Rectangular Vandermonde matrices on Chebyshev nodes. (English) Zbl 0992.15023
A rectangular Vandermonde matrix \(V=\{V_{ij}\}= \{x_i^{j-1}\}\) \((i=1,\dots, n;\;j=1,\dots, m;\;n\leq m)\) defined on the so-called Chebyshev nodes (the roots of Chebyshev polynomials of the first order) is studied, by making use of combinatorial identities from number theory [cf. A. Eisinberg, P. Pugliese, and N. Salerno, Numer. Math. 87, No. 4, 663–674 (2001; Zbl 0974.65029); A. Eisinberg, G. Franzé, and P. Puyliese, Linear Algebra Appl. 283, No. 1-3, 205–219 (1998; Zbl 0935.65016) and Numer. Math. 80, No. 1, 75–85 (1998; Zbl 0913.65022)]. The explicit factorizations \(V=HUD\) as well as \(V^+= (1/n)D^{-1}QBH^T\) are presented, where \(H\) is rectangular, \(U\) and \(Q=U^{-1}\) are triangular, \(D\) and \(B\) \((B_{1,1}= 1;\;B_{i,i}=2;\;i=2,\dots,n)\) are diagonal matrices. It is proved that the so-called condition number [cf. G. H. Golub and C. F. Van Loan, Matrix computations, 2nd ed. (1989; Zbl 0733.65016), 1. Aufl. (Zbl 0559.65011)] of the matrix \(V\) does not depend on the dimension of the nodes sets. Numerical experiments are also performed and, as a result, some numerical properties of the proposed formulae are established.
Reviewer: A.A.Bogush (Minsk)
MSC:
| 15B57 | Hermitian, skew-Hermitian, and related matrices |
| 15A12 | Conditioning of matrices |
| 65F35 | Numerical computation of matrix norms, conditioning, scaling |
| 15A09 | Theory of matrix inversion and generalized inverses |
| 15A23 | Factorization of matrices |
| 68R05 | Combinatorics in computer science |
| 93E24 | Least squares and related methods for stochastic control systems |
Keywords:
factorizations; numerical experiments; rectangular Vandermonde matrix; roots of Chebyshev polynomials; combinatorial identities; condition numberReferences:
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