Study of Algorithms and Applications of Approximate Algebra

这项研究的目的是：1)开发用于多种代数运算的近似代数算法；2)对近似代数算法进行误差分析；3)实现发展到NSL-GAL系统中的算法；4)寻找近似代数的应用。算法研究：近似因式分解算法的改进(Sasaki&Nagaka)，带误差项的Sturm多项式序列理论(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法(Noda&KAI)。误差分析：带浮点数的多元Hensel结构的抵消误差分析(Sasaki&Yamaguchi)，带浮点数的多元结式计算的抵消误差分析(Sasaki&Sato)，基于近似GCD的混合有理函数逼近的误差评估(KAI&NADA).实现：GAL(Kako&Sasaki)的近似代数计算工具，计算多项式系数线性方程多项式解的程序(Motoyoshi)，GAL与Internet的接口的开发和相应的GAL的修改(Fukui)，计算近似Grobner基的程序包(Suzuku)，利用有效数计算多元多项式方程的有效解(Suzuki)。应用：代数-数值计算在实投影平面上N线排列分类中的应用(Fukui&Sekiguchi)，混合有理函数逼近在机翼理论中出现的奇异积分方程解的应用(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).