Macsyma computation of local minimal realization of dynamical systems of which generating power series are finite
✍ Scribed by Nour Eddine Oussous
- Publisher
- Elsevier Science
- Year
- 1991
- Tongue
- English
- Weight
- 513 KB
- Volume
- 12
- Category
- Article
- ISSN
- 0747-7171
No coin nor oath required. For personal study only.
✦ Synopsis
We present here a package of Macsyrna programs, allowing the manipulation of words, and noncommutative power series over some finite alphabet.
On the basis of the works of M.Fliess and C.Reutenauer, concerning local realization of nonhnear dynamical systems, we present an algorithm allowing the computation of the local and minimal realization of finite generating power series.
We describe that algorithm in the computer algebra system Macsyma.
1. Introduction
We present here a package of Macsyma programs, allowing the manipulation of words, and noncommutative power series over some finite alphabet X. This package contains, in particular, an implementation of shuffle product of two noncommutative polynomials, left and righ~ remainder of a noncommutative polynomial by another noncommutative polynomial, production (up to some fixed degree) of the Lyndon basis of the free Lie algebra £ie(X), and the canonical "Poincar4-Birkhoff-Witt" basis of noncommutative polynomials over X.
As a development, we present a package of some programs which computes the local and minimal realization of dynamical systems of which generating power series are finite. The first version of this package was implemented with computer algebra system Macsyma on the Bull computer DPS 8 under the operating system Multics. The actual version is implemented with computer algebra system Macsyma on the workstation Sun 3/80 under the operating system Unix.
Our program can deal with all polynomials but, since it uses the shuffle product, its capacity is limited by the core size of the computer. It is. able to treat polynomials up to degree five. Now, we have a mean which allows us to implement and manipulate rational series. A continuation of this work is the realization of rational series.
We use Lyndon basis as £ie(X)-basis. Lyndon words are used to compute the local coordinate system.