n. 51
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def test_piecewise():

    # Test canonicalization
    assert Piecewise((x, x < 1), (0, True)) == Piecewise((x, x < 1), (0, True))
    assert Piecewise((x, x < 1), (0, True), (1, True)) == \
        Piecewise((x, x < 1), (0, True))
    assert Piecewise((x, x < 1), (0, False), (-1, 1 > 2)) == \
        Piecewise((x, x < 1))
    assert Piecewise((x, x < 1), (0, x < 1), (0, True)) == \
        Piecewise((x, x < 1), (0, True))
    assert Piecewise((x, x < 1), (0, x < 2), (0, True)) == \
        Piecewise((x, x < 1), (0, True))
    assert Piecewise((x, x < 1), (x, x < 2), (0, True)) == \
        Piecewise((x, Or(x < 1, x < 2)), (0, True))
    assert Piecewise((x, x < 1), (x, x < 2), (x, True)) == x
    assert Piecewise((x, True)) == x
    # False condition is never retained
    assert Piecewise((x, False)) == Piecewise(
        (x, False), evaluate=False) == Piecewise()
    raises(TypeError, lambda: Piecewise(x))
    assert Piecewise((x, 1)) == x  # 1 and 0 are accepted as True/False
    raises(TypeError, lambda: Piecewise((x, 2)))
    raises(TypeError, lambda: Piecewise((x, x**2)))
    raises(TypeError, lambda: Piecewise(([1], True)))
    assert Piecewise(((1, 2), True)) == Tuple(1, 2)
    cond = (Piecewise((1, x < 0), (2, True)) < y)
    assert Piecewise((1, cond)) == Piecewise((1, ITE(x < 0, y > 1, y > 2)))

    assert Piecewise((1, x > 0), (2, And(x <= 0, x > -1))) == Piecewise(
        (1, x > 0), (2, x > -1))

    # Test subs
    p = Piecewise((-1, x < -1), (x**2, x < 0), (log(x), x >= 0))
    p_x2 = Piecewise((-1, x**2 < -1), (x**4, x**2 < 0), (log(x**2), x**2 >= 0))
    assert p.subs(x, x**2) == p_x2
    assert p.subs(x, -5) == -1
    assert p.subs(x, -1) == 1
    assert p.subs(x, 1) == log(1)

    # More subs tests
    p2 = Piecewise((1, x < pi), (-1, x < 2 * pi), (0, x > 2 * pi))
    p3 = Piecewise((1, Eq(x, 0)), (1 / x, True))
    p4 = Piecewise((1, Eq(x, 0)), (2, 1 / x > 2))
    assert p2.subs(x, 2) == 1
    assert p2.subs(x, 4) == -1
    assert p2.subs(x, 10) == 0
    assert p3.subs(x, 0.0) == 1
    assert p4.subs(x, 0.0) == 1

    f, g, h = symbols('f,g,h', cls=Function)
    pf = Piecewise((f(x), x < -1), (f(x) + h(x) + 2, x <= 1))
    pg = Piecewise((g(x), x < -1), (g(x) + h(x) + 2, x <= 1))
    assert pg.subs(g, f) == pf

    assert Piecewise((1, Eq(x, 0)), (0, True)).subs(x, 0) == 1
    assert Piecewise((1, Eq(x, 0)), (0, True)).subs(x, 1) == 0
    assert Piecewise((1, Eq(x, y)), (0, True)).subs(x, y) == 1
    assert Piecewise((1, Eq(x, z)), (0, True)).subs(x, z) == 1
    assert Piecewise((1, Eq(exp(x), cos(z))), (0, True)).subs(x, z) == \
        Piecewise((1, Eq(exp(z), cos(z))), (0, True))

    p5 = Piecewise((0, Eq(cos(x) + y, 0)), (1, True))
    assert p5.subs(y, 0) == Piecewise((0, Eq(cos(x), 0)), (1, True))

    assert Piecewise((-1, y < 1), (0, x < 0), (1, Eq(x, 0)),
                     (2, True)).subs(x, 1) == Piecewise((-1, y < 1), (2, True))
    assert Piecewise((1, Eq(x**2, -1)), (2, x < 0)).subs(x, I) == 1

    # Test evalf
    assert p.evalf() == p
    assert p.evalf(subs={x: -2}) == -1
    assert p.evalf(subs={x: -1}) == 1
    assert p.evalf(subs={x: 1}) == log(1)

