Max-Balancing Hungarian Scalings

Hook, James and Pestana, Jennifer and Tisseur, Francoise (2015) Max-Balancing Hungarian Scalings. [MIMS Preprint]

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A Hungarian scaling is a diagonal scaling of a matrix that is typically applied along with a permutation to a sparse symmetric or nonsymmetric indefinite linear system before calling a direct or iterative solver. A Hungarian scaled and reordered matrix has all its entries of modulus less than or equal to 1 and entries of modulus 1 on the diagonal. We use max-plus algebra to characterize the set of all Hungarian scalings for a given matrix and show that max-balancing a Hungarian scaled matrix yields the most ``diagonally dominant" Hungarian scaled matrix possible with respect to some ordering. We also propose a new scaling, called centre of mass scaling, which can be seen as an approximate max-balancing Hungarian scaling and whose computation is embarrassingly parallel. Numerical experiments illustrate the increased diagonal dominance produced by max-balancing and centre of mass scaling of Hungarian scaled matrices as well as the reduced need for pivoting in Gaussian elimination with partial pivoting and the improved stability of LU factorizations without pivoting.

Item Type: MIMS Preprint
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Dr Françoise Tisseur
Date Deposited: 08 Jun 2015
Last Modified: 08 Nov 2017 18:18

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