Multivariate Pade Approximants Research Articles

Analysis of Transient Thermal Distribution in a Convective–Radiative Moving Rod Using Two-Dimensional Differential Transform Method with Multivariate Pade Approximant

The transient temperature distribution through a convective-radiative moving rod with temperature-dependent internal heat generation and non-linearly varying temperature-dependent thermal conductivity is elaborated in this investigation. Symmetries are intrinsic and fundamental features of the differential equations of mathematical physics. The governing energy equation subjected to corresponding initial and boundary conditions is non-dimensionalized into a non-linear partial differential equation (PDE) with the assistance of relevant non-dimensional terms. Then the resultant non-dimensionalized PDE is solved analytically using the two-dimensional differential transform method (2D DTM) and multivariate Pade approximant. The consequential impact of non-dimensional parameters such as heat generation, radiative, temperature ratio, and conductive parameters on dimensionless transient temperature profiles has been scrutinized through graphical elucidation. Furthermore, these graphs indicate the deviations in transient thermal profile for both finite difference method (FDM) and 2D DTM-multivariate Pade approximant by considering the forced convective and nucleate boiling heat transfer mode. The results reveal that the transient temperature profile of the moving rod upsurges with the change in time, and it improves for heat generation parameter. It enriches for the rise in the magnitude of Peclet number but drops significantly for greater values of the convective-radiative and convective-conductive parameters.

Multivariate Pade approximation for solving Fokker-Plank equations of fractional order

Multivariate Pade approximation for solving Fokker-Plank equations of fractional order

Nonlinear Biodynamic Models of the Hand-arm System and Parameters Identification using the Vibration Transmissibility or the Driving-point Mechanical Impedance

This study aims at deriving nonlinear expressions of the transmissibility and the driving-point mechanical impedance (DPMI) of two nonlinear biodynamic hand-arm models having active restoring and dissipative parameters. It aims also in computing explicitly the non-directly measurable stiffness and damping coefficients of the human hand-arm system (HAS). Multivariate Pade approximants are used to express the dependence of the HAS mechanical properties on various influencing factors. The harmonic balance method is used to derive analytical expressions of the transmissibility and the DPMI. Then, the models parameters are identified by minimizing constrained error functions between the theoretical DPMI or transmissibility and the measured data. The developed workflow is applied to three experimental data sets of Z-direction vibrations where the excitation frequency and/or the grip force are varied. Using the ISO-10068 (2012) limit DPMI values versus the excitation frequency, we derived upper and lower limits of the overall stiffness coefficient and damping ratio for the human HAS. Furthermore, the model reproduces with high accuracy experimental measurements of the transmissibility, the DPMI and the vibration power absorption.

A New Padé Approximant for the Appell Hypergeometric Function F1

In this work, we present a simple method for computing the first Appell function F1(a,b,b’;c;x,y), in some particular case. We use a new definition of the general multivariate Pade approximant which allows us to get the explicit expression of the denominator polynomial. Our approach seems to give a better precision than the Taylor’s expansion, especially near the border of the convergence area.

Numerical Solution of Fractional Partial Differential-Algebraic Equation by Adomian Decomposition Method and Multivariate Pade Approximation

In this study, Adomian Decomposition Method (ADM) and Multivariate Pade Approximation (MPA) are used to get solution of fractional partial differential algebraic equation (FPDAE).The solutions in the form of power series are obtained by using ADM first and then we get approximate solutions by means of MPA. While solving the equation, Caputo derivative is utilized. Methods are applied on a test problem. Results demonstrate that they are quite applicable.

