We present an iterative algorithm which asymptotically scales the $\infty$-norm of each row and each column of a matrix to one. This scaling algorithm preserves symmetry of the original matrix and shows fast linear convergence with an asymptotic rate of 1/2. We discuss extensions of the algorithm to the 1-norm, and by inference to other norms. For the 1-norm case, we show again that convergence is linear, with the rate dependent on the spectrum of the scaled matrix. We demonstrate experimentally that the scaling algorithm improves the conditioning of the matrix and that it helps direct solvers by reducing the need for pivoting. In particular, for symmetric matrices the theoretical and experimental results highlight the potential of the proposed algorithm over existing alternatives.


  1. sparse matrices
  2. matrix scaling
  3. equilibration

MSC codes

  1. 05C50
  2. 65F35
  3. 65F50

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Information & Authors


Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 931 - 955
ISSN (online): 1095-7162


Submitted: 25 February 2011
Accepted: 16 April 2014
Published online: 17 July 2014


  1. sparse matrices
  2. matrix scaling
  3. equilibration

MSC codes

  1. 05C50
  2. 65F35
  3. 65F50



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