Structured condition number for multiple right-hand side linear systems with parameterized quasiseparable coefficient matrix

TL;DR

This paper develops explicit formulas for structured condition numbers of multiple right-hand side linear systems with quasiseparable matrices, compares different condition number measures, and demonstrates scenarios where structured measures are significantly smaller.

Contribution

It introduces explicit expressions for structured condition numbers for quasiseparable matrices and compares them with unstructured measures, providing insights into their relationships and effectiveness.

Findings

Effective structured condition number can be much smaller than unstructured ones.

Comparison between quasiseparable and Givens-vector representations shows differences in condition numbers.

Numerical experiments validate the theoretical findings and highlight the advantages of structured measures.

Abstract

In this paper, we consider the structured perturbation analysis for multiple right-hand side linear systems with parameterized coefficient matrix. Especially, we present the explicit expressions for structured condition numbers for multiple right-hand sides linear systems with {1;1}-quasiseparable coefficient matrix in the quasiseparable and the Givens-vector representations. In addition, the comparisons of these two condition numbers between themselves, and with respect to unstructured condition number are investigated. Moreover, the effective structured condition number for multiple right-hand sides linear systems with {1;1}-quasiseparable coefficient matrix is proposed. The relationships between the effective structured condition number and structured condition numbers with respect to the quasiseparable and the Givens-vector representations are also studied. Numerical experiments show that there are situations in which the effective structured condition number can be much smaller than the unstructured ones.