Fast error-free algorithms for polynomial matrix computations

2006

1 Abstract linear systems LetD be a ring (with 1, not necessarily commutative), and letA be a leftD-

2004

The problem of determination of a minimal polynomial basis of a rational vector space is the starting point of many control analysis, synthesis and design techniques. In this paper, we propose a new algorithm for the computation of a minimal polynomial basis of the left kernel of a given polynomial matrix F (s). The proposed method exploits the structure of the left null space of generalized Wolovich or Sylvester resultants, in order to compute ef ciently row polynomial vectors that form a minimal polynomial basis of the left kernel of the given polynomial matrix. One of the advantages of the algorithm is that it can be implemented using only orthogonal transformations of constant matrices and the result is a minimal basis with orthonormal coef cients.

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.

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.

arXiv: Molecular Networks, 2018

Computation biology helps to understand all processes in organisms from interaction of molecules to complex functions of whole organs. Therefore, there is a need for mathematical methods and models that deliver logical explanations in a reasonable time. For the last few years there has been a growing interest in biological theory connected to finite fields: the algebraic modeling tools used up to now are based on Grobner bases or Boolean group. Let n variables representing gene products, changing over the time on p values. A Polynomial dynamical system (PDS) is a function which has several components; each one is a polynom with n variables and coefficient in the finite field Z/pZ that model the evolution of gene products. We propose herein a method using algebraic separators, which are special polynomials abundantly studied in effective Galois theory. This approach avoids heavy calculations and provides a first Polynomial model in linear time.

2009

A collection of algorithms implemented in Mathematica 7.0, freely available over the internet, and capable to manipulate rational functions and solve related control problems using polynomial analysis and design methods is presented. The package provides all the necessary functionality and tools in order to use the theory of \(\it \Omega-\) stable functions, and is expected to provide the necessary framework for the development of several other algorithms that solve specific control problems.

2011

This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials [1] and the exterior algebra [4] representation of polynomial matrices. The results provide a new definition for the "approximate", or "almost" zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros [2], [4] of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the "approximate decoupling polynomials".

Mathematical Theory and Modeling, 2013

Let B be a one dimensional k-subalgebra of the polynomial ring k[X] := k[X 1 , . . . , X n ], where k is a field of characteristic zero. We describe an algorithm which decides if there exists a k-derivation D on k[X] such that B = k[X] (=the kernel of the derivation D). In case B is a ring of constants the algorithm also gives such a derivation.

IEEE Transactions on Automatic Control, 1998

Currently, the three most popular commercial computer algebra systems are Mathematica, Maple and MACSYMA. These systems provide a wide variety of symbolic computation facilities for commutative algebra and contain implementations of powerful algorithms in that domain. The Gröbner Basis Algorithm, for example, is an important tool used in computation with commutative algebras and in solving systems of polynomial equations. On the other hand, most of the computation involved in linear control theory is performed on matrices, and these do not commute. A typical issue of IEEE TAC is full of linear systems and computations with their coefficient matricies A B C D's or partitions of them into block matrices. Mathematica, Maple and MACSYMA are weak in the area of non-commutative operations. They allow a user to declare an operation to be non-commutative, but provide very few commands for manipulating such operations and no powerful algorithmic tools. It is the purpose of this article to report on applications of a powerful tool: a non-commutative version of the Gröbner Basis Algorithm. The commutative version of this algorithm is implemented in most major computer algebra packages. The noncommutative version is relatively new [FMora].