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_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_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_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_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_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 + Rational(1, 4)) + (x**3 + x**2/4 + x*Rational(3, 4) + 1)*Dx + \
(x*Rational(-3, 2) + Rational(7, 4))*Dx**2 + (x**2 - x*Rational(7, 4) + Rational(1, 4))*Dx**3 + (x**2 + x/4 + S.Half)*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_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)
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_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_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_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_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_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.singular_ics = [(0, [1]), (S(1) / 2, [0])]
assert p.to_expr() == s
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
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
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_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_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
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_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_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_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_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_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
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_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_to_meijerg():
x = symbols('x')
assert hyperexpand(expr_to_holonomic(sin(x)).to_meijerg()) == sin(x)
assert hyperexpand(expr_to_holonomic(cos(x)).to_meijerg()) == cos(x)
assert hyperexpand(expr_to_holonomic(exp(x)).to_meijerg()) == exp(x)
assert hyperexpand(expr_to_holonomic(
log(x)).to_meijerg()).simplify() == log(x)
assert expr_to_holonomic(4 * x**2 / 3 + 7).to_meijerg() == 4 * x**2 / 3 + 7
assert hyperexpand(
expr_to_holonomic(besselj(2, x),
lenics=3).to_meijerg()) == besselj(2, x)
p = hyper((Rational(-1, 2), -3), (), x)
assert from_hyper(p).to_meijerg() == hyperexpand(p)
p = hyper((S.One, S(3)), (S(2), ), x)
assert (hyperexpand(from_hyper(p).to_meijerg()) -
hyperexpand(p)).expand() == 0
p = from_hyper(hyper((-2, -3), (S.Half, ), x))
s = hyperexpand(hyper((-2, -3), (S.Half, ), x))
C_0 = Symbol('C_0')
C_1 = Symbol('C_1')
D_0 = Symbol('D_0')
assert (hyperexpand(p.to_meijerg()).subs({
C_0: 1,
D_0: 0
}) - s).simplify() == 0
p.y0 = {0: [1], S.Half: [0]}
assert (hyperexpand(p.to_meijerg()) - s).simplify() == 0
p = expr_to_holonomic(besselj(S.Half, x), initcond=False)
assert (
p.to_expr() -
(D_0 * sin(x) + C_0 * cos(x) + C_1 * sin(x)) / sqrt(x)).simplify() == 0
p = expr_to_holonomic(
besselj(S.Half, x),
y0={Rational(-1, 2): [sqrt(2) / sqrt(pi),
sqrt(2) / sqrt(pi)]})
assert (p.to_expr() - besselj(S.Half, x) -
besselj(Rational(-1, 2), x)).simplify() == 0
def test_to_meijerg():
x = symbols('x')
assert hyperexpand(expr_to_holonomic(sin(x)).to_meijerg()) == sin(x)
assert hyperexpand(expr_to_holonomic(cos(x)).to_meijerg()) == cos(x)
assert hyperexpand(expr_to_holonomic(exp(x)).to_meijerg()) == exp(x)
assert hyperexpand(expr_to_holonomic(
log(x)).to_meijerg()).simplify() == log(x)
assert expr_to_holonomic(4 * x**2 / 3 + 7).to_meijerg() == 4 * x**2 / 3 + 7
assert hyperexpand(
expr_to_holonomic(besselj(2, x),
lenics=3).to_meijerg()) == besselj(2, x)
p = hyper((-S(1) / 2, -3), (), x)
assert from_hyper(p).to_meijerg() == hyperexpand(p)
p = hyper((S(1), S(3)), (S(2), ), x)
assert (hyperexpand(from_hyper(p).to_meijerg()) -
hyperexpand(p)).expand() == 0
p = from_hyper(hyper((-2, -3), (S(1) / 2, ), x))
s = hyperexpand(hyper((-2, -3), (S(1) / 2, ), x))
C_0 = Symbol('C_0')
C_1 = Symbol('C_1')
D_0 = Symbol('D_0')
assert (hyperexpand(p.to_meijerg()).subs({
C_0: 1,
D_0: 0
}) - s).simplify() == 0
