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）将所开发的算法实现到NSL-GAL系统中; 4）寻求近似代数的应用。我们为每个目的进行了以下研究。算法研究：近似因式分解算法改进（Sasaki & Nagasaka），带误差项的多项式的Sturm序列理论（Sasaki & Terui），多元多项式在奇点的分解方法及其在多元因式分解中的应用（Sasaki & Inaba），二元函数的有理函数插值方法（Noda & Kai）.误差分析：浮点数多元Hensel构造中的消去误差分析（Sasaki & Yamaguchi），分析浮点数多元结果计算中的抵消误差（Sasaki & Sato），基于近似GCD的混合有理函数逼近的误差估计（Kai & Noda）.实施：GAL中的近似代数计算工具（Kako & Sasaki），一个计算多项式系数线性方程的多项式解的程序（Motoyoshi）、GAL和因特网之间的接口的开发以及GAL的相应修改（福井）、用于计算近似Grobner基的包（Suzuku）、用于通过使用有效数来计算多元多项式方程的验证解的包（Suzuki）。代数-数值计算在真实的射影平面上N条直线排列分类中的应用（福井&关口），应用混合有理函数逼近机翼理论中出现的奇异积分方程（Noda & Kai），耦合线性方程的近似幂级数解在代数控制理论中的应用（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).