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

Summary. For univariate functions the Kronecker theorem, stating the equivalence between the existence of an infinite block in the table of Padé approximants and the approximated function being rational, is well-known. In [Lubi88] Lubinsky proved that if is not rational, then its Padé table is normal almost everywhere: for an at most countable set of points the Taylor series expansion of is such that it generates a non-normal Padé table. This implies that the Padé operator is an almost always continuous operator because it is continuous when computing a normal Padé approximant [Wuyt81]. In this paper we generalize the above results to the case of multivariate Padé approximation. We distinguish between two different approaches for the definition of multivariate Padé approximants: the general order one introduced in [Levi76, CuVe84] and the so-called homogeneous one discussed in [Cuyt84]. Received December 19, 1994     

A Padé family of iterations for the matrix sign function and related problems

In this paper we consider the Pad'e family of iterations for computing the matrix sign function and the Padé family of iterations for computing the matrix p‐sector function. We prove that all the iterations of the Padé family for the matrix sign function have a common convergence region. It completes a similar result of Kenney and Laub for half of the Padé family. We show that the iterations of the Padé family for the matrix p‐sector function are well defined in an analogous common region, depending on p. For this purpose we proved that the Padé approximants to the function (1?z)^?σ, 0<σ<1, are a quotient of hypergeometric functions whose poles we have localized. Furthermore we proved that the coefficients of the power expansion of a certain analytic function form a positive sequence and in a special case this sequence has the log‐concavity property. Copyright © 2011 John Wiley & Sons, Ltd.     

On the computation of non-normal Padé approximants

Several basic techniques are described for computing Padé approximants which are not normal. These techniques are modifications of some known methods for the computation of normal Padé approximants.     

Analytical solution of three-dimensional Navier–Stokes equations for the flow near an infinite rotating disk

In this letter, we will consider variational iteration method (VIM) and Padé approximant, for finding analytical solutions of three-dimensional viscous flow near an infinite rotating disk. The solutions is compared with the numerical (fourth-order Runge–Kutta) solution. The results illustrate that VIM–Padé is an appropriate method in solving the systems of nonlinear equations. It is predicted that VIM–Padé can have wide application in engineering problems (especially for boundary-layer and natural convection problems).     

Introduction to the improved functional epsilon algorithm

This paper introduces the improved functional epsilon algorithm. We have defined this new method in principle of the modified Aitken Δ² algorithm. Moreover, we have found that the improved functional epsilon algorithm has remarkable precision of the approximation of the exact solution and there exists a relationship with the integral Padé approximant. The use of the improved functional epsilon algorithm for accelerating the convergence of sequence of functions is demonstrated. The relationship of the improved functional epsilon algorithm with the integral Padé approximant is also demonstrated. Moreover, we illustrate the similarity between the integral Padé approximant and the modified Aitken Δ² algorithm; thus we have shown that the integral Padé approximant is a natural generalisation of modified Aitken Δ² algorithm.     

Rectangular matrix Padé approximants and square matrix orthogonal polynomials

In this paper we study Padé-type and Padé approximants for rectangular matrix formal power series, as well as the formal orthogonal polynomials which are a consequence of the definition of these matrix Padé approximants. Recurrence relations are given along a diagonal or two adjacent diagonals of the table of orthogonal polynomials and their adjacent ones. A matrix qd-algorithm is deduced from these relations. Recurrence relations are also proved for the associated polynomials. Finally a short presentation of right matrix Padé approximants gives a link between the degrees of orthogonal polynomials in right and left matrix Padé approximants in order to show that the latter are identical. This revised version was published online in June 2006 with corrections to the Cover Date.     

A new look at the Padé table and the different methods for computing its elements

In this paper a survey is given of several algorithms for the computation of the Padé table of a formal power series. Those algorithms are studied which are based on certain relationships between adjacent elements in the Padé table. A new proof for the algorithms of Baker, Longman and for Gragg's variant of the qd-algorithm is given. A variant of Watson's algorithm is derived. The techniques used in this survey give some new ideas concerning the structure of the Padé table and the different ways to compute the elements of the table.     

Division algorithms for continued fractions and the Padé table

It is known [26] that the Viskovatoff algorithm can be generalized to cover the computation of continued fractions whose successive convergents form the Padé approximants of a descending staircase or diagonal, even in the case of a non-normal Padé table. It is the intention of the author to generalize this idea to other paths of the Padé table and in this way link together some algorithms scattered in literature.     

