Sparsity-Preserving Structured Backward Error Analysis for Double Saddle Point Problems

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) and aim to assess the strong backward stability of numerical algorithms designed to find their solutions. Our investigations preserve the inherent matrix structure and sparsity pattern in the corresponding perturbation matrices and derive explicit formulae for the structured 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.