    # Test doit
    f_int = Piecewise((Integral(x, (x, 0, 1)), x < 1))
    assert f_int.doit() == Piecewise((1 / 2, x < 1))

    # Test differentiation
    f = x
    fp = x * p
    dp = Piecewise((0, x < -1), (2 * x, x < 0), (1 / x, x >= 0))
    fp_dx = x * dp + p
    assert diff(p, x) == dp
    assert diff(f * p, x) == fp_dx

    # Test simple arithmetic
    assert x * p == fp
    assert x * p + p == p + x * p
    assert p + f == f + p
    assert p + dp == dp + p
    assert p - dp == -(dp - p)

    # Test power
    dp2 = Piecewise((0, x < -1), (4 * x**2, x < 0), (1 / x**2, x >= 0))
    assert dp**2 == dp2

    # Test _eval_interval
    f1 = x * y + 2
    f2 = x * y**2 + 3
    peval = Piecewise((f1, x < 0), (f2, x > 0))
    peval_interval = f1.subs(x, 0) - f1.subs(x, -1) + f2.subs(x, 1) - f2.subs(
        x, 0)
    assert peval._eval_interval(x, 0, 0) == 0
    assert peval._eval_interval(x, -1, 1) == peval_interval
    peval2 = Piecewise((f1, x < 0), (f2, True))
    assert peval2._eval_interval(x, 0, 0) == 0
    assert peval2._eval_interval(x, 1, -1) == -peval_interval
    assert peval2._eval_interval(x, -1, -2) == f1.subs(x, -2) - f1.subs(x, -1)
    assert peval2._eval_interval(x, -1, 1) == peval_interval
    assert peval2._eval_interval(x, None, 0) == peval2.subs(x, 0)
    assert peval2._eval_interval(x, -1, None) == -peval2.subs(x, -1)

    # Test integration
    assert p.integrate() == Piecewise((-x, x < -1), (x**3 / 3 + 4 / 3, x < 0),
                                      (x * log(x) - x + 4 / 3, True))
    p = Piecewise((x, x < 1), (x**2, -1 <= x), (x, 3 < x))
    assert integrate(p, (x, -2, 2)) == 5 / 6.0
    assert integrate(p, (x, 2, -2)) == -5 / 6.0
    p = Piecewise((0, x < 0), (1, x < 1), (0, x < 2), (1, x < 3), (0, True))
    assert integrate(p, (x, -oo, oo)) == 2
    p = Piecewise((x, x < -10), (x**2, x <= -1), (x, 1 < x))
    assert integrate(p, (x, -2, 2)) == Undefined

    # Test commutativity
    assert isinstance(p, Piecewise) and p.is_commutative is True
Esempio n. 52
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def test_gumbel():
    beta = Symbol("beta", positive=True)
    mu = Symbol("mu")
    x = Symbol("x")
    y = Symbol("y")
    X = Gumbel("x", beta, mu)
    Y = Gumbel("y", beta, mu, minimum=True)
    assert density(X)(x).expand() == \
    exp(mu/beta)*exp(-x/beta)*exp(-exp(mu/beta)*exp(-x/beta))/beta
    assert density(Y)(y).expand() == \
    exp(-mu/beta)*exp(y/beta)*exp(-exp(-mu/beta)*exp(y/beta))/beta
    assert cdf(X)(x).expand() == \
    exp(-exp(mu/beta)*exp(-x/beta))
Esempio n. 53
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def test_issue_18482():
    assert limit((2 * exp(3 * x) / (exp(2 * x) + 1))**(1 / x), x, oo) == exp(1)
Esempio n. 54
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def s(x, y):
    return sp.exp(x)
Esempio n. 55
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def test_issue_17792():
    assert limit(factorial(n) / sqrt(n) * (exp(1) / n)**n, n,
                 oo) == sqrt(2) * sqrt(pi)
Esempio n. 56
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def test_issue_18992():
    assert limit(n / (factorial(n)**(1 / n)), n, oo) == exp(1)
Esempio n. 57
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def test_issue_16222():
    assert limit(exp(x), x, 1000000000) == exp(1000000000)
Esempio n. 58
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def test_issue_18378():
    assert limit(log(exp(3 * x) + x) / log(exp(x) + x**100), x, oo) == 3
Esempio n. 59
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def test_issue_14556():
    assert limit(
        factorial(n + 1)**(1 / (n + 1)) - factorial(n)**(1 / n), n,
        oo) == exp(-1)
Esempio n. 60
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def test_issue_16714():
    assert limit(((x**(x + 1) + (x + 1)**x) / x**(x + 1))**x, x,
                 oo) == exp(exp(1))