def test_multiplication_initial_condition():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx**2 + x * Dx - 1, x, 0, [3, 1])
q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1])
r = HolonomicFunction((x**4 + 14*x**2 + 60) + 4*x*Dx + (x**4 + 9*x**2 + 20)*Dx**2 + \
(2*x**3 + 18*x)*Dx**3 + (x**2 + 10)*Dx**4, x, 0, [3, 4, 2, 3])
assert p * q == r
p = HolonomicFunction(Dx**2 + x, x, 0, [1, 0])
q = HolonomicFunction(Dx**3 - x**2, x, 0, [3, 3, 3])
r = HolonomicFunction((27*x**8 - 37*x**7 - 10*x**6 - 492*x**5 - 552*x**4 + 160*x**3 + \
1212*x**2 + 216*x + 360) + (162*x**7 - 384*x**6 - 294*x**5 - 84*x**4 + 24*x**3 + \
756*x**2 + 120*x - 1080)*Dx + (81*x**6 - 246*x**5 + 228*x**4 + 36*x**3 + \
660*x**2 - 720*x)*Dx**2 + (-54*x**6 + 128*x**5 - 18*x**4 - 240*x**2 + 600)*Dx**3 + \
(81*x**5 - 192*x**4 - 84*x**3 + 162*x**2 - 60*x - 180)*Dx**4 + (-108*x**3 + \
192*x**2 + 72*x)*Dx**5 + (27*x**4 - 64*x**3 - 36*x**2 + 60)*Dx**6, x, 0, [3, 3, 3, -3, -12, -24])
assert p * q == r
p = HolonomicFunction(Dx - 1, x, 0, [2])
q = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1])
r = HolonomicFunction(2 - 2 * Dx + Dx**2, x, 0, [0, 2])
assert p * q == r
q = HolonomicFunction(x * Dx**2 + 1 + 2 * Dx, x, 0, [0, 1])
r = HolonomicFunction((x - 1) + (-2 * x + 2) * Dx + x * Dx**2, x, 0,
[0, 2])
assert p * q == r
p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 3])
q = HolonomicFunction(Dx**3 + 1, x, 0, [1, 2, 1])
r = HolonomicFunction(6 * Dx + 3 * Dx**2 + 2 * Dx**3 - 3 * Dx**4 + Dx**6,
x, 0, [1, 5, 14, 17, 17, 2])
assert p * q == r
def test_series():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx**2 + 2*x*Dx, x, 0, [0, 1]).series(n=10)
q = x - x**3/3 + x**5/10 - x**7/42 + x**9/216 + O(x**10)
assert p == q
p = HolonomicFunction(Dx - 1, x).composition(x**2, 0, [1]) # e^(x**2)
q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]) # cos(x)
r = (p * q).series(n=10) # expansion of cos(x) * exp(x**2)
s = 1 + x**2/2 + x**4/24 - 31*x**6/720 - 179*x**8/8064 + O(x**10)
assert r == s
t = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1]) # log(1 + x)
r = (p * t + q).series(n=10)
s = 1 + x - x**2 + 4*x**3/3 - 17*x**4/24 + 31*x**5/30 - 481*x**6/720 +\
71*x**7/105 - 20159*x**8/40320 + 379*x**9/840 + O(x**10)
assert r == s
p = HolonomicFunction((6+6*x-3*x**2) - (10*x-3*x**2-3*x**3)*Dx + \
(4-6*x**3+2*x**4)*Dx**2, x, 0, [0, 1]).series(n=7)
q = x + x**3/6 - 3*x**4/16 + x**5/20 - 23*x**6/960 + O(x**7)
assert p == q
p = HolonomicFunction((6+6*x-3*x**2) - (10*x-3*x**2-3*x**3)*Dx + \
(4-6*x**3+2*x**4)*Dx**2, x, 0, [1, 0]).series(n=7)
q = 1 - 3*x**2/4 - x**3/4 - 5*x**4/32 - 3*x**5/40 - 17*x**6/384 + O(x**7)
assert p == q
p = expr_to_holonomic(erf(x) + x).series(n=10)
C_3 = symbols('C_3')
q = (erf(x) + x).series(n=10)
assert p.subs(C_3, -2/(3*sqrt(pi))) == q
assert expr_to_holonomic(sqrt(x**3 + x)).series(n=10) == sqrt(x**3 + x).series(n=10)
assert expr_to_holonomic((2*x - 3*x**2)**(S(1)/3)).series() == ((2*x - 3*x**2)**(S(1)/3)).series()
assert expr_to_holonomic(sqrt(x**2-x)).series() == (sqrt(x**2-x)).series()
assert expr_to_holonomic(cos(x)**2/x**2, y0={-2: [1, 0, -1]}).series(n=10) == (cos(x)**2/x**2).series(n=10)
assert expr_to_holonomic(cos(x)**2/x**2, x0=1).series(n=10) == (cos(x)**2/x**2).series(n=10, x0=1)
assert expr_to_holonomic(cos(x-1)**2/(x-1)**2, x0=1, y0={-2: [1, 0, -1]}).series(n=10) \
== (cos(x-1)**2/(x-1)**2).series(x0=1, n=10)
def test_addition_initial_condition():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx - 1, x, 0, [3])
q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0])
r = HolonomicFunction(-1 + Dx - Dx**2 + Dx**3, x, 0, [4, 3, 2])
assert p + q == r
p = HolonomicFunction(Dx - x + Dx**2, x, 0, [1, 2])
q = HolonomicFunction(Dx**2 + x, x, 0, [1, 0])
