Decomposition Of Rational Functions Research Articles

Partial fraction decomposition for rational Pythagorean hodograph curves

All rational parametric curves with prescribed polynomial tangent direction form a vector space. Via tangent directions with rational norm, this includes the important case of rational Pythagorean hodograph curves. We study vector subspaces defined by fixing the denominator polynomial and describe the construction of canonical bases for them. We also show (as an analogy to the fraction decomposition of rational functions) that any element of the vector space can be obtained as a finite sum of curves with single roots at the denominator. Our results give insight into the structure of these spaces, clarify the role of their polynomial and truly rational (non-polynomial) curves, and suggest applications to interpolation problems.

Applications of constructed new families of generating‐type functions interpolating new and known classes of polynomials and numbers

The aim of this article is to construct some new families of generating‐type functions interpolating a certain class of higher order Bernoulli‐type, Euler‐type, Apostol‐type numbers, and polynomials. Applying the umbral calculus convention method and the shift operator to these functions, these generating functions are investigated in many different aspects such as applications related to the finite calculus, combinatorial analysis, the chordal graph, number theory, and complex analysis especially partial fraction decomposition of rational functions associated with Laurent expansion. By using the falling factorial function and the Stirling numbers of the first kind, we also construct new families of generating functions for certain classes of higher order Apostol‐type numbers and polynomials, the Bernoulli numbers and polynomials, the Fubini numbers, and others. Many different relations among these generating functions, difference equation including the Eulerian numbers, the shift operator, minimal polynomial, polynomial of the chordal graph, and other applications are given. Moreover, further remarks and comments on the results of this paper are presented.

Analysis of the transient state in a parallel circuit of the class RLβCα

The paper presents the results of the conducted analysis of transient states in a parallel circuit of the class RLβCα, supplied by an ideal current source. The considered circuit consists of a real coil Lβ and a (super)capacitor Cα, modeled as fractional-order elements. The paper proposes a method for determining the current and voltage waveforms, which uses the decomposition of rational functions into partial fractions and inverse Laplace method. By using this method, transient waveforms occurring in the system for any kind of current excitation can be determined. Two cases of the problem solutions have been considered in the paper, for real and complex poles of the rational functions, which result from the analysis of the main fractional-order differential equation, describing the considered circuit. Analytical relations describing transient state waveforms in the system have been determined for different types of current excitations: constant, monoharmonic, polyharmonic and arbitrary being an element of a Hilbert space. The obtained results have been illustrated by an examplary case of a parallel circuit of the class RLβCα. The paper is a continuation of previous studies concerning transient-state analysis in circuits with elements modeled as fractional-order (Jakubowska and Walczak 2015, 2016).

On partial fraction decompositions by repeated polynomial divisions

ABSTRACTWe present a method for finding partial fraction decompositions of rational functions with linear or quadratic factors in the denominators by means of repeated polynomial divisions. This method does not involve differentiation or solving linear equations for obtaining the unknown partial fraction coefficients, which is very suitable for either hand or machine calculations. Some illustrative examples are provided.

Analysis of the Transient State in a Series Circuit of the Class $$RL_{\beta }C_{\alpha }$$ R L β C α

Results of analysis of transient states in a series circuit of the class $${RL}_{\beta }{C}_{\alpha }$$RLβC�, supplied by an ideal voltage source, have been described in the paper. This circuit consists of a coil $${L}_{\beta }$$Lβ and a supercapacitor $${C}_{\alpha }$$C� described by fractional-order differential equations. A method for determining the current and voltage waveforms in the analyzed circuit, based on the decomposition of rational functions into partial fractions, has been described. This method allows to determine transient waveform shapes in the system for any kind of voltage excitation. Two cases of the problem solutions have been considered. The first case concerns a situation where poles of rational functions are real, and the second where rational functions have complex poles. Effective relations enabling the determination of transient waveforms in a closed form have been given. Analytical formulae describing transient state waveforms in the system for different types of voltage excitations: constant, monoharmonic, periodic and arbitrary being an element of a Hilbert space, have been determined, too. The obtained results have been illustrated by an example.

Decompositions of rational functions over real and complex numbers and a question about invariant curves

We consider the connection of functional decompositions of rational functions over the real and complex numbers, and a question about curves in the complex plane which are invariant under a rational function.

Computing the partial fraction decomposition of rational functions with irreducible quadratic factors in the denominators

In this note, a new method for computing the partial fraction decomposition of rational functions with irreducible quadratic factors in the denominators is presented. This method involves polynomial divisions and substitutions only, without having to solve for the complex roots of the irreducible quadratic polynomial or to solve a system of linear equations. Some examples of its applications are included.

Complex Partial Fraction Decompositions of Rational Functions

This work is aimed to obtain the complex partial fraction decompositions of rational functions. We express the coefficients of complex partial fraction decomposition of arbitrary rational functions in terms of the coefficients of their real partial fraction decomposition. This type of decompositions is then used to generalize the high order derivatives of such rational functions. Moreover, different applications are selected to demonstrate the applicability of introduced algorithms.

