On the sensitivity of singular and ill-Conditioned linear systems

TL;DR

This paper explores the sensitivity and stability of solutions to singular and ill-conditioned linear systems, showing that under certain data accuracy conditions, solutions are well-behaved and approximations are reliable.

Contribution

It introduces a perspective where the general solution of a singular system has bounded sensitivity and establishes conditions for accurate numerical solutions based on data accuracy.

Findings

General solutions of singular systems are Lipschitz continuous.

Accurate data leads to unique, stable solutions in the affine Grassmannian.

Backward accurate solutions approximate true solutions effectively.

Abstract

Solving a singular linear system for an individual vector solution is an ill-posed problem with a condition number infinity. From an alternative perspective, however, the general solution of a singular system is of a bounded sensitivity as a unique element in an affine Grassmannian. If a singular linear system is given through empirical data that are sufficiently accurate with a tight error bound, a properly formulated general numerical solution uniquely exists in the same affine Grassmannian, enjoys Lipschitz continuity and approximates the underlying exact solution with an accuracy in the same order as the data. Furthermore, any backward accurate numerical solution vector is an accurate approximation to one of the solutions of the underlying singular system.