Structured backward errors for block three-by-three saddle point systems with Hermitian and sparsity block matrices

In this paper, we explore the structured backward errors for a class of block three-by-three saddle point systems with Hermitian and sparsity block matrices. We derive an explicit formula for the structured backward errors under the assumption that the inherent matrix structure and sparsity pattern are maintained in the associated perturbation. Moreover, the optimal backward perturbation matrix for achieving structured backward error is constructed. Our analysis further explores the structured backward error when the sparsity structure is not preserved. Numerical experiments show that the computable formulas of structured backward errors are useful for testing the stability of practical algorithms.