A block algorithm for matrix 1-norm estimation, with an application to 1-norm pseudospectra

Higham, Nicholas J. and Tisseur, Françoise (2000) A block algorithm for matrix 1-norm estimation, with an application to 1-norm pseudospectra. SIAM Journal On Matrix Analysis And Applications, 21 (4). pp. 1185-1201. ISSN 1095-7162

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Abstract

The matrix 1-norm estimation algorithm used in LAPACK and various other software libraries and packages has proved to be a valuable tool. However, it has the limitations that it offers the user no control over the accuracy and reliability of the estimate and that it is based on level 2 BLAS operations. A block generalization of the 1-norm power method underlying the estimator is derived here and developed into a practical algorithm applicable to both real and complex matrices. The algorithm works with n × t matrices, where t is a parameter. For t=1 the original algorithm is recovered, but with two improvements (one for real matrices and one for complex matrices). The accuracy and reliability of the estimates generally increase with t and the computational kernels are level 3 BLAS operations for t > 1. The last t-1 columns of the starting matrix are randomly chosen, giving the algorithm a statistical flavor. As a by-product of our investigations we identify a matrix for which the 1-norm power method takes the maximum number of iterations. As an application of the new estimator we show how it can be used to efficiently approximate 1-norm pseudospectra.

Item Type:	Article
Uncontrolled Keywords:	matrix 1-norm, matrix norm estimation, matrix condition number, condition number estimation, $p$-norm power method, 1-norm pseudospectrum, LAPACK, level 3 BLAS
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:	Ms Lucy van Russelt
Date Deposited:	27 Jun 2006
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/321

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