Chebyshev Spectral Differentiation via Fast Fourier Transform

f(t)

′

f

(t)

error in f(t)

sum of error square vs. N

interior points

10

Consider the function

f(x)=sin(5x)

x

e

with derivative

f'(x)=sin(5x)

x

e

+5cos(5x)

x

e

. This Demonstration uses Chebyshev differentiation via Fast Fourier Transform (FFT) [1] to estimate

f'(x)

at the

(N+1)

Chebyshev–Gauss–Lobatto points. You can change the number of interior points,

(N-1)

. The error (i.e., the difference between the exact and approximate values of

f'(x)

) decreases for large values of

N

.