Generalized Pade Approximants Research Articles

Differential Quadrature Method for Fully Intrinsic Equations of Geometrically Exact Beams

In this paper, a differential quadrature method of high-order precision (DQ−Pade), which is equivalent to the generalized Pade approximation for approximating the end of a time or spatial interval, is used to solve nonlinear fully intrinsic equations of beams. The equations are a set of first-order differential equations with respect to time and space, and the explicit unknowns of the equations involve only forces, moments, velocity and angular velocity, without displacements and rotations. Based on the DQ−Pade method, the spatial and temporal discrete forms of fully intrinsic equations were derived. To verify the effectiveness and applicability of the proposed method for discretizing the fully intrinsic equations, different examples available in the literatures were considered. It was found that when using the DQ−Pade method, the solutions of the intrinsic beam equations are obviously superior to those found by some other usual algorithms in efficiency and computational accuracy.

Direct and inverse results on row sequences of generalized Padé approximants to polynomial expansions

Starting from the orthogonal and Faber polynomial expansions of a function F, we study the asymptotic behaviors of two generalized Pade approximations (orthogonal Pade approximation and Pade–Faber approximation). We obtain both direct and inverse results relating the convergence of the poles of these approximants and the singularities of F. Thereby, we obtain analogues of theorems by A. A. Gonchar and S. P. Suetin.

A generalized Padé approximation of time delay operator

This paper addresses the problem of fractional order Pade approximation for the time delay operator. Based on optimization method, a fitting polynomial with basic order α is constructed to approximate the time delay operator. This makes for two generalized Pade approximations in the form of normal and symmetric approximation, respectively. The advantage of the proposed method lies in the fact that the resulting approximate model is in the standard transform function form of any commensurate order α, making all of the existing stability analysis methods and control design tools for such systems amenable to the fractional order time delay systems. Some numerical examples are given to demonstrate the effectiveness of the proposed method.

A generalized Padé–Lindstedt–Poincaré method for predicting homoclinic and heteroclinic bifurcations of strongly nonlinear autonomous oscillators

By combining the generalized Pade approximation method and the well-known Lindstedt–Poincare method, a novel technique, referred to as the generalized Pade–Lindstedt–Poincare method, is proposed for determining homo-/heteroclinic orbits of nonlinear autonomous oscillators. First, the classical Pade approximation method is generalized. According to this generalization, the numerator and denominator of the Pade approximant are extended from polynomial functions to a series composed of any kind of continuous function, which means that the generalized Pade approximant is not limited to some certain forms, but can be constructed variously in solving different matters. Next, the generalized Pade approximation method is introduced into the Lindstedt–Poincare method’s procedure for solving the perturbation equations. Via the proposed generalized Pade–Lindstedt–Poincare method, the homo-/heteroclinic bifurcations of the generalized Helmholtz–Duffing–Van der Pol oscillator and \(\Phi ^{6}\)-Van der Pol oscillator are predicted. Meanwhile, the analytical solutions to these oscillators are also calculated. To illustrate the accuracy of the present method, the solutions obtained in this paper are compared with those of the Runge–Kutta method, which shows the method proposed in this paper is both effective and feasible. Furthermore, the proposed method can be also utilized to solve many other oscillators.

Unconditionally Stable Time-Step-Integration Algorithms Based on Hamilton's Principle

Unconditionally stable time-step-integration algorithms derived from Hamilton's principle are presented. The corresponding constrained variations are assumed to be in the form of Σα k τ k (1 - τ) to ensure vanishing variations at the ends of a time interval. The initial conditions are strongly enforced in the formulation. The order of accuracy can be 2n - 1 or 2n, where n is the number of unknown variables. The ultimate spectral radii of the algorithms can be controlled directly, The approximate solutions are in fact equivalent to the generalized Pade approximations for linear second-order differential equations. Hence, Hamilton's principle can be used to construct unconditionally stable higher-order-accurate time-step-integration algorithms directly as well, provided the appropriate constrained variations are used.