On parallel algorithms for semidiscretized parabolic partial differential equations based on subdiagonal Padé approximations

It is well known that methods for solving semidiscretized parabolic partial differential equations based on the second-order diagonal [1/1] Padé approximation (the Crank–Nicolson or trapezoidal method) can produce poor numerical results when a time discretization is imposed with steps that are “too large” relative to the spatial discretization. A monotonicity property is established for all diagonal Padé approximants from which it is shown that corresponding higher-order methods suffer a similar time step restriction as the [1/1] Padé. Next, various high-order methods based on subdiagonal Padé approximations are presented which, through a partial fraction expansion, are no more complicated to implement than the first-order implicit Euler method based on the [0/1] Padé approximation; moreover, the resulting algorithms are free of a time step restriction intrinsic to those based on diagonal Padé approximations. Numerical results confirm this when various test problems from the literature are implemented on a Multiple Instruction Multiple Data (MIMD) machine such as an Alliant FX/8. © 1993 John Wiley & Sons, Inc.     

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 Padé 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 Padé approximants, to the application of multipoint Padé 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 Padé-Hermite approximation for systems of Markov type functions.     

An algorithm for the quadratic approximation

The quadratic approximation is a three dimensional analogue of the two dimensional Padé approximation. A determinantal expression for the polynomial coefficients of the quadratic approximation is given. A recursive algorithm for the construction of these coefficients is derived. The algorithm constructs a table of quadratic approximations analogous to the Padé table of rational approximations.     

A look-ahead method for computing vector Padé-Hermite approximants

In this paper, we present an algorithm to compute vector Padé-Hermite approximants along a sequence of perfect points in the vector Padé-Hermite table. We show the connection to matrix Padé approximants. The algorithm is used to compute the solution of a block Hankel system of linear equations.     

Error analysis of Padé iterations for computing matrix invariant subspaces

The method of Padé matrix iteration is commonly used for computing matrix sign function and invariant subspaces of a real or complex matrix. In this paper, a detailed rounding error analysis is given for two classical schemes of the Pad’e matrix iteration, using basic matrix floating point arithmetics. Error estimations of computing invariant subspaces by the Padé sign iteration are also provided. Numerical experiments are given to show the numerical behaviors of the Padé iterations and the corresponding subspace computation.      

Perron fractions : an algorithm for computing the Padé table

A set of rhombus rules is given for generating the coefficients of the Perron fractions whose convergents form sawtooth sequences in the Padé table of a normal series. The coefficients of the Perron fractions for the reciprocal series are also derived, without finding the reciprocal series, and hence the Padé table for the series is obtained.     

Multivariate Padé-Bergman approximants

Summary. We define the multivariate Padé-Bergman approximants (also called Padé approximants) and prove a natural generalization of de Montessus de Ballore theorem. Numerous definitions of multivariate Padé approximants have already been introduced. Unfortunately, they all failed to generalize de Montessus de Ballore theorem: either spurious singularities appeared (like the homogeneous Padé [3,4], or no general convergence can be obtained due to the lack of consistency (like the equation lattice Padé 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 Padé 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). Received December 4, 2001 / Revised version received January 2, 2002 / Published online April 17, 2002     

A Theorem on the Convergence of Padé Approximants

We prove a theorem which gives necessary and sufficient conditions on the distribution of poles and zeros of the Padé approximants for point-by-point convergence. The special case of convergence to a function meromorphic in a disk by a sequence of Padé approximants free of extraneous poles and zeros is proven.     

On the phenomenon of spurious interpolation of elliptic functions by diagonal Padé approximants

We study diagonal Padé approximants for elliptic functions. The presence of spurious poles in the approximants not corresponding to the singularities of the original function prevents uniform convergence of the approximants in the Stahl domain. This phenomenon turns out to be closely related to the existence in the Stahl domain of points of spurious interpolation at which the Padé approximants interpolate the other branch of the elliptic function. We also investigate the behavior of diagonal Padé approximants in a neighborhood of points of spurious interpolation.     

Orthogonal rational functions and quadrature on the real half line

In this paper we generalize the notion of orthogonal Laurent polynomials to orthogonal rational functions. Orthogonality is considered with respect to a measure on the positive real line. From this, Gauss-type quadrature formulas are derived and multipoint Padé approximants for the Stieltjes transform of the measure. Convergence of both the quadrature formula and the multipoint Padé approximants is discussed.     

Multipoint matrix Padé approximant bounds on effective anisotropic transport coefficients of two-phase media

It is well known that a tensor Stieltjes function f represents an effective transport coefficient q of an inhomogeneous medium consisting of two isotropic components. In this paper, we investigate multipoint matrix Padé approximants to matrix expansions of f. We prove that matrix Padé ones to f estimate f from the top and below. Consequently the Padé approximants to q form upper and lower bounds on q. The inequalities for matrix Padé bounds on f and q are established. They reduce to the inequalities for scalar Padé ones Tokarzewski (ZAMP 61:773–780, 2010). As an illustrative example, matrix Padé estimates of an effective conductivity of a specially laminated two-phase medium are computed.     

Continued fractions as the best tool to estimate the padé approximant errors

For a Stieltjes functions the problems of the Padé polinomial constructions and the analysis of the Padé approximant errors by continued fractions are investigated. Reprinted fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 84–88.