An algorithm for computing a Padé approximant with minimal degree denominator

In this paper, a new definition of a reduced Padé approximant and an algorithm for its computation are proposed. Our approach is based on the investigation of the kernel structure of the Toeplitz matrix. It is shown that the reduced Padé approximant always has nice properties which the classical Padé approximant possesses only in the normal case. The new algorithm allows us to avoid the appearance of Froissart doublets induced by computer roundoff in the non-normal Padé table.     

Padé approximation and Apostol-Bernoulli and Apostol-Euler polynomials

Using the Padé approximation of the exponential function, we obtain recurrence relations between Apostol-Bernoulli and between Apostol-Euler polynomials. As applications, we derive some new lacunary recurrence relations for Bernoulli and Euler polynomials with gap of length 4 and lacunary relations for Bernoulli and Euler numbers with gap of length 6.     

Computing the determinants of matrix Padé approximation

This paper is devoted to a computational problem of two special determinants which appear in the construction of generalized inverse matrix Padé approximants of type [n/2k] for the given power series with matrix coefficients. The main tools to be used are well-known Schur complement theorem and Arnoldi process for skew-symmetric systems.     

Multivariate Frobenius–Padé approximants: Properties and algorithms

The aim of this paper is to construct rational approximants for multivariate functions given by their expansion in an orthogonal polynomial system. This will be done by generalizing the concept of multivariate Padé approximation. After defining the multivariate Frobenius–Padé approximants, we will be interested in the two following problems: the first one is to develop recursive algorithms for the computation of the value of a sequence of approximants at a given point. The second one is to compute the coefficients of the numerator and denominator of the approximants by solving a linear system. For some particular cases we will obtain a displacement rank structure for the matrix of the system we have to solve. The case of a Tchebyshev expansion is considered in more detail.     

A note on the initial identification of scalar component models

This paper solves an open theoretical question in the identification stage of Scalar Component Models, posed initially by Tiao and Tsay and noted by several researchers, specifically as it refers to the choice of a certain parameter present in the process and which they denote h. The theoretical concept of sure overall orders, instead of the so-called overall orders, is useful in addressing this issue. Using simple examples, we justify the need for our theoretical results. Moreover, we use a Ranks Table and its properties to complement the SCM identification stage with interesting theoretical information without adding significant calculations to the procedure initially proposed by Tiao and Tsay.     

Function-valued Padé-type approximant via E-algorithm and its applications in solving integral equations

The function-valued Padé-type approximant (FPTA) was defined in the inner product space [8]. In this work, we choose the coefficients in the Neumann power series to make the inner product with both sides a function-valued system of equations to yield a scalar system. Then we express an FPTA in the determinant form. To avoid the direct computation of the determinants, we present the E-algorithm for FPTA based on the vector-valued E-algorithm given by Brezinski [4]. The method of FPTA via E-algorithm (FPTAVEA) not only includes all previous methods but overcomes their essential difficulties. The numerical experiment for a typical integral equation [1] illustrates that the method of FPTAVEA is simpler and more effective for obtaining the characteristic values and the characteristic functions than all previous methods. In addition, this method is also applicable to other Fredholm integral equations of the second kind without explicit characteristic values and characteristic functions. A corresponding example [12] is given and the numerical result is the same as that in [12].     

A two-dimensional matrix Padé-type approximation in the inner product space

By introducing a bivariate matrix-valued linear functional on the scalar polynomial space, a general two-dimensional (2-D) matrix Padé-type approximant (BMPTA) in the inner product space is defined in this paper. The coefficients of its denominator polynomials are determined by taking the direct inner product of matrices. The remainder formula is developed and an algorithm for the numerator polynomials is presented when the generating polynomials are given in advance. By means of the Hankel-like coefficient matrix, a determinantal expression of BMPTA is presented. Moreover, to avoid the computation of the determinants, two efficient recursive algorithms are proposed. At the end the method of BMPTA is applied to partial realization problems of 2-D linear systems.     

Convergence Rates of Padé and Padé-Type Approximants

A comparison is made between Padé and Padé-type approximants. LetQ_nbe thenth orthonormal polynomial with respect to a positive measureμwith compact support inC. We show that for functions of the form[formula]wherewis an analytic function on the support ofμ, Padé-type approximants with denominatorQ_ngive a successful and, in general, better approximation procedure than Padé approximation.     

Function-valued Padé-type approximant via the formal orthogonal polynomials and its applications in solving integral equations

A kind of function-valued Padé-type approximant via the formal orthogonal polynomials (FPTAVOP) is introduced on the polynomial space and an algorithm is sketched by means of the formal orthogonal polynomials. This method can be applied to approximate characteristic values and the corresponding characteristic function of Fredholm integral equation of the second kind. Moreover, theoretical analyses show that FPTAVOP method is the most effective one for accelerating the convergence of a sequence of functions. In addition, a typical numerical example is presented to illustrate when the estimates of characteristic value and characteristic function by using this new method are more accurate than other methods.     

