Multivariate Rational Functions Research Articles

On the representability of sequences as constant terms

A constant term sequence is a sequence of rational numbers whose n-th term is the constant term of Pn(x)Q(x), where P(x) and Q(x) are multivariate Laurent polynomials. While the generating functions of such sequences are invariably diagonals of multivariate rational functions, and hence special period functions, it is a famous open question, raised by Don Zagier, to classify diagonals that are constant terms. In this paper, we provide such a classification in the case of sequences satisfying linear recurrences with constant coefficients. We also consider the case of hypergeometric sequences and, for a simple illustrative family of hypergeometric sequences, classify those that are constant terms.

Recursive Neural Network–Particle Swarm Versus Nonlinear Multivariate Rational Function Algorithms for Optimization of Biodiesel Derived from Hevea brasiliensis

Recursive Neural Network–Particle Swarm Versus Nonlinear Multivariate Rational Function Algorithms for Optimization of Biodiesel Derived from Hevea brasiliensis

Hybrid modelling of a batch separation process

Applying machine learning (ML) techniques is a complex task when the data quality is poor. Integrating first-principle models and ML techniques, namely hybrid modelling significantly supports this task. This paper introduces a novel approach to developing a hybrid model for dynamic chemical systems. The case in analysis employs one first-principle structure and two ML-based predictors. Two training approaches (serial and parallel), two optimisers (particle swarm optimisation and differential evolution) and two ML functions (multivariate rational function and polynomial) are tested. The polynomial function trained with the differential evolution showed the most accurate and robust results. The training approach does not significantly affect the hybrid model accuracy. However, the main effect of the training approach is on the robustness of the parameter predictions. The coefficients of determination (R2) on the test batches are above 0.95. In addition, it showed satisfactory extrapolation capabilities on different production scales with R2>0.9.

On efficient computation of rational {1,2}-pseudo-inverses for multivariable rational matrix-valued functions and their applications

In this article we consider the problem of the existence of rational 1,2-pseudo-inverses for rational multivariable matrix-valued functions. We prove that any rational multivariable matrix-valued function has rational 1,2-pseudo-inverse and we describe the set of all 1,2-pseudo-inverses of a given function, in terms of rational free parameters. It is shown that the 1,2-pseudo-inverse can have an effective description relative to the Moore–Penrose pseudo-inverse (that in general is not even rational), thus making it an alternative effective solution to the problem of online matrix inversion of large matrices that depend on real-time measured parameters. The rationality of the 1,2-pseudo-inverse is crucial when it should be realized in the physical real world (e.g. in inverse control), in contrast with realizations in computer algorithms (e.g. in image processing and communication systems) where the rationality is not necessary but the effective description of the pseudo-inverse becomes crucial. The results has applications in control systems, robust control, inverse control, mechanical systems, kinematic chains, kinematic networks, multi-degree-of-freedom systems, image processing, signal processing and communication systems.

Numerical optimization and positivity certificates for polynomials and rationals over simplices

Multivariate rational control functions r(x) of high degree d can be expressed in the Bernstein form defined over k-dimensional simplices (triangles). The range of rational functions can be optimized by the enclosure bound (minimum and maximum) of Bernstein coefficients. In this paper, linear and quadratic rate of convergence for the maximum error bound between the enclosure and range bounds are investigated. Furthermore, minimization and numerical identities for certifying the positivity of any given rational function over simplices are given. In order to provide global and local positivity certificates for r, the degree of Bernstein functions and number of simplex subdivision steps are estimated. Subsequently, we establish a bound for the maximum degree of Bernstein polynomials.

Life cycle improvement of serially connected batteries system by redundancy based on failure distribution analysis

The utilization of redundant batteries is seen as an effective method to promote battery life cycle. On account of the diminishing marginal effect, it is necessary to reasonably configure redundant batteries quantity according to battery characteristics. Based on failure distribution analysis, an evaluated method for serially connected batteries system is proposed to improve lifetime by redundant batteries. The optimal redundancy quantity with maximum cycle numbers can be obtained against the various number of serially connected batteries and the different shape parameter of battery failure distribution by multivariate rational function with rounding. Besides, the scheduling control strategy for serially connected batteries is presented in light of battery health and cycle times. The proposed method is suitable for different batteries type, battery station as well as other devices with similar characteristics to batteries.

