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The dual inverse scaling and squaring algorithm for the matrix logarithm
Örebro University, School of Science and Technology.ORCID iD: 0000-0002-6015-391x
Dipartimento di Matematica e Informatica, Università di Perugia, Perugia, Italy.
2022 (English)In: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 42, no 3, p. 2829-2851Article in journal (Refereed) Published
Abstract [en]

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 Pade approximant to the matrix exponential at 0. This equation is solved by a substitution technique in the style of those developed by Fasi & Iannazzo (2020, Substitution algorithms for rational matrix equations. Elect. Trans. Num. Anal., 53, 500-521). The new method is tailored to the special structure of the diagonal Pade approximants to the exponential and in terms of computational cost is more efficient than the state-of-the-art inverse scaling and squaring algorithm.

Place, publisher, year, edition, pages
Oxford University Press, 2022. Vol. 42, no 3, p. 2829-2851
Keywords [en]
primary matrix function, matrix logarithm, inverse scaling and squaring algorithm, Schur form, Pade approximant
National Category
Mathematics
Identifiers
URN: urn:nbn:se:oru:diva-99040DOI: 10.1093/imanum/drab065ISI: 000790064800001Scopus ID: 2-s2.0-85135615972OAI: oai:DiVA.org:oru-99040DiVA, id: diva2:1658634
Funder
Wenner-Gren Foundations, UPD2019-0067
Note

Funding agencies:

MathWorks

Royal Society of London European Commission

Istituto Nazionale di Alta Matematica (INdAM-GNCS Project 2019)

Available from: 2022-05-17 Created: 2022-05-17 Last updated: 2022-09-12Bibliographically approved

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Fasi, Massimiliano

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