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]

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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^(1/2^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 algorithm in the style of [M. Fasi and B. Iannazzo, MIMS EPrint 2019.8, 2019] which is tailored to the special structure of the approximants to the exponential. In terms of floating-point operations, the resulting method is cheaper 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: 27 May 2020 09:27
Last Modified: 27 May 2020 09:27
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2766

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