Pade Approximation Theory Research Articles

Numerical Stability of Runge-Kutta Methods for Differential Equations with Piecewise Constant Arguments with Matrix Coefficients

The paper discusses the analytical stability and numerical stability of differential equations with piecewise constant arguments with matrix coefficients. Firstly, the Runge-Kutta method is applied to the equation and the recurrence relationship of the numerical solution of the equation is obtained. Secondly, it is proved that the Runge-Kutta method can preserve the convergence order. Thirdly, the stability conditions of the numerical solution under different matrix coefficients are given by Pad$\acute{e}$ approximation and order star theory. Finally, the conclusions are verified by several numerical experiments.

Optimality and resonances in a class of compact finite difference schemes of high order

In this paper, we revisit the old problem of compact finite difference approximations of the homogeneous Dirichlet problem in dimension 1. We design a large and natural set of schemes of arbitrary high order, and we equip this set with an algebraic structure. We give some general criteria of convergence and we apply them to obtain two new results. On the one hand, we use Pade approximant theory to construct, for each given order of consistency, the most efficient schemes and we prove their convergence. On the other hand, we use diophantine approximation theory to prove that almost all of these schemes are convergent at the same rate as the consistency order, up to some logarithmic correction.

Inverse Polynomial Images are Always Sets of Minimal Logarithmic Capacity

In this paper, we prove that each inverse polynomial image (that is, each inverse image of an interval with respect to a polynomial mapping) is a set of minimal logarithmic capacity in a certain sense. Such sets play an important role in the theory of Pade-Approximation. The proofs are all based on the characterization theorems of Herbert Stahl.

Padé-Approximation-Based Behavioral Modeling

A new, nonlinear, frequency-domain behavioral model, based on Pade approximation theory, is presented in this paper. The new Pade model is similar in concept to the X-parameter method, but it uses a rational function instead of the X-parameter model's polynomial expression. The basic theory of the proposed Pade model and the model extraction methodology are provided. The proposed Pade approach increases the available modeling space beyond that available to the X-parameter method, and the performance of a selection of this family of models under various loading conditions is described in this paper. Along with the examples displaying the ability of the new model to predict fundamental frequency behavior, a second-harmonic example is also provided, with simulation results presented in both the frequency domain and the time domain. Based on the simulation studies and experimentally measured results, the new Pade model shows good accuracy and a high level of stability. All of these characteristics show that the proposed Pade model has high potential in RF design.

Symbol Error Rate Analysis of M-QAM with Equal Gain Combining Over A Mobile Satellite Channel

Mobile Satellite Communications (MSC) have become an essential part of the world telecommunication infrastructure. However, the systems suffer from multipath propagation effects. In this paper, error analysis of M-ary quadrature amplitude modulation (M-QAM) with Equal Gain Combiner (EGC) over mobile satellite channel was carried out. The satellite channel was modelled as the product of Rayleigh and Ricians. This was then used to develop a system model for the received signal which was simulated and evaluated in terms of Average Symbol Error Rate (ASER) using the exact closed-form expression derived from moment generating function (MGF) and Pade Approximants (PA) theory. The results showed that at 16dB, Rician factor ‘k’=0, ASER obtained are 41.83%, 18.56% and 10.81% for paths ‘L’ = 2, 3, 4 respectively. ASER values reduced as ‘k’ increased. The results are in agreement with the simulation. DOI: http://dx.doi.org/10.11591/ijece.v3i6.4343

Flue Gas Temperature Control Based on Symphony DCS System in Seawater FGD of Coal-Fired Power Plant

With the reinforcement of people’s environmental awareness, the flue gas desulphurization of coal-fired power plant becomes a necessary process. Based on the principle that the temperature of seawater FDG system determines the efficiency of desulphurization in a certain extent, the article introduces the Symphony DCS system which applied to the strategies of the temperature control and the realization methods in the seawater desulphurization. It uses the optimal control principle in the mathematical PADE approximation theory and modern control theory to optimize the temperature control that is with hysteretic. Moreover, it is not only uses C language interface providing by COMPOSER to configuration, but also gives the concrete realization methods. It has been proved be feasible by the actual inspection in the site.

