# Estimating the matrix p-norm

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

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 H\"older norm, $p$-norm, matrix norm, condition number estimation, LAPACK. MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theoryMSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Ms Lucy van Russelt 28 Jun 2006 20 Oct 2017 14:12 http://eprints.maths.manchester.ac.uk/id/eprint/328