def gauss_radau_quadrature(N, alpha=0.0, r0=None, shift=0.0, scale=1.0):
"""
Returns the N-point Laguerre-Gauss-Radau(alpha) quadrature rule over
the interval (shift,inf) For nontrivial values of shift, scale, the
weight function associated with this quadrature rule is the directly-mapped
Laguerre weight + the Jacobian factor introduced by scale.
"""
from numpy import array
from spyctral.laguerre.coeffs import recurrence_range
from spyctral.opoly1d.quad import grq as ogrq
from spyctral.common.maps import physical_scaleshift as pss
from spyctral.common.maps import standard_scaleshift as sss
if r0 is None:
r0 = 0 + shift
r0 = array(r0)
[a, b] = recurrence_range(N, alpha)
sss(r0, scale=scale, shift=shift)
temp = ogrq(a, b, r0=r0)
pss(r0, scale=scale, shift=shift)
pss(temp[0], scale=scale, shift=shift)
return temp
def gauss_radau_quadrature(N,alpha=-1/2.,beta=-1/2.,r0=None,shift=0.,scale=1.) :
# Returns the N-point Jacobi-Gauss-Radau(alpha,beta) quadrature rule over the interval
# (-scale,scale)+shift
# For nontrivial values of shift, scale, the weight function associated with
# this quadrature rule is the directly-mapped Jacobi weight + the Jacobian
# factor introduced by scale.
from numpy import array
from coeffs import recurrence_range
from spyctral.opoly1d.quad import grq as ogrq
from spyctral import chebyshev
from spyctral.common.maps import physical_scaleshift as pss
from spyctral.common.maps import standard_scaleshift as sss
if r0 is None:
r0 = [-scale+shift]
r0 = array(r0)
tol = 1e-12;
if (abs(alpha+1/2.)<tol) & (abs(beta+1/2.)<tol) & (abs(abs(r0)-1.)<tol) :
return chebyshev.quad.grq(N,r0=r0,shift=shift,scale=scale)
else :
[a,b] = recurrence_range(N,alpha,beta)
sss(r0,scale=scale,shift=shift)
temp = ogrq(a,b,r0=r0)
pss(r0,scale=scale,shift=shift)
pss(temp[0],scale=scale,shift=shift)
return temp
def grq(N,mu=0.,r0=0.,shift=0,scale=1) :
from spyctral.hermite.coeffs import recurrence_range
from spyctral.opoly1d.quad import grq as ogrq
from spyctral.common.maps import physical_scaleshift as pss
[a_s,b_s] = recurrence_range(N,mu)
[x,w] = ogrq(a_s,b_s,N)
pss(x,shift=shift,scale=scale)
w *= scale
return [x,w]