def jacobi_function(x,ns,s=1.,t=1.,scale=1.,shift=0.):
from spyctral.jacobi.eval import jpoly
from spyctral.mapjpoly.maps import x_to_r, st_to_ab
[alpha,beta] = st_to_ab(s,t)
return jpoly(x_to_r(x,scale=scale,shift=shift),
ns,alpha=alpha,beta=beta)
def fseries(theta,k,gamma=0.,delta=0.,shift=0.,scale=1.):
"""
Evaluates the generalized Szego-Fourier functions at the locations theta \in
[-pi,pi]. This function mods the inputs theta to lie in this interval and then
evaluates them. The function class is (g,d), and the function index is the
vector of integers k
"""
from numpy import array, ndarray, abs, zeros
from numpy import sin, cos, sqrt, sign
from scipy import pi
from spyctral.jacobi.eval import jpoly
from spyctral.common.maps import standard_scaleshift as sss
from spyctral.common.maps import physical_scaleshift as pss
# Preprocessing: unravelling, etc.
theta = array(theta)
theta = theta.ravel()
sss(theta,shift=shift,scale=scale)
theta = (theta+pi) % (2*pi) - pi
if type(k) != (ndarray or list):
k = [k]
k = array(k,dtype='int')
k = k.ravel()
kneq0 = k != 0
r = cos(theta)
# Evaluate polynomials and multiplication factors
p1 = jpoly(r,abs(k),delta-1/2.,gamma-1/2.).reshape([theta.size,k.size])
# Add things together
Psi = zeros([theta.size,k.size],dtype=complex)
Psi[:,~kneq0] = 1/sqrt(2)*p1[:,~kneq0]
if k[kneq0].any():
p2 = jpoly(r,abs(k[kneq0])-1,delta+1/2.,gamma+1/2.).\
reshape([theta.size,k[kneq0].size])
kmat = sign(k[kneq0])
tmat = sin(theta)
p2 = 1j*(p2.T*tmat).T*kmat
Psi[:,kneq0] = 1/2.*(p1[:,kneq0] + p2)
pss(theta,shift=shift,scale=scale)
return Psi.squeeze()
def dfseries(theta,k,gamma=0.,delta=0.,shift=0.,scale=1.):
"""
Evaluates the derivative of the generalized Szego-Fourier functions at the locations theta \in
[-pi,pi]. This function mods the inputs theta to lie in this interval and then
evaluates them. The function class is (gamma,delta), and the function index is the
vector of integers k
"""
from numpy import array, zeros, dot, any, sum
from numpy import sin, cos, sqrt, abs, sign
from scipy import pi
from spyctral.jacobi.eval import jpoly, djpoly
from spyctral.common.maps import standard_scaleshift as sss
from spyctral.common.maps import physical_scaleshift as pss
# Preprocessing: unravelling, etc.
theta = array(theta)
theta = theta.ravel()
sss(theta,shift=shift,scale=scale)
# Now theta \in [-pi,pi]
theta = (theta+pi) % (2*pi) - pi
k = array(k)
k = k.ravel()
kneq0 = k != 0
a = delta-1/2.
b = gamma-1/2.
r = cos(theta)
# Term-by-term: first the even term
dPsi = zeros([theta.size,k.size],dtype=complex)
dPsi += 1/2.*djpoly(r,abs(k),a,b)
dPsi = (-sin(theta)*dPsi.T).T
dPsi[:,~kneq0] *= sqrt(2)
# Now the odd term:
if any(kneq0):
term2 = zeros([theta.size,sum(kneq0)],dtype=complex)
term2 += (cos(theta)*jpoly(r,abs(k[kneq0])-1,a+1,b+1).T).T
term2 += (-sin(theta)**2*djpoly(r,abs(k[kneq0])-1,a+1,b+1).T).T
term2 = 1./2*term2*(1j*sign(k[kneq0]))
else:
term2 = 0.
dPsi[:,kneq0] += term2
# Transform theta back to original interval
pss(theta,shift=shift,scale=scale)
return dPsi/scale