# A symmetry preserving algorithm for matrix scaling

Philip Knight, Daniel Ruiz, Bora Ucar

Research output: Contribution to journalArticle

16 Citations (Scopus)

### 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.
Original language English 931–955 25 SIAM Journal on Matrix Analysis and Applications 35 3 17 Jul 2014 https://doi.org/10.1137/110825753 Published - 2014

### Keywords

• sparse matrices
• equilibration
• matrix scaling