Algebraic Algorithms

Our Chapter in the upcoming Volume I: Computer Science and Software Engineering of Computing Handbook (Third edition), Allen Tucker, Teo Gonzales and Jorge L. Diaz-Herrera, editors, covers Algebraic Algorithms, both symbolic and numerical, for matrix computations and root-finding for polynomials and systems of polynomials equations. We cover part of these large subjects and include basic bibliography for further study. To meet space limitation we cite books, surveys, and comprehensive articles with pointers to further references, rather than including all the original technical papers.

1986

A direct numerical method is proposed for the determination of all isolated zeros of a system of multivariate polynomial equations. By "polynomial combination", the system is reduced to a special form which may be interpreted as a multiplication table for power products modulo the system. The zeros are then formed from an ordinary eigenvalue problem for the matrix of the multiplication table. Degenerate situations may be handled by perturbing them into general form and reaching the zeros of the unperturbed system via a homotopy method.

29th IEEE Conference on Decision and Control, 1990

Matrices of pol nomials over rings and fields provide a unifying framework $r many control system design problems. These include dynamic compensator design, infinite dimensional systems, controllers for nonlinear systems, and even controllers for discrete event s stems. An important obstacle for utilizing these owerful matiematical tools in practical applications has been &e non-availability of accurate and efficient algorithms to carry through the precise error-free computations required b these algebraic methods. In this paper we develop highly ekcient, error-free a1 orithms, for most of the important computations needed in %near systems over fields or rings. We show that the structure of the underlying rings and modules is critical in designing such algorithms.

Japan Journal of Industrial and Applied Mathematics, 2003

The zeros of a polynomial can be found by computing the eigenvalues of the corresponding companion matrix. However, in the case of multiple zeros, the calculated results are usually not as accurate as those for simple zeros. This problem is discussed and a new method is presented by constructing a new companion matrix which has only simple eigenvalues. By this method, instead of calculating all the zeros simultaneously, we can calculate the distinct zeros and their multiplicities separately. Numerical examples are presented to illustrate the efficiency of the method.

We describe an Average Las Vegas algorithm to compute several zeros of polynomial systems, and analyze its complexity and Shannon Entropy. We also derive the first universal algorithm to solve this kind of systems in average running time linear in the Bézout number and polynomial in the input length for dense encoding of polynomials.

2009

Starting from algorithms introduced in [Ky M. Vu, An extension of the Faddeev's algorithms, in: Proceedings of the IEEE Multi-conference on Systems and Control on September 3-5th, 2008, San Antonio, TX] which are applicable to one-variable regular polynomial matrices, we introduce two dual extensions of the Faddeev's algorithm to one-variable rectangular or singular matrices. Corresponding algorithms for symbolic computing the Drazin and the Moore-Penrose inverse are introduced. These algorithms are alternative with respect to previous representations of the Moore-Penrose and the Drazin inverse of one-variable polynomial matrices based on the Leverrier-Faddeev's algorithm. Complexity analysis is performed. Algorithms are implemented in the symbolic computational package MATHEMATICA and illustrative test examples are presented.

Journal of Computational and Applied Mathematics, 2000

Computers & Mathematics with Applications, 2005

Chapman & Hall/CRC Applied Algorithms and Data Structures series, 2009

Contemporary Mathematics, 2011

These pages contain a short overview on the state of the art of efficient Numerical Analysis methods that solve systems of multivariate polynomial equations. We focus on the work of Steve Smale who initiated this research framework, and on the collaboration between Steve Smale and Mike Shub, which set the foundations of this approach to polynomial system-solving.