r = HolonomicFunction((-x**4 - x**3/4 - x**2 + 1/4) + (x**3 + x**2/4 + 3*x/4 + 1)*Dx + \
(-3*x/2 + 7/4)*Dx**2 + (x**2 - 7*x/4 + 1/4)*Dx**3 + (x**2 + x/4 + 1/2)*Dx**4, x, 0, [2, 2, -2, 2])
assert p + q == r
p = HolonomicFunction(Dx**2 + 4 * x * Dx + x**2, x, 0, [3, 4])
q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1])
r = HolonomicFunction((x**6 + 2*x**4 - 5*x**2 - 6) + (4*x**5 + 36*x**3 - 32*x)*Dx + \
(x**6 + 3*x**4 + 5*x**2 - 9)*Dx**2 + (4*x**5 + 36*x**3 - 32*x)*Dx**3 + (x**4 + \
10*x**2 - 3)*Dx**4, x, 0, [4, 5, -1, -17])
assert p + q == r
q = HolonomicFunction(Dx**3 + x, x, 2, [3, 0, 1])
p = HolonomicFunction(Dx - 1, x, 2, [1])
r = HolonomicFunction((-x**2 - x + 1) + (x**2 + x)*Dx + (-x - 2)*Dx**3 + \
(x + 1)*Dx**4, x, 2, [4, 1, 2, -5 ])
assert p + q == r
p = expr_to_holonomic(sin(x))
q = expr_to_holonomic(1 / x, x0=1)
r = HolonomicFunction((x**2 + 6) + (x**3 + 2*x)*Dx + (x**2 + 6)*Dx**2 + (x**3 + 2*x)*Dx**3, \
x, 1, [sin(1) + 1, -1 + cos(1), -sin(1) + 2])
assert p + q == r
C_1 = symbols('C_1')
p = expr_to_holonomic(sqrt(x))
q = expr_to_holonomic(sqrt(x**2 - x))
r = (p + q).to_expr().subs(C_1, -I / 2).expand()
assert r == I * sqrt(x) * sqrt(-x + 1) + sqrt(x)
def test_to_Sequence():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
n = symbols('n', integer=True)
_, Sn = RecurrenceOperators(ZZ.old_poly_ring(n), 'Sn')
p = HolonomicFunction(x**2 * Dx**4 + x + Dx, x).to_sequence()
q = [(HolonomicSequence(1 + (n + 2) * Sn**2 +
(n**4 + 6 * n**3 + 11 * n**2 + 6 * n) * Sn**3), 0,
1)]
assert p == q
p = HolonomicFunction(x**2 * Dx**4 + x**3 + Dx**2, x).to_sequence()
q = [
(HolonomicSequence(1 + (n**4 + 14 * n**3 + 72 * n**2 + 163 * n + 140) *
Sn**5), 0, 0)
]
assert p == q
p = HolonomicFunction(x**3 * Dx**4 + 1 + Dx**2, x).to_sequence()
q = [(HolonomicSequence(1 + (n**4 - 2 * n**3 - n**2 + 2 * n) * Sn +
(n**2 + 3 * n + 2) * Sn**2), 0, 0)]
assert p == q
p = HolonomicFunction(3 * x**3 * Dx**4 + 2 * x * Dx + x * Dx**3,
x).to_sequence()
q = [(HolonomicSequence(2 * n +
(3 * n**4 - 6 * n**3 - 3 * n**2 + 6 * n) * Sn +
(n**3 + 3 * n**2 + 2 * n) * Sn**2), 0, 1)]
assert p == q
def test_to_expr():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx - 1, x, 0, [1]).to_expr()
q = exp(x)
assert p == q
p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).to_expr()
q = cos(x)
assert p == q
p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]).to_expr()
q = cosh(x)
assert p == q
p = HolonomicFunction(2 + (4*x - 1)*Dx + \
(x**2 - x)*Dx**2, x, 0, [1, 2]).to_expr().expand()
q = 1 / (x**2 - 2 * x + 1)
assert p == q
p = expr_to_holonomic(sin(x)**2 / x).integrate((x, 0, x)).to_expr()
q = (sin(x)**2 / x).integrate((x, 0, x))
assert p == q
C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3')
p = expr_to_holonomic(log(1 + x**2)).to_expr()
q = C_2 * log(x**2 + 1)
assert p == q
p = expr_to_holonomic(log(1 + x**2)).diff().to_expr()
q = C_0 * x / (x**2 + 1)
assert p == q
p = expr_to_holonomic(erf(x) + x).to_expr()
q = 3 * C_3 * x - 3 * sqrt(pi) * C_3 * erf(x) / 2 + x + 2 * x / sqrt(pi)
assert p == q
p = expr_to_holonomic(sqrt(x), x0=1).to_expr()
assert p == sqrt(x)
assert expr_to_holonomic(sqrt(x)).to_expr() == sqrt(x)
p = expr_to_holonomic(sqrt(1 + x**2)).to_expr()
assert p == sqrt(1 + x**2)
p = expr_to_holonomic((2 * x**2 + 1)**(S(2) / 3)).to_expr()
assert p == (2 * x**2 + 1)**(S(2) / 3)
p = expr_to_holonomic(sqrt(-x**2 + 2 * x)).to_expr()
assert p == sqrt(x) * sqrt(-x + 2)
p = expr_to_holonomic((-2 * x**3 + 7 * x)**(S(2) / 3)).to_expr()
q = x**(S(2) / 3) * (-2 * x**2 + 7)**(S(2) / 3)
assert p == q
p = from_hyper(hyper((-2, -3), (S(1) / 2, ), x))
s = hyperexpand(hyper((-2, -3), (S(1) / 2, ), x))
D_0 = Symbol('D_0')
C_0 = Symbol('C_0')
assert (p.to_expr().subs({C_0: 1, D_0: 0}) - s).simplify() == 0
p.y0 = {0: [1], S(1) / 2: [0]}
