Chapter 8. Chebyshev spectral methods

8.3. CHEBYSHEV DIFFERENTIATION BY THE FFT TREFETHEN 1994 275
ALGORITHM FOR CHEBYSHEV DIFFERENTIATION
1. Given data fv j g de ned at the Chebyshev points fx j g, 0 j N, think of the same
data as being de ned at the equally spaced points f j g in [0 ].
2. (FFT) Find the coe cients fa ng of the trigonometric polynomial
that interpolates fv j g at f j g.
3. (FFT) Compute the derivative
dq
d
NX
q( )= an cos n (8:3:12)
n=0
NX
= ; nan sin n : (8:3:13)
n=0
4. Change variables to obtain the derivative with respect to x:
dq dq d
=
dx d dx =
NX
n=0
na n sin n
sin
NX
=
n=0
na n sin n
p 1;x 2
At x = 1, i.e. =0 , L'Hopital's rule gives the special values
NX
: (8:3:14)
dq
( 1) = ( 1)
dx
n=0
n n 2 an (8:3:15)
5. Evaluate the result at the Chebyshev points:
wj = dq
dx (xj ): (8:3:16)
Note that by (8.3.3), equation (8.3.12) can be interpreted as a linear combination of
Chebyshev polynomials, and by (8.3.6), equation (8.3.14) is the corresponding linear combination
of derivatives.* But of course the algorithmic content of the description above
relates to the variable, for in Steps 2 and 3, we have performed Fourier spectral di erentiation
exactly as in x7.3: discrete Fourier transform, multiply by i ,inverse discrete Fourier
transform. Only the use of sines and cosines rather than complex exponentials, and of n
instead of , has disguised the process somewhat.
*orofChebyshev polynomials Un(x) ofthe second kind.