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In this work, we have computed and implemented one-body integrals concerning Gaussian confinement potentials over Gaussian basis functions. Then, we have set an equivalence between Gaussian and Hooke atoms and we have observed that, according to singlet and triplet state energies, both systems are equivalent for large confinement depth for a series of even number of electrons n = 2, 4, 6, 8 and 10. Unlike with harmonic potentials, Gaussian confinement potentials are dissociative for small enough depth parameter; this feature is crucial in order to model phenomena such as ionization. In this case, in addition to corresponding Taylor-series expansions, the first diagonal and sub-diagonal Pade approximant were also obtained, useful to compute the upper and lower limits for the dissociation depth. Hence, this method introduces new advantages compared to others. This research was funded by Eusko Jaurlaritza (the Basque Government), through Consolidated Group Project No. IT1254-19 and IT1584-22. Technical and human support provided by IZO-SGI, SGIker (UPV/EHU, MICINN, GV/EJ, ERDF and ESF) is gratefully acknowledged.

Inequalities play a fundamental role in both theoretical and applied mathematics and contain many patterns of symmetries. In many studies, inequalities have been used to provide estimates of some functions based on the properties of their symmetry. In this paper, we present the following new asymptotic expansion related to the ordinary gamma function Γ(1+w)∼2πw(w/e)ww2+760w2−120w/2exp∑r=1∞μrwr,w→∞, with the recurrence relation of coefficients μr. Furthermore, we use Padé approximants and our new asymptotic expansion to deduce the new bounds of Γ(w) better than some of its recent ones.

We investigate the rational approximation of fractional powers of unbounded positive operators attainable with a specific integral representation of the operator function. We provide accurate error bounds by exploiting classical results in approximation theory involving Pad\'{e} approximants. The analysis improves some existing results and the numerical experiments proves its accuracy.

[ES]Las fracciones continuas han tenido un papel fundamental en el desarrollo de numerosas teorías matemáticas y actualmente siguen siendo un tema de investigación muy activo. El estudio de las fracciones continuas con coeficientes en los complejos se asienta en el análisis de las regiones donde la fracción continua converge. Esta teoría de convergencia nos permite definir funciones meromorfas como fracciones continuas a partir de su serie de potencias mediante una sucesión de aproximaciones racionales conocidas como aproximaciones de Padé. Además, se puede establecer una equivalencia entre los números reales y las fracciones continuas simples (un tipo especial de fracciones continuas con coeficientes enteros) y, mediante esta equivalencia, también se pueden estudiar numerosos problemas de teoría de números como la aproximación de números irracionales por aproximaciones racionales o la resolución de la ecuación de Pell, una ecuación diofántica, [EN]Continued fractions have played a central role in the development of manu mathematical theories and, even today, they are still a very active line of research. The study of continued fractions with complex coefficients relies o the analysis fo the regions where the continued fraction converges. This convergence theory allows us to define meromorphic functions as continued fractions fron their formal power series with the help of a sequence of rational approximations known as Padé approximants. Furthermore, there is an equivalence between real numbers and simple cotinued fractions (a special case of continued fractions with integer coefficients) and, based on this equivalence, one can study problems of number theory such as how well irrational numbers can be approximated by rational numbers or how to solve Pell's equation, a kind of Diophantine equation

In this paper, by making use of the generating function methods and Padé approximation techniques, we establish some new recurrence formulas for the Hermite polynomials and their squares. These results presented here are the corresponding extensions of some known formulas.

The rational GARCH (RGARCH) model has been proposed as an alternative GARCHmodel that captures the asymmetric property of volatility. In addition to the previously proposedRGARCH model, we propose an alternative RGARCH model called the RGARCH-Exp model thatis more stable when dealing with outliers. We measure the performance of the volatility estimationby a loss function calculated using realized volatility as a proxy for true volatility and compare theRGARCH-type models with other asymmetric type models such as the EGARCH and GJR models.We conduct empirical studies of six stocks on the Tokyo Stock Exchange and find that a volatilityestimation using the RGARCH-type models outperforms the GARCH model and is comparable toother asymmetric GARCH models.