assert p.to_expr() == s
assert expr_to_holonomic(x**5).to_expr() == x**5
assert expr_to_holonomic(2*x**3-3*x**2).to_expr().expand() == \
2*x**3-3*x**2
a = symbols("a")
p = (expr_to_holonomic(1.4 * x) * expr_to_holonomic(a * x, x)).to_expr()
q = 1.4 * a * x**2
assert p == q
p = (expr_to_holonomic(1.4 * x) + expr_to_holonomic(a * x, x)).to_expr()
q = x * (a + 1.4)
assert p == q
p = (expr_to_holonomic(1.4 * x) + expr_to_holonomic(x)).to_expr()
assert p == 2.4 * x
def test_to_expr():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx - 1, x, 0, 1).to_expr()
q = exp(x)
assert p == q
p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).to_expr()
q = cos(x)
assert p == q
p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]).to_expr()
q = cosh(x)
assert p == q
p = HolonomicFunction(2 + (4*x - 1)*Dx + \
(x**2 - x)*Dx**2, x, 0, [1, 2]).to_expr()
q = 1 / (x**2 - 2 * x + 1)
assert p == q
p = expr_to_holonomic(sin(x)**2 / x).integrate((x, 0, x)).to_expr()
q = (sin(x)**2 / x).integrate((x, 0, x))
assert p == q
C_1, C_2, C_3 = symbols('C_1, C_2, C_3')
p = expr_to_holonomic(log(1 + x**2)).to_expr()
q = C_2 * log(x**2 + 1)
assert p == q
p = expr_to_holonomic(log(1 + x**2)).diff().to_expr()
q = C_1 * x / (x**2 + 1)
assert p == q
p = expr_to_holonomic(erf(x) + x).to_expr()
q = 3 * C_3 * x - 3 * sqrt(pi) * C_3 * erf(x) / 2 + x + 2 * x / sqrt(pi)
assert p == q
p = expr_to_holonomic(sqrt(x), x0=1).to_expr()
assert p == sqrt(x)
p = expr_to_holonomic(sqrt(1 + x**2)).to_expr()
assert p == sqrt(1 + x**2)
p = expr_to_holonomic((2 * x**2 + 1)**(S(2) / 3)).to_expr()
assert p == (2 * x**2 + 1)**(S(2) / 3)
def test_integrate():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = expr_to_holonomic(sin(x)**2 / x, x0=1).integrate((x, 2, 3))
q = '0.166270406994788'
assert sstr(p) == q
p = expr_to_holonomic(sin(x)).integrate((x, 0, x)).to_expr()
q = 1 - cos(x)
assert p == q
p = expr_to_holonomic(sin(x)).integrate((x, 0, 3))
q = 1 - cos(3)
assert p == q
p = expr_to_holonomic(sin(x) / x, x0=1).integrate((x, 1, 2))
q = '0.659329913368450'
assert sstr(p) == q
p = expr_to_holonomic(sin(x)**2 / x, x0=1).integrate((x, 1, 0))
q = '-0.423690480850035'
assert sstr(p) == q
p = expr_to_holonomic(sin(x) / x)
assert p.integrate(x).to_expr() == Si(x)
assert p.integrate((x, 0, 2)) == Si(2)
p = expr_to_holonomic(sin(x)**2 / x)
q = p.to_expr()
assert p.integrate(x).to_expr() == q.integrate((x, 0, x))
assert p.integrate((x, 0, 1)) == q.integrate((x, 0, 1))
assert expr_to_holonomic(1 / x).integrate(x).to_expr() == log(x)
p = expr_to_holonomic((x + 1)**3 * exp(-x), x0=-1,
lenics=4).integrate(x).to_expr()
q = (-x**3 - 6 * x**2 - 15 * x + 6 * exp(x + 1) - 16) * exp(-x)
assert p == q
def test_addition_initial_condition():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx - 1, x, 0, 3)
q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0])
r = HolonomicFunction(-1 + Dx - Dx**2 + Dx**3, x, 0, [4, 3, 2])
assert p + q == r
p = HolonomicFunction(Dx - x + Dx**2, x, 0, [1, 2])
q = HolonomicFunction(Dx**2 + x, x, 0, [1, 0])
r = HolonomicFunction((-x**4 - x**3/4 - x**2 + 1/4) + (x**3 + x**2/4 + 3*x/4 + 1)*Dx + \
(-3*x/2 + 7/4)*Dx**2 + (x**2 - 7*x/4 + 1/4)*Dx**3 + (x**2 + x/4 + 1/2)*Dx**4, x, 0, [2, 2, -2, 2])
assert p + q == r
p = HolonomicFunction(Dx**2 + 4 * x * Dx + x**2, x, 0, [3, 4])
q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1])
r = HolonomicFunction((x**6 + 2*x**4 - 5*x**2 - 6) + (4*x**5 + 36*x**3 - 32*x)*Dx + \
(x**6 + 3*x**4 + 5*x**2 - 9)*Dx**2 + (4*x**5 + 36*x**3 - 32*x)*Dx**3 + (x**4 + \
10*x**2 - 3)*Dx**4, x, 0, [4, 5, -1, -17])
assert p + q == r
q = HolonomicFunction(Dx**3 + x, x, 2, [3, 0, 1])
p = HolonomicFunction(Dx - 1, x, 2, [1])
r = HolonomicFunction((-x**2 - x + 1) + (x**2 + x)*Dx + (-x - 2)*Dx**3 + \
