On absolute value equations associated with M-matrices and H-matrices

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics Letters Pub Date : 2025-03-19 DOI:10.1016/j.aml.2025.109550

Chun-Hua Guo

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引用次数: 0

Abstract

We consider the absolute value equation (AVE)

A x - | x | = b

, where the diagonal entries of

A \in R^{n \times n}

are all greater than 1 and

〈 A 〉 - I

is an irreducible singular

M

-matrix (

〈 A 〉

is the comparison matrix of

A

). We investigate the existence and uniqueness of solutions for the AVE. The AVE does not necessarily have a unique solution for every

b \in R^{n}

, so most of the existing convergence results for various iterative methods are not generally applicable. Moreover, the generalized Newton method may break down. We show that if the AVE has a solution

x^{*}

with at least one negative component, then the sequence generated by the generalized Gauss–Seidel iteration converges to

x^{*}

linearly for any initial vector.

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关于与[公式省略]-矩阵和[公式省略]-矩阵相关的绝对值方程

我们考虑绝对值方程（AVE） Ax−|x|=b，其中A∈Rn×n的对角线项均大于1，且< A >−I是一个不可约的奇异m矩阵（< A >是A的比较矩阵）。我们研究了AVE解的存在性和唯一性。AVE并不一定对每个b∈Rn都有唯一解，因此现有的各种迭代方法的收敛性结果大多不普遍适用。此外，广义牛顿法可能会失效。我们证明了如果AVE具有至少有一个负分量的解x *，则广义高斯-赛德尔迭代生成的序列对于任何初始向量线性收敛于x *。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applied Mathematics Letters 数学-应用数学

CiteScore

7.70

自引率

5.40%

发文量

347

审稿时长

10 days

期刊介绍： The purpose of Applied Mathematics Letters is to provide a means of rapid publication for important but brief applied mathematical papers. The brief descriptions of any work involving a novel application or utilization of mathematics, or a development in the methodology of applied mathematics is a potential contribution for this journal. This journal''s focus is on applied mathematics topics based on differential equations and linear algebra. Priority will be given to submissions that are likely to appeal to a wide audience.