Duality in Padé-type approximation

Padé and Padé-type approximants are usually defined by replacing the function (1 − xt)⁻¹ by its Hermite (that is confluent) interpolation polynomial and then applying the functional c defined by c(xⁱ) = c_i where the c_i's are the coefficients of the series to be approximated. In this paper the functional d which, applied to (1 − xt)⁻¹, gives the same Padé or Padé-type approximant as before is studied. It can be considered as the dual of the interpolation operator applied to the functional c.     

Padé approximation for a multivariate Markov transform

Methods of Padé approximation are used to analyse a multivariate Markov transform which has been recently introduced by the authors. The first main result is a characterization of the rationality of the Markov transform via Hankel determinants. The second main result is a cubature formula for a special class of measures.     

Moment methods in two point Padé approximation

In the separable Hilbert space (H, ·, ·) the following “operator moment problem” is solved: given a complex sequence (c_k)_{k ε Z} generated by a meromorphic function f, find T ε B(H) and u₀ ε H such that T^ku₀, u₀ = c_k (k ε Z). If the sequence (c_k)_{k ε Z} is “normal,” an adapted form of Vorobyev's method of moments yields a sequence of two point Padé approximants to f. A sufficient condition for convergence of this sequence of approximants is given.     

Hermite–Padé approximation and simultaneous quadrature formulas

We study Hermite–Padé approximation of the so-called Nikishin systems of functions. In particular, the set of multi-indices for which normality is known to take place is considerably enlarged as well as the sequences of multi-indices for which convergence of the corresponding simultaneous rational approximants takes place. These results are applied to the study of the convergence properties of simultaneous quadrature rules of a given function with respect to different weights.     

Shear stress analysis in axially symmetric flow using the Padé approximation technique

This paper evaluates the shear stress on the surface of a circular cylinder in axially symmetric flow within the boundary layer. An extended solution is obtained by established techniques, e.g. asymptotic series or Pohlhausen method. A Padé approximation technique is applied to the extended solution obtained. The region of validity of the solution is extended and the results are then compared and examined against known approximate Pohlhausen and series solutions. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on the determinant formulas computation of generalized inverse matrix Padé approximation

In this paper, the computation of two special determinants which appear in the construction of a generalized inverse matrix Padé approximation of type [n/2k] (described in [Linear Algebra Appl. 322 (2001) 141]) for a given power series is investigated. Here a common computational approach of determinant can not be used. The main tool to be used to do the two special determinants is the well-known Schur complement theorem.     

Padé iteration method for regularization

In this study we present iterative regularization methods using rational approximations, in particular, Padé approximants, which work well for ill-posed problems. We prove that the (k, j)-Padé method is a convergent and order optimal iterative regularization method in using the discrepancy principle of Morozov. Furthermore, we present a hybrid Padé method, compare it with other well-known methods and found that it is faster than the Landweber method. It is worth mentioning that this study is a completion of the paper [A. Kirsche, C. Böckmann, Rational approximations for ill-conditioned equation systems, Appl. Math. Comput. 171 (2005) 385–397] where this method was treated to solve ill-conditioned equation systems.     

Convergence of Multipoint Padé-type Approximants

Let μ be a finite positive Borel measure whose support is a compact subset K of the real line and let I be the convex hull of K. Let r denote a rational function with real coefficients whose poles lie in C\I and r(∞)=0. We consider multipoint rational interpolants of the function where some poles are fixed and others are left free. We show that if the interpolation points and the fixed poles are chosen conveniently then the sequence of multipoint rational approximants converges geometrically to f in the chordal metric on compact subsets of C\I.     

Optimization of two-point Padé-type approximants

In this paper, we first give characterization theorems for the best two-point Padé-type approximants (2PTAs) in the uniform norm. Secondly, we consider sequences of 2PTAs in a domain of the complex plane from the viewpoint of the asymptotic degree of convergence, and we also give conditions for geometric convergence.     

IBLU decompositions based on Padé approximants

In this paper, we introduce a new class of frequency‐filtering IBLU decompositions that use continued‐fraction approximation for the diagonal blocks. This technique allows us to construct efficient frequency‐filtering preconditioners for discretizations of elliptic partial differential equations on domains with non‐trivial geometries. We prove theoretically for a class of model problems that the application of the proposed preconditioners leads to a convergence rate of up to 1?O(h^1/4) of the CG iteration. Copyright © 2008 John Wiley & Sons, Ltd.     