(x + 1)*Dx**4, x, 2, [4, 1, 2, -5 ])
assert p + q == r
p = from_sympy(sin(x))
q = from_sympy(1 / x)
r = HolonomicFunction((x**2 + 6) + (x**3 + 2*x)*Dx + (x**2 + 6)*Dx**2 + (x**3 + 2*x)*Dx**3, \
x, 1, [sin(1) + 1, -1 + cos(1), -sin(1) + 2])
assert p + q == r
def test_multiplication_initial_condition():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx**2 + x * Dx - 1, x, 0, [3, 1])
q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1])
r = HolonomicFunction((x**4 + 14*x**2 + 60) + 4*x*Dx + (x**4 + 9*x**2 + 20)*Dx**2 + \
(2*x**3 + 18*x)*Dx**3 + (x**2 + 10)*Dx**4, x, 0, [3, 4, 2, 3])
assert p * q == r
p = HolonomicFunction(Dx**2 + x, x, 0, [1, 0])
q = HolonomicFunction(Dx**3 - x**2, x, 0, [3, 3, 3])
r = HolonomicFunction((x**8 - 37*x**7/27 - 10*x**6/27 - 164*x**5/9 - 184*x**4/9 + \
160*x**3/27 + 404*x**2/9 + 8*x + 40/3) + (6*x**7 - 128*x**6/9 - 98*x**5/9 - 28*x**4/9 + \
8*x**3/9 + 28*x**2 + 40*x/9 - 40)*Dx + (3*x**6 - 82*x**5/9 + 76*x**4/9 + 4*x**3/3 + \
220*x**2/9 - 80*x/3)*Dx**2 + (-2*x**6 + 128*x**5/27 - 2*x**4/3 -80*x**2/9 + 200/9)*Dx**3 + \
(3*x**5 - 64*x**4/9 - 28*x**3/9 + 6*x**2 - 20*x/9 - 20/3)*Dx**4 + (-4*x**3 + 64*x**2/9 + \
8*x/3)*Dx**5 + (x**4 - 64*x**3/27 - 4*x**2/3 + 20/9)*Dx**6, x, 0, [3, 3, 3, -3, -12, -24])
assert p * q == r
p = HolonomicFunction(Dx - 1, x, 0, [2])
q = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1])
r = HolonomicFunction(2 - 2 * Dx + Dx**2, x, 0, [0, 2])
assert p * q == r
q = HolonomicFunction(x * Dx**2 + 1 + 2 * Dx, x, 0, [0, 1])
r = HolonomicFunction((x - 1) + (-2 * x + 2) * Dx + x * Dx**2, x, 0,
[0, 2])
assert p * q == r
p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 3])
q = HolonomicFunction(Dx**3 + 1, x, 0, [1, 2, 1])
r = HolonomicFunction(6 * Dx + 3 * Dx**2 + 2 * Dx**3 - 3 * Dx**4 + Dx**6,
x, 0, [1, 5, 14, 17, 17, 2])
assert p * q == r
p = expr_to_holonomic(sin(x))
q = expr_to_holonomic(1 / x)
r = HolonomicFunction(x + 2 * Dx + x * Dx**2, x, 1,
[sin(1), -sin(1) + cos(1)])
assert p * q == r
def test_negative_power():
x = symbols("x")
_, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
h1 = HolonomicFunction((-1) + (x) * Dx, x)**-2
h2 = HolonomicFunction((2) + (x) * Dx, x)
assert h1 == h2
def test_HolonomicFunction_addition():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx**2 * x, x)
q = HolonomicFunction((2) * Dx + (x) * Dx**2, x)
assert p == q
p = HolonomicFunction(x * Dx + 1, x)
q = HolonomicFunction(Dx + 1, x)
r = HolonomicFunction((x - 2) + (x**2 - 2) * Dx + (x**2 - x) * Dx**2, x)
assert p + q == r
p = HolonomicFunction(x * Dx + Dx**2 * (x**2 + 2), x)
q = HolonomicFunction(Dx - 3, x)
r = HolonomicFunction((-54 * x**2 - 126 * x - 150) + (-135 * x**3 - 252 * x**2 - 270 * x + 140) * Dx +\
(-27 * x**4 - 24 * x**2 + 14 * x - 150) * Dx**2 + \
(9 * x**4 + 15 * x**3 + 38 * x**2 + 30 * x +40) * Dx**3, x)
assert p + q == r
p = HolonomicFunction(Dx**5 - 1, x)
q = HolonomicFunction(x**3 + Dx, x)
r = HolonomicFunction((-x**18 + 45*x**14 - 525*x**10 + 1575*x**6 - x**3 - 630*x**2) + \
(-x**15 + 30*x**11 - 195*x**7 + 210*x**3 - 1)*Dx + (x**18 - 45*x**14 + 525*x**10 - \
1575*x**6 + x**3 + 630*x**2)*Dx**5 + (x**15 - 30*x**11 + 195*x**7 - 210*x**3 + \
1)*Dx**6, x)
assert p+q == r
p = x**2 + 3*x + 8
q = x**3 - 7*x + 5
p = p*Dx - p.diff()
q = q*Dx - q.diff()
r = HolonomicFunction(p, x) + HolonomicFunction(q, x)
s = HolonomicFunction((6*x**2 + 18*x + 14) + (-4*x**3 - 18*x**2 - 62*x + 10)*Dx +\
(x**4 + 6*x**3 + 31*x**2 - 10*x - 71)*Dx**2, x)
assert r == s
def test_to_hyper():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx - 2, x, 0, [3]).to_hyper()
q = 3 * hyper([], [], 2*x)
assert p == q
p = hyperexpand(HolonomicFunction((1 + x) * Dx - 3, x, 0, [2]).to_hyper()).expand()
q = 2*x**3 + 6*x**2 + 6*x + 2
assert p == q
p = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1]).to_hyper()
q = -x**2*hyper((2, 2, 1), (3, 2), -x)/2 + x
assert p == q
p = HolonomicFunction(2*x*Dx + Dx**2, x, 0, [0, 2/sqrt(pi)]).to_hyper()
q = 2*x*hyper((1/2,), (3/2,), -x**2)/sqrt(pi)
assert p == q
p = hyperexpand(HolonomicFunction(2*x*Dx + Dx**2, x, 0, [1, -2/sqrt(pi)]).to_hyper())
q = erfc(x)
assert p.rewrite(erfc) == q
p = hyperexpand(HolonomicFunction((x**2 - 1) + x*Dx + x**2*Dx**2,
x, 0, [0, S(1)/2]).to_hyper())
q = besselj(1, x)
assert p == q
p = hyperexpand(HolonomicFunction(x*Dx**2 + Dx + x, x, 0, [1, 0]).to_hyper())
q = besselj(0, x)
assert p == q
def test_series():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx**2 + 2 * x * Dx, x, 0, [0, 1]).series(n=10)
q = x - x**3 / 3 + x**5 / 10 - x**7 / 42 + x**9 / 216 + O(x**10)
assert p == q
p = HolonomicFunction(Dx - 1, x).composition(x**2, 0, 1) # e^(x**2)
q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]) # cos(x)
r = (p * q).series(n=10) # expansion of cos(x) * exp(x**2)
s = 1 + x**2 / 2 + x**4 / 24 - 31 * x**6 / 720 - 179 * x**8 / 8064 + O(x**
10)
assert r == s
t = HolonomicFunction((1 + x) * Dx**2 + Dx, x, 0, [0, 1]) # log(1 + x)
r = (p * t + q).series(n=10)
s = 1 + x - x**2 + 4*x**3/3 - 17*x**4/24 + 31*x**5/30 - 481*x**6/720 +\
71*x**7/105 - 20159*x**8/40320 + 379*x**9/840 + O(x**10)
assert r == s
p = HolonomicFunction((6+6*x-3*x**2) - (10*x-3*x**2-3*x**3)*Dx + \
(4-6*x**3+2*x**4)*Dx**2, x, 0, [0, 1]).series(n=7)
q = x + x**3 / 6 - 3 * x**4 / 16 + x**5 / 20 - 23 * x**6 / 960 + O(x**7)
assert p == q
p = HolonomicFunction((6+6*x-3*x**2) - (10*x-3*x**2-3*x**3)*Dx + \
(4-6*x**3+2*x**4)*Dx**2, x, 0, [1, 0]).series(n=7)
q = 1 - 3 * x**2 / 4 - x**3 / 4 - 5 * x**4 / 32 - 3 * x**5 / 40 - 17 * x**6 / 384 + O(
x**7)
assert p == q
def test_to_Sequence_Initial_Coniditons():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
n = symbols('n', integer=True)
_, Sn = RecurrenceOperators(QQ.old_poly_ring(n), 'Sn')
p = HolonomicFunction(Dx - 1, x, 0, [1]).to_sequence()
q = [(HolonomicSequence(-1 + (n + 1)*Sn, 1), 0)]
assert p == q
p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]).to_sequence()
q = [(HolonomicSequence(1 + (n**2 + 3*n + 2)*Sn**2, [0, 1]), 0)]
assert p == q
p = HolonomicFunction(Dx**2 + 1 + x**3*Dx, x, 0, [2, 3]).to_sequence()
q = [(HolonomicSequence(n + Sn**2 + (n**2 + 7*n + 12)*Sn**4, [2, 3, -1, -1/2, 1/12]), 1)]
assert p == q
p = HolonomicFunction(x**3*Dx**5 + 1 + Dx, x).to_sequence()
q = [(HolonomicSequence(1 + (n + 1)*Sn + (n**5 - 5*n**3 + 4*n)*Sn**2), 0, 3)]
assert p == q
C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3')
p = expr_to_holonomic(log(1+x**2))
q = [(HolonomicSequence(n**2 + (n**2 + 2*n)*Sn**2, [0, 0, C_2]), 0, 1)]
assert p.to_sequence() == q
p = p.diff()
q = [(HolonomicSequence((n + 2) + (n + 2)*Sn**2, [C_0, 0]), 1, 0)]
assert p.to_sequence() == q
p = expr_to_holonomic(erf(x) + x).to_sequence()
q = [(HolonomicSequence((2*n**2 - 2*n) + (n**3 + 2*n**2 - n - 2)*Sn**2, [0, 1 + 2/sqrt(pi), 0, C_3]), 0, 2)]
assert p == q
def test_extended_domain_in_expr_to_holonomic():
x = symbols('x')
p = expr_to_holonomic(1.2 * cos(3.1 * x))
assert p.to_expr() == 1.2 * cos(3.1 * x)
assert sstr(p.integrate(x).to_expr()) == '0.387096774193548*sin(3.1*x)'
_, Dx = DifferentialOperators(RR.old_poly_ring(x), 'Dx')
p = expr_to_holonomic(1.1329138213 * x)
q = HolonomicFunction((-1.1329138213) + (1.1329138213 * x) * Dx, x, 0,
{1: [1.1329138213]})
assert p == q
assert p.to_expr() == 1.1329138213 * x
assert sstr(p.integrate((x, 1, 2))) == sstr((1.1329138213 * x).integrate(
(x, 1, 2)))
y, z = symbols('y, z')
p = expr_to_holonomic(sin(x * y * z), x=x)
assert p.to_expr() == sin(x * y * z)
assert p.integrate(x).to_expr() == (-cos(x * y * z) + 1) / (y * z)
