ACCURACY AND SPEED IN COMPUTING THE CHEBYSHEV COLLOCATION DERIVATIVE

We study several algorithms for computing the Chebyshev spectral deriv ative and compare their roundoff error. For a large number of collocat ion points, the elements of the Chebyshev differentiation matrix, if c onstructed in the usual way, are not computed accurately. A subtle cau se is found to account for the poor accuracy when computing the deriva tive by the matrix-vector multiplication method. Methods for accuratel y computing the elements of the matrix are presented and we find that if the entries of the matrix are computed accurately, the roundoff err or of the matrix-vector multiplication is as small as that of the tran sform-recursion algorithm. Furthermore, results of the CPU time usage are shown for several different algorithms for computing the derivativ e by the Chebyshev collocation method for a wide variety of two-dimens ional grid sizes on both an IBM mainframe and a Cray 2 computer. We fi nd that which algorithm is fastest on a particular machine depends not only on the grid size, but also on small details of the computer hard ware as well. For most practical grid sizes used in computation, the e ven-odd decomposition algorithm is found to be faster than the transfo rm-recursion method.