QR-factorization method for computing the greatest common divisor of polynomials with inexact coefficients

Author

Zarowski, Christopher J. ; Ma, Xiaoyan ; Fairman, Frederick W.

Author_Institution

Dept. of Electr. & Comput. Eng., Queen´´s Univ., Kingston, Ont., Canada

Volume

48

Issue

11

fYear

2000

fDate

11/1/2000 12:00:00 AM

Firstpage

3042

Lastpage

3051

Abstract

This paper presents a novel means of computing the greatest common divisor (GCD) of two polynomials with real-valued coefficients that have been perturbed by noise. The method involves the QR-factorization of a near-to-Toeplitz matrix derived from the Sylvester matrix of the two polynomials. It turns out that the GCD to within a constant factor is contained in the last nonzero row of the upper triangular matrix R in the QR-factorization of the near-to-Toeplitz matrix. The QR-factorization is efficiently performed by an algorithm due to Chun et al. (1987). A condition number estimator due to Bischof (1990) and an algorithm for rank estimation due to Zarowski (1998) are employed to account for the effects of noise

Keywords

Toeplitz matrices; matrix decomposition; noise; polynomial matrices; QR-factorization method; Sylvester matrix; condition number estimator; greatest common divisor; inexact coefficient polynomials; near-to-Toeplitz matrix; noise; quadratic residue; rank estimation; real-valued coefficients; upper triangular matrix; Arithmetic; Convergence; Decoding; Galois fields; Helium; Noise measurement; Partitioning algorithms; Polynomials; Quantization; Reed-Solomon codes;

fLanguage

English

Journal_Title

Signal Processing, IEEE Transactions on

Publisher

ieee

ISSN

1053-587X

Type

jour

DOI

10.1109/78.875462

Filename

875462