Solving ill-conditioned linear algebraic systems using methods that improve conditioning

TL;DR

This paper introduces a new method using minimal pseudoinverse matrices to improve the conditioning of ill-conditioned linear systems, leading to more stable and accurate solutions despite perturbations.

Contribution

The paper proposes a novel approach and algorithm for improving the conditioning of ill-conditioned SLAEs using minimal pseudoinverse matrices, enhancing solution stability and accuracy.

Findings

The method effectively improves matrix conditioning.

Numerical experiments confirm increased solution stability.

Higher accuracy achieved compared to traditional methods.

Abstract

We consider the solution of systems of linear algebraic equations (SLAEs) with an ill-conditioned or degenerate exact matrix and an approximate right-hand side. An approach to solving such a problem is proposed and justified, which makes it possible to improve the conditionality of the SLAE matrix and, as a result, obtain an approximate solution that is stable to perturbations of the right hand side with higher accuracy than using other methods. The approach is implemented by an algorithm that uses so-called minimal pseudoinverse matrices. The results of numerical experiments are presented that confirm the theoretical provisions of the article.