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.


  1. matrix norm estimation
  2. largest elements
  3. power method
  4. mixed subordinate norm
  5. condition number estimation

MSC codes

  1. 65F35

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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: C584 - C601
ISSN (online): 1095-7197


Submitted: 21 December 2015
Accepted: 11 August 2016
Published online: 19 October 2016


  1. matrix norm estimation
  2. largest elements
  3. power method
  4. mixed subordinate norm
  5. condition number estimation

MSC codes

  1. 65F35



Funding Information

European Research Council http://dx.doi.org/10.13039/501100000781 : 267526

Funding Information

Engineering and Physical Sciences Research Council http://dx.doi.org/10.13039/501100000266 : EP/I01912X/1

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