Kristina Schindelar

Computing diagonal form and Jacobson normal form of a matrix using Gröbner bases

In this paper we present an algorithm for the computation of a diagonal form of a matrix over non-commutative Euclidean domain over a field with the help of Grobner bases. We propose a general framework of Ore localizations of non-commutative G-algebras and show its merits and constructiveness. It allows us to handle, among others, common operator algebras with rational coefficients. We introduce the splitting of the computation of a normal form (like the Jacobson form over simple domain) for matrices over Ore localizations into the diagonalization (the computation of a diagonal form of a matrix) and the normalization (the computation of the normal form of a diagonal matrix). These ideas are also used for the computation of the Smith normal form in the commutative case. We give a special algorithm for the normalization of a diagonal matrix over the rational Weyl algebra and present counterexamples to its idea over rational shift and q-Weyl algebras. Our implementation of the algorithm in Singular:Plural relies on the fraction-free polynomial strategy, details of which will be described in the forthcoming article. It shows quite an impressive performance, compared with methods which directly use fractions. In particular, we experience quite a moderate swell of coefficients and obtain uncomplicated transformation matrices. We leave questions on the algorithmic complexity of this algorithm open, but we stress the practical applicability of the proposed method to a large class of non-commutative algebras.

Gröbner bases and behaviors over finite rings

For several decades Gröbner bases have proved useful tools for different areas in system theory, particularly multidimensional system theory. These areas range from controller design to minimal realizations of linear systems over fields. In this paper we focus on the univariate case and identify the so-called “predictable leading monomial property” as a property of a minimal Gröbner basis that is crucial in many of these areas. The property is stronger than “row reducedness”. We revisit the recently developed theory of [17] in which row reducedness is extended to polynomial matrices over the finite ring ℤ<inf>p</inf>^r (with p a prime integer and r a positive integer), which find applications in error control coding over ℤ<inf>p</inf>^r. We recast the ideas of [17] in the more general setting of Gröbner bases and derive new results on how to use minimal Gröbner bases to achieve the predictable leading monomial property over ℤ<inf>p</inf>^r. A major advantage of the Gröbner approach is that computational packages are available to compute a minimal Gröbner basis over ℤ<inf>p</inf>^r, such as the SINGULAR computer algebra system. Another advantage of the Gröbner approach is its generality with respect to the choice of ordering of polynomial vectors.

Exact linear modeling with polynomial coefficients

Given a finite set of polynomial, multivariate, and vector-valued functions, we show that their span can be written as the solution set of a linear system of partial differential equations (PDE) with polynomial coefficients. We present two different but equivalent ways to construct a PDE system whose solution set is precisely the span of the given trajectories. One is based on commutative algebra and the other one works directly in the Weyl algebra, thus requiring the use of tools from non-commutative computer algebra. In behavioral systems theory, the resulting model for the data is known as the most powerful unfalsified model (MPUM) within the class of linear systems with kernel representations over the Weyl algebra, i.e., the ring of differential operators with polynomial coefficients.

The predictable leading monomial property for polynomial vectors over a ring

The “predictable degree property”, a terminology introduced by Forney in 1970, is a property of polynomial matrices over a field F that has proven itself to be fundamentally useful for a range of applications. In this paper we strengthen this property into the “predictable leading monomial” property, and show that this PLM property is shared by minimal Gröbner bases for any positional term order (here: TOP and POT) in F[x]q. The property is useful particularly for minimal interpolationtype problems. Because of the presence of zero divisors, minimal Gröbner bases over a finite ring of the type ℤpr (where p is a prime integer and r is an integer > 1) do not have the PLM property. We show how to construct, from an ordered minimal Gröbner basis, a so-called minimal Gröbner p-basis that does have a PLM property. The parametrization of all shortest linear recurrence relations of a finite sequence over ℤpr is a type of problem for which this is useful and we include an illustrative example.

Boundary Actuation in Galerkin-Models of Linear Distributed Parameter Systems

Abstract Proper orthogonal decomposition and subsequent Galerkin-projection is a common technique to obtain low-dimensional models for distributed parameter systems. In this paper a transformation of boundary conditions into equivalent source terms for linear partial differential equations is used to set up such reduced-order models. This approach provides the means to consider boundary actuation in Galerkin-models, which is essential for the design of feedback controllers. Using the example of a one-dimensional convection-diffusion equation, the transient dynamics between the non-actuated and actuated steady state are described with such a reduced-order-model. Dynamic range and model accuracy are investigated in connection with the employed number of POD modes and the dynamics of actuation.