def gauss_lobatto_quadrature(N,alpha=-1/2.,beta=-1/2.,r0=None,shift=0.,scale=1.) :
# Returns the N-point Jacobi-Gauss-Lobatto(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 coeffs import recurrence_range
from numpy import array
from spyctral import chebyshev
from spyctral.opoly1d.quad import glq as oglq
from spyctral.common.maps import physical_scaleshift as pss
from spyctral.common.maps import standard_scaleshift as sss
if r0 is None:
r0 = array([-scale,scale])+shift
r0 = array(r0)
tol = 1e-12;
if (abs(alpha+1/2.)<tol) & (abs(beta+1/2.)<tol) :
return chebyshev.quad.glq(N,shift=shift,scale=scale)
else :
[a,b] = recurrence_range(N,alpha,beta)
sss(r0,scale=scale,shift=shift)
temp = oglq(a,b,r0=r0)
pss(r0,scale=scale,shift=shift)
pss(temp[0],scale=scale,shift=shift)
#temp[1] *= scale
return temp
def jacobi_polynomial(x,n,alpha=-1/2.,beta=-1/2.,d=0, normalization='normal',scale=1., shift=0.) :
"""
Evaluates the Jacobi polynomials of class (alpha,beta), order n (list)
at the points x (list/array). The normalization is specified by the keyword
argument. Possibilities:
'normal', 'normalized':
'monic':
"""
from numpy import array, ndarray, ones, sqrt
from coeffs import recurrence_range
from spyctral.opoly1d.eval import eval_opoly, eval_normalized_opoly
from spyctral.common.maps import physical_scaleshift as pss
from spyctral.common.maps import standard_scaleshift as sss
x = array(x)
if all([type(n) != dtype for dtype in [list,ndarray]]):
# Then it's probably an int
n = array([n])
n = array(n)
[a,b] = recurrence_range(max(n)+2,alpha=alpha,beta=beta)
# dummy function for scaling
def scale_ones(n,alpha,beta):
return ones([1,len(n)])
# Set up different normalizations
normal_fun = {'normal': eval_normalized_opoly,
'normalized': eval_normalized_opoly,
'monic': eval_opoly}
normal_scale = {'normal': scale_ones,
'normalized': scale_ones,
'monic': scale_ones}
# Shift to standard interval and use opoly 3-term recurrence
normalization = normalization.lower()
sss(x,scale=scale,shift=shift)
temp = normal_fun[normalization](x,n,a,b,d=d)
temp *= normal_scale[normalization](n,alpha,beta)
pss(x,scale=scale,shift=shift)
return temp
def gauss_quadrature(N,alpha=-1/2.,beta=-1/2.,shift=0.,scale=1.) :
# Returns the N-point Jacobi-Gauss(alpha,beta) quadrature rule over the interval
# (-scale,scale)+shift.
# The quadrature rule is *not* normalized in the sense that the affine parameters
# shift and scale are built into the new weight function for which this
# quadrature rule is valid.
from coeffs import recurrence_range
from spyctral.opoly1d.quad import gq as ogq
from spyctral import chebyshev
from spyctral.common.maps import physical_scaleshift as pss
tol = 1e-12;
if (abs(alpha+1/2.)<tol) & (abs(beta+1/2.)<tol) :
return chebyshev.quad.gq(N,shift=shift,scale=scale)
else :
[a,b] = recurrence_range(N,alpha,beta)
temp = ogq(a,b)
pss(temp[0],scale=scale,shift=shift)
return temp
