Computing the Isolated Roots by Matrix Methods

Abstract

Two main approaches are used, nowadays, to compute the roots of a zero-dimensional polynomial system. The first one involves Gröbner basis computation, and applies to any zero-dimensional system. But, it is performed withexact arithmetic and, usually, large numbers appear during the computation. The other approach is based on resultant formulations and can be performed with floating point arithmetic. However, it applies only to generic situations, leading to singular problems in several systems coming from robotics and computational vision, for instance.In this paper, reinvestigating the resultant approach from the linear algebra point of view, we handle the problem of genericity and present a new algorithm for computing the isolated roots of an algebraic variety, not necessarily of dimension zero. We analyse two types of resultant formulations, transform them into eigenvector problems, and describe special linear algebra operations on the matrix pencils in order to reduce the root computation to a non-singular eigenvector problem. This new algorithm, based on pencil decompositions, has a good complexity even in the non-generic situations and can be executed with floating point arithmetic.