The volatility of stock return does not follow the classical Brownian motion, but instead it follows a form that is closely related to fractional Brownian motion. Taking advantage of this information, the rough version of classical Heston model also known as rough Heston model has been derived as the macroscopic level of microscopic Hawkes process where it acts as a high-frequency price process. Unlike the pricing of options under the classical Heston model, it is significantly harder to price options under rough Heston model due to the large computational cost needed. Previously, some studies have proposed a few approximation methods to speed up the option computation. In this study, we calibrate five different approximation methods for pricing options under rough Heston model to SPX options, namely a third-order Padé approximant, three variants of fourth-order Padé approximant, and an approximation formula made from decomposing the option price. The main purpose of this study is to fill in the gap on lack of numerical study on real market options. The numerical experiment includes calibration of the mentioned methods to SPX options before and after the Lehman Brothers collapse.

In the present article the classical problem of electromagnetic scattering by a single homogeneous sphere is revisited. Main focus is the study of the scattering behavior as a function of the material contrast and the size parameters for all electric and magnetic resonances of a dielectric sphere. Specifically, the Pad\'e approximants are introduced and utilized as an alternative system expansion of the Mie coefficients. Low order Pad\'e approximants can give compact and physically insightful expressions for the scattering system and the enabled dynamic mechanisms. Higher order approximants are used for predicting accurately the resonant pole spectrum. These results are summarized into general pole formulae, covering up to fifth order magnetic and forth order electric resonances of a small dielectric sphere. Additionally, the connection between the radiative damping process and the resonant linewidth is investigated. The results obtained reveal the fundamental connection of the radiative damping mechanism with the maximum width occurring for each resonance. Finally, the suggested system ansatz is used for studying the resonant absorption maximum through a circuit-inspired perspective., Comment: 8 pages, 4 figures

International audience; Frequency-based methods were set up in order to circumvent the limits of classical finite element methods in fast dynamic simulations due to discretizations. In this approach the dynamic loading was shifted in the frequency domain by FFT, then treated by the Variational Theory of Complex Rays, and then the time response was reconstructed through an IFFT. This strategy proved to be very efficient due to the CPU VTCR very low cost. However in the case of a large loading spectrum this frequency-by-frequency approach could seriously degrade the computational performances of the strategy.This paper addresses this point by proposing the use of Padé approximants in order to limit the number of frequencies at which the response should be calculated. Padé approximation is applied to the overall VTCR system based on its frequency dependency. Finally, as simulations on a simple academic case and on a civil engineering structure show, this method is found to be very efficient for interpolating the frequency response functions of a complex structure. This is a key point to preserve the efficiency of the complete VTCR strategy for transient dynamic problems.

The paper discusses the formulation of high-order accurate time-stepping schemes for transient convection–diffusion problems to be combined with finite element methods of the least-squares type for a stable discretization of highly convective problems. Padé approximations of the exponential function are considered for deriving multi-stage time integration schemes involving first time derivatives only, thus easier to implement in conjunction with C0 finite elements than standard time-stepping schemes which incorporate higher-order time derivatives. After a brief discussion of the stability and accuracy properties of the multi-stage Padé schemes and having underlined the similarity between Padé and Runge–Kutta methods, the paper closes with the presentation of illustrative examples which indicate the effectiveness of the proposed methods.

Approximants to the inverse Langevin and Brillouin functions appear in diverse contexts such as polymer science, molecular dynamics simulations, turbulence modeling, magnetism, theory of rubber. The exact inverses have no analytic representations, and are typically not implemented in software distributions. Various approximants for the inverse Langevin function L-1 had been proposed in the literature. After proving asymptotic features of the inverse functions, that had apparently been overlooked in the past, we use these properties to revisit this field of ongoing research. It turns out that only a subset of existing approximations obeys the relationships. Here we are able to derive improved (or 'corrected') approximations analytically. We disqualify the classical Padé solution approach that is typically used to obtain coefficients for approximate forms, and recommend a simple rational function LFENE-1(y)/(1+y2/2) that has a maximum relative error of 1-2 orders of magnitude smaller compared with the usually employed approximations LFENE-1(y)=3y/(1-y2) (50% maximal relative error) or also LCohen-1(y)=LFENE-1(y)(1-y2/3) (4.94%), while it is exactly as efficiently implemented and convenient for analytic (both force and energy) calculations. A number of applications is worked out. Moreover is the strategy proposed in this manuscript general and can be equally applied to fitting problems when asymptotic features are known., Journal of Non-Newtonian Fluid Mechanics, 223, ISSN:0377-0257, ISSN:1873-2631

