Spectral Analysis of Block Diagonally Preconditioned Multiple Saddle-Point Matrices with Inexact Schur Complements

TL;DR

This paper provides eigenvalue bounds for preconditioned saddle-point systems with multiple blocks, including cases with inexact Schur complements, supported by numerical validation.

Contribution

It generalizes eigenvalue bounds to systems with an arbitrary number of blocks and inexact Schur complements, extending previous results.

Findings

Eigenvalue bounds are derived for block-diagonally preconditioned systems.

Numerical experiments confirm the accuracy of the eigenvalue estimates.

The analysis applies to systems with an arbitrary number of blocks.

Abstract

We derive eigenvalue bounds for symmetric block-tridiagonal multiple saddle-point systems preconditioned with block-diagonal Schur complement matrices. This analysis applies to an arbitrary number of blocks and accounts for the case where the Schur complements are approximated, generalizing the findings in [Bergamaschi et al., Linear Algebra and its Applications, 2026]. Numerical experiments are carried out to validate the proposed estimates.