Multivariate Rational Functions Research Articles

Nearly optimal algorithms for the decomposition of multivariate rational functions and the extended Lüroth Theorem

The extended Lüroth Theorem says that if the transcendence degree of K ( f 1 , … , f m ) / K is 1 then there exists f ∈ K ( X ¯ ) such that K ( f 1 , … , f m ) is equal to K ( f ) . In this paper we show how to compute f with a probabilistic algorithm. We also describe a probabilistic and a deterministic algorithm for the decomposition of multivariate rational functions. The probabilistic algorithms proposed in this paper are softly optimal when n is fixed and d tends to infinity. We also give an indecomposability test based on gcd computations and Newton’s polytope. In the last section, we show that we get a polynomial time algorithm, with a minor modification in the exponential time decomposition algorithm proposed by Gutierez–Rubio–Sevilla in 2001.

Proper stabilisation of multidimensional input/output behaviours

Most papers in multidimensional stabilisation theory use discrete signals and a suitable integral domain $${\fancyscript{S}}$$ of "SISO stable plants" which consists of multivariate rational functions over a field. In this theory a system is described by its transfer matrix H of rational functions and is called stable if the entries of H belong to $${\fancyscript{S}}$$ . In most cases the transfer matrix is assumed proper. The significance of properness in the one-dimensional theory was already recognized by Kalman and is well-known to engineers. In the multidimensional theory properness and stability of the transfer matrix often enable its interpretation as input/output (IO) map or (convolution) operator and the proof of BIBO stability. Important contributions to this multidimensional stabilisation theory, in particular for our own work, are due to Bose, Guiver, Lin, Quadrat, Shankar, Sule, Xu, Ying, Zerz et al., some of whom were, in turn, essentially influenced by the seminal papers of Desoer, Francis, Ku?era, Vidyasagar, Youla, Zames and their co-authors in the one-dimensional case and on the fractional representation approach. In a recent paper we developed a stabilisation theory for multidimensional IO behaviours both in the continuous and the discrete cases, but did neither require nor prove the properness of the given or derived transfer matrices. The transfer matrix describes only the controllable part of a system whereas its behavioural description also enables the finer discussion of the internal stability of its autonomous part which is not determined by its transfer matrix. The latter fact was one of Kalman's reasons for the development of the one-dimensional state space theory. In the present paper we treat proper stabilisation of multidimensional IO behaviors, i.e., we characterise those proper or non-proper plants which admit a proper or non-proper stabilising compensator such that the resulting closed loop system is not only stable, but also has a proper transfer matrix. Under a weak pole condition on the plant which implies its properness we also construct strictly proper stabilising compensators for a stabilisable plant. For discrete systems described by transfer matrices such a result had been proven by Lin. For the construction of stabilising compensators we describe various algorithms and demonstrate these by examples. The parametrisation of all stabilising compensators was given in our previous paper. Our results seem interesting for one-dimensional IO behaviours too.

Structure theorems for parallel integration

We introduce structure theorems that refine Liouville’s Theorem on integration in closed form for general derivations on multivariate rational function fields. By predicting the arguments of the new logarithms that can appear in integrals, as well as the denominator of its rational part, those theorems provide theoretical backing for the Risch–Norman integration method. They also generalize its applicability to non-monomial extensions, for example the Lambert W function.

Adaptive multivariate rational data fitting with applications in electromagnetics

The behavior of certain electromagnetic devices or components can be simulated with great detail in software. A drawback of these simulation models is that they are very time consuming. Since the accuracy required for the computational electromagnetic analysis is usually only 2-3 significant digits, an approximate analytic model is sometimes used instead, as noted by Lehmensiek and Meyer in 2001. The most complex model we consider here is a multivariate rational function, which interpolates a number of simulation data. The interpolating rational function is constructed in such a way that it minimizes both the truncation error and the number of simulation data since each evaluation of the simulation model is computationally costly.

Rational Functions with Prescribed Global and Local Minimizers

A family of multivariate rational functions is constructed. It has strong local minimizers with prescribed function values at prescribed positions. While there might be additional local minima, such minima cannot be global. A second family of multivariate rational functions is given, having prescribed global minimizers and prescribed interpolating data.

On Multivariate Rational Function Decomposition

In this paper we discuss several notions of decomposition for multivariate rational functions, and we present algorithms for decomposing multivariate rational functions over an arbitrary field. We also provide a very efficient method to decide if a unirational field has transcendence degree one, and in the affirmative case to compute the generator.