General order multivariate Padé approximants for Pseudo-multivariate functions. II

Explicit formulas for general order multivariate Padé approximants of pseudo-multivariate functions are constructed on specific index sets. Examples include the multivariate forms of the exponential function E ( x _ ) = ∑ j 1 , j 2 , … , j m = 0 ∞ x 1 j 1 x 2 j 2 ⋯ x m j m ( j 1 + j 2 + ⋯ + j m ) ! , \begin{equation*} E\left (\underline {x}\right ) =\sum _{j_{1},j_{2},\ldots ,j_{m}=0}^{\infty } \frac {x_{1}^{j_{1}}x_{2}^{j_{2}}\cdots x_{m}^{j_{m}}}{\left ( j_{1}+j_{2}+\cdots +j_{m}\right ) !}, \end{equation*} the logarithm function L ( x _ ) = ∑ j 1 + j 2 + ⋯ + j m ≥ 1 x 1 j 1 x 2 j 2 ⋯ x m j m j 1 + j 2 + ⋯ + j m , \begin{equation*} L(\underline {x})=\sum _{j_{1}+j_{2}+\cdots +j_{m}\geq 1}\frac { x_{1}^{j_{1}}x_{2}^{j_{2}}\cdots x_{m}^{j_{m}}}{j_{1}+j_{2}+\cdots +j_{m}}, \end{equation*} the Lauricella function F D ( m ) ( a , 1 , … , 1 ; c ; x 1 , … , x m ) = ∑ j 1 , j 2 , … , j m = 0 ∞ ( a ) j 1 + ⋯ + j m ( c ) j 1 + ⋯ + j m x 1 j 1 ⋯ x m j m , \begin{equation*} F_{D}^{\left ( m\right ) }\left ( a,1,\ldots ,1;c;x_{1},\ldots ,x_{m}\right ) =\sum _{j_{1},j_{2},\ldots ,j_{m}=0}^{\infty }\frac {\left ( a\right ) _{j_{1}+\cdots +j_{m}}}{\left ( c\right ) _{j_{1}+\cdots +j_{m}}} x_{1}^{j_{1}}\cdots x_{m}^{j_{m}}, \end{equation*} and many more. We prove that the constructed approximants inherit the normality and consistency properties of their univariate relatives. These properties do not hold in general for multivariate Padé approximants. A truncation error upperbound is also given.

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).

On the fast solution of Toeplitz-block linear systems arising in multivariate approximation theory

When constructing multivariate Pade approximants, highly structured linear systems arise in almost all existing definitions [10]. Until now little or no attention has been paid to fast algorithms for the computation of multivariate Pade approximants, with the exception of [17]. In this paper we show that a suitable arrangement of the unknowns and equations, for the multivariate definitions of Pade approximant under consideration, leads to a Toeplitz-block linear system with coefficient matrix of low displacement rank. Moreover, the matrix is very sparse, especially in higher dimensions. In Section 2 we discuss this for the so-called equation lattice definition and in Section 3 for the homogeneous definition of the multivariate Pade approximant. We do not discuss definitions based on multivariate generalizations of continued fractions [12, 25], or approaches that require some symbolic computations [6, 18]. In Section 4 we present an explicit formula for the factorization of the matrix that results from applying the displacement operator to the Toeplitz-block coefficient matrix. We then generalize the well-known fast Gaussian elimination procedure with partial pivoting developed in [14, 19], to deal with a rectangular block structure where the number and size of the blocks vary. We do not aim for a superfast solver because of the higher risk for instability. Instead we show how the developed technique can be combined with an easy interval arithmetic verification step. Numerical results illustrate the technique in Section 5.

Explicit construction of general multivariate Pad� approximants to an Appell function

Properties of Pade approximants to the Gauss hypergeometric function 2F1(a,b;c;z) have been studied in several papers and some of these properties have been generalized to several variables in [6]. In this paper we derive explicit formulae for the general multivariate Pade approximants to the Appell function F1(a,1,1;a+1;x,y)=∑ i,j=0 ∞ (axiyj/(i+j+a)), where a is a positive integer. In particular, we prove that the denominator of the constructed approximant of partial degree n in x and y is given by \(q(x,y)=(-1)^{n}{{m+n+a}\choose n}F_{1}(-m-a,-n,-n;-m-n-a;x,y)\) , where the integer m, which defines the degree of the numerator, satisfies m≥n+1 and m+a≥2n. This formula generalizes the univariate explicit form for the Pade denominator of 2F1(a,1;c;z), which holds for c>a>0 and only in half of the Pade table. From the explicit formulae for the general multivariate Pade approximants, we can deduce the normality of a particular multivariate Pade table.

