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arXiv · 2507.04971

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

Yuezhi Wang·Gwi Soo Kim·Jie Meng
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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.

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Yuezhi Wang, Gwi Soo Kim, Jie Meng. 2025-07-07. Theoretical analysis and numerical solution to a vector equation $Ax-\|x\|_1x=b$. https://arxiv.org/abs/2507.04971

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