## 2006.145: A block algorithm for matrix 1-norm estimation, with an application to 1-norm pseudospectra

2006.145:
Nicholas J. Higham and Françoise Tisseur
(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

Full text available as:

PDF - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader 359 Kb |

DOI: 10.1137/S0895479899356080

## 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 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis |

MIMS number: | 2006.145 |

Deposited By: | Miss Louise Stait |

Deposited On: | 27 June 2006 |

Download Statistics: last 4 weeks

Repository Staff Only: edit this item