Gröbner Bases and Numerical Analysis

Summary

Abstract

By concentrating on system solving, numerical interpolation, integration, and differentiation, we show the use of Gröbner bases in numerical analysis. The ideas of the factorizing Gröbner algorithm, of system solving by solving Eigenproblems, of computing interpolation polynomials with algorithms for computing Gröbner bases, and of constructing numerical integration and differentiation formulas by Gröbner bases are presented. A short section on Gröbner bases computation using floating point arithmetics is included.

Introduction

In the nineties, there is an increasing interest in combining symbolic and numerical methods. This can be seen at diverse instances. There are now international symposia supported by organizations from both sides, and the number of contributes displaying the symbolic - numerical interplay is increasing. Other examples are the facilities of using floating point arithmetics and simple numerical procedures in Computer Algebra Systems on the one hand and the (eventually partly) integration of Computer Algebra Systems into numerical software packages on the other hand. The most prominent example is here the migration of the Computer Algebra System AXIOM to NAG, the Numerical Algorithm Group.

Many interesting results have been obtained by combining symbolic and numerical methods, like in polynomial continuation the avoiding of solution paths diverging to infinity by means of concepts from toric ideals or like the numerical solving of systems of polynomial equations using resultants, see for instance Canny and Manocha (1993).