Study of Algorithms and Applications of Approximate Algebra

近似代数算法及应用研究

基本信息

批准号：
15300002
负责人：
SASAKI Tateaki
金额：
$ 7.36万
依托单位：
University of Tsukuba
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

项目摘要

The purposes of this research are, 1) to establish and analyze new concepts, 2) to develop new algorithms, 3) to stabilize existing algorithms, and 4) applications to science and engineering.Results on 1) Concepts of approximately singular and approximate non-conjugateness are introduced (Sasaki). The denominator factors appearing in extended Hensel factors are clarified, and the properties of convergence and many-valuedness of extended Hensel series are investigated numerically (Sasaki ＆ student).Results on 2) A stable method for computing Grobner bases with floating-point numbers is proposed (Sasaki-Kako). A new method is found for multivariate polynomial factorization (Sasaki ＆ student). A semi-algebraic method is proposed to separate close-root clusters of univariate polynomial (Sasaki-Kako). A very tight error-bound formula is derived for numerical roots of univariate polynomial (Sasaki). A theory of recursive subresultants is developed for separating the real roots of univariate … More polynomial (Terui). A simultaneous iterative formula of arbitrary degree of convergence is derived for symbolic Newton's method (Terui). A method of drawing two-dimensional pseudovariety is developed (Kai et al.). A method of constructing nearest polynomials of degrees up to 4 is developed (Noda-Kai et al.). A method for approximate indefinite integral of rational functions with parameters is proposed (Kai-Noda ＆ student).Results on 3) The reason of occurrence of large errors in the computation of floating-point Grobner base is clarified (Sasaki-Kako). As for ill-conditioned cases in computing approximate GCDs of multivariate as well as univariate polynomials, several techniques to stabilize PRS-type algorithms are proposed (Sasaki ＆ student). By analyzing the univariate rational-function approximation with floating-point numbers, clarified is the reason of appearance of unnecessary poles and a stabilization method based on the Pade approximation is proposed (Kai-Noda ＆ students).Results on 4) As for the 8-line arrangement problem, a complete classification of the arrangements is attained, after many experiments of generating 8-line arrangements (Fukui et al.). Less

这项研究的目的是：1）建立和分析新概念，2）开发新算法，3）稳定现有算法，以及4）4）对科学和工程的应用。对1）引入了近似单数和近似非混合性的概念（Sasaki）。阐明了扩展的Hensel因子中出现的分母因子，并且分别研究了收敛性和扩展的Hensel系列的多元值（Sasaki＆Students）。对2）进行计算具有浮点数数量的Grobner碱基的稳定方法（Sasaki-Kako）。发现了一种新方法，用于多元多项式分解（Sasaki＆Student）。提出了一种半代数方法来分离单变量多项式（Sasaki-Kako）的近距离群簇。对于单变量多项式（Sasaki）的数值根，得出了一个非常紧密的误差公式。递归替代者的理论是为了分离单变量的真正根源……更多项式（TERUI）。对于符号牛顿的方法（TERUI），得出了一个任意收敛程度的简单迭代公式。开发了一种绘制二维伪动力的方法（Kai等人）。开发了一种构造最接近4度的最接近多项式的方法（Noda-kai等人）。提出了一种与参数的无限函数无限整合的方法（Kai-Noda＆Student）。3）阐明了浮点数Grobner基础计算中发生较大误差的原因（Sasaki-kako）。至于计算多变量和单变量多项式的近似GCD中的情况不佳的情况，提出了几种稳定PRS型算法的技术（Sasaki＆Student）。通过分析与浮点数数量的单变量合理功能近似，澄清是出现不必要的两极的原因，并提出了基于宫颈近似的稳定方法（Kai-Noda＆Student on 4）。较少的

项目成果

期刊论文数量（77）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Tighter bounds of errors of numerical roots

数值根误差的更严格界限

DOI：
发表时间：
2007
期刊：
Japan J. Indus. Appl. Math. Vol 24
影响因子：
0
作者：
S. Tadokoro;Y. Yamaguti;I. Tsuda and H. Fujii;M. Yamazato;M. Takeda;T. Sasaki
通讯作者：
T. Sasaki

L.Zhi, M.T.Noda, H.Kai, W.Wu: "Hybrid method for computing the nearest singular polynomials"Jpn J.Ind.App.Math.. (to appear). (2004)

L.Zhi、M.T.Noda、H.Kai、W.Wu：“计算最近奇异多项式的混合方法”Jpn J.Ind.App.Math..（即将出现）。