Pade-type Approximants Research Articles

ON APPROXIMATION OF PERIODIC ATEB-FUNCTIONS

Two versions of approximation formulae for periodic Ateb-sine and Ateb-cosine in the first quarter of their common period are proposed. The first version is a Pade type approximation derived when constructing analytical solution of corresponding integral equation by iteration method with transforming the power series into a closed sum by Shanks’ formula. Two iteration approximations are considered. The first one is more concise but of worse approximation accuracy which deteriorates with increasing the argument value. To improve the approximation accuracy a hybrid approximation is proposed when the values of the Ateb-functions in the beginning (for the cosine) and in the end (for the sine) of the quarter period are computed by a separate formula obtained a priory by the asymptotic method. The comparison analysis of the approximate and exact values of the special functions indicates the error of the approximation proposed to be less than one per cent. The second variant of approximation is by replacing the periodic Ateb-functions by trigonometric functions of specific argument. The arguments are chosen so that the values of the special functions are exact at specific points of the quarter period. Five such collocation points are introduced in the paper. To implement this version of approximation a separate table of the values of the periodic Ateb-functions at the collocation points is compiled. The computational examples presented in the paper show the approximate values of the special functions obtained by the second version of approximation to have a good accuracy.

КОЛИВАННЯ ІМПУЛЬСНО НАВАНТАЖЕНОГО ОСЦИЛЯТОРА ЗІ СТЕПЕНЕВИМ ПОЗИЦІЙНИМ ТЕРТЯМ

Nonstationary oscillations of the oscillator with nonlinear positional friction caused by an instantaneous force pulse are described. The power dependence of the positional friction force on the displacement of the system, which generalizes the known models, is accepted. The corresponding dynamics problems were solved precisely by the method of adding and approximated by the method of energy balance. In the study, using periodic Ateb-functions, an exact analytical solution of the nonlinear differential equation of motion was constructed. Compact formulas for calculating oscillation ranges and half-cycle durations are derived. It is shown that the decrease in the amplitude of oscillations, as well as under the action of the force of linear viscous resistance, follows the law of geometric progression. The denominator of the progression is less than one and depends on the positional friction constants, in particular on the nonlinearity index. Thus, we have not only a decrease in the amplitude of oscillations, but also an increase in the durations of half-cycles, which is characteristic of nonlinear systems with a rigid force characteristic. Approximate displacement calculations use Pade-type approximations for periodic Ateb-functions. The error of these approximations is less than one percent. From the obtained analytical relations, as separate cases, the known dependences covered in the theory of oscillations for linear positional friction follow. It is shown that even in the case of nonlinear positional friction the process of oscillations caused by an instantaneous momentum has many oscillations and is not limited in time. In the case of power positional friction, the oscillation ranges of the pulse-loaded oscillator can be calculated by elementary formulas. The calculation of displacements in time is associated with the use of periodic Ateb-functions, the values of which are not difficult to determine by known asymptotic formulas. Calculations confirm that the obtained approximate formula does not give large errors. In order to verify the adequacy of the obtained analytical solutions, numerical computer integration of the original nonlinear differential equation of motion was performed. The results of the calculation, which lead to analytical and numerical solutions of the Cauchy problem, are well matched.

Generalized Moment Representations and Multivariate Multipoint Padé Approximants

Dzyadyk’s method of generalized moment representations is used to construct and study bivariate twopoint Pade-type approximants.

Some unusual results on extrapolation methods

This paper is devoted to properties of sequence transformations and the corresponding recursive algorithms for their implementation, which were never considered before. We first give necessary conditions that are satisfied if the transformed sequence converges faster than the initial one. These conditions can be used for deciding if a method is worth to be used. They also serve as the basis for defining criteria for stopping the acceleration algorithm when the best possible precision is obtained. Then, prescribing the transformed sequence, we show how to obtain the initial sequence which produces it via the transformation or via its recursive algorithm. These results show that almost any behavior is possible for the transformed sequence. A similar problem about Pade-type approximants is studied.

Numerical Solution of Time-Dependent Problems with a Fractional-Power Elliptic Operator

An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves a fractional power of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, standard two-level schemes are used. The approximate solution at a new time-level is obtained as a solution of a discrete problem with the fractional power of the elliptic operator. A Pade-type approximation is constructed on the basis of special quadrature formulas for an integral representation of the fractional power elliptic operator using explicit schemes. A similar approach is applied in the numerical implementation of implicit schemes. The results of numerical experiments are presented for a test two-dimensional problem.

