The authors discuss Chebyshev collocation methods and study several algorithms for computing Chebyshev spectral derivatives. Then they describe a preconditioning method for reducing the roundoff error. By means of a statistical approach they estimate the minimum possible roundoff error.
Using different algorithms they obtain some results on the accuracy of computing. The numerical errors associated with computing the elements of the differentiation matrix are described. They find out that if the entries of the matrix are computed accurately, then the roundoff error of the matrix-vector multiplication is as small as that obtained by the transform-recursion algorithm. For most practical grid sizes used in computations, the even-odd decomposition algorithm is found to be faster than the transform-recursion method.