Ws. Don et A. Solomonoff, ACCURACY AND SPEED IN COMPUTING THE CHEBYSHEV COLLOCATION DERIVATIVE, SIAM journal on scientific computing, 16(6), 1995, pp. 1253-1268
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.