p.singular_ics = [(0, [1]), (S(1) / 2, [0])]
assert (hyperexpand(p.to_meijerg()) - s).simplify() == 0
p = expr_to_holonomic(besselj(S(1) / 2, x), initcond=False)
assert (
p.to_expr() -
(D_0 * sin(x) + C_0 * cos(x) + C_1 * sin(x)) / sqrt(x)).simplify() == 0
p = expr_to_holonomic(
besselj(S(1) / 2, x),
singular_ics=((S(-1) / 2, [sqrt(2) / sqrt(pi),
sqrt(2) / sqrt(pi)]), ))
assert (p.to_expr() - besselj(S(1) / 2, x) -
besselj(S(-1) / 2, x)).simplify() == 0
def test_to_meijerg():
x = symbols('x')
assert hyperexpand(expr_to_holonomic(sin(x)).to_meijerg()) == sin(x)
assert hyperexpand(expr_to_holonomic(cos(x)).to_meijerg()) == cos(x)
assert hyperexpand(expr_to_holonomic(exp(x)).to_meijerg()) == exp(x)
assert hyperexpand(expr_to_holonomic(log(x)).to_meijerg()).simplify() == log(x)
assert expr_to_holonomic(4*x**2/3 + 7).to_meijerg() == 4*x**2/3 + 7
assert hyperexpand(expr_to_holonomic(besselj(2, x), lenics=3).to_meijerg()) == besselj(2, x)
p = hyper((-S(1)/2, -3), (), x)
assert from_hyper(p).to_meijerg() == hyperexpand(p)
p = hyper((S(1), S(3)), (S(2), ), x)
assert (hyperexpand(from_hyper(p).to_meijerg()) - hyperexpand(p)).expand() == 0
p = from_hyper(hyper((-2, -3), (S(1)/2, ), x))
s = hyperexpand(hyper((-2, -3), (S(1)/2, ), x))
C_0 = Symbol('C_0')
C_1 = Symbol('C_1')
D_0 = Symbol('D_0')
assert (hyperexpand(p.to_meijerg()).subs({C_0:1, D_0:0}) - s).simplify() == 0
p.y0 = {0: [1], S(1)/2: [0]}
assert (hyperexpand(p.to_meijerg()) - s).simplify() == 0
p = expr_to_holonomic(besselj(S(1)/2, x), initcond=False)
assert (p.to_expr() - (D_0*sin(x) + C_0*cos(x) + C_1*sin(x))/sqrt(x)).simplify() == 0
p = expr_to_holonomic(besselj(S(1)/2, x), y0={S(-1)/2: [sqrt(2)/sqrt(pi), sqrt(2)/sqrt(pi)]})
assert (p.to_expr() - besselj(S(1)/2, x) - besselj(S(-1)/2, x)).simplify() == 0
def test_expr_to_holonomic():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = expr_to_holonomic((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 = expr_to_holonomic(1/(1+x**2)**2)
q = HolonomicFunction(4*x + (x**2 + 1)*Dx, x, 0, [1])
assert p == q
p = expr_to_holonomic(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 = expr_to_holonomic(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 = expr_to_holonomic(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 = expr_to_holonomic(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 = expr_to_holonomic(cosh(x)*x)
q = HolonomicFunction((-x**2 + 2) -2*x*Dx + x**2*Dx**2, x, 0, [0, 1])
assert p == q
p = expr_to_holonomic(besselj(2, x))
q = HolonomicFunction((x**2 - 4) + x*Dx + x**2*Dx**2, x, 0, [0, 0])
assert p == q
p = expr_to_holonomic(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 = expr_to_holonomic(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 = expr_to_holonomic(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 = expr_to_holonomic(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_expr()
q = log(x)/2 - Ci(2*x)/2 + Ci(2)/2
assert p == q
p = expr_to_holonomic(x**(S(1)/2), x0=1)
q = HolonomicFunction(x*Dx - 1/2, x, 1, [1])
assert p == q
p = expr_to_holonomic(sqrt(1 + x**2))
q = HolonomicFunction((-x) + (x**2 + 1)*Dx, x, 0, [1])
assert p == q
assert (expr_to_holonomic(sqrt(x) + sqrt(2*x)).to_expr()-\
(sqrt(x) + sqrt(2*x))).simplify() == 0
assert expr_to_holonomic(3*x+2*sqrt(x)).to_expr() == 3*x+2*sqrt(x)
p = expr_to_holonomic((x**4+x**3+5*x**2+3*x+2)/x**2, lenics=3)
q = HolonomicFunction((-2*x**4 - x**3 + 3*x + 4) + (x**5 + x**4 + 5*x**3 + 3*x**2 + \
2*x)*Dx, x, 0, {-2: [2, 3, 5]})
assert p == q
p = expr_to_holonomic(1/(x-1)**2, lenics=3, x0=1)
q = HolonomicFunction((2) + (x - 1)*Dx, x, 1, {-2: [1, 0, 0]})
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)**Rational(2, 3)).to_expr()
assert p == (2 * x**2 + 1)**Rational(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)**Rational(2, 3)).to_expr()
q = x**Rational(2, 3) * (-2 * x**2 + 7)**Rational(2, 3)
assert p == q
p = from_hyper(hyper((-2, -3), (S.Half, ), x))
