def gauss_laguerre(n):
d = mp.matrix(mp.arange(1, 2 * n, 2))
e = mp.matrix(mp.arange(-1, -n - 1, -1))
z = mp.eye(n)[0, :]
tridiag_eigen(mp, d, e, z)
z = z.apply(lambda x: x**2)
return d, z.T
def gauss_hermite(n):
d = mp.matrix([mp.mpf('0.0') for _ in range(n)])
e = [mp.sqrt(k / 2) for k in mp.arange(1, n)]
e.append(mp.mpf('0.0'))
e = mp.matrix(e)
z = mp.eye(n)[0, :]
tridiag_eigen(mp, d, e, z)
z = mp.sqrt(mp.pi) * z.apply(lambda x: x**2)
return d, z.T
def gauss_legendre(n):
d = mp.matrix([mp.mpf('0.0') for _ in range(n)])
e = [k / mp.sqrt(4 * k**2 - 1) for k in mp.arange(1, n)]
e.append(mp.mpf('0.0'))
e = mp.matrix(e)
z = mp.eye(n)[0, :]
tridiag_eigen(mp, d, e, z)
z = 2 * z.apply(lambda x: x**2)
return d, z.T
def gauss_genlaguerre(n, alpha):
d = mp.matrix([i + alpha for i in mp.arange(1, 2 * n, 2)])
e = [-mp.sqrt(k * (k + alpha)) for k in mp.arange(1, n)]
e.append(mp.mpf('0.0'))
e = mp.matrix(e)
z = mp.eye(n)[0, :]
tridiag_eigen(mp, d, e, z)
z = mp.gamma(alpha + 1) * z.apply(lambda x: x**2)
return d, z.T
def gauss_gegenbauer(n, alpha):
d = mp.matrix([mp.mpf('0.0') for _ in range(n)])
e = [
mp.sqrt(k * (k + 2 * alpha - 1) / ((2 * k + 2 * alpha - 1)**2 - 1))
for k in mp.arange(1, n)
]
e.append(mp.mpf('0.0'))
e = mp.matrix(e)
z = mp.eye(n)[0, :]
tridiag_eigen(mp, d, e, z)
z = 2**(2 * alpha) * mp.beta(alpha + 1 / 2,
alpha + 1 / 2) * z.apply(lambda x: x**2)
return d, z.T
def _scheme_from_rc_mpmath(alpha, beta):
# Create vector cut of the first value of beta
n = len(alpha)
b = mp.zeros(n, 1)
for i in range(n - 1):
b[i] = mp.sqrt(beta[i + 1])
z = mp.zeros(1, n)
z[0, 0] = 1
d = mp.matrix(alpha)
tridiag_eigen(mp, d, b, z)
# nx1 matrix -> list of mpf
x = numpy.array([mp.mpf(sympy.N(xx, mp.dps)) for xx in d])
w = numpy.array([mp.mpf(sympy.N(beta[0], mp.dps)) * mp.power(ww, 2) for ww in z])
return x, w
def gauss_jacobi(n, alpha, beta):
d = mp.matrix([(beta**2 - alpha**2) / (2 * k + alpha + beta) /
(2 * k + alpha + beta - 2) for k in mp.arange(1, n + 1)])
e = [
k * (k + alpha) * (k + beta) * (k + alpha + beta) /
((2 * k + alpha + beta)**2 - 1) for k in mp.arange(1, n)
]
e = [
2 / (2 * k + alpha + beta) * mp.sqrt(x)
for k, x in zip(mp.arange(1, n), e)
]
e.append(mp.mpf('0.0'))
e = mp.matrix(e)
z = mp.eye(n)[0, :]
tridiag_eigen(mp, d, e, z)
z = 2**(alpha + beta + 1) * mp.beta(alpha + 1,
beta + 1) * z.apply(lambda x: x**2)
return d, z.T
def _gauss_from_coefficients_mpmath(alpha, beta, decimal_places):
mp.dps = decimal_places
# Create vector cut of the first value of beta
n = len(alpha)
b = mp.zeros(n, 1)
for i in range(n - 1):
# work around <https://github.com/fredrik-johansson/mpmath/issues/382>
x = beta[i + 1]
if isinstance(x, numpy.int64):
x = int(x)
b[i] = mp.sqrt(x)
z = mp.zeros(1, n)
z[0, 0] = 1
d = mp.matrix(alpha)
tridiag_eigen(mp, d, b, z)
# nx1 matrix -> list of mpf
x = numpy.array([mp.mpf(sympy.N(xx, decimal_places)) for xx in d])
w = numpy.array([
mp.mpf(sympy.N(beta[0], decimal_places)) * mp.power(ww, 2) for ww in z
])
return x, w