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 >= 1 values a ij or a ij  for an m x n matrix A, along with their locations in the matrix. The matrix is accessed using only matrixvector or matrixmatrix 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 x t matrices, where t > 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 matrixvector products for random matrices and give a class of counterexamples. Our numerical experiments show that for p = 1 the algorithm usually converges in just two iterations, requiring the equivalent of 4t matrixvector 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 reallife datasets show that the algorithm is highly effective in practice.
Item Type:  MIMS Preprint 

Additional Information:  To appear in the SIAM Journal on Scientific Computing. 
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:  18 Aug 2016 
Last Modified:  08 Nov 2017 18:18 
URI:  http://eprints.maths.manchester.ac.uk/id/eprint/2495 
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Estimating the Largest Elements of a Matrix. (deposited 21 Dec 2015)
 Estimating the Largest Elements of a Matrix. (deposited 18 Aug 2016) [Currently Displayed]
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