Title

On the sensitivity of singular and ill-conditioned linear systems

Document Type

Article

Publication Date

1-1-2019

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.

DOI

10.1137/18M1197990

Publication Title

SIAM Journal on Matrix Analysis and Applications

Volume Number

40

Issue Number

3

First Page

918

Last Page

942

ISSN

08954798

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