Pade Method Research Articles

Calculation of hydrogen molecule energy levels using the moment constant summability method with specialized weight

The moment constant summability method with specialized weight for the summation of divergent series of the perturbation theory is considered in this work. The method is applied to the calculation of vibrational-rotational energy levels of a two-atom hydrogen molecule. It was tested earlier only on model problems. Some energy levels obtained by the Pade, Pade-Borel, and Pade-Hermit methods and ab initio are considered for comparison. The calculations have been carried out for the ground state and first ten excited vibrational states.

Analytical continuation of imaginary axis data for optical conductivity

We compare different methods for performing analytical continuation of spectral data from the imaginary time or frequency axis to the real frequency axis for the optical conductivity sigma(omega). We compare the maximum entropy (MaxEnt), singular value decomposition (SVD), sampling and Pade methods for analytical continuation. We also study two direct methods for obtaining sigma(0). For the MaxEnt approach we focus on a recent modification. The data are split up in batches, a separate MaxEnt calculation is done for each batch and the results are averaged. For the problems studied here, we find that typically the SVD, sampling and modified MaxEnt methods give comparable accuracy, while the Pade approximation is usually less reliable.

Pade Method to Painleve Equations

A class of special solutions of Painleve/Garnier systems arising as the Backlund or Schlesinger transformations of the Riccati solutions is known. In the past several years, the corresponding τ-functions have been explicitly computed and expressed as certain specialization of the Schur functions with rectangle shape partitions. In this note, we will give a simple and direct derivation of these solutions. Our method is based on the Pade approximation and its intrinsic relation to iso-monodromy deformations.

Improved Method for Creating Time-Domain Unsteady Aerodynamic Models

Multidisciplinary aeroservoelastic interactions are studied by the combination of knowledge acquired in two main disciplines: aeroelasticity and servocontrols. In aeroelasticity, the doublet lattice method is used to calculate the unsteady aerodynamic forces for a range of reduced frequencies and Mach numbers on a business aircraft in the subsonic flight regime by use of NASTRAN software. For aeroservoelasticity studies, there is the need to conceive methods for these unsteady aerodynamic forces conversions from frequency into Laplace domain. A new method different from classical methods is presented, in which Chebyshev polynomials theories and their orthogonality properties are applied. In this paper, a comparison between flutter results expressed in terms of flutter speeds and frequencies obtained with our method with flutter results obtained with classical Pade and least squares methods is presented for a business aircraft at one Mach number and a range of reduced frequencies. It has been found that results obtained with our method are better in terms of average error than results obtained with the two classical methods here presented.

On An Equal Fourth-Order-Accurate Temporal/Spatial Scheme for the Convection-Diffusion Equation

In this article, the convection-diffusion equation is discretized using the Pade method for the temporal derivative term and the wavenumber-extended method for the spatial derivative term. These temporal and spatial approximations result in two explicit equations and two implicitly coupled equations. To construct an equal-order scheme for the solution obtained at n Δt, both temporal/spatial derivatives are approximated to render fourth-order accuracy without using solutions obtained previously at (n − 2)Δt, (n − 3)Δt, etc. When approximating the first-order derivative term, it is essential to take the upwind nodal points into consideration. For revealing the dispersion and dissipation natures of the proposed scheme, both von Neumann (Fourier) and dispersion analyses were conducted. We validate the proposed method by solving several problems that are amenable to exact solutions. Results with theoretical rates of convergence are obtained for each of the one- and two-dimensional problems investigated.

Computation of Fractional Order Derivative and Integral via Power Series Expansion and Signal Modelling

The three techniques of s-to-z transform, power series expansion (PSE) and signal modelling are combined to develop a new procedure for efficiently computing the fractional order derivatives and integrals of discrete-time signals. A mapping function between the s-plane and the z-plane is first chosen, and then a PSE of this mapping function raised to fractional order is performed to get the desired infinite impulse response of the ideal digital fractional operator. Finally, the desired impulse response is modelled as the impulse response of a linear invariant system whose rational transfer function is determined using deterministic signal modelling techniques. Three non-iterative techniques, namely Pade, Prony and Shanks’ methods have been considered in this paper. Using Al-Alaoui’s rule as s-to-z transform, computation examples show that both Prony and Shanks’ method can achieve more accurate fractional differentiation and integration than Pade method which is equivalent to continued fraction expansion technique.