Broadband physics-based modeling of microwave passive devices through frequency mapping

We present a new computer-aided modeling methodology to develop physics-based models for passive components. We coherently integrate full-wave electromagnetic simulators, artificial neural networks, multivariable rational functions, dimensional analysis, and frequency mapping to establish broadband models. We consider both frequency-independent and frequency-dependent empirical models. Frequency mapping is used to develop the frequency-dependent empirical models. Useful properties of the frequency mapping are presented and utilized in the modeling process. We also consider the transformation from frequency-dependent models into frequency-independent ones in addition to the passivity of the frequency-dependent empirical model. We illustrate the modeling process through various examples, including a microstrip right angle bend, a microstrip via, a microstrip double-step (to be used as a basic element of constructing a model for nonuniform or tapered microstrip transmission lines), and a coplanar waveguide step junction. © 2001 John Wiley & Sons, Inc. Int J RF and Microwave CAE 11: 156–170, 2001.

A Simple Least-Squares Design of M-D IIR Filters with Fixed Separable Denominator Based on Multivariate Division Algorithm

In this paper, we propose a simple algorithmic solution to the best approximation problem of finding the nearest multivariate rational function, with a fixed separable denominator polynomial, from a given multivariate polynomial, where the numerator polynomial is desired to minimize the integral of the squared error over the distinguished boundary of the unit polydisc. The proposed algorithm does not require any numerical integration or numerical root finding technique because this is realized based on the standard multivariate division algorithm. A simple observation of the proposed algorithm leads to an ideal membership problem characterizing the solution to the problem. A relation of this characterization and a multivariate generalization of the Walsh's Theorem is also discussed with another ideal membership problem derived by applying a corollary of the Hilbert Nullstellensatz to the Walsh's Theorem. Although the discussion to derive the latter ideal membership problem seems to be roundabout, such a characterization would be useful for further generalization, for example to some weighted least-squares approximation. Numerical examples demonstrate the practical applicability of the proposed method to design problems of multidimensional IIR filters.

On the Froissart phenomenon in multivariate homogeneous Padé approximation

In univariate Pade approximation we learn from the Froissart phenomenon that Pade approximants to perturbed Taylor series exhibit almost cancelling pole–zero combinations that are unwanted. The location of these pole–zero doublets was recently characterized for rational functions by the so‐called Froissart polynomial. In this paper the occurrence of the Froissart phenomenon is explored for the first time in a multivariate setting. Several obvious questions arise. Which definition of Pade approximant is to be used? Which multivariate rational functions should be investigated? When considering univariate projections of these functions, our analysis confirms the univariate results obtained so far in [13], under the condition that the noise is added after projection. At the same time, it is apparent from section 4 that for the unprojected multivariate Froissart polynomial no conjecture can be formulated yet.

A Matrix Approach to Low-Order Uncertainty Modeling of Real Parameters

Robust control system analysis and design is based on an uncertainty description which separates the uncertain system from the nominal system. Low-order models of this structure are difficult to formulate for many practical problems involving uncertainty in the physical parameters of the system. This paper presents a matrix-based method of directly constructing a low-order uncertainty model for real parameter problems involving multivariate rational functions of the uncertain parameters.

The use of multivariable rational functions for nonlinear data representation and classification

Multivariable rational functions provide a richly nonlinear representation which also incorporates strong intervariable nonlinear interactions, within a reasonable number of adjustable parameters. They are thus a convenient means of representing arbitrary relationships, even in circumstances where these are quantized or discontinuous. They are mathematically well-defined, and readily fitted using either special or general least squares techniques. The use of these functions in representing complex data is illustrated in examples which include pattern recognition and physical property prediction.

Model reduction of multidimensional linear shift-invariant recursive systems using Padé techniques

This paper describes a very flexible “general order” multivariate Pade approximation technique for the model reduction of a multidimensional linear shift-invariant recursive system, i.e., a system characterized by a multivariate rational transfer function. The technique presented allows full control of the regions of support in numerator and denominator of the reduced system and also admits a nonbranched continued fraction representation for an easy realization of the model. The method presented here overcomes some of the problems of related approaches to model reduction of multidimensional linear recursive systems. Different rational approximants can be introduced to compute the reduced model, but a drawback is that these approximants are not always readily available in continued fraction form for immediate implementation of the reduced system. Also multibranched continued fractions can be used to approximate the transfer function, but it was pointed out that the regions of support of numerator and denominator blow up rapidly as one considers successive convergents. Both these problems are overcome here.

