Approximating the logarithm of a matrix to specified accuracy. (English) Zbl 0989.65057

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

SIAM J. Matrix Anal. Appl. 22, No. 4, 1112-1125 (2001).

From the authors’ 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.

Reviewer: Michael Tsatsomeros (Pullman)

MSC:

65F30	Other matrix algorithms (MSC2010)
15A24	Matrix equations and identities

Keywords:

matrix logarithm; Padé approximation; inverse scaling and squaring method; matrix square root; Denman-Beavers iteration; matrix multiplication; matrix inversion; LU factorization

Software:

Matlab