Effective computations with 2-variable polynomial matrices in MATHEMATICA (original) (raw)

New package for effective polynomial computation in Mathematica

This report describes our work on implementation of effective numerical routines for polynomials and polynomial matrices in the MATHEMATICA software. Such operations are recalled during the controller design process if the so called polynomial or algebraic design methods are employed. This research is also motivated by the fact that MATHEMATICA developers pay attention to control engineers needs and produce the Control Systems Professional package for use with MATH-EMATICA and, as we believe, a set of routines for algebraic approach could conveniently complement the existing bunch of programs primarily intended for state-space representations.

SINGULAR 2-2 – A Computer Algebra System for Polynomial Computations

Singular is a specialized computer algebra system for polynomial computations with emphasize on the needs of commutative algebra, alge-braic geometry, and singularity theory. Singular's main computational objects are polynomials, ideals and modules over a large variety of rings. Singular features one of the fastest and most general implementations of various algorithms for computing standard resp. Gröbner bases. The new, upcoming version 2-2 includes also algorithms for a wide class of non-commutative algebras (Plural) and the possiblity for dynamic extension of the program at run-time (dynamic modules). Furthermore, it provides multivariate polynomial factorization, resultant, characteristic set and gcd computations, syzygy and free-resolution computations, numerical root– finding, visualisation, and many more related functionalities.

On fast evaluation of bivariate polynomials at equispaced arguments

IEEE Transactions on Signal Processing, 1992

The initial value problem arising in the recursive evaluation of a 2-D polynomial at equispaced points is treated in detail; the results facilitate efficient implementation of Bose's recursive algorithm. A comparison has been made of the computational complexity with that involved in a direct computation, and some general observations have been made for an alternative scheme proposed by Nie and Unbehauen.

Fast error-free algorithms for polynomial matrix computations

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.

Max – a Program System for Symbolic Manipulation of Polynomial Matrices

Computer Aided Design in Control and Engineering Systems, 1986

MAX is an interpreted programming language for polynomial matrix manipulations. It contains especially those operations which are frequently needed in linear control. MAX is written using the C programming language under the VAX!VMS operating system. In this paper the design , implementation and available operations of MAX are described.

Numerical performance of the matrix pencil algorithm computing the greatest common divisor of polynomials and comparison with other matrix-based methodologies

Journal of Computational and Applied Mathematics, 1996

This paper presents a new numerical algorithm for the computation of the greatest common divisor (GCD) of several polynomials, based on system-theoretic properties. The specific algorithm, characterizes the GCD as the output decoupling zero polynomial of an appropriate linear system associated with the given polynomial set. The computation of the GCD is thus reduced to specifying a nonzero entry of a vector forming the compound matrix of a matrix pencil directly produced from the associated linear system. A detailed description of the implementation of the algorithm is presented and analytical proofs of its stability are also developed. The MATLAB code of the algorithm is also described in the appendix. Pro(S) = [pl(S),...,pm(S)] t = [P0,Pl,"" ,Pd] ed(s) = Pined(s), where Pm~ ~m× (a+ 1), ea(s) = [1, s, ..., se] t. By GCD {~m,d} --~b(s) we shall denote the GCD of the set. {~d} denotes any set of polynomials of N[s] of maximal degree d' <~ d, d fixed. Notation 2. (1) Qp,n denotes the set of strictly increasing sequences of p integers (1 ~< p ~< n) chosen from 1, 2, ..., n. If c~, [3 ~ Qp,, we say that ~ precedes [3 (~ < [3), if there exists an integer t (1 ~< t ~< p) for which ~1 = [31, ..., c~t-t = [3t-1, c~t = [3t, where ~ = [3i denote the elements of ~, [3, respectively.

On the computation of the minimal polynomial of a two-variable polynomial matrix

The Fourth International Workshop on Multidimensional Systems, 2005. NDS 2005., 2005

The main contribution of this work is to provide an algorithm for the computation of the minimal polynomial of a two variable polynomial matrix, based on the solution of linear matrix equations. The whole theory is implemented via an illustrative example.

Numerical Solution of Bivariate and Polyanalytic Polynomial Systems

SIAM Journal on Numerical Analysis, 2014

Finding the real solutions of a bivariate polynomial system is a central problem in robotics, computer modeling and graphics, computational geometry and numerical optimization. We propose an efficient and numerically robust algorithm for solving bivariate and polyanalytic polynomial systems using a single generalized eigenvalue decomposition. In contrast to existing eigen-based solvers, the proposed algorithm does not depend on Gröbner bases or normal sets, nor does it require computing eigenvectors or solving additional eigenproblems to recover the solution. The method transforms bivariate systems into polyanalytic systems and then uses resultants in a novel way to project the variables onto the real plane associated with the two variables. Solutions are returned counting multiplicity and their accuracy is maximized by means of numerical balancing and Newton-Raphson refinement. Numerical experiments show that the proposed algorithm consistently recovers a higher percentage of solutions and is at the same time significantly faster and more accurate than competing double precision solvers.

Direct Computation of Operational Matrices for Polynomial Bases

Mathematical Problems in Engineering, 2010

Several numerical methods for boundary value problems use integral and differential operational matrices, expressed in polynomial bases in a Hilbert space of functions. This work presents a sequence of matrix operations allowing a direct computation of operational matrices for polynomial bases, orthogonal or not, starting with any previously known reference matrix. Furthermore, it shows how to obtain the reference matrix for a chosen polynomial base. The results presented here can be applied not only for integration and differentiation, but also for any linear operation.

Extensions of Faddeev's algorithms to polynomial matrices

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.