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 dhermite_function(x,ns,mu=0.,scale=1.,shift=0.):
from spyctral.hermite.weights import dsqrt_weight, sqrt_weight
from numpy import array, zeros, exp,sqrt
from spyctral.common.maps import physical_scaleshift as pss
from spyctral.common.maps import standard_scaleshift as sss
ns = array(ns)
N = ns.size
mu = float(mu)
ps = hermite_polynomial(x,ns,mu,scale=scale,shift=shift)
dps = hermite_polynomial(x,ns,mu,d=1,scale=scale,shift=shift)
sss(x,shift=shift,scale=scale)
#x = (x-shift)/scale
#w = x**(mu)*exp(-1/2.*x**2)
w = sqrt_weight(x,mu=mu)
dw = dsqrt_weight(x,mu=mu)
pss(x,shift=shift,scale=scale)
#dw = dsqrt_weight(x,mu=mu,shift=shift,scale=scale)
# yay product rule
ps = (dps.T*w).T + (ps.T*dw).T
# I DON'T UNDERSTAND THIS (part of hermite_function)
#ps /= sqrt(scale)
# WARNING: THIS CAUSES PROBLEMS FOR VERY VERY LARGE N: it eliminates
# nan's
wflags = (w==0)
ps[wflags,:] = 0.
return ps/scale
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 weight(r,alpha=-1/2.,beta=-1/2.,shift=0.,scale=1.):
from spyctral.common.maps import standard_scaleshift as sss
from spyctral.common.maps import physical_scaleshift as pss
sss(r,shift=shift,scale=scale)
wfun = (1-r)**alpha*(1+r)**beta
wfun *= scale
pss(r,shift=shift,scale=scale)
return wfun
def weight(x,mu=0.,shift=0.,scale=1.):
from numpy import abs, exp
from spyctral.common.maps import physical_scaleshift as pss
from spyctral.common.maps import standard_scaleshift as sss
sss(x,shift=shift,scale=scale)
#xt = (x-shift)/scale
weight = abs(x)**(2*mu)*exp(-x**2)
pss(x,shift=shift,scale=scale)
return weight
def dsqrt_weight(x,mu=0.,shift=0.,scale=1.):
from spyctral.common.maps import physical_scaleshift as pss
from spyctral.common.maps import standard_scaleshift as sss
from numpy import exp, abs
sss(x,shift=shift,scale=scale)
#xt = (x-shift)/scale
w = exp(-x**2/2)*(mu*abs(x)**(mu-1) - x*abs(x)**mu)
pss(x,shift=shift,scale=scale)
return w/scale
def weight(theta,gamma=0.,delta=0.,shift=0.,scale=1.):
from numpy import array, cos
from spyctral.common.maps import standard_scaleshift as sss
from spyctral.common.maps import physical_scaleshift as pss
theta = array(theta)
sss(theta,scale=scale,shift=shift)
w = ((1-cos(theta))**delta)*((1+cos(theta))**gamma)
pss(theta,scale=scale,shift=shift)
return w
def gq(N,mu=0.,shift=0.,scale=1.):
from spyctral.hermite.coeffs import recurrence_range
from spyctral.opoly1d.quad import gq as ogq
from spyctral.common.maps import physical_scaleshift as pss
[a_s,b_s] = recurrence_range(N,mu)
[x,w] = ogq(a_s,b_s)
pss(x,scale=scale,shift=shift)
w *= scale
return [x,w]
def gq(N,s=1.,t=1.,scale=1.,shift=0.):
from spyctral.common.maps import physical_scaleshift as pss
from spyctral.mapjpoly.maps import st_to_ab, r_to_x
from spyctral.jacobi.quad import gq
[alpha,beta]=st_to_ab(s,t)
[r,w] = gq(N,alpha=alpha,beta=beta)
x = r_to_x(r)
#x *= scale
#x += shift
pss(x,scale=scale,shift=shift)
return [x,w]
def hermite_polynomial(x,n,mu=0.,d=0,shift=0.,scale=1.):
from numpy import max, sqrt
from spyctral.hermite.coeffs import recurrence_range
from spyctral.opoly1d.eval import eval_normalized_opoly
from spyctral.common.maps import physical_scaleshift as pss
from spyctral.common.maps import standard_scaleshift as sss
N = max(n)
[a,b] = recurrence_range(N+1,mu)
sss(x,shift=shift,scale=scale)
temp = eval_normalized_opoly(x,n,a,b,d)
pss(x,shift=shift,scale=scale)
return temp/sqrt(scale)
def weight(x,alpha=0.,shift=0.,scale=1.):
"""
Returns the weight function for Laguerre orthogonal polynomials with
parameter specifications alpha, shift, and scale.
