Computing the weighted geometric mean of two large-scale matrices and its inverse times a vector

Fasi, Massimiliano and Iannazzo, Bruno (2016) Computing the weighted geometric mean of two large-scale matrices and its inverse times a vector. [MIMS Preprint]

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Abstract

We investigate different approaches for the computation of the action of the weighted geometric mean of two large-scale positive definite matrices on a vector. We derive several algorithms, based on numerical quadrature and the Krylov subspace, and compare them in terms of convergence speed and execution time. By exploiting an algebraic relation between the weighted geometric mean and its inverse, we show how these methods can be used for the solution of large linear system whose coefficient matrix is a weighted geometric mean. We derive two novel algorithms, based on Gauss�Jacobi quadrature, and tailor an existing technique based on contour integration. On the other hand, we adapt several existing Krylov subspace techniques to the computation of the weighted geometric mean. According to our experiments, both classes of algorithms perform well on some problems but there is no clear winner, while some problem-dependent recommendations are provided.

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: Massimiliano Fasi
Date Deposited: 22 May 2016
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2474

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