Numeric and Symbolic Methods for Polynomial Manipulation

基本的符号、数值和几何计算及其在图形、几何建模、机器人、视觉和工程问题中的应用都需要符号和数字操作系统的算法。本研究采用三重方法来解决这些问题：(1)开发更好的符号结果计算算法，并使用结果和矩阵计算来求解非线性多项式方程。算法是在精确算术和浮点算术的背景下开发的。它涉及到用矩阵和行列式更好地表述结果，以及利用矩阵公式使用符号和数值算法。(2)通过使用这些应用中出现的特定多项式系统，使这些算法专门应用于计算机图形学，几何建模和机器人技术。(3)开发一个例程库ELIMPACK，用于符号结果计算和多项式方程的求根（精确和浮点算术）。该软件包对符号计算、数值计算、几何应用和工程界都有很大的实用价值。这些结果也有助于理解用多项式方程描述的几何问题的复杂性。

Algorithms for symbolic and numeric manipulation of systems are needed for basic symbolic, numeric and geometric computations and their application to problems in graphics, geometric modeling, robotics, vision and engineering. This research takes a three-fold approach to these problems using multipolynomial resultants: (1) Develop better algorithms for symbolic resultant computation and solving nonlinear polynomial equations using resultants and matrix computations. Algorithms are being developed in the context of exact arithmetic as well as floating point arithmetic. It involves better formulations of resultants in terms of matrices and determinants and use of symbolic and numeric algorithms making use of the matrix formulation. (2) Specialize these algorithms to applications in computer graphics, geometric modeling and robotics by making use of the specific polynomial systems arising in these applications. (3) Develop a library of routines, ELIMPACK, for symbolic resultant computation and finding roots of polynomial equations (in exact and floating point arithmetic). This package is of great utility to symbolic computation, numerical computation, geometric applications and the engineering community. The results also help in understanding the complexity of geometric problems described in terms of polynomial equations.