Weighting parameters for time-step integration algorithms with predetermined coefficients

In this paper, the effect of using the predetermined coefficients in constructing time-step integration algorithms is investigated. Both first- and second-order equations are considered. The approximate solution is assumed to be in a form of polynomial in the time domain. It can be related to the truncated Taylor's series expansion of the exact solution. Therefore, some of the coefficients can be predetermined from the known initial conditions. If there are m predetermined coefficients and r unknown coefficients in the approximate solution, the unknowns can be solved by the weighted residual method. The weighting parameter method is used to investigate the resultant algorithm characteristics. It is shown that the formulation is consistent with a minimum order of accuracy m+r. The maximum order of accuracy achievable is m+2r. Unconditionally stable algorithms exist if m⩽r for first-order equations and m+1⩽r for second-order equations. Hence, the Dahlquist's theorem is generalized. Algorithms equivalent to the Pade approximations and unconditionally stable algorithms equivalent to the generalized Pade approximations are constructed. The corresponding weighting parameters and weighting functions for the Pade and generalized Pade approximations are given explicitly. Copyright © 2000 John Wiley & Sons, Ltd.

Summation of asymptotic expansions of multiple-valued functions using algebraic approximants: Application to anharmonic oscillators

The divergent Rayleigh-Schrodinger perturbation expansions for energy eigenvalues of cubic, quartic, sextic and octic oscillators are summed using algebraic approximants. These approximants are generalized Pade approximants that are obtained from an algebraic equation of arbitrary degree. Numerical results indicate that given enough terms in the asymptotic expansion the rate of convergence of the diagonal staircase approximant sequence increases with the degree. Different branches of the approximants converge to different branches of the function. The success of the high-degree approximants is attributed to their ability to model the function on multiple sheets of the Riemann surface and to reproduce the correct singularity structure in the limit of large perturbation parameter. An efficient recursive algorithm for computing the diagonal approximant sequence is presented.

Asymptotic Two-Point Approximations by Rational Function

Continuous functions in 0 ≤ s ≤ ∞ with asymptotic power expansions both for t → +0 and t → +∞ are approximated by rational functions, being generalized Pade approximants both for s → +0 and s → +∞. The existence and the uniform convergence of such approximants are proved. As examples, piecewise rational functions, square roots of rational functions, and a modified error function are considered.

Diagonally implicit general linear methods for ordinary differential equations

We investigate some classes of general linear methods withs internal andr external approximations, with stage orderq and orderp, adjacent to the class withs=r=q=p considered by Butcher. We demonstrate that interesting methods exist also ifs+1=r=q, p=q orq+1,s=r+1=q, p=q orq+1, ands=r=q, p=q+1. Examples of such methods are constructed with stability function matching theA-acceptable generalized Pade approximations to the exponential function.

On the convergence of generalized Padé approximants

Letf be an analytic function with all its singularities in a compact set\(E_f \subset \bar C\) of (logarithmic) capacity zero. The function may have branch points. The convergence of generalized (multipoint) Pade approximants to this type of function is investigated. For appropriately selected schemes of interpolation points, it is shown that close-to-diagonal sequences of generalized Pade approximants converge in capacity tof in a certain domain that can be characterized by the property of the minimal condenser capacity. Using a pole elimination procedure, another set of rational approximants tof is derived from the considered generalized Pade approximants. These new approximants converge uniformly on a given continuum\(V \subset \bar C\backslash E_f\) with a rate of convergence that has been conjectured to be best possible. The continuumV is assumed not to divide the complex plane.

Harmonic balance methods and the theory of generalized Padé approximants

The appropriateness of the theory of generalized Pade approximants is demonstrated for some of the aspects of the harmonic balance method, as discussed in recent work by R. E. Mickens.

Product and sum separable functions

Tests for the product and sum separability of multivariable functions are given. A product (sum) separable function is a multivariable function that can be written as a product (sum) of functions in disjoint sets of variables. Generalized Pade approximation is used to derive the individual functions in disjoint sets of variables in the case of a sum separable function.

Application of generalized Padé approximants to the special function evaluation problem

A new approach to generalized Pade approximants is presented. Its main interest lies in the fact that it avoids instability and/or round-off errors which occur in traditional algorithms for evaluating Pade approximants. Moreover, the approximants are readily obtained. Applications concerned with the tabulation of some transcendental functions in their critical domain (Weber and Bessel functions) illustrate the power of the method.

Generalized Pade Approximation

Approximation by minimizing the magnitude of the complex error phasor in the steady state is considered and relationships to conventional magnitude-only and phase-only methods are discussed. The particular case of the Taylor approximation about zero frequency is shown to be equivalent to the Taylor approximation in terms of the complex variable. Generalized examples are given which relate to rational approximations for transistor alpha, resistance-capacitance transmission lines, and the Z transform.