Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle
Loading...
Authors
Gragg, William B.
Subjects
Toeplitz matrices
unitary Hessenberg matrices
Szegő polynomials
unitary Hessenberg matrices
Szegő polynomials
Advisors
Date of Issue
1993
Date
1993
Publisher
Elsevier Science Publishers B.V.
Language
Abstract
We show that the well-known Levinson algorithm for computing the inverse Cholesky factorization of positive
definite Toeplitz matrices can be viewed as a special case of a more general process. The latter process
provides a very efficient implementation of the Arnoldi process when the underlying operator is isometric.
This is analogous with the case of Hermitian operators where the Hessenberg matrix becomes tridiagonal and
results in the Hermitian Lanczos process. We investigate the structure of the Hessenberg matrices in the
isometric case and show that simple modifications of them move all their eigenvalues to the unit circle. These
eigenvalues are then interpreted as abscissas for analogs of Gaussian quadrature, now on the unit circle
instead of the real line. The trapezoidal rule appears as the analog of the Gauss-Legendre formula.
Type
Article
Description
Series/Report No
Department
Applied Mathematics
Organization
Naval Postgraduate School (U.S.)
Identifiers
NPS Report Number
Sponsors
National Science Foundation
Funder
National Science Foundation
Format
16 p.
Citation
Gragg, William B. "Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle." Journal of Computational and Applied Mathematics 46.1 (1993): 183-198.
Distribution Statement
Rights
This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.