MultivariateApart: Generalized partial fractions

We present a package to perform partial fraction decompositions of multivariate rational functions. The algorithm allows to systematically avoid spurious denominator factors and is capable of producing unique results also when being applied to terms of a sum separately. The package is designed to work in Mathematica, but also provides interfaces to the Form and Singular computer algebra systems. Program summaryProgram title:MultivariateApartCPC Library link to program files:https://doi.org/10.17632/zbt9tfpkgv.1Developer's repository link:https://gitlab.msu.edu/vmante/multivariateapartLicensing provisions: GPLv3Programming language: Mathematica (Wolfram Language)Nature of problem: Partial fraction decomposition is widely used in particle physics to bring rational functions into a unique form. Treating the multivariate case by applying the univariate method iteratively risks the introduction of spurious singularities. Here, we formulate an algorithm that handles non-linear multivariate denominators, avoids the introduction of spurious denominators, and aims at providing good performance for typical applications in particle physics. It can be used to obtain unique results also when decomposing individual terms of a sum separately, independent of the details of the input form. A variant of the approach allows to reach a Leĭnartas' decomposition as the output form.Solution method: We reformulate the problem of calculating a partial fraction decomposition into the problem of reducing a polynomial with respect to a specific ideal. The polynomial reduction is based on a Gröbner basis calculation and guarantees a unique result for the partial fraction decomposition. We provide a complete implementation as a Mathematica package. Optionally, the included interfaces to Form and Singular can be used to speed up the computation.Additional comments including restrictions and unusual features: Our algorithm does not introduce new singularities that were not presented in the input. If, however, the input contains spurious singularities, our package can be used in such a way that the cancellation of these singularities is not guaranteed. By default, the main MultivariateApart function of our package avoids this problem by first putting terms in a sum over a common denominator, and canceling factors in the result.

On sparse interpolation of rational functions and gcds

In this note, we present a variant of a probabilistic algorithm by Cuyt and Lee for the sparse interpolation of multivariate rational functions. We also present an analogous method for the computation of sparse gcds.

Hidden Symmetries of Weighted Lozenge Tilings

We study the weighted partition function for lozenge tilings, with weights given by multivariate rational functions originally defined by Morales, Pak and Panova (2019) in the context of the factorial Schur functions. We prove that this partition function is symmetric for large families of regions. We employ both combinatorial and algebraic proofs.

The asymptotics of reflectable weighted walks in arbitrary dimension

Gessel and Zeilberger generalized the reflection principle to handle walks confined to Weyl chambers, under some restrictions on the allowable steps. For some models that are invariant under the Weyl group action, they express the counting function for the walks with fixed starting point and ending point as a constant term in the Taylor series expansion of a rational function. Here we focus on the simplest case, the Weyl group A1d, which corresponds to walks in the first orthant Nd taking steps from a subset of {±1,0}d that is invariant under reflection across any axis. The principle novelty here is the incorporation of weights on the steps and the main result is a very general theorem giving asymptotic enumeration formulas for walks that end anywhere in the orthant. The formulas are determined by singularity analysis of multivariable rational functions, an approach that has already been successfully applied in numerous related cases.

Effective coefficient asymptotics of multivariate rational functions via semi-numerical algorithms for polynomial systems

The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several variables (ACSV) makes it possible to compute asymptotic expansions. We consider these methods from the point of view of effectivity. In particular, given a rational function, ACSV requires one to determine a (generically) finite collection of points that are called critical and minimal. Criticality is an algebraic condition, meaning it is well treated by classical methods in computer algebra, while minimality is a semi-algebraic condition describing points on the boundary of the domain of convergence of a multivariate power series. We show how to obtain dominant asymptotics for the diagonal coefficient sequence of multivariate rational functions under some genericity assumptions using symbolic-numeric techniques. To our knowledge, this is the first completely automatic treatment and complexity analysis for the asymptotic enumeration of rational functions in an arbitrary number of variables.

