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

本研究的目的是，1) 建立和分析新概念，2) 开发新算法，3) 稳定现有算法，4) 在科学和工程中的应用。结果 1) 近似奇异和近似非的概念- 引入了共轭性（Sasaki）。阐明了扩展 Hensel 因子中出现的分母因子，并对扩展 Hensel 级数的收敛性和多值性进行了数值研究（Sasaki）。 2）结果提出了一种用浮点数计算 Grobner 基的稳定方法（Sasaki-Kako）。发现了一种用于多元多项式分解的新方法（Sasaki ＆ Student）提出了一种半代数方法。分离单变量多项式的近根簇（Sasaki-Kako），导出了单变量多项式数值根的非常严格的误差界限公式。（Sasaki）开发了一种递归子结果理论，用于分离单变量多项式的实根（Terui）。为符号牛顿法（Terui）导出了任意收敛度的联立迭代公式。开发了维度伪变性（Kai 等人）。开发了一种构造最多 4 次的最近多项式的方法（Noda-Kai 等人）。提出了带参数的有理函数的近似不定积分方法（Kai-Noda＆student）。3）结果阐明了浮点Grobner基计算中出现大误差的原因（Sasaki-Kako）。在计算多变量和单变量多项式的近似 GCD 时的病态情况下，提出了几种稳定 PRS 型算法的技术（Sasaki 和学生）。浮点数有理函数逼近，阐明了出现不必要极点的原因，并提出了基于 Pade 逼近的稳定方法（Kai-Noda ＆学生）。4）关于 8 行排列问题的结果，经过多次生成 8 行排列的实验，获得了排列的完整分类（Fukui 等人）。

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

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