Quasi-orthogonality of the best approximant sets

A concept of orthogonality on the normed linear space was introduced by Birkhoff. We shall define the quasi-orthogonal sets in best approximant sets and also some results on best approximation will be obtained.     

Solving nonlinear partial differential equations using the modified variational iteration Padé technique

In this paper, the modified variational iteration method (MVIM) is reintroduced with the enhancement of Padé approximants to lengthen the interval of convergence of VIM or MVIM when used alone in solving nonlinear problems. KdV, mKdV, Burger's and Lax's equations are used as examples to illustrate the effectiveness and convenience of the proposed technique.     

On Padé-type model order reduction of J-Hermitian linear dynamical systems

A simple, yet powerful approach to model order reduction of large-scale linear dynamical systems is to employ projection onto block Krylov subspaces. The transfer functions of the resulting reduced-order models of such projection methods can be characterized as Padé-type approximants of the transfer function of the original large-scale system. If the original system exhibits certain symmetries, then the reduced-order models are considerably more accurate than the theory for general systems predicts. In this paper, the framework of J-Hermitian linear dynamical systems is used to establish a general result about this higher accuracy. In particular, it is shown that in the case of J-Hermitian linear dynamical systems, the reduced-order transfer functions match twice as many Taylor coefficients of the original transfer function as in the general case. An application to the SPRIM algorithm for order reduction of general RCL electrical networks is discussed.     

Generic approximation of functions by their Padé approximants

In the present article we obtain generic approximations, under sharp conditions, of holomorphic functions on arbitrary open sets by sequences of their Padé approximants. Similar results hold for functions smooth on the boundary of their domain of definition. In addition, the approximation is valid simultaneously with respect to all centers of expansion.     

A class of multi‐step difference schemes by using Padé approximant

In this paper, we present a method for the construction of a class of multi‐step finite differences schemes for solving arbitrary order linear two‐point boundary value problems. The construction technique is based on Padé approximant. It is easy to derive multi‐step difference schemes, and it includes many existing schemes as its special cases. Numerical experiments show that the proposed schemes are flexible and convergent. Copyright © 2013 John Wiley & Sons, Ltd.     

A duality-like transformation in WZNW models inspired by dual riemannian globally symmetric spaces

We investigate transformations on the group manifold element and gauged fields on two different kinds of gauged WZNW models and thus obtain a duality-like transformation between chiral- and vector-gauged WZNW models with null gauged subgroups that exactly converts the chiral-gauged WZNW action to vector-gauged WZNW action and vice versa. These duality-like transformations correspond to the duality in Riemannian globally symmetric spaces.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 3, pp. 413–421, June, 1995.     

Strict positivstellensätze for matrix polynomials with scalar constraints

We extend Krivine’s strict positivstellensätz for usual (real multivariate) polynomials to symmetric matrix polynomials with scalar constraints. The proof is an elementary computation with Schur complements. Analogous extensions of Schmüdgen’s and Putinar’s strict positivstellensätz were recently proved by Hol and Scherer using methods from optimization theory.     

Cauchy–Schwarz and Kantorovich type inequalities for scalar and matrix moment sequences

First we present various scalar inequalities that extends the classical Cauchy–Schwarz and Kantorovich inequalities. Some of these extensions are based on the moment problem and the Hölder and Minkowski inequalities. These results are then extended to the matrix case. Many well-known inequalities are recovered ans new ones are obtained.     

Stancu‐type generalization of Dunkl analogue of Szász–Kantorovich operators

The purpose of the paper is to introduce Stancu‐type linear positive operators generated by Dunkl generalization of exponential function. We present approximation properties with the help of well‐known Korovkin‐type theorem and weighted Korovkin‐type theorem and also acquire the rate of convergence in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre's K‐functional, and second‐order modulus of continuity by Dunkl analogue of Szász operators. Copyright © 2015 John Wiley & Sons, Ltd.     

Padé-Type Approximants of Markov and Meromorphic Functions

We study the question of convergence of Padé and Padé-type approximants to functions meromorphic in a domain. As an example we investigate in detail the case of functions of the formwhereμ(z) is a Markov function.     

On moments based Padé approximations of ruin probabilities

In this paper, we investigate the quality of the moments based Padé approximation of ultimate ruin probabilities by exponential mixtures. We present several numerical examples illustrating the quick convergence of the method in the case of Gamma processes. While this is not surprising in the completely monotone case (which holds when the shape parameter is less than 1), it is more so in the opposite case, for which we improve even further the performance by a fix-up which may be of special importance due to its potential use in the four moments Gamma approximation.We also review the connection of the exponential mixtures approximation to Padé approximation, orthogonal polynomials, and Gaussian quadrature. These connections may turn out useful for providing rates of convergence.