p = expr_to_holonomic(sin(x * y + z), x=x).integrate(x).to_expr()
q = (cos(z) - cos(x * y + z)) / y
assert p == q
a = symbols('a')
p = expr_to_holonomic(a * x, x)
assert p.to_expr() == a * x
assert p.integrate(x).to_expr() == a * x**2 / 2
D_2, C_1 = symbols("D_2, C_1")
p = expr_to_holonomic(x) + expr_to_holonomic(1.2 * cos(x))
p = p.to_expr().subs(D_2, 0)
assert p - x - 1.2 * cos(1.0 * x) == 0
p = expr_to_holonomic(x) * expr_to_holonomic(1.2 * cos(x))
p = p.to_expr().subs(C_1, 0)
assert p - 1.2 * x * cos(1.0 * x) == 0
def test_from_sympy():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = from_sympy((sin(x) / x)**2)
q = HolonomicFunction(8*x + (4*x**2 + 6)*Dx + 6*x*Dx**2 + x**2*Dx**3, x, 0, \
[1, 0, -2/3])
assert p == q
p = from_sympy(1 / (1 + x**2)**2)
q = HolonomicFunction(4 * x + (x**2 + 1) * Dx, x, 0, 1)
assert p == q
p = from_sympy(exp(x) * sin(x) + x * log(1 + x))
q = HolonomicFunction((2*x**3 + 10*x**2 + 20*x + 18) + (-2*x**4 - 10*x**3 - 20*x**2 \
- 18*x)*Dx + (2*x**5 + 6*x**4 + 7*x**3 + 8*x**2 + 10*x - 4)*Dx**2 + \
(-2*x**5 - 5*x**4 - 2*x**3 + 2*x**2 - x + 4)*Dx**3 + (x**5 + 2*x**4 - x**3 - \
7*x**2/2 + x + 5/2)*Dx**4, x, 0, [0, 1, 4, -1])
assert p == q
p = from_sympy(x * exp(x) + cos(x) + 1)
q = HolonomicFunction((-x - 3)*Dx + (x + 2)*Dx**2 + (-x - 3)*Dx**3 + (x + 2)*Dx**4, x, \
0, [2, 1, 1, 3])
assert p == q
assert (x * exp(x) + cos(x) + 1).series(n=10) == p.series(n=10)
p = from_sympy(log(1 + x)**2 + 1)
q = HolonomicFunction(
Dx + (3 * x + 3) * Dx**2 + (x**2 + 2 * x + 1) * Dx**3, x, 0, [1, 0, 2])
assert p == q
p = from_sympy(erf(x)**2 + x)
q = HolonomicFunction((8*x**4 - 2*x**2 + 2)*Dx**2 + (6*x**3 - x/2)*Dx**3 + \
(x**2+ 1/4)*Dx**4, x, 0, [0, 1, 8/pi, 0])
assert p == q
p = from_sympy(cosh(x) * x)
q = HolonomicFunction((-x**2 + 2) - 2 * x * Dx + x**2 * Dx**2, x, 0,
[0, 1])
assert p == q
p = from_sympy(besselj(2, x))
q = HolonomicFunction((x**2 - 4) + x * Dx + x**2 * Dx**2, x, 0, [0, 0])
assert p == q
p = from_sympy(besselj(0, x) + exp(x))
q = HolonomicFunction((-x**2 - x/2 + 1/2) + (x**2 - x/2 - 3/2)*Dx + (-x**2 + x/2 + 1)*Dx**2 +\
(x**2 + x/2)*Dx**3, x, 0, [2, 1, 1/2])
assert p == q
p = from_sympy(sin(x)**2 / x)
q = HolonomicFunction(4 + 4 * x * Dx + 3 * Dx**2 + x * Dx**3, x, 0,
[0, 1, 0])
assert p == q
p = from_sympy(sin(x)**2 / x, x0=2)
q = HolonomicFunction((4) + (4 * x) * Dx + (3) * Dx**2 + (x) * Dx**3, x, 2,
[
sin(2)**2 / 2,
sin(2) * cos(2) - sin(2)**2 / 4,
-3 * sin(2)**2 / 4 + cos(2)**2 - sin(2) * cos(2)
])
assert p == q
p = from_sympy(log(x) / 2 - Ci(2 * x) / 2 + Ci(2) / 2)
q = HolonomicFunction(4*Dx + 4*x*Dx**2 + 3*Dx**3 + x*Dx**4, x, 0, \
[-log(2)/2 - EulerGamma/2 + Ci(2)/2, 0, 1, 0])
assert p == q
p = p.to_sympy()
q = log(x) / 2 - Ci(2 * x) / 2 + Ci(2) / 2
assert p == q
def test_expr_in_power():
x, n = symbols("x n")
Q = QQ[n].get_field()
_, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx')
h1 = HolonomicFunction((-1) + (x) * Dx, x)**(n - 3)
h2 = HolonomicFunction((-n + 3) + (x) * Dx, x)
assert h1 == h2
def test_beta():
a, b, x = symbols("a b x", positive=True)
e = x**(a - 1) * (-x + 1)**(b - 1) / beta(a, b)
Q = QQ[a, b].get_field()
h1 = expr_to_holonomic(e, x, domain=Q)
_, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx')
h2 = HolonomicFunction((a + x * (-a - b + 2) - 1) + (x**2 - x) * Dx, x)
assert h1 == h2
def test_gamma():
a, b, x = symbols("a b x", positive=True)
e = b**(-a) * x**(a - 1) * exp(-x / b) / gamma(a)
Q = QQ[a, b].get_field()
h1 = expr_to_holonomic(e, x, domain=Q)
_, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx')
h2 = HolonomicFunction((-a + 1 + x / b) + (x) * Dx, x)
assert h1 == h2
def test_from_sympy():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = from_sympy((sin(x) / x)**2)
q = HolonomicFunction(24*x + (24*x**2 + 12)*Dx + (4*x**3 + 24*x)*Dx**2 + \