International audience; This paper presents a new non-overlapping domain decomposition method for the time harmonic Maxwell's equations, whose effective convergence is quasi-optimal. These improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the Magnetic-to-Electric operator. A convergence theorem of the algorithm is established and numerical results validating the new approach are presented.

This article presents and discusses the general features and aspects regarding the electromagnetic scattering by a small core-shell plasmonic sphere. First, the thickness effects on the plasmonic resonances are presented in the electrostatic (Rayleigh) limit, utilizing the MacLaurin expansion of the Mie coefficients of hollow scatterers. Several aspects regarding the core effects are given, illustrating the enabling mechanisms and peculiarities of its resonant scattering response on it electrostatic limit. The electrodynamic aspects of the scattering process are revealed through the newly introduced Pad\'e expansion of the Mie coefficients. Additionally we expose how the core material affects the dynamic mechanisms, such as the dynamic depolarization and radiative damping. The described method can be expanded for other type of resonances and canonical shapes, while the general characteristics presented here are expected to stimulate further studies regarding the functionalities of the core-shell scatterers., Comment: 8 pages, 3 figures, under review to IEEE TAP

Differential matrix Riccati equations (DMREs) enable to model many physical systems appearing in different branches of science, in some cases, involving very large problem sizes. In this paper, we propose an adaptive algorithm for time-invariant DMREs that uses a piecewise-linearized approach based on the Pade approximation of the matrix exponential. The algorithm designed is based upon intensive use of matrix products and linear system solutions so we can seize the large computational capability that modern graphics processing units (GPUs) have on these types of operations using CUBLAS and CULATOOLS libraries (general purpose GPU), which are efficient implementations of BLAS and LAPACK libraries, respectively, for NVIDIA $$\copyright $$ © GPUs. A thorough analysis showed that some parts of the algorithm proposed can be carried out in parallel, thus allowing to leverage the two GPUs available in many current compute nodes. Besides, our algorithm can be used by any interested researcher through a friendly MATLAB $$\copyright $$ © interface.

The exact expression for the maximum tension of a pendulum string is used to obtain a closed-form approximate expression for the solution of a simple pendulum in terms of elementary functions. This approximate solution has a rational harmonic expression and depends on an unknown function, which must be determined. This unknown function is expanded using the Padé approximant and two new parameters are introduced which are determined by means of a term-by-term comparison of the power series expansion for the approximate maximum tension with the corresponding series for the exact one. Using this approach, accurate approximate analytical expressions for the periodic solution are obtained. We also compared the Fourier series expansions of the approximate solutions and the exact ones. This allowed us to compare the coefficients for the different harmonics in these solutions. We also compared the approximate and exact solutions as a function of time for several oscillation amplitudes. Finally, in this procedure we used some of the approximate expressions for the simple pendulum frequency which can be found in the bibliography; however, the procedure can be applied using other approximate frequencies. This work was supported by the Generalitat Valenciana of Spain under the project PROMETEO/2011/021 and by the ‘Vicerrectorado de Tecnología e Innovación Educativa’ of the University of Alicante, Spain, under the project GITE-09006-UA.

