C++ Codes of Implicit Lu Algorithms for Absdll01

TL;DR

The paper addresses solving linear systems $Ax=b$ where the coefficient matrix $A$ may be regular, singular, or ill-conditioned by leveraging ABS methods and implicit LU algorithms built around the Abaffian matrix. It introduces three C++/VC++ implementations—iLUa (no pivoting), iLUaPivotC (column pivoting), and iLUaPivotR (row pivoting)—and provides a main program that demonstrates their use within a DLL-based library ABSDLL. The work details algorithmic structures including implicit factorization $A_i^T P_i = L_i$ and associated null-space relations, as well as practical considerations such as pivoting and component ordering. Numerical tests on matrices such as the Micchelli-Fiedler family suggest efficiency and robustness of the proposed ABS-based implicit LU solvers for both determined and underdetermined systems, enabling practical use in C++ environments.

Abstract

This report is devoted to some C++ codes implementing the implicit LU class algorithms for solving linear determined, and undetermined systems with $n$ variables and $m$ equations. A main program used in part of the numerical test is given in the last section.