Structured Backward Error Analysis for Double Saddle Point Problems

TL;DR

This paper develops structured backward error analysis for double saddle point problems, preserving matrix structure and sparsity, to evaluate the stability of numerical algorithms solving these problems.

Contribution

It introduces explicit formulas for structured backward errors and minimal perturbations for DSPPs, linking them to least squares problems with equality constraints.

Findings

Structured BE formulas effectively assess algorithm stability.

Explicit minimal perturbation matrices are derived.

Numerical experiments confirm the stability analysis.

Abstract

Backward error (BE) analysis emerges as a powerful tool for assessing the backward stability and strong backward stability of numerical algorithms. In this paper, we explore structured BEs for a class of double saddle point problems (DSPPs), aiming to assess the strong backward stability of numerical algorithms devised to find their solution. Our investigations preserve the inherent matrix structure and sparsity pattern in the corresponding perturbation matrices and derive explicit formulae for the structure BEs. Moreover, we provide formulae for the structure-preserving minimal perturbation matrices for which the structured BE is attained. Utilizing the relationship between the DSPP and the least squares problem with equality constraints (LSE), we derive the sparsity-preserving BE formula for LSE within our framework. Numerical experiments are performed to test the strong backward stability of various numerical algorithms.