Approximating the logarithm of a matrix to specified accuracy

Sheung Hun Cheng; Nicholas J. Higham; Charles S. Kenney; Alan J. Laub

doi:10.1137/S0895479899364015

Approximating the logarithm of a matrix to specified accuracy

Sheung Hun Cheng, Nicholas J. Higham, Charles S. Kenney, Alan J. Laub

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	1112-1125
Number of pages	13
Journal	SIAM Journal on Matrix Analysis and Applications
Volume	22
Issue number	4
DOIs	https://doi.org/10.1137/S0895479899364015
Publication status	Published - Jan 2001

Keywords

Denman-Beavers iteration
Inverse scaling and squaring method
Matrix logarithm
Matrix square root
Padé approximation

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10.1137/S0895479899364015

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title = "Approximating the logarithm of a matrix to specified accuracy",

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 Pad{\'e} 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 Pad{\'e} 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.",

keywords = "Denman-Beavers iteration, Inverse scaling and squaring method, Matrix logarithm, Matrix square root, Pad{\'e} approximation",

author = "Cheng, {Sheung Hun} and Higham, {Nicholas J.} and Kenney, {Charles S.} and Laub, {Alan J.}",

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AU - Cheng, Sheung Hun

AU - Higham, Nicholas J.

AU - Kenney, Charles S.

AU - Laub, Alan J.

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Y1 - 2001/1

N2 - 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.

AB - 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.

KW - Denman-Beavers iteration

KW - Inverse scaling and squaring method

KW - Matrix logarithm

KW - Matrix square root

KW - Padé approximation

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DO - 10.1137/S0895479899364015

M3 - Article

SN - 0895-4798

VL - 22

SP - 1112

EP - 1125

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 4

ER -

Approximating the logarithm of a matrix to specified accuracy

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Keywords

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