A theoretical method for obtaining Padé type approximation to temperature integrals via the Stieltjes integral

The normal and generalized temperature integrals originate from the use of the Arrhenius and Harcourt–Esson equations in non-isothermal chemical kinetics analysis of solid materials, where approximate expressions to them are to be required. The aim of the present theoretical exploration is to provide both useful formulas and new insights to this problem; to the end, we address this with an extended view and new method. First, we derive an analytical connection between the two temperature integrals in terms of the Stieltjes integral. Secondly, based on this finding, a unified approach to obtaining approximate expressions of them is developed, which makes use of mathematical relationships between the Stieltjes integral, Stieltjes series and Pade approximant and leads to new Pade type approximations to the two integrals. Lastly, an error analysis and comparison of the proposed approximations along with published ones is performed, showing that they are accurate enough and applicable in thermal analysis.

Determination of Commercial Silicon Diode (4007IN) Parameters from the I–V and P-V Characteristics

A simple method is presented for determining the four diode parameters of the single exponential model of a silicon commercial diode, namely: Is reverse saturation current, n diode ideality factor, Rs series resistance and Gp shunt conductance. The current-voltage and power-voltage curves of the corresponding commercial diode are calculated and simulated via the exact explicit analytical solutions. By the use of the Lambert W function in its Pade-type approximation and the the FindFit function in the Mathematica software package, the four parameters were extracted. The fitted current-voltage, power-voltage curves and the corresponding experimental data are in good agreement. Also we provided the Pade-type approximation expression for Lambert W function.

Padé -Type Approximants for Some Classes of Multivariate Functions

We extend Dzyadyk’s method of generalized moment representations to the multidimensional case and, on this basis, construct and investigate the Pade-type approximants for some classes of multivariate functions.

Modeling and Analysis of On-Chip Single and H-tree Distributed RLC Interconnects

This paper presents novel methods for modeling and analysis of on-chip Single and H-tree distributed resistance inductance capacitance interconnects. The matrix pade-type approximation and scaling and squaring methods are employed for the numerical estimation of delay in single interconnect, and H-tree interconnects. The proposed models, which are based on these methods, provide rational function approximation for obtaining a passive interconnect model. Multiple single input single output model approximated transfer functions are developed for H-tree interconnects structure. With the equivalent reduced order lossy interconnect transfer functions, finite ramp responses are obtained, and line delay is estimated for various line lengths, input ramp rise times, source resistances, parasitic capacitances and load capacitances. In order to demonstrate the accuracy of proposed models, the estimated 50 % delay values are compared with the standard HSPICE W-element model and are found to be in good agreement. The proposed models worst case 50 % delay errors of single interconnect are 0.27 and 0.24 % respectively, while the worst case 50 % delay errors of H-tree structure are 5.73 and 3.94 % respectively.

MULTIVARIATE HOMOGENEOUS MATRIX PADE-TYPE APPROXIMANTS AND PADE APPROXIMANTS

Matrix Pade-type approximants and Pade approximants has been studied by several authors in the scalar case. Some definitions in multivari- ate case have been introduced. In the current work, we define a multivariate homogeneous matrix-type approximants of which generating polynomials have shift. We consider higher-order approximants and introduce multivariate ho- mogeneous matrix Pade approximants. We derive some recurrence relations for generating polynomials and associated polynomials. Finally, numerical example is given to illustrate our results.

Parameters Extraction of the Au/SnO 2 -Si(n)/Al p-n Junction Solar Cell Using Lambert W Function

The paper present a simple method for the extraction of solar cells parameters with a single diode circuit model from its dark characteristic and considering the series and shunt resistances. In order to extract the four parameters of the solar cell, an optimized technique is presented by using a combination of Lambert W function in its Pade-type approximation and the FindFit function in the Mathematica software package. By using the exact explicit analytical solutions, the current-voltage curves of the Au/SnO2-Si(n)/Al p-n junction solar cell are calculated and simulated. The results have the good agreement between the fitted current-voltage curves and the experimental data of the corresponding solar cell. A closed form expression for the Lambert W function based on a Pade-type approximation is also provided.

Padé-type approximants for generalized Euler transforms

We prove that sequences generated by the generalized Euler's transform can be considered as Pade-type approximants obtained by Hermite interpolation of the generating function u ? ( 1 + xu ) ? 1 $u \rightarrow (1+xu)^{-1}$ at the endpoints of the interval 0 , 1 $[0,1]$ . A first natural extension is then proposed by considering Hermite interpolation at multiple points of larger intervals.