Analytic approximation of the blow‐up time for nonlinear differential equations by the ADM–Padé technique

We present a new approach to calculate analytic approximations of blow‐up solutions and their critical blow‐up times. Our approach applies the Adomian decomposition–Padé method to quickly and easily compute the critical blow‐up times, which comprises the Adomian decomposition method combined with the Padé approximants technique. We validate our new approach with a variety of numerical examples, including nonlinear ODEs, systems of nonlinear ODEs, and nonlinear PDEs. Furthermore, our new method is shown to be more convenient than prior art that relies on compound discretized algorithms. Copyright © 2013 John Wiley & Sons, Ltd.     

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 convergence of bivariate two-point Padé-type approximants

Some choices of denominators are given which ensure the geometrical convergence of certain convergence of bivariate two-point Padé-type approximants to functions being holomorphic on certain domains     

Stieltjes representation of the 3D Bruggeman effective medium and Padé approximation

The paper deals with Bruggeman effective medium approximation (EMA) which is often used to model effective complex permittivity of a two-phase composite. We derive the Stieltjes integral representation of the 3D Bruggeman effective medium and use constrained Padé approximation method introduced in [39] to numerically reconstruct the spectral density function in this representation from the effective complex permittivity known in a range of frequencies. The problem of reconstruction of the Stieltjes integral representation arises in inverse homogenization problem where information about the spectral function recovered from the effective properties of the composite, is used to characterize its geometric structure. We present two different proofs of the Stieltjes analytical representation for the effective complex permittivity in the 3D Bruggeman effective medium model: one proof is based on direct calculation, the other one is the derivation of the representation using Stieltjes inversion formula. We show that the continuous spectral density in the integral representation for the Bruggeman EMA model can be efficiently approximated by a rational function. A rational approximation of the spectral density is obtained from the solution of a constrained minimization problem followed by the partial fractions decomposition. We show results of numerical rational approximation of Bruggeman continuous spectral density and use these results for estimation of fractions of components in a composite from simulated effective permittivity of the medium. The volume fractions of the constituents in the composite calculated from the recovered spectral function show good agreement between theoretical and predicted values.     

Addison-type series representation for the Stieltjes constants

The Stieltjes constants γk(a) appear in the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(s,a) about its only pole at s=1. We generalize a technique of Addison for the Euler constant γ=γ₀(1) to show its application to finding series representations for these constants. Other generalizations of representations of γ are given.     

Quadratic Hermite–Padé polynomials associated with the exponential function

The asymptotic behavior of quadratic Hermite–Padé polynomials associated with the exponential function is studied for n→∞. These polynomials are defined by the relation

(*)

p_n(z)+q_n(z)e^z+r_n(z)e^2z=O(z³ⁿ⁺²) as z→0,

where O(·) denotes Landau's symbol. In the investigation analytic expressions are proved for the asymptotics of the polynomials, for the asymptotics of the remainder term in (*), and also for the arcs on which the zeros of the polynomials and of the remainder term cluster if the independent variable z is rescaled in an appropriate way. The asymptotic expressions are defined with the help of an algebraic function of third degree and its associated Riemann surface. Among other possible applications, the results form the basis for the investigation of the convergence of quadratic Hermite–Padé approximants, which will be done in a follow-up paper.     

B-splines and Hermite–Padé approximants to the exponential function

This paper is the continuation of a work initiated in [P. Sablonnière, An algorithm for the computation of Hermite–Padé approximations to the exponential function: divided differences and Hermite–Padé forms. Numer. Algorithms 33 (2003) 443–452] about the computation of Hermite–Padé forms (HPF) and associated Hermite–Padé approximants (HPA) to the exponential function. We present an alternative algorithm for their computation, based on the representation of HPF in terms of integral remainders with B-splines as Peano kernels. Using the good properties of discrete B-splines, this algorithm gives rise to a great variety of representations of HPF of higher orders in terms of HPF of lower orders, and in particular of classical Padé forms. We give some examples illustrating this algorithm, in particular, another way of constructing quadratic HPF already described by different authors. