def compute_gauss_jacobi_points(a, b, m):
"""Computes the m roots of P_{m}^{a,b} on [-1,1] by Newton's method.
The initial guesses are the Chebyshev points. Algorithm
implemented in Python from the pseudocode given by Karniadakis and
Sherwin"""
x = []
eps = 1.e-8
max_iter = 100
for k in range(0, m):
r = -math.cos((2.0 * k + 1.0) * math.pi / (2.0 * m))
if k > 0:
r = 0.5 * (r + x[k - 1])
j = 0
delta = 2 * eps
while j < max_iter:
s = 0
for i in range(0, k):
s = s + 1.0 / (r - x[i])
f = jacobi.eval_jacobi(a, b, m, r)
fp = jacobi.eval_jacobi_deriv(a, b, m, r)
delta = f / (fp - f * s)
r = r - delta
if math.fabs(delta) < eps:
break
else:
j = j + 1
x.append(r)
return x
def compute_gauss_jacobi_points(a, b, m):
"""Computes the m roots of P_{m}^{a,b} on [-1,1] by Newton's method.
The initial guesses are the Chebyshev points. Algorithm
implemented in Python from the pseudocode given by Karniadakis and
Sherwin"""
x = []
eps = 1.e-8
max_iter = 100
for k in range(0, m):
r = -math.cos((2.0 * k + 1.0) * math.pi / (2.0 * m))
if k > 0:
r = 0.5 * (r + x[k - 1])
j = 0
delta = 2 * eps
while j < max_iter:
s = 0
for i in range(0, k):
s = s + 1.0 / (r - x[i])
f = jacobi.eval_jacobi(a, b, m, r)
fp = jacobi.eval_jacobi_deriv(a, b, m, r)
delta = f / (fp - f * s)
r = r - delta
if math.fabs(delta) < eps:
break
else:
j = j + 1
x.append(r)
return x
def compute_gauss_jacobi_rule(a, b, m):
xs = compute_gauss_jacobi_points(a, b, m)
a1 = math.pow(2, a + b + 1)
a2 = math.gamma(a + m + 1)
a3 = math.gamma(b + m + 1)
a4 = math.gamma(a + b + m + 1)
a5 = math.factorial(m)
a6 = a1 * a2 * a3 / a4 / a5
ws = [a6 / (1.0 - x**2.0) / jacobi.eval_jacobi_deriv(a, b, m, x)**2.0
for x in xs]
return xs, ws
def compute_gauss_jacobi_rule(a, b, m):
xs = compute_gauss_jacobi_points(a, b, m)
a1 = math.pow(2, a + b + 1)
a2 = math.gamma(a + m + 1)
a3 = math.gamma(b + m + 1)
a4 = math.gamma(a + b + m + 1)
a5 = math.factorial(m)
a6 = a1 * a2 * a3 / a4 / a5
ws = [a6 / (1.0 - x**2.0) / jacobi.eval_jacobi_deriv(a, b, m, x)**2.0
for x in xs]
return xs, ws