arXiv 1005.12522010-05-07math.NAcs.DScs.MScs.NAcs.NE

通用算法、半环数学与并行计算

Universal algorithms, mathematics of semirings and parallel computations

J. V. Poncelet Laboratory (UMI 2615 CNRS)（J.V.庞塞莱实验室）；D. V. Skobeltsyn Research Institute for Nuclear Physics, Moscow State University（莫斯科国立大学D.V.斯科别尔岑核物理研究所）；Physics Faculty, Moscow State University（莫斯科国立大学物理系）；A. A. Kharkevich Institute for Information Transmission Problems（A.A.哈尔凯维奇信息传输问题研究所）

G. L. Litvinov, V. P. Maslov, A. Ya. Rodionov, A. N. Sobolevski

备注：: 36 pages, 1 figure. To appear in Springer Lecture Notes in Computational Science and Engineering.

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AI总结：

本文综述半环数学在数值分析和计算中的应用，探讨通用算法与半环数学的关系，并指出半环线性化对并行计算的便利性。

摘要：

这是一篇关于半环数学在数值分析和计算中应用的综述论文。讨论了通用算法和泛型程序的概念，考察了这些概念与半环数学之间的关系。简要介绍了半环数学（包括幂等和热带数学）。指出了在优化问题、幂等线性代数和区间分析中的具体应用。已知某些非线性问题（尤其是优化问题）在具有幂等加法的适当半环上变为线性（所谓的幂等叠加原理）。这种半环上的线性性质便于并行计算。

英文摘要：

This is a survey paper on applications of mathematics of semirings to numerical analysis and computing. Concepts of universal algorithm and generic program are discussed. Relations between these concepts and mathematics of semirings are examined. A very brief introduction to mathematics of semirings (including idempotent and tropical mathematics) is presented. Concrete applications to optimization problems, idempotent linear algebra and interval analysis are indicated. It is known that some nonlinear problems (and especially optimization problems) become linear over appropriate semirings with idempotent addition (the so-called idempotent superposition principle). This linearity over semirings is convenient for parallel computations.

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