Partial Fraction Decomposition Research Articles

LinApart: Optimizing the univariate partial fraction decomposition

We present LinApart, a routine designed for efficiently performing the univariate partial fraction decomposition of large symbolic expressions. Our method is based on an explicit closed formula for the decomposition of rational functions with fully factorized denominators. We provide an implementation in the Wolfram Mathematica language, which can lead to very significant performance gains over the built-in Apart command. Furthermore, a C language library implementing the core functionality and suitable for interfacing with other software is also provided. Both codes are made available at https://github.com/fekeshazy/LinApart.

Rational Approximations for the Oscillatory Two-Parameter Mittag–Leffler Function

The two-parameter Mittag–Leffler function Eα,β is of fundamental importance in fractional calculus, and it appears frequently in the solutions of fractional differential and integral equations. However, the expense of calculating this function often prompts efforts to devise accurate approximations that are more cost-effective. When α>1, the monotonicity property is largely lost, resulting in the emergence of roots and oscillations. As a result, current rational approximants constructed mainly for α∈(0,1) often fail to capture this oscillatory behavior. In this paper, we develop computationally efficient rational approximants for Eα,β(−t), t≥0, with α∈(1,2). This process involves decomposing the Mittag–Leffler function with real roots into a weighted root-free Mittag–Leffler function and a polynomial. This provides approximants valid over extended intervals. These approximants are then extended to the matrix Mittag–Leffler function, and different implementation strategies are discussed, including using partial fraction decomposition. Numerical experiments are conducted to illustrate the performance of the proposed approximants.

A Numerical Scheme for Time-Space Fractional diffusion Models

Time-space fractional models are proved to be highly efficient in characterizing anomalous diffusion in many intricate systems. This study introduces a numerical scheme employing a matrix transform technique to discretize the fractional spectral Laplacian and subsequently applying generalized exponential time differencing to the semi-discrete system. The resulting second-order scheme is implemented efficiently using rational approximations for the two-parameter Mittag-Leffler function and the partial fraction decomposition of these approximations. The advantage of this approach is its applicability to different types of homogeneous boundary conditions including Robin boundary conditions. Numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed numerical scheme.

A fast third order algorithm for two dimensional inhomogeneous fractional parabolic partial differential equations

A computationally fast third order numerical algorithm is developed for inhomogeneous parabolic partial differential equations. The algorithm is based on a third order method developed by using a rational approximation with single Gaussian quadrature pole to avoid complex arithmetic and to achieve high efficiency and accuracy. Difficulties with computational efficiency and accuracy are addressed using partial fraction decomposition technique. Third order accuracy and convergence of the method is proved analytically and verified numerically. Several classical as well as more challenging fractional and distributed order inhomogeneous problems are considered to perform numerical experiments. Computational efficiency of the method is demonstrated through central processing unit (CPU) time and is given in the convergence tables.

Pfd-parallel, a Singular/GPI-Space package for massively parallel multivariate partial fractioning

Multivariate partial fractioning is a powerful tool for simplifying rational function coefficients in scattering amplitude computations. Since current research problems lead to large sets of complicated rational functions, performance of the partial fractioning as well as size of the obtained expressions are a prime concern. We develop a large scale parallel framework for multivariate partial fractioning, which implements and combines an improved version of Leinartas' algorithm and the MultivariateApart algorithm. Our approach relies only on open source software. It combines parallelism over the different rational function coefficients with parallelism for individual expressions. The implementation is based on the Singular/GPI-Space framework for massively parallel computer algebra, which formulates parallel algorithms in terms of Petri nets. The modular nature of this approach allows for easy incorporation of future algorithmic developments into our package. We demonstrate the performance of our framework by simplifying expressions arising from current multiloop scattering amplitude problems. We also provide a Mathematica interface to our package, which is available on https://github.com/Wu-Zihao/PfdParallelM. Program summaryProgram Title: pfd-parallelCPC Library link to program files:https://doi.org/10.17632/r99gdp9v76.1Developer's repository link:https://github.com/singular-gpispace/pfd-parallelLicensing provisions: GPLv3Programming language: Singular language, GPI-Space Petri netNature of problem: In scattering amplitude computation, we often encounter complicated rational functions, which may be simplified using the multivariate partial fraction decomposition. With the consideration of the increasing complexity of the problems, an efficient implementation of such a method is needed.Solution method: We present the package pfd-parallel, which is a large-scale parallelized framework for multivariate partial fractioning. Our package relies only on open source software. It combines different algorithms and provides parallelization based on the Singular/GPI-Space framework [1–3]. The package combines both the improved Leinartars' algorithm [4], as well as the MultivariateApart algorithm [5], and combines parallelism over the different rational function coefficients and parallelism for individual expressions. Using this approach cutting-edge computations can be handled in an efficient way.Additional comments including restrictions and unusual features: The software, including all dependencies like Singular, GPI-2, GPI-Space, and the Singular/GPI-Space framework, is distributed via the supercomputing package manager Spack, which allows for convenient installation of scientific software, in particular in HPC environments. The code has been tested on Centos 7 and 8, Ubuntu 18.04 LTS and 20.04 LTS.