Reconstructing rational functions with FireFly

We present the open-source C++ library FireFly for the reconstruction of multivariate rational functions over finite fields. We discuss the involved algorithms and their implementation. As an application, we use FireFly in the context of integration-by-parts reductions and compare runtime and memory consumption to a fully algebraic approach with the program Kira. Program summaryProgram title:FireFlyProgram files doi:http://dx.doi.org/10.17632/nzgxdwwt8k.1Licensing provisions: GNU General Public License 3Programming language:C++External routines/libraries used: GNU GMP[1], FLINT[2]Nature of problem: The interpolation of an unknown rational function, called black box, from only its evaluations can be used in many physical contexts where algebraic calculations fail due to memory and time restrictions.Solution method: The black-box function is evaluated at different points over a finite field. These points are then used by interpolation algorithms[3,4] to obtain the analytic form of the black-box function. The members of a finite field are promoted to Q using a rational reconstruction algorithm[5,6].Restrictions: The CPU time and the available RAM

On the Extended Hensel Construction and its application to the computation of real limit points

The Extended Hensel Construction (EHC) is a procedure which, for an input bivariate polynomial with complex coefficients, can serve the same purpose as the Newton-Puiseux algorithm, and, for the multivariate case, can be seen as an effective variant of Jung-Abhyankar Theorem. We show that the EHC requires only linear algebra and univariate polynomial arithmetic. We deduce complexity estimates and report on a software implementation together with experimental results. This work is motivated and illustrated by two applications. The first one is the computation of real branches of space curves. The second one is the computation of limits of real multivariate rational function.For the latter, we present an algorithm for determining the existence of the limit of a real multivariate rational function q at a given point p which is an isolated zero of the denominator of q. When the limit exists, the algorithm computes it, without making any assumption on the number of variables. A process, which extends the work of Cadavid, Molina and Vélez, reduces the multivariate setting to computing limits of bivariate rational functions. By using regular chain theory and triangular decomposition of semi-algebraic systems, we avoid the computation of singular loci and the decomposition of algebraic sets into irreducible components.

FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs

Complex algebraic calculations can be performed by reconstructing analytic results from numerical evaluations over finite fields. We describe FiniteFlow, a framework for defining and executing numerical algorithms over finite fields and reconstructing multivariate rational functions. The framework employs computational graphs, known as dataflow graphs, to combine basic building blocks into complex algorithms. This allows to easily implement a wide range of methods over finite fields in high-level languages and computer algebra systems, without being concerned with the low-level details of the numerical implementation. This approach sidesteps the appearance of large intermediate expressions and can be massively parallelized. We present applications to the calculation of multi-loop scattering amplitudes, including the reduction via integration-by-parts identities to master integrals or special functions, the computation of differential equations for Feynman integrals, multi-loop integrand reduction, the decomposition of amplitudes into form factors, and the derivation of integrable symbols from a known alphabet. We also release a proof-of-concept C++ implementation of this framework, with a high-level interface in Mathematica.

On the positivity of the Gillis–Reznick–Zeilberger rational function

In this note we provide further evidence for a conjecture of Gillis, Reznick, and Zeilberger on the positivity of the diagonal coefficients of a multivariate rational function. Kauers had proven this conjecture for up to 6 variables using computer algebra. We present a variation of his approach that allows us to prove positivity of the coefficients up to 17 variables using symbolic computation.

