Levinson-type algorithms for polynomial fitting and for Cholesky and Q factors of Hankel and Vandermonde matrices

Author

Porsani, Milton José ; Ulrych, Tadeusz J.

Author_Institution

Programa de Pesquisa e Pos-Graduacao em Geofisica, Univ. Federal da Bahia, Salvador, Brazil

Volume

43

Issue

1

fYear

1995

fDate

1/1/1995 12:00:00 AM

Firstpage

63

Lastpage

70

Abstract

This paper presents Levinson (1947)-type algorithms for (i) polynomial fitting (ii) obtaining a Q decomposition of Vandermonde matrices and a Cholesky factorization of Hankel matrices (iii) obtaining the inverse of Hankel matrices. The algorithm for the least-squares solution of Hankel systems of equations requires 3n²+9n+3 multiply and divide operation (MDO). The algorithm for obtaining an orthogonal representation of an (m×n) Vandermonde matrix X and computing the Cholesky factors F of Hankel matrices requires 5mn+n²+2n-3m MDO, and the algorithm for generating the inverse of Hankel matrices requires 3(n²+n-2)/2 MDO. Our algorithms have been tested by means of fitting of polynomials of various orders and Fortran versions of all subroutines are provided in the Appendix

Keywords

Hankel matrices; least squares approximations; matrix inversion; polynomials; Cholesky factorization; Cholesky factors; Fortran; Hankel equations; Hankel matrices; Levinson-type algorithms; Q decomposition; Q factors; Vandermonde matrices; algorithm; inverse Hankel matrices; least-squares solution; multiply and divide operation; orthogonal representation; polynomial fitting; subroutines; Algorithms; Equations; Error correction; Geophysics; Matrix decomposition; Polynomials; Q factor; Reflectivity; Testing; Wiener filter;

fLanguage

English

Journal_Title

Signal Processing, IEEE Transactions on

Publisher

ieee

ISSN

1053-587X

Type

jour

DOI

10.1109/78.365286

Filename

365286