Structured Backward Errors of Sparse Generalized Saddle Point Problems with Hermitian Block Matrices

TL;DR

This paper derives structured backward errors for sparse generalized saddle point problems with Hermitian block matrices, providing optimal perturbations and stability assessments for numerical methods.

Contribution

It introduces the derivation of structured backward errors that preserve sparsity and Hermitian structure, along with optimal perturbations for generalized saddle point problems.

Findings

Structured backward errors are reliably computed for GSPPs.

Optimal backward perturbations are constructed explicitly.

Numerical experiments confirm the stability analysis.

Abstract

In this paper, we derive the structured backward error (BE) for a class of generalized saddle point problems (GSPP) by preserving the sparsity pattern and Hermitian structures of the block matrices. Additionally, we construct the optimal backward perturbation matrices for which the structured BE is achieved. Our analysis also examines the structured BE in cases where the sparsity pattern is not maintained. Through numerical experiments, we demonstrate the reliability of the derived structured BEs and the corresponding optimal backward perturbations. Additionally, the derived structured BEs are used to assess the strong backward stability of numerical methods for solving the GSPP.