On the computation of rational solutions of linear integro-differential equations with polynomial coefficients

We develop the first algorithm for computing rational solutions of scalar integro-differential equations with polynomial coefficients. It starts by finding the possible poles of a rational solution. Then, bounding the order of each pole and solving an algebraic linear system, we compute the singular part of rational solutions at each possible pole. Finally, using partial fraction decomposition, the polynomial part of rational solutions is obtained by computing polynomial solutions of a non-homogeneous scalar integro-differential equation with a polynomial right-hand side. The paper is illustrated by examples where the computations are done with our Maple implementation.

On compact explicit formulas of the partial fraction decomposition and applications

On compact explicit formulas of the partial fraction decomposition and applications

Removing singularities in the Bateman radioactive transformation equations using partial fraction decomposition

This paper deals with the problem of singularities caused by repeated decay constants in the Bateman equations for nuclide concentrations in the radioactive transformation chains. Using partial fraction decomposition with the Bateman equations, the general solution for the nuclide concentrations in the radioactive transformation linear chains with repeated decay constants has been derived. The derived equations are similar to the solutions obtained by other authors, with the difference being the improved calculation scheme. A comparative analysis was conducted to assess the computational efficiency in determining concentration changes within transmutation chains, particularly in the presence of repetitive reaction rates. The investigation involved evaluating several methodologies.

Trigonometric Sums and Riemann Zeta Function

ABSTRACTSeveral trigonometric sums are examined by means of partial fraction decompositions. Their generating functions will be evaluated in closed forms and the related asymptotic formulae in terms of Riemann zeta function will be established.

The art of Schwinger and Feynman parametrizations

We present a derivation of the Schwinger parametrization and the Feynman parametrization in detail and their elementary applications. Although the parametrizations are essential to computing the loop integral arising in relativistic quantum field theory, their detailed derivations are not presented in usual textbooks. Beginning with an integral representation of the unity, we derive the Schwinger parametrization by performing multiple partial derivatives and utilizing the analyticity of the gamma function. The Feynman parametrization is derived by the partial-fraction decomposition and the change of variables introducing an additional delta function. Through the extensive employment of the analyticity of a complex function, we show the equivalence of those parametrizations. As applications of the parametrizations, we consider the combinatorial factor arising in the Feynman parametrization integral and the multivariate beta function. The combinatorial factor corresponds to an elementary integral embedded in the time-ordered product of the Dyson series in the time-dependent perturbation theory. We believe that the derivation presented here can be a good pedagogical example that students enhance their understanding of complex variables and train the use of the Dirac delta function in coordinate transformation.

Calculating Specific Integrals by Using Residue Theorem

The aim of this paper is to solve the some specific integrals such as , , , and . The traditional method is that people normally decompose this polynomial into several partial fractions first. This process involves adding it all up, expanding brackets, and doing matrices computation, which takes too many steps of calculation. The partial fraction part requires using Euler’s formula and large amounts of expanding brackets to prove that the multiplication of those partial denominators is equal to the denominator given in the original equation. Once making one little mistake, the next process will be all nonsense. Therefore, introducing complex analysis and residue theorem can result in considerably fewer calculation steps than the traditional method does in order to calculate integral, as opposed to completing hundreds of steps of partial fraction decomposition and substitution. To solve the first integral: , we can rewrite Cauchy’s residue theorem in a new form, by considering residue at infinity , .Then, we can get the coefficient of and hence compute the integral. For the second and third integral: , and , we exploit the reciprocal function of Taylor polynomials, writing and in a new form to get the coefficient of Then we can compute the integrals by using the residue theorem.

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.

Determining the Inverse of a Vandermonde Matrix Using Partial Fraction Decomposition Method

One method that can be used to calculate the inverse of a matrix is ​​the adjoin method. In this method, the process begins by calculating the value of the determinant and adjoin of a matrix. This study discusses a method for calculating the inverse, especially on the Vandermonde matrix using partial fraction decomposition. The advantage of this method is that it can calculate the inverse of a matrix, without the need to calculate the value of the determinant and adjoin of a matrix. The steps taken are to define a rational function and then write it in the form of a partial fraction, then by using a formula to calculate the coefficient of a partial fraction, a formula is derived to calculate the inverse of the Vandermonde matrix. After obtaining the formula for calculating the inverse, then comparing the results of the inverse calculation of the Vandermonde matrix using the partial fraction decomposition method with the adjoin method. The results obtained a formula to calculate the inverse of the Vandermonde matrix, V-1=WxA. Based on the case examples given, it can be concluded that the results of the inverse calculations performed using the partial fraction decomposition method are the same as the results of the calculations performed using the adjoin method. However, the calculations performed using the partial fraction decomposition method are more effective and efficient than using the adjoin method

Partial-fraction decomposition of a rational function and its application

In this paper, by using the residue method of complex analysis, we obtain an explicit partial fraction decomposition for the general rational function frac{x^{M}}{(x+1)^{lambda}_{n}} (M is any nonnegative integer, λ and n are any positive integers). As applications, we deduce the corresponding algebraic identities and combinatorial identities which are the corresponding extensions of Chu’ results. We also give some explicit formulas of Apostol-type polynomials and harmonic Stirling numbers of the second kind.