Rational approximation of analytic functions

The paper provides an overview of the author’s contribution to the theory of constructive rational approximations of analytic functions. The results presented are related to the convergence theory of Pade approximants and of more general rational interpolation processes, which significantly expand the classical theory’s framework of continuous fractions, to inverse problems in the theory of Pade approximants, to the application of multipoint Pade approximants (solutions of Cauchy-Jacobi interpolation problem) in explorations connected with the rate of Chebyshev rational approximation of analytic functions and to the asymptotic properties of Pade-Hermite approximation for systems of Markov type functions.

A Rational Approach to the Resonance Region

Resonance Saturation in QCD can be understood in the large- N c limit from the mathematical theory of Pade Approximants to meromorphic functions.

A rational approach to resonance saturation in large-NcQCD

We point out that resonance saturation in QCD can be understood in the large-Nc limit from the mathematical theory of Pade Approximants to meromorphic functions. These approximants are rational functions which encompass any saturation with a finite number of resonances as a particular example, explaining several results which have appeared in the literature. We review the main properties of Pade Approximants with the help of a toy model for the two-point correlator, paying particular attention to the relationship among the Chiral Expansion, the Operator Product Expansion and the resonance spectrum. In passing, we also comment on an old proposal made by Migdal in 1977 which has recently attracted much attention in the context of AdS/QCD models. Finally, we apply the simplest Pade Approximant to the correlator in the real case of QCD. The general conclusion is that a rational approximant may reliably describe a Green's function in the Euclidean, but the same is not true in the Minkowski regime due to the appearance of unphysical poles and/or residues.

Large-NcQCD and Pade approximant theory

In the large-${N}_{c}$ limit of QCD, the expansion of the vacuum polarization at low energies determines the whole function at any arbitrarily large (but finite) energy. This result is an immediate consequence of the theory of Pade approximants to Stieltjes functions.

Results on Maximally Flat Fractional-Delay Systems

The two classes of maximally flat finite-impulse response (FIR) and all-pass infinite-impulse response (IIR) fractional-sample delay systems are thoroughly studied. New expressions for the transfer functions are derived and mathematical properties revealed. Our contributions to the FIR case include a closed-form formula for the Farrow structure, a three-term recurrence relation based on the interpolation algorithm of Neville, a concise operator-based formula using the forward shift operator, and a continued fraction representation. Three types of structures are developed based on these formulas. Our formula for the Farrow structure enhances the existing contributions by Valimaki, and by Vesma and Sarama/spl uml/ki on the subsystems of the structure. For the IIR case, it is rigorously proved, using the theory of Pade approximants, that the continued fraction formulation of Tassart and Depalle yields all-pass fractional delay systems. It is also proved that the maximally flat all-pass fractional-delay systems are closely related to the Lagrange interpolation. It is shown that these IIR systems can be characterized using Thiele's rational interpolation algorithm. A new formula for the transfer function is derived based on the Thiele continued fractions. Finally, a new class of maximally flat FIR fractional-sample delay systems that exhibit an almost all-pass magnitude response is proposed. The systems possess a maximally flat group-delay response at the end frequencies 0 and /spl pi/, and are characterized by a closed-form formula. Their main advantage over the classical FIR Lagrange interpolators is the improved magnitude response characteristics.

Solving the optimal PWM problem for single-phase inverters

In this paper, the basic algebraic properties of the optimal PWM problem for single-phase inverters are revealed. Specifically, it is shown that the nonlinear design equations given by the standard mathematical formulation of the problem can be reformulated, and that the sought solution can be found by computing the roots of a single univariate polynomial P(x), for which algorithms are readily available. Moreover, it is shown that the polynomials P(x) associated with the optimal PWM problem are orthogonal and can therefore be obtained via simple recursions. The reformulation draws upon the Newton identities, Pade approximation theory, and properties of symmetric functions. As a result, fast O(n log/sup 2/ n) algorithms are derived that provide the exact solution to the optimal PWM problem. For the PWM harmonic elimination problem, explicit formulas are derived that further simplify the algorithm.

Applications of the Hypergeometric Method to the Generalized Ramanujan-Nagell Equation

In this paper, we refine work of Beukers, applying results from the theory of Pade approximation to (1 − z)1/2 to the problem of restricted rational approximation to quadratic irrationals. As a result, we derive effective lower bounds for rational approximation to \(\sqrt m\) (where m is a positive nonsquare integer) by rationals of certain types. Forexample, we have $$\left| {\sqrt 2 - \frac{p}{q}} \right| \gg q^{ - 1.47} {\text{ and }}\left| {\sqrt 3 - \frac{p}{q}} \right| \gg q^{ - 1.62}$$ provided q is a power of 2 or 3, respectively. We then use this approach to obtain sharp bounds for the number of solutions to certain families of polynomial-exponential Diophantine equations. In particular, we answer a question of Beukers on the maximal number of solutions of the equation x2 + D = pn where D is a nonzero integer and p is an odd rational prime, coprime to D.

