# Estimating the Largest Elements of a Matrix

Higham, Nicholas J. and Relton, Samuel D. (2015) Estimating the Largest Elements of a Matrix. [MIMS Preprint] There is a more recent version of this item available. PDF paper.pdf Download (371kB)

## Abstract

We derive an algorithm for estimating the largest p Ã�Â¢Ã¢Â�Â°Ã�Â¥ 1 values a ij or |a ij | for an m Ã�Â�Ã¢Â�Â� n matrix A, along with their locations in the matrix. The matrix is accessed using only matrixÃ�Â¢Ã¢Â�Â¬Ã¢Â�Â� vector or matrixÃ�Â¢Ã¢Â�Â¬Ã¢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 Ã�Â�Ã¢Â�Â� t matrices, where t Ã�Â¢Ã¢Â�Â°Ã�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 matrixÃ�Â¢Ã¢Â�Â¬Ã¢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 matrixÃ�Â¢Ã¢Â�Â¬Ã¢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 matrix norm estimation, largest elements, power method, mixed subordinate norm, condition number estimation MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Dr Samuel Relton 21 Dec 2015 08 Nov 2017 18:18 http://eprints.maths.manchester.ac.uk/id/eprint/2424

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