Multivariate Two-Point Padé-Type and Two-Point Padé Approximants

In his paper the notions of two-point Pade-type and two-point Pade approximants are generalized for multivariate functions, with a generating denominator polynomial of general form. The multivariate two-point Pade approximant can be expressed as a ratio of two determinants and computed recursively using the E-algorithm. A comparison is made with previous definitions by other authors using particular generating denominator polynomials. The last section contains some convergence results.

Multivariate Padé-Bergman approximants

We define the multivariate Pade-Bergman approximants (also called $A^2_p$ Pade approximants) and prove a natural generalization of de Montessus de Ballore theorem. Numerous definitions of multivariate Pade approximants have already been introduced. Unfortunately, they all failed to generalize de Montessus de Ballore theorem: either spurious singularities appeared (like the homogeneous Pade [3,4], or no general convergence can be obtained due to the lack of consistency (like the equation lattice Pade type [3]). Recently a new definition based on a least squares approach shows its ability to obtain the desired convergence [6]. We improve this initial work in two directions. First, we propose to use Bergman spaces on polydiscs as a natural framework for stating the least squares problem. This simplifies some proofs and leads us to the multivariate $A^2_p$ Pade approximants. Second, we pay a great attention to the zero-set of multivariate polynomials in order to find weaker (although natural) hypothesis on the class of functions within the scope of our convergence theorem. For that, we use classical tools from both algebraic geometry (Nullstellensatz) and complex analysis (analytic sets, germs).

Efficient FDTD modeling of irises/slots in microwave structures and its application to the design of combline filters

A new methodology is proposed for the computationally efficient, numerically stable, and accurate finite difference time-domain (FDTD) simulation of microwave structures with electrically thin irises and slots. The proposed method is based on the hybridization of Yee's standard FDTD scheme with Pade approximations of the electromagnetic properties of the irises/slots. Through the use of rigorous modal expansions for the description of the fields in the waveguide sections formed by the irises and slots, highly accurate rational function approximations of their transmission and reflection properties are obtained. These transfer functions are then incorporated directly in the FDTD algorithm through their corresponding z-transform expressions. Combined with the matrix valued multivariate Pade approximation, the proposed method provides a suitable design tool for combline filters and manifold multiplexes for satellite and wireless communication systems. Results are presented to demonstrate the validity and accuracy of the proposed methodology.

Multivariate orthogonal polynomials, homogeneous Padé approximants and Gaussian cubature

The connection between orthogonal polynomials, Pade approximants and Gaussian quadrature is well known and will be repeated in section 1. In the past, several generalizations to the multivariate case have been suggested for all three concepts [4,6,9,...], however without reestablishing a fundamental and clear link. In sections 2 and 3 we will elaborate definitions for multivariate Pade and Pade-type approximation, multivariate polynomial orthogonality and multivariate Gaussian integration in order to bridge the gap between these concepts. We will show that the new m-point Gaussian cubature rules allow the exact integration of homogeneous polynomials of degree 2m−1, in any number of variables. A numerical application of the new integration rules can be found in sections 4 and 5.

Exploring multivariate Padé approximants for multiple hypergeometric series

We investigate the approximation of some hypergeometric functions of two variables, namely the Appell functions Fi, i = 1,...,4, by multivariate Pade approximants. Section 1 reviews the results that exist for the projection of the Fi onto ϰ=0 or y=0, namely, the Gauss function 2F1(a, b; c; z), since a great deal is known about Pade approximants for this hypergeometric series. Section 2 summarizes the definitions of both homogeneous and general multivariate Pade approximants. In section 3 we prove that the table of homogeneous multivariate Pade approximants is normal under similar conditions to those that hold in the univariate case. In contrast, in section 4, theorems are given which indicate that, already for the special case F1(a, b, b′; c; x; y) with a = b = b′ = 1 and c = 2, there is a high degree of degeneracy in the table of general multivariate Pade approximants. Section 5 presents some concluding remarks, highlighting the difference between the two types of multivariate Pade approximants in this context and discussing directions for future work.

