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 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 diagonal dominance. Our method uses max-balancing, which minimizes the largest off-diagonal entries in the 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.
Highlights
A Hungarian scaling is a two-sided diagonal scaling of a matrix that is applied along with a permutation P to a linear system Ax = b, with A ∈ Cn×n and b ∈ Cn, yielding (1.1)H = P D1AD2, Hy = P D1b, x = D2y, where D1, D2 ∈ Rn×n are diagonal and nonsingular
We have introduced max-balanced Hungarian scaling, which is applied to a matrix A ∈ Cn×n in two stages
In Theorem 3.6 we proved that max-balancing preserves the properties of a Hungarian scaling, so that M satisfies |mij| ≤ 1 and |mii| = 1 for i = 1, . . . , n as well as (5.1)
Summary
Theorem 3.8 states that the max-balanced Hungarian scaling of a matrix is optimal with respect to a particular measure of diagonal dominance. For any irreducible A ∈ Cn×n there exists a unique max-balanced matrix M diagonally similar to A, M = diag(dmax)−1A diag(dmax), where the scaling parameter dmax ∈ Rn+ is unique up to scalar multiple. Let H ∈ Cn×n be an irreducible Hungarian matrix, and let dmax ∈ Rn be such that M = diag(dmax)−1H diag(dmax) is the max-balanced scaling of H.
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