Partial Fraction Decomposition Research Articles

Identities involving rational sums by inversion and partial fraction decomposition

Identities appearing recently in: J. L. Diaz-Barrero, J. Gibergans-Baguena P. G. Popescu: Some identities involving rational sums. Appl. Anal. Discrete Math., 1 (2007), 397-402, are treated by inverting them; the resulting sums are evaluated using partial fraction decomposition, following Wenchang Chu: A binomial coefficient identity associated with Beukers’ conjecture on Apery numbers. Electron. J. Combin., 11 (1): Note 15, 3 pp. (electronic), 2004. This approach produces a general formula, not only special cases.

Partial fraction decompositions and trigonometric sum identities

The partial fraction decomposition method is explored to establish several interesting trigonometric function identities, which may have applications to the evaluation of classical multiple hypergeometric series, trigonometric approximation and interpolation.

Partial fractions decompositions of some rational functions

In this paper there are presented three new methods of partial fractions decomposition of some rational functions. These methods can be applied to generate the general numerical algorithms and to calculations by hand.

Telescoping partial fractions decompositions, the little q-Jacobi functions of complex order, and the nonterminating q-Saalschütz sum

We use telescoping partial fractions decompositions to give new proofs of the orthogonality property and the normalization relation for the little q-Jacobi polynomials, and the q-Saalschutz sum. In [20], we followed the development [19] of Schur functions for partitions with complex parts, and we showed that there exist natural little q-Jacobi functions of complex order which satisfy extensions of the orthogonality property and normalization relation of the little q-Jacobi polynomials, and that these two results follow from and together imply the nonterminating form of the q-Saalschutz sum. Writing the q-Pochhammer symbol of complex order as a ratio of infinite products in the usual way, we obtain new telescoping partial fractions decomposition proofs of our results [20] for the little q-Jacobi functions of complex order. We give several new proofs of the q-Saalschutz sum and its nonterminating form.

High order smoothing schemes for inhomogeneous parabolic problems with applications in option pricing

AbstractA new family of numerical schemes for inhomogeneous parabolic partial differential equations is developed utilizing diagonal Padé schemes combined with positivity–preserving Padé schemes as damping devices. We also develop a split version of the algorithm using partial fraction decomposition to address difficulties with accuracy and computational efficiency in solving and to implement the algorithms in parallel. Numerical experiments are presented for several inhomogeneous parabolic problems, including pricing of financial options with nonsmooth payoffs.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

Pricing exotic options with L-stable Padé schemes

In this paper we develop a strongly stable ( L-stable) and highly accurate method for pricing exotic options. The method is based on Padé schemes and also utilizes partial fraction decomposition to address issues regarding accuracy and computational efficiency. Due to non-smooth payoffs, which cause discontinuities in the solution (or its derivatives), standard A-stable methods are prone to produce large and spurious oscillations in the numerical solutions which would mislead to estimating options accurately. The proposed method does not suffer these drawbacks while being easy to implement on concurrent processors. Numerical results are presented for digital options, butterfly spread and barrier options in one and two assets. In addition, the methods are tested on the Heston stochastic volatility model.

Summation formulae on trigonometric functions

Trigonometric sums over the angles equally distributed on the upper half plane are investigated systematically. Their generating functions and explicit formulae are established through the combination of the formal power series method and partial fraction decompositions.

Series expansions for the third incomplete elliptic integral via partial fraction decompositions

We find convergent double series expansions for Legendre's third incomplete elliptic integral valid in overlapping subdomains of the unit square. Truncated expansions provide asymptotic approximations in the neighborhood of the logarithmic singularity ( 1 , 1 ) if one of the variables approaches this point faster than the other. Each approximation is accompanied by an error bound.

Overlap Integrals Between Irregular Solid Harmonics and STOs Via the Fourier Transform Methods

In this paper, a unified analytical and numerical treatment of overlap integrals between Slater type orbitals (STOs) and irregular Solid Harmonics (ISH) with different screening parameters is presented via the Fourier transform method. Fourier transform of STOs is probably the simplest to express of overlap integrals. Consequently, it is relatively easy to express the Fourier integral representations of the overlap integrals as finite sums and infinite series of STOs, ISHs, Gegenbauer, and Gaunt coefficients. The another mathematical tools except for Fourier transform have used partial-fraction decomposition and Taylor expansions of rational functions. Our approach leads to considerable simplification of the derivation of the previously known analytical representations for the overlap integrals between STOs and ISHs with different screening parameters. These overlap integrals have also been calculated for extremely large quantum numbers using Gegenbauer, Clebsch-Gordan and Binomial coefficients. The accuracy of the numerical results is quite high for the quantum numbers of Slater functions, irregular solid harmonic functions and for arbitrary values of internuclear distances and screening parameters of atomic orbitals.