Current Awareness in NMR in Biomedicine

Current Awareness in NMR in Biomedicine

Aerodynamic Forces Approximations using the Chebyshev Method for Closed-Loop Aero-servoelasticity Studies

The approximation of unsteady generalized aerodynamic forces from the frequency domain into the Laplace domain acting on a Fly-By-Wire aircraft presents an important challenge in the aero-servoelasticity area. The aerodynamic forces in the reduced-frequency domain are approximated in the Laplace domain, to be able to study the effects of the control laws on the flexible aircraft structure. In this paper, we present a new method for the approximation of the generalized aerodynamic forces by use of Chebyshev polynomials and their orthogonality properties. A comparison of this new method with the Pade method used to calculate an approximation of the generalized aerodynamic forces from the frequency domain into the Laplace domain is presented. This comparison shows that this new method gives excellent results with respect to the Pade method and is applied on the Aircraft Test Model from NASA Dryden Flight Research Center.

Quadratic Padé approximants and the intruder state problem of multireference perturbation methods

Simple and quadratic Pade resummation methods are applied to high-order series from multireference many-body perturbation theory (MR-MBPT) calculations using various partitioning schemes (Moller–Plesset, Epstein–Nesbet, and forced degeneracy) to determine their efficacy in resumming slowly convergent or divergent series. The calculations are performed for the ground and low-lying excited states of (i) CH2, (ii) BeH2 at three geometries, and (iii) Be, for which full configuration interaction (CI) calculations are available for comparison. The 49 perturbation series that are analyzed include those with oscillatory and monotonic divergence and convergence, including divergences that arise from either frontdoor or backdoor intruder states. Both the simple and quadratic Pade approximations are found to speed the convergence of slowly convergent or divergent series. However, the quadratic Pade method generally outperforms the simple Pade resummation. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Accelerating FEM and BEM acoustic solutions

Reducing time-to-market, reducing development and production costs and increasing acoustic comfort, means less physical prototyping and more predictive modeling. However, predictive acoustic and vibro-acoustic models using FEM and BEM methods have often had limited usefulness, partly due to the long times taken to get the substantial amounts of results needed for engineering design optimization. In this paper, recently-developed technologies are presented which accelerate acoustic solutions. An array of approaches is presented, using finite, infinite and boundary element methods, which are based on re-usable Modal Acoustic Transfer Vectors, Pade methods for rapid frequency-sweep solutions, domain decomposition, finite element iterative solvers and multi-processor 'netsolvers'. These technologies make for timely and effective acoustic predictions, which are also accurate. They tackle a wide range of applications, such as engine acoustics and other machinery noise radiation, interior vehicle acoustics and component vibro-acoustics. They enable the design of practical solutions and are effective in reducing time-to-market and development costs. The fundamentals of various methods, their deployment in software and a selection of practical applications, timing benchmarks and case studies are presented.

Numerical study of axisymmetric dielectric resonators

In this paper, we present an effective approach to study the behavior of axisymmetric dielectric resonators. This approach is based on the finite-difference time-domain method and two accompanying techniques in order to increase the accuracy and decrease the computational cost. These two techniques are the fast-Fourier-transform approach and the Pade method. This method is used to obtain the resonant frequencies and quality factors for several resonators. Comparisons are made to show the utility of the method.

Reliability of Lanczos-Type Product Methods from Perturbation Theory

Lanczos methods are remarkably successful for the iterative solution of large, sparse linear systems. The Richardson iteration generates a vector sequence whose convergence to the solution of the linear system can be accelerated using Pade methods. We consider methods of Van der Vorst, Gutknecht and Zhang in the context of vector Pade approximation. This formulation helps to analyse the cause of minimisation breakdown and to avoid it.

