chebfftPy.py

from numpy import pi,cos,arange,array, flipud,\
     real,zeros, sqrt
from numpy.fft import fft,ifft

def chebfft(v):
    '''Chebyshev differentiation via fft.
       Ref.: Trefethen's 'Spectral Methods in MATLAB' book.
    '''
    N = len(v)-1
    if N == 0:
        w = 0.0 # only when N is even!
        return w
    x  = cos(pi*arange(0,N+1)/N)
    ii = arange(0,N)
    V = flipud(v[1:N]); V = list(v) + list(V);
    U = real(fft(V))
    b = list(ii); b.append(0); b = b + list(arange(1-N,0));
    w_hat = 1j*array(b)
    w_hat = w_hat * U
    W = real(ifft(w_hat))
    w = zeros(N+1)
    w[1:N] = -W[1:N]/sqrt(1-x[1:N]**2)
    w[0] = sum(ii**2*U[ii])/N + 0.5*N*U[N]
    w[N] = sum((-1)**(ii+1)*ii**2*U[ii])/N + \
              0.5*(-1)**(N+1)*N*U[N]
    return w