A Generalized Hilbert Matrix Problem and Confluent Chebyshev--Vandermonde Systems

Abstract

This paper deals with fast solution of the generalized Hilbert matrix problem and confluent Chebyshev--Vandermonde systems. First, two methods for the generalized Hilbert matrix problem are presented. One is for the case where the points involved in the generalized Hilbert matrices satisfy a TH-relation introduced in the present paper, which include equidistant points, clustered points, and Chebyshev points. The approach is based on an O(n log n) fast multiplication of a Toeplitz plus Hankel matrix with a vector. The other method is to reduce the generalized Hilbert matrix problem to products of confluent Vandermonde-like matrices and dual confluent Vandermonde-like matrices with vectors by using J-matches and links of polynomials. Second, two strategies for confluent Chebyshev--Vandermonde systems are considered. Based on the result of the solution of confluent Vandermonde-like systems, the solution of confluent Chebyshev--Vandermonde systems for Chebyshev $\sigma$-points, i.e., the zeros of $T(\lambda)-\sigma$ with $|\sigma|<1$, where $T(\lambda)$ is the Chebyshev polynomial of the first kind, is reduced to fast Fourier transforms (FFT) or to sine or cosine transforms by special choices of J-matches and links of Chebyshev polynomials, and hence we obtain some O(n log n) algorithms for the systems. The solution of Chebyshev--Vandermonde systems is also reduced to the generalized Hilbert matrix problem by using J-matches, links of Chebyshev polynomials, and the inversion of a class of generalized Hilbert matrices. This yields an O(n log n) algorithm for Chebyshev--Vandermonde systems for another class of practical points. Third, the results obtained are applied to related problems, for example, confluent Chebyshev--Vandermonde systems for near Chebyshev $\sigma$-points, Hermite interpolation in terms of Chebyshev polynomials, and a class of generalized Hilbert systems. Finally, numerical examples show quite accurate results even for large systems of equations.