A POLYNOMIAL INTERPOLATION PROCESS AT QUASI-CHEBYSHEV NODES WITH THE FFT

Interpolation polynomial p(n) at the Chebyshev nodes cos pi j/n (0 <= j <= n) for smooth functions is known to converge fast as n -> infinity. The sequence {p(n)} is constructed recursively and efficiently in O(n log(2) n) flops for each p(n) by using the FFT, where n is increased geometrically, n = 2(i) (i = 2, 3, ... ), until an estimated error is within a given tolerance of epsilon. This sequence {2(j)}, however, grows too fast to get p(n) of proper n, often a much higher accuracy than epsilon being achieved. To cope with this problem we present quasi-Chebyshev nodes (QCN) at which {p(n)} can be constructed efficiently in the same order of flops as in the Chebyshev nodes by using the FFT, but with a increasing at a slower rate. We search for the optimum set in the QCN that minimizes the maximum error of {p(n)}. Numerical examples illustrate the error behavior of {p(n)} with the optimum nodes set obtained.