The dual inverse scaling and squaring algorithm for the matrix logarithm

Fasi, Massimiliano and Iannazzo, Bruno (2020) The dual inverse scaling and squaring algorithm for the matrix logarithm. [MIMS Preprint] (Unpublished)

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Abstract

The inverse scaling and squaring algorithm computes the logarithm of a square matrix A by evaluating a rational approximant to the logarithm at the matrix B:=A^{2^{-s}} for a suitable choice of s. We introduce a dual approach and approximate the logarithm of B by solving the rational equation r(X)=B, where r is a diagonal Padé approximant to the matrix exponential at 0. This equation is solved by a substitution technique in the style of those developed in (Fasi & Iannazzo, Elect. Trans. Num. Anal., 53 (2020), pp. 500--521). The new method is tailored to the special structure of the diagonal Padé approximants to the exponential, and in terms of computational cost is more efficient than the state-of-the-art inverse scaling and squaring algorithm.

Item Type: MIMS Preprint
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: Mr Massimiliano Fasi
Date Deposited: 15 May 2021 07:21
Last Modified: 15 May 2021 07:21
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2810

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