VDR decomposition of Chebyshev-Vandermonde matrices with the Arnoldi Process

Abstract

This paper introduces the VDR decomposition of Chebyshev-Vandermonde matrices, where V represents an ordinary Vandermonde matrix, D is diagonal, and R is upper triangular. Our motivation for this work stems from the study by Brubeck et al. [Vandermonde with Arnoldi. SIAM Rev. 2021;63(2):405–415]. We explore the VDR decomposition and combine it with a QR decomposition of V for Chebyshev-Vandermonde matrices. Furthermore, we demonstrate that the upper triangular factor in the QR decomposition can be recursively obtained from the upper Hessenberg matrix in the Arnoldi process. To compare our approach with the results obtained using the default QR code in Matlab, we present experimental results. Finally, we provide an algorithm for solving linear systems with Chebyshev-Vandermonde matrices as coefficient matrices, and experimental results for an upper bound on the relative error by applying an a posteriori error analysis for the computed solutions of the systems.