Approximating the logarithm of a matrix to specified accuracy

Cheng, Sheung Hun and Higham, Nicholas J and Kenney, Charles S and Laub, Alan J (2001) Approximating the logarithm of a matrix to specified accuracy. SIAM Journal On Matrix Analysis And Applications, 22 (4). pp. 1112-1125. ISSN 1095-7162

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Abstract

The standard inverse scaling and squaring algorithm for computing the matrix logarithm begins by transforming the matrix to Schur triangular form in order to facilitate subsequent matrix square root and Padé approximation computations. A transformation-free form of this method that exploits incomplete Denman--Beavers square root iterations and aims for a specified accuracy (ignoring roundoff) is presented. The error introduced by using approximate square roots is accounted for by a novel splitting lemma for logarithms of matrix products. The number of square root stages and the degree of the final Padé approximation are chosen to minimize the computational work. This new method is attractive for high-performance computation since it uses only the basic building blocks of matrix multiplication, LU factorization and matrix inversion.

Item Type:	Article
Uncontrolled Keywords:	matrix logarithm, Pade approximation, inverse scaling and squaring method, matrix square root, Denman--Beavers iteration
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Ms Lucy van Russelt
Date Deposited:	27 Jun 2006
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/318

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