We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian,$H(\beta)=-d^2/dx^2+x^2+i\sqrt{\beta}x^3$,for $\beta$ in the cut plane $\C_c:=\C\backslash (-\infty, 0)$. Moreover, we prove that the spectrum consists of the perturbative eigenvalues $\{E_n(\beta)\}_{n\geq 0}$ labeled by the constant number $n$ of nodes of the corresponding eigenfunctions. In addition, for all $\beta\in\C_c$, $E_n(\beta)$ can be computed as the Stieltjes-Pad\'e sum of its perturbation series at $\beta=0$. This also gives an alternative proof of the fact that the spectrum of $H(\beta)$ is real when $\beta $ is a positive number. This way, the main results on the repulsive PT-symmetric and on the attractive quartic oscillators are extended to the cubic case., Comment: 23 pages, 3 figures

PurposeThe purpose of this paper is to propose an approximate method for solving a fractional population growth model in a closed system. The fractional derivatives are described in the Caputo sense.Design/methodology/approachThe approach is based on the differential transform method. The solutions of a fractional model equation are calculated in the form of convergent series with easily computable components.FindingsThe diagonal Padé approximants are effectively used in the analysis to capture the essential behavior of the solution.Originality/valueIllustrative examples are included to demonstrate the validity and applicability of the technique.

In this paper, Shooting Type Laplace–Adomian Decomposition Algorithm (STLADA), is applied to some boundary value problems with one of the boundary conditions at infinity. The analytic solution obtained by using this method converges rapidly, highly effective in terms of accuracy and very close to the exact solution. The solution is compared with the existing results. Shooting Type Laplace–Adomian Decomposition Algorithm is an efficient method for solving these types of problems occurring especially in fluid mechanics.

As a continuation of the well known connection between the theory of orthogonal polynomials on the unit circle and the interval [ − 1 , 1 ] , in this paper properties concerning error and convergence of certain rational approximants associated with the measures d μ ( t ) and d σ ( θ ) = | d μ ( cos θ ) | supported on [ − 1 , 1 ] and the unit circle respectively are deduced. Numerical illustrations are also given.

Boundary conditions in an unbounded domain, i.e. boundary condition at infinity, pose a problem in general for the numerical solution methods. The aim of this study is to overcome this difficulty by using Padé approximation with the differential transform method (DTM) on a form of classical Blasius equation. The obtained results are compared with, for the first time, the ones obtained by using a modified form of Adomian decomposition method (ADM). Furthermore, in order to see the consistency of solutions, they are also compared with the ones obtained by using variational iteration method (VIM).

In this paper, we apply the modified Laplace decomposition method (MLDM) to obtain series solutions of the boundary layer equation. The technique is based on the application of Laplace transform to boundary layers in fluid mechanics. The nonlinear terms can be easily handled by the use of Adomian polynomials. The obtained series solution is combined with the diagonal Pade approximants to handle the boundary condition at infinity. Comparison of the present solution is made with the existing solution and excellent agreement is noted.

We obtain results on the convergence of Padé approximants of Stieltjes-type meromorphic functions and the relative asymptotics of orthogonal polynomials on unbounded intervals. These theorems extend some results given by Guillermo López in this direction substituting the Carleman condition in his theorems by the determination of the corresponding moment problem. © 2009 Elsevier Inc. All rights reserved.

The problem of steady, laminar, thermal Marangoni convection flow of non-Newtonian power law fluid along a horizontal surface with variable surface temperature is studied. The partial differential equations are transformed into ordinary differential equations by using a suitable similarity transformation and analytical approximate solutions are obtained by an efficient transformation, asymptotic expansion and Pad? approximants technique. The effects of power law index and Marangoni number on velocity and temperature profiles are examined and discussed.

Conference on Approximation and extrapolation of Convergent and Divergent Sequences and Series Luminy, FRANCE, SEP 28-OCT 02, 2009, Commonly used techniques to study non-perturbative aspects of the strong interactions have a deep connection with rational approximants, and in particular with Pack approximants to meromorphic functions. However, only recently this connection has been acknowledged and efforts at fully exploiting it are only starting. In this article I will briefly review the most prominent techniques used in non-perturbative strong interactions with special emphasis on its relation with Pade approximants. I will then concentrate on a set of open problems outside the scope of these conventional techniques where Pade approximants might be extremely useful.

