Verification methods for dense and sparse systems of equations
Verification methods for dense and sparse systems of equations
复制标题
稠密和稀疏方程组的验证方法
DOI：
10.15480/882.325
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发表时间：
1994
期刊：
影响因子：
--
通讯作者：
["S. Rump
中科院分区：
文献类型：
--
作者：
["S. Rump
在本文中，我们描述了稠密和大型稀疏线性和非线性方程组的验证方法。所描述的大多数方法都是作者发展起来的。文中还提到了其他方法，但并不打算概述现有的方法。许多结果都以类似的形式发表在研究论文或书籍上。在这本专著中，我们想对这一主题的一些基本概念进行简明而紧凑的处理。此外，许多新的结果都包含在其他地方没有发表的文章中。其中包括以下内容。给出了区间矩阵正则性的一个新判别法。结果表明，对于矩阵类，它是明显更好的。给出了连续函数的包含定理，这些函数不一定是可微的。非线性函数W.r.t.的一些推广。使用的点x̃可以是斜率、雅可比或其他。通过(I)使用斜率而不是雅可比，(Ii)改进超越函数的斜率，(Iii)分两步证明在小范围内的存在和在大区间内的唯一性，从而允许在更广泛的域中证明唯一性并显著提高速度，(Iv)使用Einzelschrittverfahren，(V)计算差值的包含，从而实现了更窄的包含和更广泛的适用性(显著更宽的输入容差)。近似解。给出了具有参数依赖的输入区间问题的求解方法，给出了内包含法和外包含法。介绍了改善内夹杂物质量的方法。给出了(I)M-矩阵、(Ii)对称正定、(Iii)对称、(Iv)一般展开矩阵的参数化稀疏非线性系统的方法。作者所在的研究所开发了一个快速间隔图书馆，与现有图书馆相比，速度要快得多。
In this paper we describe verification methods for dense and large sparse systems of linear and nonlinear equations. Most of the methods described have been developed by the author. Other methods are mentioned, but it is not intended to give an overview over existing methods. Many of the results are published in similar form in research papers or books. In this monograph we want to give a concise and compact treatment of some fundamental concepts of the subject. Moreover, many new results are included not being published elsewhere. Among them are the following. A new test for regularity of an interval matrix is given. It is shown that it is significantly better for classes of matrices. Inclusion theorems are formulated for continuous functions not necessarily being differentiable. Some extension of a nonlinear function w.r.t. a point x̃ is used which may be a slope, Jacobian or other. More narrow inclusions and a wider range of applicability (significantly wider input tolerances) are achieved by (i) using slopes rather than Jacobians, (ii) improvement of slopes for transcendental functions, (iii) a two-step approach proving existence in a small and uniqueness in a large interval thus allowing for proving uniqueness in much wider domains and significantly improving the speed, (iv) use of an Einzelschrittverfahren, (v) computing an inclusion of the difference w.r.t. an approximate solution. Methods for problems with parameter dependent input intervals are given yielding inner and outer inclusions. An improvement of the quality of inner inclusions is described. Methods for parametrized sparse nonlinear systems are given for expansion matrix being (i) M-matrix, (ii) symmetric positive definite, (iii) symmetric, (iv) general. A fast interval library having been developed at the author’s institute is presented being significantly faster compared to existing libraries.