# A symmetry preserving algorithm for matrix scaling

Knight, Philip and Ruiz, Daniel and Ucar, Bora (2014) A symmetry preserving algorithm for matrix scaling. SIAM Journal on Matrix Analysis and Applications, 35 (3). 931–955. ISSN 0895-4798 (https://doi.org/10.1137/110825753)

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## Abstract

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.

#### ORCID iDs

Knight, Philip ORCID: https://orcid.org/0000-0001-9511-5692, Ruiz, Daniel and Ucar, Bora;
• Item type: Article 49349 DateEvent2014Published17 July 2014Published Online16 April 2014Accepted sparse matrices, equilibration, matrix scaling, Mathematics, Analysis Science > Mathematics Faculty of Science > Mathematics and Statistics Pure Administrator 24 Sep 2014 14:05 21 Jan 2023 04:07 Journal or Publication https://strathprints.strath.ac.uk/id/eprint/49349