Two widely used methods for computing matrix exponentials and matrix logarithms are, respectively, the scaling and squaring and the inverse scaling and squaring. Both methods become effective when combined with Padé approximation. This paper deals with the computation of exponentials of skew-symmetric matrices and logarithms of orthogonal matrices. Our main goal is to improve these two methods by exploiting the special structure of skew-symmetric and orthogonal matrices. Geometric features of the matrix exponential and logarithm and extensions to the special Euclidean group of rigid motions are also addressed.

Using the Padé approximation of the exponential function, we obtain recurrence relations between Apostol–Bernoulli and between Apostol–Euler polynomials. As applications, we derive some new lacunary recurrence relations for Bernoulli and Euler polynomials with gap of length 4 and lacunary relations for Bernoulli and Euler numbers with gap of length 6.

In this paper we propose a reliable algorithm for the solution of nonlinear oscillators. Our algorithm is based upon the homotopy perturbation method (HPM), Laplace transforms, and Padé approximants. This modified homotopy perturbation method (MHPM) utilizes an alternative framework to capture the periodic behavior of the solution, which is characteristic of oscillator equations, and to give a good approximation to the true solution in a very large region. The current results are compared with those derived from the established Runge–Kutta method in order to verify the accuracy of the MHPM. It is shown that there is excellent agreement between the two sets of results. Results also show that the numerical scheme is very effective and convenient for solving strongly nonlinear oscillators.

In this paper, we apply the variational iteration method using He’s polynomials for finding the analytical solution of unsteady flow of gas through a porous medium. The proposed method is an elegant combination of He’s variational iteration and the homotopy perturbation methods. The suggested algorithm is quite efficient and is practically well suited for use in such problems. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. The diagonal Pade approximants are effectively used in the analysis to capture the essential behavior of the solution and to determine the initial slope. A clear advantage of this technique over the decomposition method is that no calculation of Adomian’s polynomials is needed.

This paper presents an efficient numerical algorithm for approximate solutions of a fractional population growth model in a closed system. The time-fractional derivative is considered in the Caputo sense. The algorithm is based on Adomian’s decomposition approach and the solutions are calculated in the form of a convergent series with easily computable components. Then the Pade approximants are effectively used in the analysis to capture the essential behavior of the population u(t) of identical individuals.

The application of Adomian's decomposition method to partial differential equations, when the exact solution is not reached, demands the use of truncated series. But the solution's series may have small convergence radius and the truncated series may be inaccurate in many regions. In order to enlarge the convergence domain of the truncated series, Padé approximants (PAs) to the Adomian's series solution have been tested and applied to partial and ordinary differential equations, with good results. In this paper, PAs, both in x and t directions, applied to the truncated series solution given by Adomian's decomposition technique for Burgers equation, are tested. Numerical and graphical illustrations show that this technique can improve the accuracy and enlarge the domain of convergence of the solution. It is also shown in this paper, that the application of Adomian's method to the ordinary differential equations set arising from the discretization of the spatial derivatives by finite differences, the so-called method of lines, may reduce the convergence domain of the solution's series.

We give a necessary and sufficient condition for an algebraic ODE to have a rational type general solution. For a first order autonomous ODE F = 0 , we give an exact degree bound for its rational solutions, based on the connection between rational solutions of F = 0 and rational parametrizations of the plane algebraic curve defined by F = 0 . For a first order autonomous ODE, we further give a polynomial time algorithm for computing a rational general solution if it exists based on the computation of Laurent series solutions and Pade approximants. Experimental results show that the algorithm is quite efficient.

We give a survey of recent generalizations of orthogonal polynomials. That includes multidimensional (matrix and vector orthogonal polynomials) and multivariate versions, multipole (orthogonal rational functions) variants, and extensions of the orthogonality conditions (multiple orthogonality). Most of these generalizations are inspired by the applications in which they are applied. We also give a glimpse of these applications, which are usually generalizations of applications where classical orthogonal polynomials also play a fundamental role: moment problems, numerical quadrature, rational approximation, linear algebra, recurrence relations, and random matrices. (c) 2004 Elsevier B.V. All rights reserved. ispartof: Journal of Computational and Applied Mathematics vol:179 issue:1 pages:57-95 ispartof: location:NORWAY, Roros status: published

Szegö polynomials and Szegö quadrature on the unit circle are studied for the Fejér kernel. Connections with orthogonal polynomials on the line and Padé approximants are obtained.

