Study of Algorithms and Applications of Approximate Algebra

本研究的目的是：1）建立和分析新的概念，2）开发新的算法，3）稳定现有的算法，4）在科学和工程中的应用。阐明了扩展的Hensel因子中的分母因子，数值研究了扩展的Hensel级数的收敛性和多值性（Sasaki & student）.结果2）给出了计算浮点数Grobner基的稳定方法（Sasaki-Kako）.本文提出了一种多元多项式因式分解的新方法。提出了一种分离一元多项式闭根簇的半代数方法。给出了一元多项式数值根的严格误差界公式。建立了一种用于分离一元真实的根的递归子结式理论 ...更多信息 多项式（Terui）。本文导出了符号牛顿法（Terui法）的一个任意收敛阶的联立迭代公式. Kai等人发展了一种绘制二维伪簇的方法。开发了一种构造次数最多为4的最近多项式的方法（Noda Kai等人）。本文提出了含参数有理函数不定积分的一种近似方法（Kai-Noda & Student），并得到了以下结果：（3）阐明了浮点Grobner基计算中出现较大误差的原因（Sasaki-Kako）。至于病态的情况下，在计算近似GCD的多元以及单变量多项式，提出了几种技术来稳定PRS型算法（佐佐木和学生）。通过对浮点数的一元有理函数逼近的分析，阐明了出现不必要极点的原因，并提出了基于Pade逼近的稳定化方法（Kai-Noda & Students）。结果：4）对于8线排列问题，通过多次生成8线排列的实验（福井等），得到了排列的完整分类。少

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