Electronic Transactions on Numerical Analysis (ETNA)

Volume 3, pp. 39-49, 1995.

On graded QR decompositions of products of matrices

G. W. Stewart

Abstract

This paper is concerned with the singular values and vectors of a product $M_{m} = A_{1} A_{2} \dots A_{m}$ of matrices of order $n$ . The chief difficulty with computing them directly from $M_{m}$ is that with increasing $m$ the ratio of the small to the large singular values of $M_{m}$ may fall below the rounding unit, so that the former are computed inaccurately. The solution proposed here is to compute recursively the factorization $M_{m} = Q R P^{T}$ , where $Q$ is orthogonal, $R$ is a graded upper triangular, and $P^{T}$ is a permutation.

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Key words

QR~decomposition, singular value decomposition, graded matrix, matrix product.

AMS subject classifications

65F30.

ETNA articles which cite this article

Vol. 5 (1997), pp. 29-47 David E. Stewart: A new algorithm for the SVD of a long product of matrices and the stability of products

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