In the application of Pad\'{e} methods to signal processing a basic problem is to take into account the effect of measurement noise on the computed approximants. Qualitative deterministic noise models have been proposed which are consistent with experimental results. In this paper the Pad\'{e} approximants to the $Z$-transform of a complex Gaussian discrete white noise process are considered. Properties of the condensed density of the Pad\'{e} poles such as circular symmetry, asymptotic concentration on the unit circle and independence on the noise variance are proved. An analytic model of the condensed density of the Pad\'{e} poles for all orders of the approximants is also computed. Some Montecarlo simulations are provided.

In this work we propose three different procedures for vector-valued rational interpolation of a function F(z), where F:C→CN, and develop algorithms for constructing the resulting rational functions. We show that these procedures also cover the general case in which some or all points of interpolation coalesce. In particular, we show that, when all the points of interpolation collapse to the same point, the procedures reduce to those presented and analyzed in an earlier paper (J. Approx. Theory 77 (1994) 89) by the author, for vector-valued rational approximations from Maclaurin series of F(z). Determinant representations for the relevant interpolants are also derived.

We describe methods for the derivation of strong asymptotics for the denominator polynomials and the remainder of Padé approximants for a Markov function with a complex and varying weight. Two approaches, both based on a Riemann–Hilbert problem, are presented. The first method uses a scalar Riemann–Hilbert boundary value problem on a two-sheeted Riemann surface, the second approach uses a matrix Riemann–Hilbert problem. The result for a varying weight is not with the most general conditions possible, but the loss of generality is compensated by an easier and transparent proof.

In this paper, high-order prediction–correction algorithms are presented to solve unilateral contact problems. These algorithms are built up using the asymptotic numerical method which associates a perturbation technique with a numerical one as the finite element method.

RésuméDans cet article, nous énonçons et résolvons des problèmes d'approximation de Padé nouveaux et très généraux dont les solutions s'expriment à l'aide de séries hypergéométriques : par spécialisation, ces séries permettent de retrouver l'irrationalité de ζ(3), d'une infinité de ζ(2n+1), n entier ⩾1, et essentiellement tous les résultats de cette nature déjà présents dans la littérature. Nous présentons également deux nouvelles applications diophantiennes de notre méthode.AbstractIn this paper, we present and solve some very general new Padé approximant problems, whose solutions can be expressed with hypergeometric series. These series appear in the proofs of the irrationality of ζ(3), of infinitely many ζ(2n+1), and in essentially all results of this kind in the literature. We also prove two new Diophantine results with this method.

The constructive solution of the strong Stieltjes moment problem can be linked with positive Perron-Caratheodory continued fractions (PC-fractions) which are contradictions of positive Thron continued fractions (T-fractions). Their approximants are two-point Pade approximants for a related function. The multi-point moment problem similarly leads to positive multi-point Pade continued fractions (MP-fraction) and positive extended multi-point Pade continued fractions (EMP-fraction) whose approximants are multi-point Pade approximants. These relationships are explored also in the situation where the positivity condition is dropped. editors: Bonan-Hamada, Cathy and Gustafson, Phil ispartof: The Rocky Mountain journal of mathematics vol:33 issue:2 pages:609-627 ispartof: location:Mesa State College, Grand Junction, CO status: published

11 pages, no figures.-- MSC2000 codes: 42C05, 41A21. MR#: MR1858277 (2002h:42043) Zbl#: Zbl 1009.42016 We study the asymptotic behavior of orthogonal polynomials inside the unit circle for a subclass of measures that satisfy Szegö's condition. We give a connection between such behavior and a Montessus de Ballore-type theorem for Szegö–Padé rational approximants of the corresponding Szegö function. The research of the second author (G.L.L.) was partially supported by Dirección General de Enseñanza Superior under Grant PB 96-0120-C03-01. The research of the third author (E.B.S.) was supported, partly by NSF research Grant DMS-9801677. Publicado

