Comments on finite termination of the generalized Newton method for absolute value equations

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY Accounts of Chemical Research Pub Date : 2024-05-20 DOI:10.1007/s11590-024-02121-0

Chun-Hua Guo

引用次数: 0

Abstract

We consider the generalized Newton method (GNM) for the absolute value equation (AVE) \(Ax-|x|=b\). The method has finite termination property whenever it is convergent, no matter whether the AVE has a unique solution. We prove that GNM is convergent whenever \(\rho (|A^{-1}|)<1/3\). We also present new results for the case where \(A-I\) is a nonsingular M-matrix or an irreducible singular M-matrix. When \(A-I\) is an irreducible singular M-matrix, the AVE may have infinitely many solutions. In this case, we show that GNM always terminates with a uniquely identifiable solution, as long as the initial guess has at least one nonpositive component.

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关于绝对值方程广义牛顿法有限终止的评论

我们考虑了绝对值方程（AVE） \(Ax-|x|=b\)的广义牛顿法（GNM）。无论绝对值方程是否有唯一解，只要该方法收敛，它就具有有限终止特性。我们证明，只要 \(\rho (|A^{-1}|)<1/3\), GNM 就是收敛的。我们还针对 \(A-I\) 是非奇异 M 矩阵或不可还原奇异 M 矩阵的情况提出了新的结果。当 \(A-I\) 是不可还原的奇异 M 矩阵时，AVE 可能有无穷多个解。在这种情况下，我们证明了只要初始猜测至少有一个非正分量，GNM 总是以一个唯一可识别的解结束。

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

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.

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