Jordan-Hölder theorem for imprimitivity systems and maximal decompositions of rational functions

In this paper we prove several results about the lattice of imprimitivity systems of a permutation group containing a cyclic subgroup with at most two orbits. As an application we generalize the first Ritt theorem about functional decompositions of polynomials, and some other related results. Besides, we discuss examples of rational functions, related to finite subgroups of Aut(ℂℙ1), for which the first Ritt theorem fails to be true.

Prime and composite Laurent polynomials

In the paper [J. Ritt, Prime and composite polynomials, Trans. Amer. Math. Soc. 23 (1922) 51–66] Ritt constructed the theory of functional decompositions of polynomials with complex coefficients. In particular, he described explicitly polynomial solutions of the functional equation f ( p ( z ) ) = g ( q ( z ) ) . In this paper we study the equation above in the case where f , g , p , q are holomorphic functions on compact Riemann surfaces. We also construct a self-contained theory of functional decompositions of rational functions with at most two poles generalizing the Ritt theory. In particular, we give new proofs of the theorems of Ritt and of the theorem of Bilu and Tichy.

On the functional decomposition of multivariate laurent polynomials (abstract only)

Determining whether a univariate polynomial may be written as the functional composition of two others of lower degree is a question that has been studied since at least the work of Ritt [1]. Algorithms by Barton and Zippel [2] and then by Kozen and Landau [3] have been incorporated in many computer algebra systems. Generalizations have been studied for functional decomposition of rational functions [4], algebraic functions [5], multivariate polynomials [6] and univariate Laurent polynomials [7]. We explore the functional decomposition problem for multivariate Laurent polynomials, considering the case f = g o h where g is univariate and h may be multivariate. We present an algorithm to find such a decomposition if it exists. The algorithm proceeds as follows: First, a variable weighting is chosen to make the weighted degree zero term in f constant. The positive degree and negative degree parts of h are then reconstructed separately, in a manner similar to that of Kozen and Landau, but by treating the homogeneous collections of terms by grade rather than individual monomials. Then terms of the univariate polynomial g are reconstructed degree by degree using a generic univariate projection of h.

On the decomposition of rational functions

Let f : = p / q ∈ K ( x ) be a rational function in one variable. By Lüroth’s theorem, the collection of intermediate fields K ( f ) ⊊ L ⊊ K ( x ) is in bijection with inequivalent proper decompositions f = g ∘ h , with g , h ∈ K ( x ) of degrees ≥ 2 . In [Alonso, Cesar, Gutierrez, Jaime, Recio, Tomas, 1995. A rational function decomposition algorithm by near-separated polynomials. J. Symbolic Comput. 19, 527–544] an algorithm is presented to calculate such a function decomposition. In this paper we describe a simplification of this algorithm, avoiding expensive solutions of linear equations. A MAGMA implementation shows the efficiency of our method. We also prove some indecomposability criteria for rational functions, which were motivated by computational experiments.

Symbolic additive decomposition of rational functions

The problem of additive decomposition of rational functions minimizing the degree of the remainder's denominator with the additional minimization of the degree of the remainder's numerator is considered. An algorithm that solves this problem under certain additional assumptions is proposed. This algorithm is implemented in Maple.

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.

Algorithms for Function Fields

Let K/Q(t) be a finite extension. We describe algorithms for computingsubfields and automorphisms of K/Q(t). As an application we give an algorithm for finding decompositions of rational functions in Q(α). We also present an algorithm which decides if an extension L/Q(t) is a subfield of K. In case [K: Q(t)] = [L: Q(t)] we obtain a Q(t)-isomorphism test. Furthermore, we describe an algorithm which computes subfields of the normal closure of K/Q(t).

A note on separated factors of separated polynomials

By introducing the concept of near-separated polynomial we extend to rational functions a theorem of Fried and MacRae (1969) concerning separated factors of separated polynomials. Our result allows to solve several problems about the decomposition of rational functions by means of factorization of bivariate polynomials.

A Rational Function Decomposition Algorithm by Near-separated Polynomials

In this paper we present an algorithm for decomposing rational functions over an arbitrary coefficient field. The algorithm requires exponential time, but is more efficient in practice than the previous ones, including the polynomial time algorithm. Moreover, our algorithm is easier to implement. We also present some applications of rational function decomposition: (1) faithful re-parameterizing unfaithfully parameterized curves, (2) computing intermediate fields in a simple purely transcendental field extension K , and (3) providing a birationality test for subfields of K ( x). Several examples are computed using an implementation of our algorithm using MAPLE V.

Decomposition of the infinite companion and interpolation

Formulae are given expressing the infinite companion matrix of a polynomial p as the sum of smaller matrices corresponding to the individual factors if p = p 1 ... p m . Connections with interpolation problems and partial fraction decompositions of rational functions are explained. The results are intended as a tool for representing Hankel matrices as sums of simpler Hankel matrices corresponding to the factors of the compatible polynomial. Explicit formulae are given for the inverse of the Chinese remainder operator and for the inverse of the confluent Vandermonde matrix.

New methods of rational function decomposition with applications in active network synthesis†

With the growing importance of active RC synthesis, it has become necessary to decompose a given rational function in a certain manner. In this paper three new methods of decomposition of a real rational function with no restriction on the numerator order, relative to the denominator order, are presented and their applications illustrated