A note on indefinite matrix splitting and preconditioning

Journal article

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 polyno8 mial 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.

Authors: Wathen, A

• Publication status: In press
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Linear Algebra and its Applications
• Publication date: 2025-04-23
• Acceptance date: 2025-04-07
• DOI: 10.1016/j.laa.2025.04.004
• EISSN: 1873-1856
• ISSN: 0024-3795
• Language: English
• Keywords: iterative methods for linear systems; indefinite matrices; matrix splitting; precon ditioning; smoothing