Efficient Computer Algorithms for Symbolic Mathematics

关于有效串行与并行的几个问题 计算机代数和符号的算法设计和开发 数学被认为。 高效处理器的设计 上一般与特殊线性方程组的并行算法 一个抽象的，可能是模块化的，域将被调查，以及 设计了一个有效的顺序算法来计算 一个隐式给定方阵的特征多项式， 函数，计算这个矩阵与向量的乘积。 一 Goldwasser-Kilian/Atkin拉斯维加斯的计算机实现 通过改进几个辅助函数，提高了整数素性检验的性能 测试中使用的数论算法。 的算法 由黑盒给出的多元多项式的因式分解 他们的评估计划也将在一个网络上实施， 工作站。 此外，寻找更有效的版本， 希尔伯特不可约型定理将继续下去。 与新兴应用相关的两个新课题的研究 的符号计算。 受几何设计的启发， 计算代数方程不可约分量的算法 由多项式定义隐式方程给出的曲线或曲面 将开发浮点系数。 攻击 计算机代数方法，问题是计算近似 有理二元多项式的因式分解。 最后，分支 代数算术之外，特殊函数的使用，如 平方根，指数，或多项式，在直线程序 将被调查。 特别是，一个分析的直- 线程序将制定，并为此转换理论 模型将被研究。

Several problems on the subject of efficient sequential and parallel algorithm design and development for computer algebra and symbolic mathematics are considered. The design of a processor-efficient parallel algorithm for solving general and special linear systems over an abstract, possibly modular, domain will be investigated, as well as the design of an efficient sequential algorithm for computing the characteristic polynomial of an implicitly given square matrix by a function that computes the product of this matrix with a vector. A computer implementation of the Goldwasser-Kilian/Atkin Las Vegas integer primality test will be enhanced by improving several auxiliary number theoretical algorithms used in the test. An algorithm for factoring multivariate polynomials that are given by black box programs for their evaluation will also be implemented on a network of workstations. Furthermore, a search for more effective versions of Hilbert irreducibility-type theorems will continue. Research on two new topics that are related to emerging applications of symbolic computation is suggested. Motivated by geometric design, an algorithm for computing the irreducible components of an algebraic curve or surface given by its polynomial defining implicit equation with floating point coefficients will be developed. Attacked by computer algebra methods, the problem is to compute the approximate factorization of a rational bivariate polynomial. Finally, branching out of algebraic arithmetic, the usage of special function, such as squareroots, exponentials, or logarithms, in the straight-line program will be investigated. In particular, a model of an analytic straight- line program will be formulated, and a transformation theory for this model will be investigated.