Max-Balanced Hungarian Scalings

James Hook, Jennifer Pestana, Francoise Tisseur, Jonathan Hogg

Research output: Contribution to journalArticle

2 Citations (Scopus)
12 Downloads (Pure)

Abstract

A Hungarian scaling is a diagonal scaling of a matrix that is typically applied along with a permutation to a sparse linear system before calling a direct or iterative solver. A matrix that has been Hungarian scaled and reordered has all entries of modulus less than or equal to 1 and entries of modulus 1 on the diagonal. An important fact that has been largely overlooked by the previous research into Hungarian scaling of linear systems is that a given matrix typically has a range of possible Hungarian scalings, and direct or iterative solvers may behave quite differently under each of these scalings. Since standard algorithms for computing Hungarian scalings return only one scaling, it is natural to ask whether a superior performing scaling can be obtained by searching within the set of all possible Hungarian scalings. To this end we propose a method for computing a Hungarian scaling that is optimal from the point of view of a measure of diagonal dominance. Our method uses max-balancing, which minimizes the largest off-diagonal entries in the scaled and permuted matrix. Numerical experiments illustrate the increased diagonal dominance produced by max-balanced Hungarian scaling as well as the reduced need for row interchanges in Gaussian elimination with partial pivoting and the improved stability of LU factorizations without pivoting. We additionally find that applying the max-balancing scaling before computing incomplete LU preconditioners improves the convergence rate of certain iterative methods. Our numerical experiments also show that the Hungarian scaling returned by the HSL code MC64 has performance very close to that of the optimal max-balanced Hungarian scaling, which further supports the use of this code in practice.

Original languageEnglish
Pages (from-to)320-346
Number of pages27
JournalSIAM Journal On Matrix Analysis and Applications (SIMAX)
Volume40
Issue number1
Early online date26 Feb 2019
DOIs
Publication statusPublished - 2019

Keywords

  • Diagonal dominance
  • Diagonal scaling
  • Hungarian scaling
  • Incomplete LU preconditioner
  • Linear systems of equations
  • Max-balancing
  • Max-plus algebra
  • Sparse matrices

ASJC Scopus subject areas

  • Analysis

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