def test_derivative_expansion(self):
"""
Expansion coefficients of derivatives.
"""
alpha = 10*np.random.rand(1)[0]
beta = 10*np.random.rand(1)[0]
J = JacobiPolynomials(alpha=alpha, beta=beta)
N = 13
K = 11
x, w = J.gauss_quadrature(2*N)
V = J.eval(x, range(K+1))
for s in range(4):
C = J.derivative_expansion(s, N, K)
Vd = J.eval(x, range(N+1), d=s)
C2 = Vd.T @ np.diag(w) @ V
reserror = np.linalg.norm(C-C2)
msg = "Failed for (s, alpha, beta)=({0:d}, {1:1.6f}, {2:1.6f})".format(s, alpha, beta)
delta = 1e-8
self.assertAlmostEqual(reserror, 0, delta=delta, msg=msg)
def test_gq_modification_global(self):
""" gq_modification for a global integral
Testing of gq_modification on an interval [-1,1] with specified
integrand.
"""
alpha = -1. + 6 * np.random.rand()
beta = -1. + 6 * np.random.rand()
J = JacobiPolynomials(alpha=alpha, beta=beta)
delta = 1e-8
N = 10
G = np.zeros([N, N])
for n in range(N):
for m in range(N):
# Integrate the entire integrand
integrand = lambda x: J.eval(x, m).flatten() * J.eval(
x, n).flatten() * jacobi_weight_normalized(x, alpha, beta)
G[n, m] = quad.gq_modification(integrand,
-1,
1,
ceil(1 + (n + m) / 2.),
gamma=[alpha, beta])
errstr = 'Failed for (N,alpha,beta) = ({0:d}, {1:1.6f}, {2:1.6f})'.format(
N, alpha, beta)
self.assertAlmostEqual(np.linalg.norm(G - np.eye(N), ord=np.inf),
0,
delta=delta,
msg=errstr)
def test_ratio(self):
""" Evaluation of orthogonal polynomial ratios. """
alpha = -1. + 10*np.random.rand(1)[0]
beta = -1. + 10*np.random.rand(1)[0]
J = JacobiPolynomials(alpha=alpha, beta=beta)
N = int(np.ceil(60*np.random.rand(1)))
x = (1 + 5*np.random.rand(1)) * (1 + np.random.rand(50))
y = (1 + 5*np.random.rand(1)) * (-1 - np.random.rand(50))
x = np.concatenate([x, y])
P = J.eval(x, range(N+1))
rdirect = np.zeros([x.size, N+1])
rdirect[:, 0] = P[:, 0]
rdirect[:, 1:] = P[:, 1:]/P[:, :-1]
r = J.r_eval(x, range(N+1))
delta = 1e-6
errs = np.abs(r-rdirect)
i, j = np.where(errs > delta)[:2]
if i.size > 0:
errstr = 'Failed for alpha={0:1.3f}, beta={1:1.3f}, n={2:d}, x={3:1.6f}'.format(alpha, beta, j[0], x[i[0]])
else:
errstr = ''
self.assertAlmostEqual(np.linalg.norm(errs, ord=np.inf), 0, delta=delta, msg=errstr)
def test_gq(self):
"""Gaussian quadrature integration accuracy"""
alpha = -1. + 10*np.random.rand(1)[0]
beta = -1. + 10*np.random.rand(1)[0]
J = JacobiPolynomials(alpha=alpha, beta=beta)
N = int(np.ceil(60*np.random.rand(1)))
x, w = J.gauss_quadrature(N)
w /= w.sum() # Force probability measure
V = J.eval(x, range(2*N))
integrals = np.dot(w, V)
integrals[0] -= V[0, 0] # Exact value
self.assertAlmostEqual(np.linalg.norm(integrals, ord=np.inf), 0.)
def test_idist_legendre(self):
"""Evaluation of Legendre induced distribution function."""
J = JacobiPolynomials(alpha=0., beta=0.)
n = int(np.ceil(25*np.random.rand(1))[0])
M = 25
x = -1. + 2*np.random.rand(M)
# JacobiPolynomials method
F1 = J.idist(x, n)
y, w = J.gauss_quadrature(n+1)
# Exact: integrate density
F2 = np.zeros(F1.shape)
for xind, xval in enumerate(x):
yquad = (y+1)/2.*(xval+1) - 1.
F2[xind] = np.dot(w, J.eval(yquad, n)**2) * (xval+1)/2
self.assertAlmostEqual(np.linalg.norm(F1-F2, ord=np.inf), 0.)