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Approximating The Logarithm Of A Matrix To Specified Accuracy

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


. 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 is presented that exploits incomplete Denman--Beavers square root iterations and aims for a specified accuracy. 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. Key words. matrix logarithm, Pad'e approximation, inverse scaling and squaring method, matrix square root, Denman--Beavers iteration AMS subject classifications. 65F30 1..

Year: 1999
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