Estimating the Condition Number of f(A)b

Deadman, Edvin (2014) Estimating the Condition Number of f(A)b. [MIMS Preprint]

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Abstract

New algorithms are developed for estimating the condition number of $f(A)b$, where $A$ is a matrix and $b$ is a vector. The condition number estimation algorithms for $f(A)$ already available in the literature require the explicit computation of matrix functions and their Fr\'{e}chet derivatives and are therefore unsuitable for the large, sparse $A$ typically encountered in $f(A)b$ problems. The algorithms we propose here use only matrix-vector multiplications. They are based on a modified version of the power iteration for estimating the norm of the Fr\'{e}chet derivative of a matrix function, and work in conjunction with any existing algorithm for computing $f(A)b$. The number of matrix-vector multiplications required to estimate the condition number is proportional to the square of the number of matrix-vector multiplications required by the underlying $f(A)b$ algorithm. We develop a specific version of our algorithm for estimating the condition number of $e^Ab$, based on the algorithm of Al-Mohy and Higham [SIAM J. Matrix Anal. Appl., 30(4):1639--1657, 2009]. Numerical experiments demonstrate that our condition estimates are reliable and of reasonable cost.

Item Type: MIMS Preprint
Uncontrolled Keywords: matrix function; matrix exponential; condition number estimation; Frechet derivative; power iteration; block 1-norm estimator; Python
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: Dr Edvin Deadman
Date Deposited: 01 Dec 2014
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2200

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