SHORTER PROOF OF THE RIESZ INTERPOLATION FORMULA FOR TRIGONOMETRIC POLYNOMIALS

By means of partial fraction decompositions, a shorter proof is presented for the important interpolation formula of trigonometric polynomials discovered by Riesz (1914).

Tapir: A tool for topologies, amplitudes, partial fraction decomposition and input for reductions

The demand for precision predictions in the field of high energy physics has dramatically increased over recent years. Experiments conducted at the LHC, as well as precision measurements at the intensity frontier such as Belle II require equally precise theoretical predictions to make full use of the acquired data. To match the experimental precision, second-, third- and, for certain quantities, even higher-order calculations in perturbative quantum field theory are required.To facilitate such calculations, computer software automating as many steps as possible is required. Yet, each calculation poses different challenges and thus, a high level of configurability is required. In this context we present tapir: a tool for identification, manipulation and minimization of Feynman integral families. It is designed to integrate in toolchains based on the computer algebra system FORM, the use of which is common practice in the field. tapir can be used to reduce the complexity of multi-loop problems with cut-filters, topology mapping, partial fraction decomposition and alike. Program summaryProgram Title:tapirCPC Library link to program files:https://doi.org/10.17632/ptc9t46xyn.1Developer's repository link:https://gitlab.com/tapir-devs/tapirLicensing provisions: GPLv3Programming language:python 3, C++Nature of problem: Multi-loop computations require the automatization of a large number of different tasks related to Feynman integral topologies. Among them are the identification and minimization of integral topologies, partial fraction decomposition of topologies in the case of linearly dependent propagators as well as mapping scalar products of loop momenta to scalar functions.Solution method: The minimization of topologies is performed by comparison of their respective Nickel indices [1], even further minimization utilizes Pak's algorithm [2]. To efficiently map scalar products of loop momenta to scalar functions FORM [3] code is generated.Additional comments including restrictions and unusual features: Minimization based on Pak's algorithm slows down for many lines and scales. A coarser minimization using the Nickel indices, however, is still possible.

New families of mock theta functions and partial fraction decomposition

In 2000, Andrews challenged mathematicians in the 21st century to elucidate the overlap between classes of q-series and modular forms. Motivated by Andrews' challenge, seeking q-hypergeometric series which are mock theta functions in the modern sense and finding their alternative representations have attracted the attention of many mathematicians in recent years. In this paper, we use partial fraction decomposition to give rise to several families of two-parameter mock theta functions and express them in the form of Appell-Lerch sums. In particular, we establish some identities involving several families of mock theta functions and Appell-Lerch sums which imply three identities proved by Cui and Gu.

On the rational approximation of Markov functions, with applications to the computation of Markov functions of Toeplitz matrices

We investigate the problem of approximating the matrix function f(A) by r(A), with f a Markov function, r a rational interpolant of f, and A a symmetric Toeplitz matrix. In a first step, we obtain a new upper bound for the relative interpolation error 1 − r/f on the spectral interval of A. By minimizing this upper bound over all interpolation points, we obtain a new, simple, and sharp a priori bound for the relative interpolation error. We then consider three different approaches of representing and computing the rational interpolant r. Theoretical and numerical evidence is given that any of these methods for a scalar argument allows to achieve high precision, even in the presence of finite precision arithmetic. We finally investigate the problem of efficiently evaluating r(A), where it turns out that the relative error for a matrix argument is only small if we use a partial fraction decomposition for r following Antoulas and Mayo. An important role is played by a new stopping criterion which ensures to automatically find the degree of r leading to a small error, even in presence of finite precision arithmetic.

The exponential matrix: an explicit formula by an elementary method

We show an explicit formula, with a quite easy deduction, for the exponential matrix $e^{tA}$ of a real and finite square matrix $A$ (and for complex ones also). The elementary method developed avoids Jordan canonical form, eigenvectors, resolution of any linear system, matrix inversion, polynomial interpolation, complex integration, functional analysis, and generalized Fibonacci sequences. The basic tools are the Cayley-Hamilton theorem and the method of partial fraction decomposition. Two examples are given. We also show that such method applies to algebraic operators on infinite dimensional real Banach spaces.

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.