10*x**2*Dx**3 + x**3*Dx**4, x, 1, [-cos(2)/2 + 1/2, -1 + cos(2) + sin(2), \
-4*sin(2) - cos(2) + 3, -12 + 14*sin(2)])
assert p == q
p = from_sympy(1 / (1 + x**2)**2)
q = HolonomicFunction(4 * x + (x**2 + 1) * Dx, x, 0, 1)
assert p == q
def test_gaussian():
mu, x = symbols("mu x")
sd = symbols("sd", positive=True)
Q = QQ[mu, sd].get_field()
e = sqrt(2) * exp(-(-mu + x)**2 / (2 * sd**2)) / (2 * sqrt(pi) * sd)
h1 = expr_to_holonomic(e, x, domain=Q)
_, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx')
h2 = HolonomicFunction((-mu / sd**2 + x / sd**2) + (1) * Dx, x)
assert h1 == h2
def test_from_hyper():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = hyper([1, 1], [S(3)/2], x**2/4)
q = HolonomicFunction((4*x) + (5*x**2 - 8)*Dx + (x**3 - 4*x)*Dx**2, x, 1, [2*sqrt(3)*pi/9, -4*sqrt(3)*pi/27 + 4/3])
r = from_hyper(p)
assert r == q
p = from_hyper(hyper([1], [S(3)/2], x**2/4))
q = HolonomicFunction(-x + (-x**2/2 + 2)*Dx + x*Dx**2, x)
x0 = 1
y0 = '[sqrt(pi)*exp(1/4)*erf(1/2), -sqrt(pi)*exp(1/4)*erf(1/2)/2 + 1]'
assert sstr(p.y0) == y0
assert q.annihilator == p.annihilator
def test_DifferentialOperatorEqPoly():
x = symbols('x', integer=True)
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
do = DifferentialOperator([x**2, R.base.zero, R.base.zero], R)
do2 = DifferentialOperator([x**2, 1, x], R)
assert not do == do2
# polynomial comparison issue, see https://github.com/sympy/sympy/pull/15799
# should work once that is solved
# p = do.listofpoly[0]
# assert do == p
p2 = do2.listofpoly[0]
assert not do2 == p2
def test_diff():
x, y = symbols('x, y')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(x*Dx**2 + 1, x, 0, [0, 1])
assert p.diff().to_expr() == p.to_expr().diff().simplify()
p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0])
assert p.diff(x, 2).to_expr() == p.to_expr()
p = expr_to_holonomic(Si(x))
assert p.diff().to_expr() == sin(x)/x
assert p.diff(y) == 0
C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3')
q = Si(x)
assert p.diff(x).to_expr() == q.diff()
assert p.diff(x, 2).to_expr().subs(C_0, -S(1)/3) == q.diff(x, 2).simplify()
assert p.diff(x, 3).series().subs({C_3:-S(1)/3, C_0:0}) == q.diff(x, 3).series()
def test_DifferentialOperator():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
assert Dx == R.derivative_operator
assert Dx == DifferentialOperator([R.base.zero, R.base.one], R)
assert x * Dx + x**2 * Dx**2 == DifferentialOperator([0, x, x**2], R)
assert (x**2 + 1) + Dx + x * \
Dx**5 == DifferentialOperator([x**2 + 1, 1, 0, 0, 0, x], R)
assert (x * Dx + x**2 + 1 - Dx * (x**3 + x))**3 == (-48 * x**6) + \
(-57 * x**7) * Dx + (-15 * x**8) * Dx**2 + (-x**9) * Dx**3
p = (x * Dx**2 + (x**2 + 3) * Dx**5) * (Dx + x**2)
q = (2 * x) + (4 * x**2) * Dx + (x**3) * Dx**2 + \
(20 * x**2 + x + 60) * Dx**3 + (10 * x**3 + 30 * x) * Dx**4 + \
(x**4 + 3 * x**2) * Dx**5 + (x**2 + 3) * Dx**6
assert p == q
def test_from_meijerg():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = from_meijerg(meijerg(([], [S(3)/2]), ([S(1)/2], [S(1)/2, 1]), x))
q = HolonomicFunction(x/2 - 1/4 + (-x**2 + x/4)*Dx + x**2*Dx**2 + x**3*Dx**3, x, 1, \
[1/sqrt(pi), 1/(2*sqrt(pi)), -1/(4*sqrt(pi))])
assert p == q
p = from_meijerg(meijerg(([], []), ([0], []), x))
q = HolonomicFunction(1 + Dx, x, 0, [1])
assert p == q
p = from_meijerg(meijerg(([1], []), ([S(1)/2], [0]), x))
q = HolonomicFunction((x + 1/2)*Dx + x*Dx**2, x, 1, [sqrt(pi)*erf(1), exp(-1)])
assert p == q
p = from_meijerg(meijerg(([0], [1]), ([0], []), 2*x**2))
q = HolonomicFunction((3*x**2 - 1)*Dx + x**3*Dx**2, x, 1, [-exp(-S(1)/2) + 1, -exp(-S(1)/2)])
assert p == q
def test_integrate():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = expr_to_holonomic(sin(x)**2 / x, x0=1).integrate((x, 2, 3))