"""
from spyctral.common.maps import standard_scaleshift as sss
from spyctral.common.maps import physical_scaleshift as pss
from numpy import exp
sss(x,shift=shift,scale=scale)
wfun = exp(-x)*x**alpha
wfun *= scale
pss(x,shift=shift,scale=scale)
return wfun
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
def sqrt_weight_bias(theta,gamma=0.,delta=0.,shift=0.,scale=1.):
from numpy import cos, sin, exp
from spyctral.common.maps import standard_scaleshift as sss
from spyctral.common.maps import physical_scaleshift as pss
from scipy import pi
from scipy import power as pw
sss(theta,scale=scale,shift=shift)
phase = exp(1j*(gamma+delta)/2.*(pi-theta))
w = phase*( pw(sin(theta/2.),delta) * \
pw(cos(theta/2.),gamma)) *\
2**((gamma+delta)/2.)
pss(theta,scale=scale,shift=shift)
return w
def gauss_quadrature(N, alpha=0.0, shift=0.0, scale=1.0):
"""
Returns the N-point Laguerre-Gauss(alpha) quadrature rule over the interval
(shift,inf). 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 spyctral.laguerre.coeffs import recurrence_range
from spyctral.opoly1d.quad import gq as ogq
from spyctral.common.maps import physical_scaleshift as pss
[a, b] = recurrence_range(N, alpha)
temp = ogq(a, b)
pss(temp[0], scale=scale, shift=shift)
return temp
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 weighted_fseries(theta,k,gamma=0.,delta=0.,shift=0.,scale=1.):
from spyctral.common.maps import standard_scaleshift as sss
from spyctral.common.maps import physical_scaleshift as pss
from numpy import sqrt, array
from weights import sqrt_weight_bias as wsqrt_bias
theta = array(theta)
theta = theta.ravel()
sss(theta,shift=shift,scale=scale)
psi = fseries(theta,k,gamma=gamma,delta=delta,shift=0.,scale=1.)
phi = (wsqrt_bias(theta,gamma=gamma,delta=delta,shift=0.,scale=1.)*psi.T).T
pss(theta,shift=shift,scale=scale)
# Scaling:
return phi/sqrt(scale)
def dsqrt_weight_bias(theta,gamma=0.,delta=0.,shift=0.,scale=1.):
from numpy import cos, sin, exp, abs
from spyctral.common.maps import standard_scaleshift as sss
from spyctral.common.maps import physical_scaleshift as pss
from scipy import pi
sss(theta,scale=scale,shift=shift)
phase = exp(1j*(gamma+delta)/2.*(pi-theta))*\
2**((gamma+delta-4)/2.)
if delta != 0.:
phase *= abs(sin(theta/2.))**(delta-1)
if gamma != 0.:
phase *= cos(theta/2.)**(gamma-1)
w = phase*( delta*(1+exp(-1j*theta)) -\
gamma*(1+exp(1j*theta)))
pss(theta,scale=scale,shift=shift)
return w/scale
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
def dsqrt_weight(x,alpha=0., shift=0., scale=1.):
"""
Returns the derivative of the square root of the weight function for
Laguerre orthogonal polynomials with parameter specifications alpha, shift,
and scale.
"""
from spyctral.common.maps import standard_scaleshift as sss
from spyctral.common.maps import physical_scaleshift as pss
from numpy import exp, sqrt
sss(x,shift=shift,scale=scale)
if abs(alpha)<1e-8:
dwfun = -1/2*exp(-x/2)
else:
dwfun = (alpha/2)*x**(alpha/2-1)*exp(-x/2) + \
x**(alpha/2)*-1/2*exp(-x/2)
dwfun *= sqrt(scale)
dwfun /= scale # Jacobian factor
pss(x,shift=shift,scale=scale)
return dwfun
def pgq(N,mu=0.,scale=1.,shift=0.):
from numpy import array, zeros, exp
from spyctral.common.maps import physical_scaleshift as pss
from spyctral.common.maps import standard_scaleshift as sss
mu = float(mu) # ???
#[x,w] = gq(N,mu,scale=scale,shift=shift)
[x,w] = gq(N,mu=mu)
#sss(x,shift=shift,scale=scale)
#xt = (x-shift)/scale
w /= x**(2*mu)*exp(-x**2)
pss(x,shift=shift,scale=scale)
# I DON'T UNDERSTAND THIS
#w *= scale**2
w *= scale
return [x,w]
def gq(N,gamma=0.,delta=0.,shift=0.,scale=1.):
from numpy import arccos, hstack, isnan
from spyctral.jacobi.quad import gq as jgq
from spyctral.jacobi.quad import grq as jgrq
from spyctral.common.maps import physical_scaleshift as pss
N = int(N)
tol = 1e-8
if (N%2)==0:
[r,wr] = jgq(N/2,delta-1/2.,gamma-1/2.)
r = r.squeeze()
wr = wr.squeeze()
temp = arccos(r[::-1])
wr = wr[::-1]
theta = hstack((-temp[::-1],temp))
w = hstack((wr[::-1],wr))
pss(theta,scale=scale,shift=shift)
return [theta,w]
else:
[r,wr] = jgrq((N+1)/2,delta-1/2.,gamma-1/2.,r0=1.)
r = r.squeeze()
wr = wr.squeeze()
temp = arccos(r[::-1])
# Silly arccos machine epsilon crap
temp[isnan(temp)] = 0.
theta = hstack((-temp[::-1],temp[1:]))
wr = wr[::-1]
wr[0] *= 2
w = hstack((wr[::-1],wr[1:]))
pss(theta,scale=scale,shift=shift)
return [theta,w]
def r_to_x(r,scale=1.,shift=0.):
from numpy import sqrt
temp = r/sqrt(1-r**2)
pss(temp,scale=scale,shift=shift)
return temp