Based on the general properties of the Yukawa and Hulthen potentials, that they have Coulombic behaviour for small r and are exponentially damped for large r, we have deduced the threshold and asymptotic properties of their energy eigenfunctions, We have then developed simple wavefunctions incorporating these properties for the lowest and the first excited states for all I values, which give quite accurate energies for a wide range of the screening parameters and which give a clear, physical insight into their structure. The quality of these wavefunctions is analysed in terms of a virial relation, a threshold relation and a local energy test.

The general form of Taylor's theorem for a function f : K→K , where K is the real line or the complex plane, gives the formula, f=Pn+Rn, where Pn is the Newton interpolating polynomial computed with respect to a confluent vector of nodes, and Rn is the remainder. Whenever f′≠0, for each m=2,…,n+1, we describe a “determinantal interpolation formula”, f=Pm,n+Rm,n, where Pm,n is a rational function in x and f itself. These formulas play a dual role in the approximation of f or its inverse. For m=2, the formula is Taylor's and for m=3 is a new expansion formula and a Pade approximant. By applying the formulas to Pn, for each m⩾2, P m,m−1 ,…,P m,m+n−2 is a set of n rational approximations that includes Pn, and may provide a better approximation to f, than Pn. Hence each Taylor polynomial unfolds into an infinite spectrum of rational approximations. The formulas also give an infinite spectrum of rational inverse approximations, as well as a fundamental k-point iteration function Bm(k), for each k⩽m, defined as the ratio of two determinants that depend on the first m−k derivatives. Application of our formulas have motivated several new results obtained in sequel papers: (i) theoretical analysis of the order of B m (k) , k=1,…,m, proving that it ranges from m to the limiting ratio of generalized Fibonacci numbers of order m; (ii) computational results with the first few members of Bm(k) indicating that they outperform traditional root finding methods, e.g., Newton's; (iii) a novel polynomial rootfinding method requiring only a single input and the evaluation of the sequence of iteration functions Bm(1) at that input. This amounts to the evaluation of special Toeplitz determinants that are also computable via a recursive formula; (iv) a new strategy for general root finding; (v) new formulas for approximation of π,e, and other special numbers.

In this paper, the modal analysis model, which is made up of a linear combination of complex exponential functions, is considered. The problem of estimating the number of modes as well as the frequencies, decays, amplitudes, and phases and their variability is afforded by means of the condensed distribution of the poles of Pade approximants to the Z-transform of the data. This provides a unified framework in which all of these problems can be solved in an almost automatic way. It is experimentally proved that the results favorably compare with that provided by well-established methods.

9 pages, no figures.-- MSC2000 codes: 41A21, 65D15. MR#: MR1690593 (2000f:41020) Zbl#: Zbl 0943.41007 The block structure of the Padé table associated with a formal power series is well known. We study the analytic properties of the given power series in the case that as we travel along a row of the corresponding table, we encounter blocks of increasing size. Thus, we extend to row sequences of Padé approximants some classical results due to Hadamard and Ostrowski related with the overconvergence of subsequences of Taylor polynomials and the analytic properties of the limit function under the presence of gaps in the power series. Research by second author (G.L.L.) partially supported by RG-297 Maths/LA from Third World Academy of Science. Publicado

A new definition of multivariate Padé approximation is introduced, which is a natural generalization of the univariate Padé approximation and consists in replacing the exact interpolation problem by a least squares interpolation. This new definition allows a straightforward extension of the Montessus de Ballore theorem to the multivariate case. Except for the particular case of the so-called homogeneous Padé approximants, this extension has up to now been impossible to obtain in the classical formulation of the multivariate Padé approximation. Besides, the least squares formulation can also be applied to the univariate case, and provides an alternative to the classical Padé interpolation.

