Structured Backward Errors for Special Classes of Saddle Point Problems with Applications

TL;DR

This paper develops structured backward error analysis for saddle point problems, incorporating sparsity and special matrix structures, and applies these methods to evaluate the stability of numerical algorithms.

Contribution

It introduces a novel framework for structured backward errors that accounts for sparsity and matrix structures like circulant and Toeplitz, with minimal perturbation matrices.

Findings

Structured BEs are computed for various matrix classes.

The methods preserve sparsity and structure in perturbations.

Numerical experiments confirm the effectiveness of the approach.

Abstract

In the realm of numerical analysis, the study of structured backward errors (BEs) in saddle point problems (SPPs) has shown promising potential for development. However, these investigations overlook the inherent sparsity pattern of the coefficient matrix of the SPP. Moreover, the existing techniques are not applicable when the block matrices have circulant, Toeplitz, or symmetric-Toeplitz structures and do not even provide structure preserving minimal perturbation matrices for which the BE is attained. To overcome these limitations, we investigate the structured BEs of SPPs when the perturbation matrices exploit the sparsity pattern as well as circulant, Toeplitz, and symmetric-Toeplitz structures. Furthermore, we construct minimal perturbation matrices that preserve the sparsity pattern and the aforementioned structures. Applications of the developed frameworks are utilized to compute BEs for the weighted regularized least squares problem. Finally, numerical experiments are performed to validate our findings, showcasing the utility of the obtained structured BEs in assessing the strong backward stability of numerical algorithms.