Error analysis of a partial pivoting method for structured matrices

TL;DR

This paper analyzes the numerical stability of the GKO method for structured matrices, revealing conditions that cause large error growth and proposing a modification to improve stability.

Contribution

It provides a detailed rounding error analysis of the GKO method for Toeplitz and Cauchy matrices and introduces a modified algorithm with improved stability.

Findings

Error growth depends on auxiliary generator vectors.

Large generator growth can cause significant error amplification.

A modified pivoting strategy can reduce error growth.

Abstract

Many matrices that arise in the solution of signal processing problems have a special displacement structure. For example, adaptive filtering and direction-of-arrival estimation yield matrices of Toeplitz type. A recent method of Gohberg, Kailath and Olshevsky (GKO) allows fast Gaussian elimination with partial pivoting for such structured matrices. In this paper, a rounding error analysis is performed on the Cauchy and Toeplitz variants of the GKO method. It is shown the error growth depends on the growth in certain auxiliary vectors, the generators, which are computed by the GKO algorithms. It is also shown that in certain circumstances, the growth in the generators can be large, and so the error growth is much larger than would be encountered with normal Gaussian elimination with partial pivoting. A modification of the algorithm to perform a type of row-column pivoting is proposed; it may ameliorate this problem.