A new class of multivariate Pade approximants is introduced. When dealing with two variables x and y , the approach consists in applying the Pade approximation with respect to y to the coefficients of the Pade approximation with respect to x . This technique has a natural extension to n variables, and is well adapted to programming in any usual language. The advantage of the method is that the algorithm uses only univariate Pade approximation. As a consequence, only small systems need to be solved, and the computation of the approximant is much faster: if M is the number of derivatives with respect to x and y , the algorithm requires O( M 4 ) operations instead of O( M 6 ) when matrix methods are used for solving the coefficients problem, or O( M 3 ) operations instead of O( M 4 ) if staircase methods are used. Another important feature is that the reliable methods which have been developed for choosing the degrees of the numerator and the denominator in the univariate case can be straightforwardly used in the multivariate case.

For population genetics models with recombination, obtaining an exact, analytic sampling distribution has remained a challenging open problem for several decades. Recently, a new perspective based on asymptotic series has been introduced to make progress on this problem. Specifically, closed-form expressions have been derived for the first few terms in an asymptotic expansion of the two-locus sampling distribution when the recombination rate $\rho$ is moderate to large. In this paper, a new computational technique is developed for finding the asymptotic expansion to an arbitrary order. Computation in this new approach can be automated easily. Furthermore, it is proved here that only a finite number of terms in the asymptotic expansion is needed to recover (via the method of Pad\'{e} approximants) the exact two-locus sampling distribution as an analytic function of $\rho$; this function is exact for all values of $\rho\in[0,\infty)$. It is also shown that the new computational framework presented here is flexible enough to incorporate natural selection., Comment: Published in at http://dx.doi.org/10.1214/11-AAP780 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Sequences of polynomials that occur as denominators in the two point Paté table for two series expansions are considered in the special case when the series coefficients are solutions of a strong symmetric Stieltjes moment problem. The continued fractions whose convergents generate these polynomials as denominators are presented, together with determinant representations for the polynomials and the continued fraction coefficients. The log-normal distribution is used as an example.

In the present paper, the Pade approximants technique is applied to the study of plane jets. In order to verify the applicability of this technique in jet flows, two problems having different flow field characteristics have been chosen as test cases. In the first case, the fluid-dynamic field of a jet emerging in a medium in motion at constant velocity is analysed; in the second case, the external medium is at rest, but together with the jet, is subjected to the external force of a transverse magnetic field. In both cases, the jet issues in a medium having its same properties. The laminar boundary layer equations have been solved by first expanding the stream function in a Taylor series in terms of a suitable power of the abscissa, and then by analytically continuing the series by means of the Pade approximants. The accuracy of the results has been tested by the comparison of the velocity distributions obtained by Pade approximants with those calculated by an asymptotic solution holding for large values of the abscissa in the first case, and with the analytical solution in the second case.

In this Tesis we investigate the approximation of vector functions by vector rational function that generalizes Padé approximants. We consider two types of approximants: the simultaneous Hermite-Padé approximants, which are constructed by mean of interpolation criterion and Fourier-Padé approximants based on Fourier series expansions in terms of a system of orthogonal polynomials. The results obtained in terms of generalize to the vector case results well known for the scalar case due to R. of Montessus of Ballore, A.A. Gonchar, S.P. Suetin, P.R. Graves-Morris, and E. B. Saff., En la tesis se investiga la aproximación de funciones vectoriales mediante vectores de fracciones racionales que generalizan los llamados aproximantes de Padé correspondientes al caso de la aproximación de una función escalar. Se consideran dos tipos de aproximantes: los aproximantes simultaneos Hermite-Padé, que se construyen mediante criterios interpolatorios y los aproximantes Fourier-Padé basados en desarrollos en serie de Fourier a partir de un sistema de polinomios ortogonales.Los resultados obtenidos generalizan al caso de la aproximación vectorial resultados muy conocidos para el caso escalar debidos a R. de Montessus de Ballore, A.A. Gonchar, S.P. Suetin, P.R. Graves-Morris, y E.B. Sa ff.