q = '0.166270406994788'
assert sstr(p) == q
p = expr_to_holonomic(sin(x)).integrate((x, 0, x)).to_expr()
q = 1 - cos(x)
assert p == q
p = expr_to_holonomic(sin(x)).integrate((x, 0, 3))
q = 1 - cos(3)
assert p == q
p = expr_to_holonomic(sin(x) / x, x0=1).integrate((x, 1, 2))
q = '0.659329913368450'
assert sstr(p) == q
p = expr_to_holonomic(sin(x)**2 / x, x0=1).integrate((x, 1, 0))
q = '-0.423690480850035'
assert sstr(p) == q
p = expr_to_holonomic(sin(x) / x)
assert p.integrate(x).to_expr() == Si(x)
assert p.integrate((x, 0, 2)) == Si(2)
p = expr_to_holonomic(sin(x)**2 / x)
q = p.to_expr()
assert p.integrate(x).to_expr() == q.integrate((x, 0, x))
assert p.integrate((x, 0, 1)) == q.integrate((x, 0, 1))
assert expr_to_holonomic(1 / x, x0=1).integrate(x).to_expr() == log(x)
p = expr_to_holonomic((x + 1)**3 * exp(-x), x0=-1).integrate(x).to_expr()
q = (-x**3 - 6 * x**2 - 15 * x + 6 * exp(x + 1) - 16) * exp(-x)
assert p == q
p = expr_to_holonomic(cos(x)**2 / x**2, y0={
-2: [1, 0, -1]
}).integrate(x).to_expr()
q = -Si(2 * x) - cos(x)**2 / x
assert p == q
p = expr_to_holonomic(sqrt(x**2 + x)).integrate(x).to_expr()
q = (x**Rational(3, 2) * (2 * x**2 + 3 * x + 1) -
x * sqrt(x + 1) * asinh(sqrt(x))) / (4 * x * sqrt(x + 1))
assert p == q
p = expr_to_holonomic(sqrt(x**2 + 1)).integrate(x).to_expr()
q = (sqrt(x**2 + 1)).integrate(x)
assert (p - q).simplify() == 0
p = expr_to_holonomic(1 / x**2, y0={-2: [1, 0, 0]})
r = expr_to_holonomic(1 / x**2, lenics=3)
assert p == r
q = expr_to_holonomic(cos(x)**2)
assert (r * q).integrate(x).to_expr() == -Si(2 * x) - cos(x)**2 / x
def test_to_Sequence_Initial_Coniditons():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
n = symbols('n', integer=True)
_, Sn = RecurrenceOperators(QQ.old_poly_ring(n), 'Sn')
p = HolonomicFunction(Dx - 1, x, 0, 1).to_sequence()
q = HolonomicSequence(-1 + (n + 1)*Sn, 1)
assert p == q
p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]).to_sequence()
q = HolonomicSequence(1 + (n**2 + 3*n + 2)*Sn**2, [0, 1])
assert p == q
p = HolonomicFunction(Dx**2 + 1 + x**3*Dx, x, 0, [2, 3]).to_sequence()
q = HolonomicSequence(n + Sn**2 + (n**2 + 7*n + 12)*Sn**4, [2, 3, -1, -1/2])
assert p == q
p = HolonomicFunction(x**3*Dx**5 + 1 + Dx, x).to_sequence()
q = HolonomicSequence(1 + (n + 1)*Sn + (n**5 - 5*n**3 + 4*n)*Sn**2)
assert p == q
def test_integrate():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = from_sympy(sin(x)**2 / x, x0=1).integrate((x, 2, 3))
q = '0.166270406994788'
assert sstr(p) == q
p = from_sympy(sin(x)).integrate((x, 0, x)).to_sympy()
q = 1 - cos(x)
assert p == q
p = from_sympy(sin(x)).integrate((x, 0, 3))
q = '1.98999246812687'
assert sstr(p) == q
p = from_sympy(sin(x) / x, x0=1).integrate((x, 1, 2))
q = '0.659329913368450'
assert sstr(p) == q
p = from_sympy(sin(x)**2 / x, x0=1).integrate((x, 1, 0))
q = '-0.423690480850035'
assert sstr(p) == q
def test_extended_domain_in_expr_to_holonomic():
x = symbols('x')
p = expr_to_holonomic(1.2 * cos(3.1 * x), domain=RR)
assert p.to_expr() == 1.2 * cos(3.1 * x)
assert sstr(p.integrate(x).to_expr()) == '0.387096774193548*sin(3.1*x)'
_, Dx = DifferentialOperators(RR.old_poly_ring(x), 'Dx')
p = expr_to_holonomic(1.1329138213 * x, domain=RR, lenics=2)
q = HolonomicFunction((-1.1329138213) + (1.1329138213 * x) * Dx, x, 0,
[0, 1.1329138213])
assert p == q
assert p.to_expr() == 1.1329138213 * x
assert sstr(p.integrate((x, 1, 2))) == sstr((1.1329138213 * x).integrate(
(x, 1, 2)))
y, z = symbols('y, z')
p = expr_to_holonomic(sin(x * y * z), x=x, domain=ZZ[y, z])
assert p.to_expr() == sin(x * y * z)
assert p.integrate(x).to_expr() == (-cos(x * y * z) + 1) / (y * z)
p = expr_to_holonomic(sin(x * y + z), x=x,
domain=ZZ[y, z]).integrate(x).to_expr()
q = (cos(z) - cos(x * y + z)) / y
assert p == q