Partial Fraction Decomposition by Division

In this note, we present a method for the partial fraction decomposition of two algebraic functions: (i) fix)/iax + b)1 and (\i)fix)/ipx2 + qx + r) where fix) is a polynomial of degree n, t is a positive integer, and px2 + qx + r is an irreducible (g2 < 4pr) quadratic polynomial. Our algorithm is relatively simple in comparison with those given elsewhere [1, 2, 3, 4, 5, 6, 7, 8]. The essence of the method is to use repeated division to re-express the numerator polynomial in powers of the normalized denominator. Then upon further divisions, we obtain a sum of partial fractions in the form Ai/iax 4b)1 or iBjX + Cj)/ipx2 + qx + r)j for the original function. For (i), we let c = b/a, and express fix) as follows:

Partial Fraction Decomposition and Correlation Sequence in 2D Systems

In this paper, results of the one-dimensional (1D) digital filtering are extended to the two-dimensional (2D) case. It introduces a technique and an algorithm for the computation of the product H(z1,z2)H(z1-1z2-1). The technique is used to find a minimum phase transfer function of a 2D system such that the previous product matches a given correlation sequence. The algorithm requires less arithmetic operations than the traditional methods. The former is based on a matrix formulation of the product, which is used to investigate the 2D partial fraction decomposition (PFD) and stability.

A common generalization of some identities

By means of partial fraction decomposition, this paper provides an algebraic identity from which a lot of interesting identities follow.

804 部分分数展開法を利用したモデルマッチ法によるPID制御系の設計(機械力学・計測制御V)

804 部分分数展開法を利用したモデルマッチ法によるPID制御系の設計(機械力学・計測制御V)

A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apéry numbers

By means of partial fraction decomposition, an algebraic identity on rational function is established. Its limiting case leads us to a harmonic number identity, which in turn has been shown to imply Beukers' conjecture on the congruence of Apéry numbers.

Smoothing with positivity‐preserving Padé schemes for parabolic problems with nonsmooth data

AbstractWe introduce a new class of higher order numerical schemes for parabolic partial differential equations that are more robust than the well‐known Rannacher schemes. The new family of algorithms utilizes diagonal Padé schemes combined with positivity‐preserving Padé schemes instead of first subdiagonal Padé schemes. We utilize a partial fraction decomposition to address problems with accuracy and computational efficiency in solving the higher order methods and to implement the algorithms in parallel. Optimal order convergence for nonsmooth data is proved for the case of a self‐adjoint operator in Hilbert space as well as in Banach space for the general case. Numerical experiments support the theorems, including examples in pricing options with nonsmooth payoff in financial mathematics. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

A Fast Algorithm for MacMahon's Partition Analysis

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.

Demonstration of a Robust Pseudogap in a Three-Dimensional Correlated Electronic System

We outline a partial-fractions decomposition method for determining the one-particle spectral function and single-particle density of states of a correlated electronic system on a finite lattice in the non self-consistent T-matrix approximation to arbitrary numerical accuracy, and demonstrate the application of these ideas to the attractive Hubbard model. We then demonstrate the effectiveness of a finite-size scaling ansatz which allows for the extraction of quantities of interest in the thermodynamic limit from this method. In this approximation, in one or two dimensions, for any finite lattice or in the thermodynamic limit, a pseudogap is present and its energy diverges as T c is approached from above; this is an unphysical manifestation of using an approximation that predicts a spurious phase transition in one or two dimensions. However, in three dimensions one expects the transition predicted by the approximation to represent a true continuous phase transition, and whether or not a pseudo gap exists in the thermodynamic limit in three dimensions remains an open question. We have applied our method to the attractive Hubbard model on a three-dimensional simple cubic lattice, and find, similar to previous work, that for intermediate coupling a prominent pseudogap is found in the single-particle density of states, and this gap persists over a large temperature range. In addition, we also show that for weak coupling (an on-site Hubbard energy equal to one quarter the bandwidth) a pseudogap is also present. The pseudogap energy at the transition temperature is almost a factor of three larger than the T = 0 BCS gap for intermediate coupling, whereas for weak coupling the pseudogap and T = 0 BCS gap energies are essentially equal. These results show that a pseudogap due to superconducting fluctuations occurs in three dimensions even in the weak-coupling limit.

An exact solution for a normal point source problem

This paper presents an exact algebraic solution for a normal point source problem on elastic half-space due to impact pulse by adopting the Cagniard-Hoop method and improving Pekeris’ partial fraction decomposition means.

A Numerical Introduction to Partial Fractions

This article introduces a numerical analogue to partial fraction decomposition. The reasons explaining why the numerical process works are different from those behind the algebraic procedure. (See page 150 of [1] for a sketch of a proof of the algebraic theorem.) However, the analogue is so similar-and so much more intuitive-that most students come away from it better equipped to face partial fraction decomposition with confidence and authority. The decomposition of a rational expression N(x)/D(x) whose numerator has degree smaller than its denominator involves three primary steps [2]: (a) Express D(x) as a product of its prime factors, say

Evaluation of generalised quadratic cost functionals for linear systems with many commensurate time delays

A general analytic method for the evaluation of generalised quadratic integrals of the form Jμ,ν(σ)=∫0∞tμexp (σt)(dνe(t)/dtν)2dt, where μ and ν are integers and e(t) is the error variable of a control system having multiple commensurate time delays is presented. More precisely, the authors derive an analytic solution to the problem of evaluating Jμ,ν(σ) from the equivalent frequency-domain complex integral. The method is based on using a mathematically rigorous procedure to perform partial fraction decompositions of the integrand of the complex integral. It involves solving sets of symbolic linear equations and the factorisation of 1-D polynomials. The results presented for the evaluation of complex integrals involving the delay operator e−τs are of direct interest to the parametric optimisation and balanced realisation and reduction of linear time-delay systems.