Theoretical analysis and numerical solution to a vector equation $Ax-\|x\|_1x=b$

TL;DR

This paper investigates the properties of a specific vector equation involving an invertible M-matrix and the 1-norm, proving solution existence and analyzing fixed-point and doubling algorithms.

Contribution

It provides a theoretical foundation for the equation's solutions and introduces effective fixed-point, Newton, and doubling algorithms with convergence analysis.

Findings

Existence and uniqueness of nonnegative solutions are established.

Proposed algorithms demonstrate effective convergence in numerical experiments.

Abstract

Theoretical and computational properties of a vector equation $Ax-\|x\|_1x=b$ are investigated, where $A$ is an invertible $M$-matrix and $b$ is a nonnegative vector. Existence and uniqueness of a nonnegative solution is proved. Fixed-point iterations, including a relaxed fixed-point iteration and Newton iteration, are proposed and analyzed. A structure-preserving doubling algorithm is proved to be applicable in computing the required solution, the convergence is at least linear with rate 1/2. Numerical experiments are performed to demonstrate the effectiveness of the proposed algorithms.