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)在科学和工程上的应用。阐明了推广的Hensel因子中出现的分母因子，并用数值方法研究了推广的Hensel级数的收敛和多值性质(Sasaki和Student)。结果2)提出了一种稳定的浮点数Grobner基的计算方法(Sasaki-Kako)。给出了多元多项式因式分解的一种新方法(Sasaki&Student)。提出了一种分离一元多项式(Sasaki-Kako)闭根簇的半代数方法。给出了一元多项式(Sasaki)数值根的一个很紧的误差界公式。发展了一元…实根分离的递归子结式理论更多项式(Terui)。给出了符号牛顿法(TERUI)任意收敛阶的同时迭代公式。发展了一种绘制二维伪变种的方法(KAI等人)。发展了一种构造最接近的4次多项式的方法(Noda-kai等人)。给出了一种求含参数有理函数近似不定积分的方法(Kai-Noda和Student)。结果(3)澄清了浮点Grobner基计算中误差较大的原因(Sasaki-Kako)。对于计算多元和一元多项式的近似GCDS的病态情况，提出了几种稳定PRS型算法的技巧(Sasaki和Student)。通过对浮点数一元有理函数逼近的分析，阐明了出现不必要极点的原因，并提出了一种基于Pade逼近的稳定化方法(Kai-Noda和Students)。结果4)对于8行排列问题，经过多次产生8行排列的实验，得到了一个完整的排列分类(Fukui等)。较少

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