Stable Routh-Padé-type approximation in model reduction of interval systems

An interval Pade-type approximation is introduced and then Routh-Pade-type method (IRPTM) is presented to model reduction in interval systems. The denominator in reduced model is obtained from the stable Routh table, and its numerator is constructed by the interval Pade-type definition. Compared to the existing Routh-Pade method, IRPTM does not need to solve linear interval equations. Hence, we do not have to compute interval division in the process. Moreover, theoretical analysis shows that IRPTM has smaller computational cost than that of Routh-Pade method. A typical numerical example is given to illustrate our method.

Least-squares function-valued Padé-type approximation and its application in solving integral equations

To solve the Fredholm integral equations of the second kind, a new notion of the least-squares orthogonal polynomials of function-valued Pade-type approximation is introduced. On the basis of the error formula, the least-squares function-valued Pade-type approximation is constructed. Their existence and uniqueness are studied. A recursive computation formula of the least-squares function-valued Pade-type approximation is given. In the end, an example is given to show that the method is effective and stable.

On $q$ -analoques of divergent and exponential series

We shall consider linear independence measures for the values of the functions $D_a(z)$ and $E_a(z)$ given below, which can be regarded as $q$-analogues of Euler's divergent series and the usual exponential series. For the $q$-exponential function $E_q(z)$, our main result (Theorem 1) asserts the linear independence (over any number field) of the values 1 and $E_q(\alpha_j)(j = 1,...,m)$ together with its measure having the exponent $\mu = O(m)$, which sharpens the known exponent $\mu = O(m^2)$ obtained by a certain refined version of Siegel's lemma (cf. [$\mathbf{1}$]). Let p be a prime number. Then Theorem 1 further implies the linear independence of the $p$- adic numbers $\prod_{n=0}^\infty(1 + kp^n), (k = 0,1,...,p-1)$, over $\boldsymbol{Q}$ with its measure having the exponent $\mu < 2p$. Our proof is based on a modification of Maier's method which allows to construct explicit Pade type approximations (of the second kind) for certain $q$-hypergeometric series.

Function-valued partial Padé-type approximation method for estimating eigenvalues of Fredholm integral equation of the second kind

A new function-valued partial Pade-type approximation was introduced in the polynomial space, and an explicit determinant formula was derived by means of some orthogonal polynomials. This method can be applied to estimating surplus eigenvalues of the Fredholm integral equation of the second kind when its partial eigenvalues have been known, and at the same time, it can be applied to solving the approximating solution of the given equation.

Generalized Homogeneous Multivariate Matrix PadÉ-Type Approximants and PadÉ Approximants

Generalized homogeneous multivariate matrix Pade-type approximants (GHMPTA) and Pade approximants are studied in ways similar to those of Brezinski and Kida in the scalar cases. By choosing an arbitrary monic bivariate scalar polynomial from the triangular form as the generating one of the approximant, we discuss their several typical important properties and study the connection between generalized homogeneous bivariate matrix Pade-type approximants and Pade approximants. The arguments given in detail in two variables can be extended directly to the case of d variables (d ges 2).

Generalized Multivariate Rectangular Matrix Pad&amp;#x00C9;-Type Approximants

A new method for the construction of generalized multivariate rectangular matrix Pade-type approximants on a rectangular grid is introduced in this paper, by choosing an arbitrary monic bivariate polynomial as the generating one of the approximant. We discuss their several typical important properties and study the connection between bivariate rectangular matrix Pade-type approximants and Pade approximants. The arguments given in detail in two variables can extend directly to the case of d variables (d ges 2).

Stable multi-dimensional model reduction and IIR filter design

It is well-known that model reduction techniques applied to stable multi-dimensional Linear Shift-Invariant (LSI) systems with Infinite-extent Impulse Response (IIR) do not necessarily guarantee a stable reduced system. Several conditions exist to check stability a posteriori. In this paper we outline a new technique that guarantees, a priori, that the system or filter is stable. In Section 1 we establish the necessary notation and definitions to deal with multi-dimensional systems and filters. Section 2 introduces the technique of multivariate Pade-type approximation and deals with stability. In Section 3 we illustrate the use of the newly proposed technique in filter design.

General Nondiagonal Cubic Hermite-Padé Approximation to the Exponential Function

General nondiagonal cubic Hermite-Pade approximation to the exponential function with coefficient polynomials of degree at most n, m, s, l, respectively is considered. Explicit formulas and differential equations are obtained for the coefficient polynomials. An exact asymptotic expression is obtained for the error function and it is also shown that these generalized Pade-type approximations can be used to asymptotically minimize the expressions on the unit disk.