本文整理汇总了Python中sympy.holonomic.HolonomicFunction类的典型用法代码示例。如果您正苦于以下问题：Python HolonomicFunction类的具体用法？Python HolonomicFunction怎么用？Python HolonomicFunction使用的例子？那么恭喜您, 这里精选的类代码示例或许可以为您提供帮助。
在下文中一共展示了HolonomicFunction类的8个代码示例，这些例子默认根据受欢迎程度排序。您可以为喜欢或者感觉有用的代码点赞，您的评价将有助于系统推荐出更棒的Python代码示例。
示例1: test_series
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)示例2: test_to_hyper
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), (2, 3), -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示例3: test_HolonomicFunction_addition
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示例4: test_to_Sequence_Initial_Coniditons
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示例5: test_to_expr
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示例6: test_diff
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()示例7: test_evalf_rk4
def test_evalf_rk4():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
# log(1+x)
p = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1])
# path taken is a straight line from 0 to 1, on the real axis
r = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1]
s = '0.693146363174626' # approx. equal to log(2) i.e. 0.693147180559945
assert sstr(p.evalf(r)[-1]) == s
# path taken is a traingle 0-->1+i-->2
r = [0.1 + 0.1*I]
for i in range(9):
r.append(r[-1]+0.1+0.1*I)
for i in range(10):
r.append(r[-1]+0.1-0.1*I)
# close to the exact solution 1.09861228866811
# imaginary part also close to zero
s = '1.09861574485151 + 1.36082967699958e-7*I'
assert sstr(p.evalf(r)[-1]) == s
# sin(x)
p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1])
s = '0.90929463522785 + 1.52655665885959e-16*I'
assert sstr(p.evalf(r)[-1]) == s
# computing sin(pi/2) using this method
# using a linear path from 0 to pi/2
r = [0.1]
for i in range(14):
r.append(r[-1] + 0.1)
r.append(pi/2)
s = '0.999999895088917' # close to 1.0 (exact solution)
assert sstr(p.evalf(r)[-1]) == s
# trying different path, a rectangle (0-->i-->pi/2 + i-->pi/2)
# computing the same value sin(pi/2) using different path
r = [0.1*I]
for i in range(9):
r.append(r[-1]+0.1*I)
for i in range(15):
r.append(r[-1]+0.1)
r.append(pi/2+I)
for i in range(10):
r.append(r[-1]-0.1*I)
# close to 1.0
s = '1.00000003415141 + 6.11940487991086e-16*I'
assert sstr(p.evalf(r)[-1]) == s
# cos(x)
p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0])
# compute cos(pi) along 0-->pi
r = [0.05]
for i in range(61):
r.append(r[-1]+0.05)
r.append(pi)
# close to -1 (exact answer)
s = '-0.999999993238714'
assert sstr(p.evalf(r)[-1]) == s
# a rectangular path (0 -> i -> 2+i -> 2)
r = [0.1*I]
for i in range(9):
r.append(r[-1]+0.1*I)
for i in range(20):
r.append(r[-1]+0.1)
for i in range(10):
r.append(r[-1]-0.1*I)
p = HolonomicFunction(Dx**2 + 1, x, 0, [1,1]).evalf(r)
s = '0.493152791638442 - 1.41553435639707e-15*I'
assert sstr(p[-1]) == s示例8: test_evalf_euler
def test_evalf_euler():
x = symbols('x')
R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx')
# log(1+x)
p = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1])
# path taken is a straight line from 0 to 1, on the real axis
r = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1]
s = '0.699525841805253' # approx. equal to log(2) i.e. 0.693147180559945
assert sstr(p.evalf(r, method='Euler')[-1]) == s
# path taken is a traingle 0-->1+i-->2
r = [0.1 + 0.1*I]
for i in range(9):
r.append(r[-1]+0.1+0.1*I)
for i in range(10):
r.append(r[-1]+0.1-0.1*I)
# close to the exact solution 1.09861228866811
# imaginary part also close to zero
s = '1.07530466271334 - 0.0251200594793912*I'
assert sstr(p.evalf(r, method='Euler')[-1]) == s
# sin(x)
p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1])
s = '0.905546532085401 - 6.93889390390723e-18*I'
assert sstr(p.evalf(r, method='Euler')[-1]) == s
# computing sin(pi/2) using this method
# using a linear path from 0 to pi/2
r = [0.1]
for i in range(14):
r.append(r[-1] + 0.1)
r.append(pi/2)
s = '1.08016557252834' # close to 1.0 (exact solution)
assert sstr(p.evalf(r, method='Euler')[-1]) == s
# trying different path, a rectangle (0-->i-->pi/2 + i-->pi/2)
# computing the same value sin(pi/2) using different path
r = [0.1*I]
for i in range(9):
r.append(r[-1]+0.1*I)
for i in range(15):
r.append(r[-1]+0.1)
r.append(pi/2+I)
for i in range(10):
r.append(r[-1]-0.1*I)
# close to 1.0
s = '0.976882381836257 - 1.65557671738537e-16*I'
assert sstr(p.evalf(r, method='Euler')[-1]) == s
# cos(x)
p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0])
# compute cos(pi) along 0-->pi
r = [0.05]
for i in range(61):
r.append(r[-1]+0.05)
r.append(pi)
# close to -1 (exact answer)
s = '-1.08140824719196'
assert sstr(p.evalf(r, method='Euler')[-1]) == s
# a rectangular path (0 -> i -> 2+i -> 2)
r = [0.1*I]
for i in range(9):
r.append(r[-1]+0.1*I)
for i in range(20):
r.append(r[-1]+0.1)
for i in range(10):
r.append(r[-1]-0.1*I)
p = HolonomicFunction(Dx**2 + 1, x, 0, [1,1]).evalf(r, method='Euler')
s = '0.501421652861245 - 3.88578058618805e-16*I'
assert sstr(p[-1]) == s