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)**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_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_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_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)**Rational(
1, 3)).series() == ((2 * x - 3 * x**2)**Rational(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).together() == (cos(x)**2 / x**2).series(n=10, x0=1).together()
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_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, Rational(-1,
3)).cancel() == q.diff(
x, 2).cancel()
assert p.diff(x, 3).series().subs({
C_3: Rational(-1, 3),
C_0: 0
}) == q.diff(x, 3).series()
def test_HolonomicFunction_multiplication():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx + x + x * Dx**2, x)
q = HolonomicFunction(x * Dx + Dx * x + Dx**2, x)
r = HolonomicFunction((8*x**6 + 4*x**4 + 6*x**2 + 3) + (24*x**5 - 4*x**3 + 24*x)*Dx + \
(8*x**6 + 20*x**4 + 12*x**2 + 2)*Dx**2 + (8*x**5 + 4*x**3 + 4*x)*Dx**3 + \
(2*x**4 + x**2)*Dx**4, x)
assert p * q == r
p = HolonomicFunction(Dx**2 + 1, x)
q = HolonomicFunction(Dx - 1, x)
r = HolonomicFunction((2) + (-2) * Dx + (1) * Dx**2, x)
assert p * q == r
p = HolonomicFunction(Dx**2 + 1 + x + Dx, x)
q = HolonomicFunction((Dx * x - 1)**2, x)
r = HolonomicFunction((4*x**7 + 11*x**6 + 16*x**5 + 4*x**4 - 6*x**3 - 7*x**2 - 8*x - 2) + \
(8*x**6 + 26*x**5 + 24*x**4 - 3*x**3 - 11*x**2 - 6*x - 2)*Dx + \
(8*x**6 + 18*x**5 + 15*x**4 - 3*x**3 - 6*x**2 - 6*x - 2)*Dx**2 + (8*x**5 + \
10*x**4 + 6*x**3 - 2*x**2 - 4*x)*Dx**3 + (4*x**5 + 3*x**4 - x**2)*Dx**4, x)
assert p * q == r
p = HolonomicFunction(x * Dx**2 - 1, x)
q = HolonomicFunction(Dx * x - x, x)
r = HolonomicFunction((x - 3) + (-2 * x + 2) * Dx + (x) * Dx**2, x)
assert p * q == r
def test_HolonomicFunction_composition():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx - 1, x).composition(x**2 + x)
r = HolonomicFunction((-2 * x - 1) + Dx, x)
assert p == r
p = HolonomicFunction(Dx**2 + 1, x).composition(x**5 + x**2 + 1)
r = HolonomicFunction((125*x**12 + 150*x**9 + 60*x**6 + 8*x**3) + (-20*x**3 - 2)*Dx + \
(5*x**4 + 2*x)*Dx**2, x)
assert p == r
p = HolonomicFunction(Dx**2 * x + x, x).composition(2 * x**3 + x**2 + 1)
r = HolonomicFunction((216*x**9 + 324*x**8 + 180*x**7 + 152*x**6 + 112*x**5 + \
36*x**4 + 4*x**3) + (24*x**4 + 16*x**3 + 3*x**2 - 6*x - 1)*Dx + (6*x**5 + 5*x**4 + \
x**3 + 3*x**2 + x)*Dx**2, x)
assert p == r
p = HolonomicFunction(Dx**2 + 1, x).composition(1 - x**2)
r = HolonomicFunction((4 * x**3) - Dx + x * Dx**2, x)
assert p == r
p = HolonomicFunction(Dx**2 + 1, x).composition(x - 2 / (x**2 + 1))
r = HolonomicFunction((x**12 + 6*x**10 + 12*x**9 + 15*x**8 + 48*x**7 + 68*x**6 + \
72*x**5 + 111*x**4 + 112*x**3 + 54*x**2 + 12*x + 1) + (12*x**8 + 32*x**6 + \
24*x**4 - 4)*Dx + (x**12 + 6*x**10 + 4*x**9 + 15*x**8 + 16*x**7 + 20*x**6 + 24*x**5+ \
15*x**4 + 16*x**3 + 6*x**2 + 4*x + 1)*Dx**2, x)
assert p == r