A note on indefinite matrix splitting and preconditioning

TL;DR

This paper investigates the limitations of matrix splitting methods for indefinite systems, showing that preserving inertia is essential for contractive stationary iterations, impacting preconditioning and multigrid smoothing.

Contribution

It proves that no splitting matrix can produce a contractive stationary iteration for indefinite matrices unless inertia is exactly preserved, revealing fundamental constraints.

Findings

No splitting matrix can lead to a contractive stationary iteration unless inertia is preserved.

Implications for preconditioning indefinite systems in scientific computing.

Constraints on smoothing techniques in multigrid methods for indefinite matrices.

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.