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A note on indefinite matrix splitting and preconditioning

Andy Wathen

TL;DR

It is proved that no splitting matrix can lead to a contractive stationary iteration unless the inertia is exactly preserved, which has consequences for preconditioning for indefinite systems and smoothing for multigrid as it further describes.

Abstract

The solution of systems of linear(ized) equations lies at the heart of many problems in Scientific Computing. In particular for systems of large dimension, iterative methods are a primary approach. Stationary iterative methods are generally based on a matrix splitting, whereas for polynomial iterative methods such as Krylov subspace iteration, the splitting matrix is the preconditioner. The smoother in a multigrid method is generally a stationary or polynomial iteration. Here we consider real symmetric indefinite and complex Hermitian indefinite coefficient matrices and prove that no splitting matrix can lead to a contractive stationary iteration unless the inertia is exactly preserved. This has consequences for preconditioning for indefinite systems and smoothing for multigrid as we further describe.

A note on indefinite matrix splitting and preconditioning

TL;DR

It is proved that no splitting matrix can lead to a contractive stationary iteration unless the inertia is exactly preserved, which has consequences for preconditioning for indefinite systems and smoothing for multigrid as it further describes.

Abstract

The solution of systems of linear(ized) equations lies at the heart of many problems in Scientific Computing. In particular for systems of large dimension, iterative methods are a primary approach. Stationary iterative methods are generally based on a matrix splitting, whereas for polynomial iterative methods such as Krylov subspace iteration, the splitting matrix is the preconditioner. The smoother in a multigrid method is generally a stationary or polynomial iteration. Here we consider real symmetric indefinite and complex Hermitian indefinite coefficient matrices and prove that no splitting matrix can lead to a contractive stationary iteration unless the inertia is exactly preserved. This has consequences for preconditioning for indefinite systems and smoothing for multigrid as we further describe.

Paper Structure

This paper contains 4 sections, 3 theorems, 17 equations.

Table of Contents

  1. Introduction and Main Result
  2. Avoidance of crossing
  3. Conclusions
  4. Acknowledgments

Key Result

Lemma 1

If matrices $A$ and $M$ with the same dimension are real symmetric and invertible with different inertia, then $M^{-1}A$ has at least one negative real eigenvalue and thus the stationary iteration (statit) can not be contractive.

Theorems & Definitions (6)

  • Lemma
  • Proof 1
  • Proposition
  • Proof 2
  • Corollary
  • Proof 3