Kronecker type theorems, normality and continuity of the multivariate Padé operator

For univariate functions the Kronecker theorem, stating the equivalence between the existence of an infinite block in the table of Pade approximants and the approximated function \(f\) being rational, is well-known. In [Lubi88] Lubinsky proved that if \(f\) is not rational, then its Pade table is normal almost everywhere: for an at most countable set of points the Taylor series expansion of \(f\) is such that it generates a non-normal Pade table. This implies that the Pade operator is an almost always continuous operator because it is continuous when computing a normal Pade approximant [Wuyt81]. In this paper we generalize the above results to the case of multivariate Pade approximation. We distinguish between two different approaches for the definition of multivariate Pade approximants: the general order one introduced in [Levi76, CuVe84] and the so-called homogeneous one discussed in [Cuyt84].

The need for knowledge and reliability in numeric computation: Case study of multivariate Pad� approximation

In this paper we review and link the numeric research projects carried out at the Department of Mathematics and Computer Science of the University of Antwerp since 1978. Results have and are being obtained in various areas. A lot of effort has been put in the theoretical investigation of the multivariate Pade approximation problem using different definitions (see Sections 3 and 7). The numerical implementation raises two delicate issues. First, there is the need to see the wood for the trees again: switching from one to many variables greatly increases the number of choices to be made on the way (see Sections 1 and 5). Second, there is the typical problem of breakdown when computing ratios of determinants: the added value of interval arithmetic combined with defect correction turns out to be significant (see Sections 2 and 4). In Section 6 these two problems are thoroughly illustrated and the interested reader is taken by the hand and guided through a typical computation session. On the way some open problems are indicated which motivate us to continue our research mainly in the area of gathering and offering more knowledge about the problem domain on one hand, and improving the arithmetic tools and numerical routines for a reliable computation of the approximants on the other hand.

Convergence in measure of multivariate Pade approximants

Convergence conclusions of Pade approximants in the univariate case can be found in various papers. However, results in the multivariate case are few. A. Cuyt seems to be the only one who discusses convergence for multivariate Pade approximants, she gives in [2] a de Montessus de Bollore type theorem. In this paper, we will discuss the zero set of a real multivariate polynomial, and present a convergence theorem in measure of multivariate Pade approximant. The proof technique used in this paper is quite different from that used in the univariate case.

Model reduction of multidimensional linear shift-invariant recursive systems using Padé techniques

This paper describes a very flexible “general order” multivariate Pade approximation technique for the model reduction of a multidimensional linear shift-invariant recursive system, i.e., a system characterized by a multivariate rational transfer function. The technique presented allows full control of the regions of support in numerator and denominator of the reduced system and also admits a nonbranched continued fraction representation for an easy realization of the model. The method presented here overcomes some of the problems of related approaches to model reduction of multidimensional linear recursive systems. Different rational approximants can be introduced to compute the reduced model, but a drawback is that these approximants are not always readily available in continued fraction form for immediate implementation of the reduced system. Also multibranched continued fractions can be used to approximate the transfer function, but it was pointed out that the regions of support of numerator and denominator blow up rapidly as one considers successive convergents. Both these problems are overcome here.

Multivariate Pad� approximation

Multivariate Pade approximation is considered in the framework of the theory developed by A. Cuyt (1982). The denominators are determined from determinants of matrices whose entries are homogeneous polynomials. The main difference to the univariate case is a typical shift of the degree of the polynomials. Singularities atx=0 are analyzed since it is the rule rather than the exception that simultaneous zeros of the numerator and the denominator cannot be removed.

Multivariate Padé approximants revisited

Several definitions of multivariate Pade approximants have been introduced during the last decade. We will here consider all types of definitions based on the choice that the coefficients in numerator and denominator of the multivariate Pade approximant are defined by means of a linear system of equations. In this case a determinant representation for the multivariate Pade approximant exists. We will show that a general recursive algorithm can be formulated to compute a multivariate Pade approximant given by any definition of this type. Here intermediate results in the recursive computation scheme will also be multivariate Pade approximants. Up to now such a recursive computation of multivariate Pade approximants only seemed possible in some special cases.