Study of Algorithms and Applications of Approximate Algebra

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)在科学和工程上的应用。1)引入了近似奇异和近似非共轭的概念（Sasaki）。阐明了扩展Hensel因子中出现的分母因子，数值研究了扩展Hensel级数的收敛性和多值性（Sasaki & student）。2)提出了一种计算浮点数Grobner基的稳定方法（Sasaki-Kako）。找到了一种新的多元多项式分解方法（Sasaki & student）。提出了一种分离单变量多项式（Sasaki-Kako）密根聚类的半代数方法。导出了单变量多项式（Sasaki）数值根的一个非常严格的误差界公式。提出了一种分离单变量…More多项式实根的递归子结果理论。对符号牛顿法（Terui）导出了任意收敛度的同时迭代公式。开发了一种绘制二维伪变种的方法（Kai等）。开发了一种构造最接近次多项式的方法（Noda-Kai et al.）。提出了一种带参数有理函数的近似不定积分方法（Kai-Noda & student）。3)明确了浮点格罗布纳基计算中出现较大误差的原因（Sasaki-Kako）。对于计算多变量和单变量多项式近似gcd的病态情况，提出了几种稳定prs型算法的技术（Sasaki & student）。通过对浮点数单变量有理函数逼近的分析，阐明了不必要极点出现的原因，提出了一种基于Pade逼近的镇定方法（Kai-Noda & students）。4)对于8线排列问题，经过多次生成8线排列的实验，得到了完整的排列分类（Fukui et al.）。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