s = hyperexpand(hyper((-2, -3), (S.Half, ), 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.Half: [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_expr_to_holonomic():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = expr_to_holonomic((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 = expr_to_holonomic(1 / (1 + x**2)**2)
q = HolonomicFunction(4 * x + (x**2 + 1) * Dx, x, 0, 1)
assert p == q
p = expr_to_holonomic(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 = expr_to_holonomic(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 = expr_to_holonomic(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 = expr_to_holonomic(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 = expr_to_holonomic(cosh(x) * x)
q = HolonomicFunction((-x**2 + 2) - 2 * x * Dx + x**2 * Dx**2, x, 0,
[0, 1])
assert p == q
p = expr_to_holonomic(besselj(2, x))
q = HolonomicFunction((x**2 - 4) + x * Dx + x**2 * Dx**2, x, 0, [0, 0])
assert p == q
p = expr_to_holonomic(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 = expr_to_holonomic(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 = expr_to_holonomic(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 = expr_to_holonomic(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_expr()
q = log(x) / 2 - Ci(2 * x) / 2 + Ci(2) / 2
assert p == q
p = expr_to_holonomic(x**(S(1) / 2), x0=1)
q = HolonomicFunction(x * Dx - 1 / 2, x, 1, 1)
assert p == q
p = expr_to_holonomic(sqrt(1 + x**2))
q = HolonomicFunction((-x) + (x**2 + 1) * Dx, x, 0, 1)
assert p == q
def test_expr_to_holonomic():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
p = expr_to_holonomic((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, Rational(-2, 3)])
assert p == q
p = expr_to_holonomic(1 / (1 + x**2)**2)
q = HolonomicFunction(4 * x + (x**2 + 1) * Dx, x, 0, [1])
assert p == q
p = expr_to_holonomic(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 + Rational(5, 2))*Dx**4, x, 0, [0, 1, 4, -1])
assert p == q
p = expr_to_holonomic(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 = expr_to_holonomic(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 = expr_to_holonomic(erf(x)**2 + x)
q = HolonomicFunction((8*x**4 - 2*x**2 + 2)*Dx**2 + (6*x**3 - x/2)*Dx**3 + \
(x**2+ Rational(1, 4))*Dx**4, x, 0, [0, 1, 8/pi, 0])
assert p == q
p = expr_to_holonomic(cosh(x) * x)
q = HolonomicFunction((-x**2 + 2) - 2 * x * Dx + x**2 * Dx**2, x, 0,
[0, 1])
assert p == q
p = expr_to_holonomic(besselj(2, x))
q = HolonomicFunction((x**2 - 4) + x * Dx + x**2 * Dx**2, x, 0, [0, 0])
assert p == q
p = expr_to_holonomic(besselj(0, x) + exp(x))
q = HolonomicFunction((-x**2 - x/2 + S.Half) + (x**2 - x/2 - Rational(3, 2))*Dx + (-x**2 + x/2 + 1)*Dx**2 +\
(x**2 + x/2)*Dx**3, x, 0, [2, 1, S.Half])
assert p == q
p = expr_to_holonomic(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 = expr_to_holonomic(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 = expr_to_holonomic(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_expr()
q = log(x) / 2 - Ci(2 * x) / 2 + Ci(2) / 2
assert p == q
p = expr_to_holonomic(x**S.Half, x0=1)
q = HolonomicFunction(x * Dx - S.Half, x, 1, [1])
assert p == q
p = expr_to_holonomic(sqrt(1 + x**2))
q = HolonomicFunction((-x) + (x**2 + 1) * Dx, x, 0, [1])
assert p == q
assert (expr_to_holonomic(sqrt(x) + sqrt(2*x)).to_expr()-\
(sqrt(x) + sqrt(2*x))).simplify() == 0
assert expr_to_holonomic(3 * x +
2 * sqrt(x)).to_expr() == 3 * x + 2 * sqrt(x)
p = expr_to_holonomic((x**4 + x**3 + 5 * x**2 + 3 * x + 2) / x**2,
lenics=3)
q = HolonomicFunction((-2*x**4 - x**3 + 3*x + 4) + (x**5 + x**4 + 5*x**3 + 3*x**2 + \
2*x)*Dx, x, 0, {-2: [2, 3, 5]})
assert p == q
p = expr_to_holonomic(1 / (x - 1)**2, lenics=3, x0=1)
q = HolonomicFunction((2) + (x - 1) * Dx, x, 1, {-2: [1, 0, 0]})
assert p == q
a = symbols("a")
p = expr_to_holonomic(sqrt(a * x), x=x)
assert p.to_expr() == sqrt(a) * sqrt(x)
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**(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_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
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