Interpolation Macdonald polynomials and Cauchy-type identities

Let Sym denote the algebra of symmetric functions and Pμ(⋅;q,t) and Qμ(⋅;q,t) be the Macdonald symmetric functions (recall that they differ by scalar factors only). The (q,t)-Cauchy identity∑μPμ(x1,x2,…;q,t)Qμ(y1,y2,…;q,t)=∏i,j=1∞(xiyjt;q)∞(xiyj;q)∞ expresses the fact that the Pμ(⋅;q,t)'s form an orthogonal basis in Sym with respect to a special scalar product 〈⋅,⋅〉q,t. The present paper deals with the inhomogeneous interpolation Macdonald symmetric functionsIμ(x1,x2,…;q,t)=Pμ(x1,x2,…;q,t)+lower degree terms. These functions come from the N-variate interpolation Macdonald polynomials, extensively studied in the 90s by Knop, Okounkov, and Sahi. The goal of the paper is to construct symmetric functions Hμ(⋅;q,t) with the biorthogonality property〈Iμ(⋅;q,t),Hν(⋅;q,t)〉q,t=δμν. These new functions live in a natural completion Symˆ⊃Sym. As a corollary one obtains a new Cauchy-type identity in which the interpolation Macdonald polynomials are paired with certain multivariate rational symmetric functions. The degeneration of this identity in the Jack limit (q,t)=(q,qk)→(1,1) is also described.

The Bulirsch‐Stoer algorithm for multivariate rational interpolation

The Neville‐type algorithms are widely used in engineering and sciences. As an analog of Neville's algorithm that deals with univariate polynomial interpolation, Bulirsch‐Stoer algorithm is a classical one for univariate rational interpolations. It can be applied to calculating the value of the interpolating function at the given point or recovering rational functions. In this paper, we generalize the algorithm to multivariate cases with two versions for different situations. These two generalizations are recursive algorithms. The first one is suitable to calculate the value of the interpolating function and the other one can be applied to recovering multivariate rational functions from accurate measurements. Some two‐variable examples illustrate that, if we recover the rational functions with higher degrees, the second generalization is superior to Thiele‐Thiele continued fraction and two‐variable Löwner matrix methods.

Data‐driven model order reduction of quadratic‐bilinear systems

SummaryWe introduce a data‐driven model order reduction approach that represents an extension of the Loewner framework for linear and bilinear systems to the case of quadratic‐bilinear (QB) systems. For certain types of nonlinear systems, one can always find an equivalent QB model without performing any approximation whatsoever. An advantage of the Loewner framework is that information about the redundancy of the given data is explicitly available, by means of the singular values of the Loewner matrices. This feature is also valid for the proposed generalization. As for the linear and bilinear cases, these matrices can be directly computed by solving Sylvester equations. We begin by defining generalized higher‐order transfer functions for QB systems. These multivariate rational functions play an important role in the model order reduction process. We construct reduced‐order systems for which the associated transfer functions match those corresponding to the original system at selected tuples of interpolation points. Another benefit of the proposed approach is that it is data‐driven oriented, in the sense that one would only need computed or measured samples to construct a reduced‐order QB system. We illustrate the practical applicability of the proposed method by means of several numerical experiments.

Certified non-conservative tests for the structural stability of discrete multidimensional systems

In this paper, we present new computer algebra based methods for testing the structural stability of n-D discrete linear systems (with n >= 2). More precisely, we show that the standard characterization of the structural stability of a multivariate rational transfer function (namely, the denominator of the transfer function does not have solutions in the unit polydisc of \C^n) is equivalent to the fact that a certain system of polynomials does not have real solutions. We then use state-of-the-art computer algebra algorithms to check this last condition, and thus the structural stability of multidimensional systems.

Multivariate Rational Response Surface Approximation of Nodal Displacements of Truss Structures

Polynomial-basis response surface method has some shortcomings for truss structures in structural optimization, concluding the low fitting accuracy and the great computational effort. Based on the theory of approximation, a response surface method based on Multivariate Rational Function basis (MRRSM) is proposed. In order to further reduce the computational workload of MRRSM, focusing on the law between the cross-sectional area and the nodal displacements of truss structure, a conjecture that the determinant of the stiffness matrix and the corresponding elements of adjoint matrix involved in displacement determination are polynomials with the same order as their respective matrices, each term of which is the product of cross-sectional areas, is proposed. The conjecture is proved theoretically for statically determinate truss structure, and is shown corrected by a large number of statically indeterminate truss structures. The theoretical analysis and a large number of numerical examples show that MRRSM has a high fitting accuracy and less computational effort. Efficiency of the structural optimization of truss structures would be enhanced.