On efficiency of critical-component method for solving singular and ill-posed systems of linear algebraic equations

TL;DR

This work addresses solving degenerate and ill-posed linear systems by the critical-component method, focusing on tri- and two-diagonal matrices. It introduces a stable, non-iterative decomposition that reduces problems to well-posed subsystems and yields a stable pseudoinverse representation, enabling a unique minimal-norm pseudosolution $Z^+ = A^+F$ even when $\det A=0$. A central theorem provides a direct, stable construction of the pseudosolution for $C_3X=Y$ with $X^+=(E+\Omega)\stackrel{\circ}{X}$ and $C^+_3=\stackrel{\circ}{B}+\Omega\stackrel{\circ}{B}$, along with explicit discrepancy bounds. Numerical experiments on 278 problems show MCC and MCS typically achieve higher accuracy than standard approaches (SVD, QR, GS, TRM), validating the method’s robustness for ill-posed and degenerate systems, with runtime trade-offs depending on matrix structure.

Abstract

Results are expoundd for the investigation of efficiency of the critical-component method for solving degenerate and ill-posed systems of linear algebraic equations