Study of Algorithms and Applications of Approximate Algebra

本研究的目的是：1)为许多代数运算开发近似代数算法；2)对近似代数算法进行误差分析；3)将开发的算法实现到NSL-GAL系统中；4)寻求近似代数的应用。我们针对每个目的进行了以下研究。算法研究：近似分解算法的改进（Sasaki & Nagasaka），带有误差项的多项式Sturm序列理论（Sasaki & Terui），多元多项式在奇点处的分解方法及其在多元分解中的应用（Sasaki & Inaba），二元函数的有理函数插值方法（Noda & Kai）。误差分析：基于浮点数的多元Hensel构造的消去误差分析（Sasaki & Yamaguchi），基于浮点数的多元结果计算的消去误差分析（Sasaki & Sato），基于近似GCD的混合有理函数近似的误差评估（Kai & Noda）。实现：GAL中的近似代数计算工具（Kako & Sasaki），计算多项式系数线性方程的多项式解的程序（Motoyoshi）， GAL与Internet之间的接口开发以及GAL的相应修改（Fukui），计算近似Grobner基的包（Suzuku），使用有效数计算多元多项式方程的验证解的包（Suzuki）。应用：将代数-数值计算应用于实数投影平面上N条线排列的分类（Fukui & Sekiguchi），将混合有理函数近似应用于机翼理论中出现的奇异积分方程（Noda & Kai），将耦合线性方程的近似幂级数解应用于代数控制理论（Kitamoto）。

The purposes of this research are, 1) to develop approximate algebraic algorithms for many algebraic operations, 2) to perform error analysis of approximate algebraic algorithms, 3) to implement the algorithms developed into NSL-GAL system, and 4) to seek for applications of approximate algebra. We performed the following researches for each purpose.Algorithm study : improvement of approximate factorization algorithm (Sasaki & Nagasaka), theory of Sturm sequence of polynomial with error terms (Sasaki & Terui), decomposition method of multivariate polynomial at a singular point and its application to multivariate factorization (Sasaki & Inaba), rational function interpolation method for bivariate functions (Noda & Kai).Error analysis : analysis of cancellation errors in multivariate Hensel construction with floating-point numbers (Sasaki & Yamaguchi), analysis of cancellation errors in multivariate resultant computation with floating-point numbers (Sasaki & Sato), evaluation of errors in hybrid rational function approximation based on approximate GCD (Kai & Noda).Implementation : facilities for approximate algebraic computation in GAL (Kako & Sasaki), a program for calculating polynomial solutions of linear equations with polynomial coefficients (Motoyoshi), development of an interface between GAL and Internet and corresponding modification of GAL (Fukui), a package for computing approximate Grobner basis (Suzuku), a package for computing validated solutions of multivariate polynomial equation by using effective numbers (Suzuki).Applications : application of algebraic-numeric computation to classification of N lines arrangement on real projective plane (Fukui & Sekiguchi), application of hybrid rational function approximation to singular integral equations appearing in wing theory (Noda & Kai), application of approximate power series solutions of coupled linear equations to algebraic control theory (Kitamoto).