scholarly journals h-Adic Polynomials and Partial Fraction Decomposition of Proper Rational Functions over R or C

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Kwang Hyun Kim ◽  
Xin Zhang
Keyword(s):  
Commutative Algebra ◽  
Computer Algebra System ◽  
Rational Functions ◽  
Partial Fraction ◽  
Decomposition Technique ◽  
Taylor Polynomial ◽  
Simple Method ◽  
Recursive Computation ◽  
Algebraic Operations ◽  
Partial Fraction Decomposition

The partial fraction decomposition technique is very useful in many areas including mathematics and engineering. In this paper we present a new and simple method on the partial fraction decomposition of proper rational functions which have completely factored denominators over R or C. The method is based on a recursive computation of the h-adic polynomial in commutative algebra which is a generalization of the Taylor polynomial. Since its computation requires only simple algebraic operations, it does not require a computer algebra system to be programmed.

Parallel Solution of Linear Systems

2016 ◽  
Vol 6 (3) ◽  
pp. 278-289
Author(s):  
Sidi-Mahmoud Kaber ◽  
Amine Loumi ◽  
Philippe Parnaudeau
Keyword(s):  
Linear Systems ◽  
Numerical Algorithms ◽  
Parallel Architecture ◽  
Decomposition Methods ◽  
Partial Fraction ◽  
Domain Decomposition Methods ◽  
Simple Method ◽  
Parallel Solution ◽  
Partial Fraction Decomposition ◽  
Programming Paradigms

AbstractComputational scientists generally seek more accurate results in shorter times, and to achieve this a knowledge of evolving programming paradigms and hardware is important. In particular, optimising solvers for linear systems is a major challenge in scientific computation, and numerical algorithms must be modified or new ones created to fully use the parallel architecture of new computers. Parallel space discretisation solvers for Partial Differential Equations (PDE) such as Domain Decomposition Methods (DDM) are efficient and well documented. At first glance, parallelisation seems to be inconsistent with inherently sequential time evolution, but parallelisation is not limited to space directions. In this article, we present a new and simple method for time parallelisation, based on partial fraction decomposition of the inverse of some special matrices. We discuss its application to the heat equation and some limitations, in associated numerical experiments.

scholarly journals A Closed Character Formula for Symmetric Powers of Irreducible Representations

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Stavros Kousidis
Keyword(s):  
Irreducible Representation ◽  
Rational Functions ◽  
Character Formula ◽  
Irreducible Representations ◽  
Partition Functions ◽  
Partial Fraction ◽  
Residue Type ◽  
Symmetric Powers ◽  
International Audience ◽  
Partial Fraction Decomposition

International audience We prove a closed character formula for the symmetric powers $S^N V(λ )$ of a fixed irreducible representation $V(λ )$ of a complex semi-simple Lie algebra $\mathfrak{g}$ by means of partial fraction decomposition. The formula involves rational functions in rank of $\mathfrak{g}$ many variables which are easier to determine than the weight multiplicities of $S^N V(λ )$ themselves. We compute those rational functions in some interesting cases. Furthermore, we introduce a residue-type generating function for the weight multiplicities of $S^N V(λ )$ and explain the connections between our character formula, vector partition functions and iterated partial fraction decomposition. Nous établissons une formule fermée pour le caractère de la puissance symétrique $S^N V(λ )$ d'une représentation irréductible $V(λ )$ d'une algèbre de Lie semi-simple complexe$\mathfrak{g}$, en utilisant des décompositions en fractions partielles. Cette formule exprime ce caractère en termes de fractions rationnelles en $r$ variables, où $r$ est le rang de $\mathfrak{g}$. Ces fractions sont plus faciles à déterminer que les multiplicités de la décomposition de $S^N V(λ )$ elles-mêmes. Nous calculons ces fonctions rationnelles dans quelques cas intéressants. Nous introduisons par ailleurs une fonction génératrice de type résidu pour les multiplicités de $S^N V(λ )$ et relions notre formule aux fonctions de partitions vectorielles et aux décompositions itérées en fractions partielles.

A simple method to test the existence of 2-D partial fraction decomposition

1984 ◽  
Vol 6 (2) ◽  
pp. 153-160 ◽  
Author(s):  
D.Raghu Rami Reddy ◽  
P.S. Reddy ◽  
M.N.S. Swamy
Keyword(s):  
Partial Fraction ◽  
Simple Method ◽  
Partial Fraction Decomposition

Evaluation of sums involving Gaussian q-binomial coefficients with rational weight functions

2016 ◽  
Vol 12 (02) ◽  
pp. 495-504 ◽  
Author(s):  
Emrah Kiliç ◽  
Helmut Prodinger
Keyword(s):  
Weight Function ◽  
Binomial Coefficients ◽  
Weight Functions ◽  
Partial Fraction ◽  
Decomposition Technique ◽  
Lucas Numbers ◽  
Finite Products ◽  
Partial Fraction Decomposition ◽  
Gaussian Formula

We consider sums of the Gaussian [Formula: see text]-binomial coefficients with a parametric rational weight function. We use the partial fraction decomposition technique to prove the claimed results. We also give some interesting applications of our results to certain generalized Fibonomial sums weighted with finite products of reciprocal Fibonacci or Lucas numbers.

scholarly journals A Fast Algorithm for MacMahon's Partition Analysis

10.37236/1811 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Guoce Xin
Keyword(s):  
Fast Algorithm ◽  
Special Class ◽  
Laurent Series ◽  
Rational Functions ◽  
Constant Term ◽  
Partial Fraction ◽  
Running Time ◽  
Partition Analysis ◽  
Several Variables ◽  
Partial Fraction Decomposition

This paper deals with evaluating constant terms of a special class of rational functions, the Elliott-rational functions. The constant term of such a function can be read off immediately from its partial fraction decomposition. We combine the theory of iterated Laurent series and a new algorithm for partial fraction decompositions to obtain a fast algorithm for MacMahon's Omega calculus, which (partially) avoids the "run-time explosion" problem when eliminating several variables. We discuss the efficiency of our algorithm by investigating problems studied by Andrews and his coauthors; our running time is much less than that of their Omega package.

scholarly journals Two-loop leading colour QCD corrections to $$ q\overline{q} $$ → γγg and qg → γγq

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Bakul Agarwal ◽  
Federico Buccioni ◽  
Andreas von Manteuffel ◽  
Lorenzo Tancredi
Keyword(s):  
Numerical Code ◽  
Tree Level ◽  
Partial Fraction ◽  
Integration By Parts ◽  
Analytic Expressions ◽  
Partial Fraction Decomposition ◽  
Loop Amplitudes

Abstract We present the leading colour and light fermionic planar two-loop corrections for the production of two photons and a jet in the quark-antiquark and quark-gluon channels. In particular, we compute the interference of the two-loop amplitudes with the corresponding tree level ones, summed over colours and polarisations. Our calculation uses the latest advancements in the algorithms for integration-by-parts reduction and multivariate partial fraction decomposition to produce compact and easy-to-use results. We have implemented our results in an efficient C++ numerical code. We also provide their analytic expressions in Mathematica format.

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