## 2015.116: Estimating the Largest Elements of a Matrix

2015.116:
Nicholas J. Higham and Samuel D. Relton
(2016)
*Estimating the Largest Elements of a Matrix.*
SIAM Journal on Scientific Computing, 38 (5).
C584-C601.

*This is the latest version of this eprint.*

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DOI: 10.1137/15M1053645

## Abstract

We derive an algorithm for estimating the largest $p \geq 1$ values $a_{ij}$ or $|a_{ij}|$ for an $m \times n$ matrix $A$, along with their locations in the matrix. The matrix is accessed using only matrix--vector or matrix--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 $n \times t$ matrices, where $t \geq p$ 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--vector 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 matrix--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.

Read More: http://epubs.siam.org/doi/abs/10.1137/15M1053645

Item Type: | Article |
---|---|

Uncontrolled Keywords: | matrix norm estimation, largest elements, power method, mixed subordinate norm, condition number estimation |

Subjects: | MSC 2000 > 65 Numerical analysis |

MIMS number: | 2015.116 |

Deposited By: | Dr Samuel Relton |

Deposited On: | 28 October 2016 |

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- Estimating the Largest Elements of a Matrix (deposited 28 October 2016)
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