The mmatrix toolbox: componentwise accurate algorithms for M-matrices with triplet representation

2026-07-03

AI Summary

This work addresses the susceptibility of ill-conditioned M-matrices to subtractive cancellation in componentwise high-precision computations by introducing a novel approach based on a triplet representation. By integrating componentwise high-precision arithmetic with the structural properties of M-matrices, the authors present—for the first time—high-precision variants of the GTH algorithm, including non-blocking, recursive, and blocked formulations, all encapsulated within an object-oriented MATLAB interface. The resulting toolbox supports a range of operations such as linear system solving, LU factorization, Schur complement computation, matrix square roots, singular value decomposition, and solutions to nonsymmetric algebraic Riccati equations. Even under severe ill-conditioning, the framework preserves componentwise numerical accuracy, thereby substantially enhancing computational reliability.

Abstract

We introduce the mmatrix toolbox, a Matlab software package for componentwise accurate computations with M-matrices described through left or right triplet representations. The core of the toolbox is a Fortran implementation, in the Lapack style, of the unblocked, recursive, and blocked versions of the GTH algorithm and its applications for the accurate computation of the solution of linear systems with M-matrix coefficient and nonnegative right-hand side, the LU factorization of an M-matrix and its inverse. These algorithms avoid subtractive cancellation; this property ensures high componentwise accuracy even for ill-conditioned problems. The toolbox contains also accurate algorithms for related problems, such as computing the Schur complement, the singular values, the square root of an M-matrix, and the solution of nonsymmetric algebraic Riccati equations associated with M-matrices. The Matlab interface is based on an object-oriented implementation allowing one to use standard Matlab operations on M-matrices with triplet representation.

Problem

• M-matrices
• componentwise accuracy
• triplet representation
• linear systems
• LU factorization

Innovation

• componentwise accuracy
• M-matrices
• triplet representation
• GTH algorithm
• subtractive cancellation