{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:29:47Z","timestamp":1787329787889,"version":"build-2736575974"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[1998,1]]},"abstract":"<jats:p>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 $|\\sigma| &lt; 1$, 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.<\/jats:p>","DOI":"10.1137\/s0895479896307221","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"253-276","source":"Crossref","is-referenced-by-count":0,"title":["A Generalized Hilbert Matrix Problem and Confluent Chebyshev--Vandermonde Systems"],"prefix":"10.1137","volume":"19","author":[{"given":"Hao","family":"Lu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"Chr. Baker, M. Derakhshan, Fast generation of quadrature rules with some special properties, NATO Adv. Sci. Inst. Ser. C Math. Phys. 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