Study of Algorithms and Applications of Approximate Algebra

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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