Specification of autocorrelated disturbances in non-linear regression model

Non-linear regression models are encountered quite often in various business and economic data analysis. In the case of time dependent regression models the autocorrelations in error is almost inevitable. Following Kumar (1987) this paper discusses a new method based on the theory of Pade approximation for the specification of error structure in the context of non-linear regression model. It has been shown through simulation studies that the method can be successfully used to specify the error structure, which follows an ARMA model. An application of the above method on a real data set, which follows a non-linear model, is also given.

An interpolation approximation for the GI/G/1 queue based on multipoint Padé approximation

The performance evaluation of many complex manufacturing, communication and computer systems has been made possible by modeling them as queueing systems. Many approximations used in queueing theory have been drawn from the behavior of queues in light and heavy traffic conditions. In this paper, we propose a new approximation technique, which combines the light and heavy traffic characteristics. This interpolation approximation is based on the theory of multipoint Pade approximation which is applied at two points: light and heavy traffic. We show how this can be applied for estimating the waiting time moments of the GI/G/1 queue. The light traffic derivatives of any order can be evaluated using the MacLaurin series analysis procedure. The heavy traffic limits of the GI/G/1 queue are well known in the literature. Our technique generalizes the previously developed interpolation approximations and can be used to approximate any order of the waiting time moments. Through numerical examples, we show that the moments of the steady state waiting time can be estimated with extremely high accuracy under all ranges of traffic intensities using low orders of the approximant. We also present a framework for the development of simple analytical approximation formulas.

Existence and uniqueness results for neural network approximations

Some approximation theoretic questions concerning a certain class of neural networks are considered. The networks considered are single input, single output, single hidden layer, feedforward neural networks with continuous sigmoidal activation functions, no input weights but with hidden layer thresholds and output layer weights. Specifically, questions of existence and uniqueness of best approximations on a closed interval of the real line under mean-square and uniform approximation error measures are studied. A by-product of this study is a reparametrization of the class of networks considered in terms of rational functions of a single variable. This rational reparametrization is used to apply the theory of Pade approximation to the class of networks considered. In addition, a question related to the number of local minima arising in gradient algorithms for learning is examined.

New sets of stability criteria as a generalization of Markov's stability theorem

In this paper new sets of algebraic stability criteria for linear time-invariant continuous-time systems are presented as a generalization of Markov's stability theorem. They are expressed as a series of inequalities for the Hankel determinants whose elements consist of the two kinds of Taylor coefficients associated with the characteristic polynomial of the systems. Results are derived that have close relations to the continued-fraction expansion and Pade approximation theory.

Hydrodynamic radius and intrinsic viscosity of polystyrene in the crossover region from θ to good‐solvent conditions

AbstractThe radius of gyration Rg, the hydrodynamic radius Rh, and the intrinsic viscosity [η] have been measured for polystyrene in cyclohexane (at the θ temperature and 44.5°C), 2‐butanone (at 30°C), and toluene (at 30°C) in the molecular weight range 2.38 × 105 ≤ Mw ≤ 5.47 × 106 to elucidate the excluded‐volume effects on dynamical behavior in dilute polymer solutions in the crossover region from θ to good‐solvent conditions. The results are compared with theoretical predictions of current thermal blob theories and the Pade approximant theory of Tanaka. It is found that the ratio of Rh/Rg decreases with an increase in the excluded‐volume effect, following the prediction of the simple blob theory, but that its magnitude is about 15% higher than the theoretical value. Experimental variation of [η] with Rh and/or Rg lies in between predictions of the scaling law and the Pade approximant. The concentration dependence of the diffusion coefficient is also compared with predictions of current theories.

A two-dimensional analogue of Pade approximant theory

The author introduces a formulation for two-dimensional Pade approximant theory, and proves a theorem on convergence under the stringent assumption of uniform boundedness. Finally he discusses some general problems connected with rational functions in several variable theory.

Integral equations for scattering by deuterons using unitary padé approximants

The application of the theory of Pade approximants to the three-body problem is suggested and studied. Integral equations are written for the first diagonal approximant to the multiple-scattering series.