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

A. S. Leonov

TL;DR

The paper tackles solving ill-conditioned SLAEs with an approximate right-hand side by introducing a scheme that constructs a conditioned,近 minimal pseudoinverse-based matrix $\tilde{A}_h$ via diagonal scaling guided by functions $x_k(h)$. By solving a discrepancy-type equation for $h(\delta)$ and forming $z_\delta = \tilde{A}_{h(\delta)}^+ u_\delta$, the method yields a convergent approximation to the normal pseudo-solution as the perturbation vanishes, while simultaneously improving the system's conditioning. The central instantiation, the MPMI method, demonstrates substantial conditioning gains and higher solution accuracy compared with TSVD and Tikhonov regularization in numerical experiments, particularly for highly ill-conditioned problems. This approach provides a practically robust alternative for stable SLAE solution under data perturbations, with explicit mechanisms to reduce effective rank and condition number without requiring detailed error levels for the perturbations. The results have implications for large-scale, ill-conditioned linear systems common in scientific computing and inverse problems.

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.

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

TL;DR

The paper tackles solving ill-conditioned SLAEs with an approximate right-hand side by introducing a scheme that constructs a conditioned,近 minimal pseudoinverse-based matrix via diagonal scaling guided by functions . By solving a discrepancy-type equation for and forming , the method yields a convergent approximation to the normal pseudo-solution as the perturbation vanishes, while simultaneously improving the system's conditioning. The central instantiation, the MPMI method, demonstrates substantial conditioning gains and higher solution accuracy compared with TSVD and Tikhonov regularization in numerical experiments, particularly for highly ill-conditioned problems. This approach provides a practically robust alternative for stable SLAE solution under data perturbations, with explicit mechanisms to reduce effective rank and condition number without requiring detailed error levels for the perturbations. The results have implications for large-scale, ill-conditioned linear systems common in scientific computing and inverse problems.

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.
Paper Structure (6 sections, 6 theorems, 37 equations, 1 figure)

This paper contains 6 sections, 6 theorems, 37 equations, 1 figure.

Table of Contents

  1. Introduction
  2. On some methods for stable solution of SLAEs
  3. Auxiliary statements
  4. Detailed formulation of the problem, proposed solution scheme and its justification
  5. A specific method that implements the proposed scheme for solving SLAE
  6. Numerical experiments

Key Result

Lemma 1

Let the matrices $\tilde{A}_h$ satisfy the relation $\left\Vert \tilde{A}_{h}^{+}\right\Vert \leq \left\Vert \bar{A}^{+}\right\Vert$, $\forall h,~0\leq h<\left\Vert \bar{A}^{+}\right\Vert ^{-1}$. Then: A) $\mathrm{Rg}\tilde{A}_{h}=\mathrm{Rg}\bar{A}$; B) $\left\| \tilde{A}_{h}^{+}-\bar{A}^{+}\right\

Figures (1)

  • Figure 1: A) To Sect.\ref{['sec4']}. Matrix singular values \ref{['Poisson']}. B) To Sect.\ref{['Sec4']}. Solution $h(\delta)$ of the equation $\bar{\beta}_{\delta }(h)=\frac{b_\delta}{\left\Vert \bar{u}\right\Vert _{2}}$ c $b_{\delta }=\sqrt{\mu _{\delta }^{2}+\delta ^{2}}$ for SLAE with matrix \ref{['Poisson']}.

Theorems & Definitions (8)

  • Lemma 1
  • Corollary 1
  • Corollary 2
  • Example 1
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3