Gröbner basis solutions of constrained interpolation problems (original) (raw)

Gr�bner basis solutions of constrained interpolation problems

Linear Algebra Appl, 2002

This paper extends the previous work of the authors on recursive Gröbner basis techniques in coding theory, Padé approximation, partial realization, interpolation, and modelling discrete-time behaviours. We present a general algorithm, applicable to a wide range of constrained interpolation problems in coding theory and systems theory, including list decoding and M-Padé approximation.

A Gröbner basis technique for Padé approximation

Journal of Symbolic Computation, 1992

We consider solving for a and b the congruence a ~ bh mod I, where a, b and h are (multivariable) polynomials and I is a polynomial ideal. This is a generalization of the well-known problem of Pad6 approximation of which decoding Hensel codes is a special case. We show how Gr~bner bases of modules may be used to generalize the Euclidean algorithm method of solution of the 1-variable problem.

List decoding codes on Garcia–Stictenoth tower using Gröbner basis

Journal of Symbolic Computation, 2009

An account of the interpolation and the root-finding steps of list decoding of one-point codes is given. The interpolation step is reduced to the problem of finding the minimal element of the Gröbner basis of a submodule of a free module over a polynomial ring of one variable. The procedure for root-finding of the interpolation polynomial going modulo a large degree place is described from the tower point of view.

Numerical computation of Gröbner bases

Proceedings of CASC2004 ( …, 2004

In this paper we deal with the problem of numerical computation of Gröbner bases of zero-dimensional polynomial systems. It is well known that the computation of a Gröbner basis cannot be generally executed in floating-point arithmetic by a standard approach. This, however, would be highly desirable for practical applications. We present an approach for computing Gröbner bases numerically. It is an elaboration of the idea of a stabilized Gröbner basis computation initially proposed by Hans Stetter. Our implementation of the algorithm based on the presented results is available online.

A new algorithm for a recursive construction of the minimal interpolation space

Mathematics and Computers in Simulation, 2009

In this work, we introduce a new interpolation algorithm, based on a recursive method for computing Lagrange interpolants. This algorithm allows to construct recursively the minimal interpolation space (see [1]) with respect to a finite set of points. We also extend this recursive method to the osculatory interpolation problem.

A recursive method for computing interpolants

Journal of Computational and Applied Mathematics, 2008

In this paper, we describe a recursive method for computing interpolants defined in a space spanned by a finite number of continuous functions in R d. We apply this method to construct several interpolants such as spline interpolants, tensor product interpolants and multivariate polynomial interpolants. We also give a simple algorithm for solving a multivariate polynomial interpolation problem and constructing the minimal interpolation space for a given finite set of interpolation points.

Numerical Grobner Bases and Syzygies: an Interval Approach

In Grobner bases computation a general open question is how to guide calculations coping with numerical coecients and/or not exact input data. It may happen that, due to error accumulation or insucient working precision, the result is not one theoretically expects. The basis may have more or less polynomials, a dierent number of solutions, a zero set with wrong multiplicity, and so on. Augmenting precision we may overcome algorithmic errors, but we don't know in advance how much it should be, and a trial-and-error approach is often the only way. Coping with initial errors is an even more dicult task. In this work the combined use of syzygies and interval arithmetic is proposed as a technique to decide at each critical point of the algorithm what to do.

Ritt’s Algorithm, Gröbner basis and discretization

In this paper, the application of commutative and differential algebra to system theory is presented. In particular, two alternatives for solving a continuous-time system discretization problem are described. The purpose is to obtain a discrete-time version of Fliess' generalized canonical observability form.

Floating-Point Gröbner Basis Computation with Ill-conditionedness Estimation

Lecture Notes in Computer Science, 2008

Computation of Gröbner bases of polynomial systems with coefficients of floating-point numbers has been a serious problem in computer algebra for a long time; the computation often becomes very unstable and people did not know how to remove the instability. Recently, the present authors clarified the origin of instability and presented a method to remove the instability. Unfortunately, the method is very time-consuming and not practical. In this paper, we first investigate the instability much more deeply than in the previous paper, then we give a theoretical analysis of the term cancellation which causes large errors, in various cases. On the basis of this analysis, we propose a practical method for computing the Gröbner bases with coefficients of floating-point numbers. The method utilizes multiple precision floating-point numbers, and it removes the drawbacks of the previous method almost completely. Furthermore, we present a method of estimating the ill-conditionedness of the input system.

Gröbner Bases for Problem Solving in Multidimensional Systems

Multidimensional Systems and Signal Processing - MULTIDIMENSION SYST SIGN PROC, 2001

The objective here is to underscore recent usage of the algorithmic theory of Gröbner bases in multidimensional systems since that possibility was highlighted about fifteen years back. The main contribution here focuses on the constructive aspects of the solution, known to exist, of the two-band multidimensional IIR perfect reconstruction problem using Gröbner bases. Other recent research results on the subject with future prospects are also briefly cited.