Multi-Use "Plug-And-Play" Software Packages for Black Box and Inexact Symbolic Objects

该项目研究设计高效的算法，在软件包中实现算法，并使符号计算系统的非专业用户可以访问程序。 将构造符号对象的黑盒表示和包含不精确（即浮点数据）的符号对象的包。 这些软件包一般被编程为具有抽象底层算术的C++模板类;它们可以使用各种快速库进行编译，用于基本字段，浮点和多项式运算。 一个服务器/客户端接口无缝地将软件包连接到所有广泛使用的通用符号系统，如Maple和Mathetmatica。 并行执行的实现算法将得到促进。 黑盒对象存储为函数。 例如：黑盒多项式是将变量的值作为输入并在该给定点处对多项式求值的过程;黑盒矩阵是将任意向量作为输入并计算矩阵乘以向量积的过程。 FoxBox系统是一个计算最大公约数和因式分解黑盒多项式的软件包。 其目的是消除FoxBox中的算法瓶颈，并添加黑盒线性代数。 为了提高速度，该项目侧重于有限域上的算法。 将开发各种通用系统的高效服务器/客户端桥接代码。 该项目还将调查如何不准确（例如， 浮点）数据可以在符号计算的过程中处理。 符号中浮点系数的允许，即，参数化模型，是从物理世界的问题的象征性的方法至关重要。 此外，浮点运算比精确运算快，特别是对于代数数。 本文介绍了几种数值模型，如后验迭代改进和摄动灵敏度分析， 输入数据，将被考虑。 Toeplitz矩阵秩的问题，多项式复根的位置，并在许多变量分解复杂的多项式将被调查。 一个即插即用的符号/数字包的设计将进行研究。

This project conducts research in the design of efficient algorithms, their implementation in software packages, and in making the programs accessible to non-specialist users of symbolic computation systems. Packages for the black box representation of symbolic objects and for symbolic objects containing imprecise, that is, floating point data will be constructed. The packages are generically programmed as C++ template classes with abstract underlying arithmetics; they can be compiled with a variety of fast libraries for the basic field, floating point, and polynomial operations. A server/client interface seamlessly attaches the packages to all widely-used general purpose symbolic systems such as Maple and Mathetmatica. Parallel execution of the implemented algorithms will be facilitated. Black box objects are stored as functions. For instance: a black box polynomial is a procedure that takes values for the variables as input and evaluates the polynomial at that given point; a black box matrix is a procedure that takes an arbitrary vector as input and computes the matrix times vector product. The FoxBox system is a package for computing greatest common divisors and factoring black box polynomials. The aim is to eliminate algorithmic bottlenecks in FoxBox and add black box linear algebra. For sake of speed, the project focuses on algorithms over finite fields. Efficient server/client bridge code to a variety of general purpose systems will be developed. The project will also investigate how inexact (e.g., floating point) data can be handled in the course of a symbolic computation. The allowance of floating point coefficients in a symbolic, i.e., parameterized model, is crucial for a symbolic approach to problems from the physical world. Moreover, floating point arithmetic is faster than exact arithmetic, especially for algebraic numbers. Several numerical models, such as a-posteriori iterative improvement and sensitivity analysis for perturbed input data, will be considered. The problems of Toeplitz matrix rank, polynomial complex root location, and factoring complex polynomials in many variables will be investigated. The design of a plug-and-play symbolic/numeric package will be studied.