Abs Algorithms for Linear Equations and Abspack

TL;DR

The paper surveys ABS algorithms (notably the scaled ABS class with Abaffians) for solving linear systems, linear Diophantine equations, KT equations, and matrix equations, and introduces ABSPACK as a FORTRAN-based implementation project. It details structural identities such as $H_{i+1}A^TV_i=0$, $H_{i+1}^TW_i=0$, and the implicit factorization $V_i^TA_i^TP_i=L_i$, enabling explicit inverses (e.g., $A^{-1}=PL^{-1}V^T$ when $m=n$) and a general solution form $x=x_{m+1}+H_{m+1}^T q$ under strong nonsingularity of $Q=V^TAH_1^TW$, with the GILU subclass embedding LU with column pivoting. The authors derive ABS methods for Diophantine and matrix equations, extend iterative Bodon-ABS schemes, and report numerical results showing that Modified Huang offers high accuracy and competitive speed versus LAPACK/SVD-based solvers, especially for rank-deficient problems. They argue that ABS solvers provide high-accuracy, low-storage computation across KT and matrix problems and outline ABSPACK's trajectory toward block implementations and eigenvalue-type applications. Overall, the work demonstrates ABS methods as a versatile framework for a broad set of linear and nonlinear problems with practical software implications.

Abstract

We present the main results obtained during a research on ABS methods in the framework of the project Analisi Numerica e Matematica Computazionale.