Estimating the Largest Elements of a Matrix

Higham, Nicholas J. and Relton, Samuel D. (2015) Estimating the Largest Elements of a Matrix. [MIMS Preprint]

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Abstract

We derive an algorithm for estimating the largest p �¢â�°�¥ 1 values a ij or |a ij | for an m ��â�� n matrix A, along with their locations in the matrix. The matrix is accessed using only matrix�¢â�¬â�� vector or matrix�¢â�¬â��matrix products. For p = 1 the algorithm estimates the norm A M := max i,j |a ij | or max i,j a ij . The algorithm is based on a power method for mixed subordinate matrix norms and iterates on n ��â�� t matrices, where t �¢â�°�¥ p is a parameter. For p = t = 1 we show that the algorithm is essentially equivalent to rook pivoting in Gaussian elimination; we also obtain a bound for the expected number of matrix�¢â�¬â��vector products for random matrices and give a class of counter-examples. Our numerical experiments show that for p = 1 the algorithm usually converges in just two iterations, requiring the equivalent of 4t matrix�¢â�¬â��vector products, and for t = 2 the algorithm already provides excellent estimates that are usually within a factor 2 of the largest element and frequently exact. For p > 1 we incorporate deflation to improve the performance of the algorithm. Experiments on real-life datasets show that the algorithm is highly effective in practice.

Item Type: MIMS Preprint
Uncontrolled Keywords: matrix norm estimation, largest elements, power method, mixed subordinate norm, condition number estimation
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Dr Samuel Relton
Date Deposited: 21 Dec 2015
Last Modified: 08 Nov 2017 18:18
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2424

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