A rational function arithmetic and simplification system in common Lisp

We describe a multivariate rational function (ratio of polynomials) arithmetic simplification system written in Common Lisp. These programs form one module in a sequence of packages written with an aim to develop an efficient, freely-distributed computer algebra system. In this report, we discuss the representation of rational functions in the system and the programs for rational addition, multiplication, and exponentiation. We also describe a package that converts between expressions in Lisp prefix form and the representation of rational expressions in the system.

Error formulas for multivariate rational interpolation and Pade´approximation

The univariate error formulas for Pade´approximants and rational interpolants, which are repeated in Section 2, are generalized to the multivariate case in Section 4. We deal with “general order” multivariate Pade´approximants and rational interpolants, where the numerator and denominator polynomials as well as the equations expressing the approximation order, can be chosen by the user of these multivariate rational functions.

On boundary implications of stability and positivity properties of multidimensional systems

Multidimensional generalizations of various 1-D results on the robustness of Hurwitz, Schur, and positivity properties of polynomials and rational functions are considered. More specifically, the convexity property of the stable region in the coefficient space of multivariable polynomials is studied. Multidimensional generalizations of Kharitonov-type results are reviewed, and further extensions, including that of the 1-D edge theorem, are discussed. Interval positivity properties of multivariate rational functions are characterized in terms of ratios of a finite number of Kharitonov-type polynomials constructed from the extreme values of the intervals of perturbation.

Computing with polynomials given byblack boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators

Algorithms are developed that adopt a novel implicit representation for multivariate polynomials and rational functions with rational coefficients, that of black boxes for their evaluation. We show that within this representation the polynomial greatest common divisor and factorization problems, as well as the problem of extracting the numerator and denominator of a rational function, can all be solved in random polynomial-time. Since we can convert black boxes efficiently to sparse format, problems with sparse solutions, e.g., sparse polynomial factorization and sparse multivariate rational function interpolation, are also in random polynomial time. Moreover, the black box representation is one of the most space efficient implicit representations that we know. Therefore, the output programs can be easily distributed over a network of processors for further manipulation, such as sparse interpolation.

GCF extraction from multivariable polynomials using indirect Padé approximation

The greatest common factor (GCF) is extracted from multi-variable polynomials using an ‘indirect’ method for computing generalised Padé-type approximations of multivariable rational functions. The algorithm is computationally easy and conceptually very simple. An example is given to illustrate the method.

On non-essential singularities of the second kind in multivariable rational functions

Two new methods to test for the existence of non-essential singularities of the second kind on the distinguished boundary of the unit polydisk, are presented. If such singularities exist, they can also be explicitly found. Both methods are shown to be much simpler, computationally, than previously known methods.

A bound on the multiplicative efficiency of iteration

For a convergent sequence { x i } generated by x i+1 =( x i , x i−1 ,…, x i−d+1 ), define the multiplicative efficiency measure E to be (log 2 p )/ M , where p is the order of convergence and M is the number of multiplications or divisions needed to compute . Then, ifis any multivariate rational function, E ≤1. Since E =1 for the sequence { x i } generated by x i+1 = x i 2 + x i −1/4 with the limit −1/2, the bound on E is sharp. Let P M denote the maximal order for a sequence generated by an iteration with M multiplications. Then P M ≤2 M for all positive integers M . Moreover this bound is sharp.

On Euclid's Algorithm and the Theory of Subresultants

Abstract : A key ingredient for systems which perform symbolic computer manipulation of multivariate rational functions are efficient algorithms for calculating polynomial greatest common divisors. Euclid's algorithm cannot be used directly because of problems with coefficient growth. The search for better methods leads naturally to the theory of subresultants. This paper presents an elementary treatment of the theory of subresultants, and examines the relationship of the subresultants of a given pair of polynomials to their polynomial remainder sequence as determined by Euclid's algorithm. This relation is expressed in the fundamental theorem of this paper. The results are essentially the same as those of Collins but the presentation is briefer, simpler, and somewhat more general. The fundamental theorem finds further applications in the proof that the modular algorithm for polynomial GCD terminates. (Author)