The project aims to develop robust numerical methods and a high quality software package PolynPak for solving three fundamental algebraic problems: the univariate polynomial AGCD (approximate greatest common divisor), the multivariate polynomial AGCD, and accurate multiplicity-identification/root-finding. All three problems are under a common assumption in application that the given data are empirical and may contain errors from measurement and rounding-off. The methodology in this project consists of a two-stage approach and the theory that, while GCD and multiple roots are ill-posed/ill-conditioned under arbitrary perturbation, they are remarkably insensitive when perturbations are structure-preserving. Therefore, the ill-posedness can be removed by reformulating the problem in a least squares setting under a structural constraint after calculating the structure of AGCD and multiplicity. The approach in this project may also be applicable to other ill-posed problems in numerical computation.
This research is carried out in the fields of computer algebra and numerical analysis where the mission is to provide the scientific and industrial community with reliable algorithms and software for solving mathematical problems. Since polynomial is one of the most fundamental models in applied mathematics with a wide range of applications in areas such as economic equilibria, chemical reaction, image processing/restoration, to name a few, developing robust algorithms and software for polynomial algebra, as the objective of this project, may have profound impact in those applications and in scientific computing.