Harmonic oscillations in a quasi-relativistic regime

A square-root anharmonic oscillator potential is analysed as a model which simulates a quasi-relativistic squeezing of the harmonic oscillator spectrum. Several eligible (namely, perturbative, variational, Hill-determinant and Riccati - Pade) methods of construction of its bound states are compared.

Charge‐exchange in the nonadiabatic Na/W gas‐surface reaction using a diabatic representation

AbstractA theoretical description of near‐resonant charge exchange is presented for gas‐surface collisions, based on the eikonal formalism, short wavelength approximation, and diabatic representation. The dynamics of the charge transfer is analyzed for the Na/W system. Accurate model diabatic potential surfaces are used to compute the charge exchange during scattering and desorption. The Pade method is applied to approximate the charge exchange probability. The derivations, developments, and comparisons of approximations of the charge‐exchange probability are discussed. These approximations are useful in experimental surface analysis to estimate the integral value of diabatic coupling between the atomic and ionic states. The frequency of oscillation during the charge‐exchange process is also discussed. Model diabatic potentials in analytic forms for the Na/W system are reported. © 1995 John Wiley & Sons, Inc.

Convergence properties of wide-angle techniques

We apply Lanczos and Pade methods to a series of field propagation techniques and examine the relative advantages of each procedure. >

Model reduction via least-squares Padé simplification of squared-magnitude functions

Simplification of squared-magnitude functions using the least-squares Pade (LS-Pade) method is proposed as a new procedure for model order-reduction. One of the advantages of this method is that it overcomes the jω-axis problem encountered in model simplification by means of squared-magnitude functions (SMF). This problem arises whenever poles of the simplified SMF are on the imaginary axis since in such cases spectral factorization cannot be applied to obtain the reduced-order model. It is shown that the standard LS-Pade algorithm can be directly applied to SMF in the simplification of transfer functions. Besides solving the jω-axis problem from a practical point of view, the proposed procedure usually yields better models even in cases when no poles are on the imaginary axis. Two numerical examples are included to illustrate the main points of the new procedure

Efficient implementation of a 2nd derivative method for stiff ODEs

Pade or Obrechkoff methods based on diagonal or subdiagonal Pade approximations for the exponential function combine high accuracy and good stability; but the occurrence of higher derivatives makes their efficient implementation difficult. For the 3rd order subdiagonal 2nd derivative method, we have found an implementation which has resulted in a competitive code for moderately large, strongly stiff systems, under moderate accuracy requirements.

The least squares Padé method for model reduction

Abstract This work describes a novel method for the reduction of S1SO continuous models. The proposed method, the least squares Pade (LS-Pade) method, provides reduced models which incorporate some information about the original model over the mid-frequency range. Consequently, the present method gives better results than the classical Pade and most of the related methods for a very small additional cost

General structure of the Thomas-Fermi equation of state.

We derive the behavior of the Thomas-Fermi equation of state near both the hot-curve'' (the limit as {ital T}{r arrow}{infinity} for a fixed de Broglie length) and the cold-curve'' (the zero-temperature isotherm) by using perturbation methods. We derive efficient computational formulas for the coefficients in these expansions using Pade methods. We are able to meld all of these results into a structurally correct representation of the Thomas-Fermi fluid pressure. We indicate how this representational structure can be used to construct the equation of state for realistic materials over large ranges of temperature and pressure. We also discuss the computation of the degree of ionization with and without the exchange correction.

A simple and flexible method for model simplification

Proposed is a simple and flexible method of solving the model-simplification problems connected with rational and irrational s-transfer systems, as well as discrete-time systems. Using an algebraic approach whereby the selection of appropriate values for k and m results in a close approximation of the simplified model to the original system and a reduction in instability, the method can easily be performed in an interactive computer program. Its ease of computation and applicability to a larger class of systems give it an adantage over the multipoint Pade method and most other stability-based methods. Three examples are given demonstrating the method's effectiveness and applicability.