def stiff_overhead(N,alpha=-1/2.,beta=-1/2.,scale=1.):
from jfft import rmatrix_entries
from numpy import arange
from coeffs import zetan
zetas = zetan(arange(N),alpha=alpha,beta=beta)/scale
Rs = rmatrix_entries(N-1,alpha,beta,1,1)
return [zetas[1:],Rs]
def jacobi_polynomial_derivative(x,n,alpha=-1/2.,beta=-1/2.,normalization='normal',scale=1.,shift=0.):
"""
Evaluates the derivative of the Jacobi polynomial with specified
normalization. Although this is possible using jacobi_polynomial, this
function evaluates the polynomial of order (alpha+1,beta+1) and scales it
accordingly.
"""
from coeffs import zetan
from numpy import array
zetas = zetan(array(n),alpha=alpha,beta=beta,normalization=normalization)/scale
return jacobi_polynomial(x,array(n)-1,alpha+1.,beta+1.,normalization=normalization,\
shift=shift,scale=scale)*zetas
def stiff_apply(F,alpha=-1/2.,beta=-1/2.,scale=1.):
"""
Applies the modal stiffness matrix to the coefficients of the L2 normalized
polynomials. Makes use of the recurrence constant zetan in addition to the
sparse representation of the connection coefficients Is an O(N) operation
"""
from jfft import rmatrix_invert
from numpy import arange,hstack, array
from coeffs import zetan
N = F.size
# Input F is of class (alpha,beta). Take the derivative by promoting
# basis functions to (alpha+1,beta+1) using zetan:
zetas = zetan(arange(N),alpha=alpha,beta=beta)/scale
# Now demote the (alpha+1,beta+1) coefficients back down to
# (alpha,beta)
filler = array([0.])
return hstack((rmatrix_invert(zetas[1:]*F[1:],alpha,beta,1,1),filler))