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2006.150: Estimating the matrix p-norm

2006.150: Nicholas J. Higham (1992) Estimating the matrix p-norm. Numerische Mathematik, 62. pp. 539-555. ISSN 0945-3245

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DOI: 10.1007/BF01396242


The H\"older $p$-norm of an $m \times n$ matrix has no explicit representation unless $p=1$,~$2$ or $\infty$. It is shown here that the $p$-norm can be estimated reliably in $O(mn)$ operations. A generalization of the power method is used, with a starting vector determined by a technique with a condition estimation flavour. The algorithm nearly always computes a $p$-norm estimate correct to the specified accuracy, and the estimate is always within a factor $n^{1-1/p}$ of $\|A\|_p$. As a by-product, a new way is obtained to estimate the 2-norm of a rectangular matrix; this method is more general and produces better estimates in practice than a similar technique of Cline, Conn and Van Loan.

Item Type:Article
Uncontrolled Keywords:H\"older norm, $p$-norm, matrix norm, condition number estimation, LAPACK.
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2006.150
Deposited By:Miss Louise Stait
Deposited On:28 June 2006

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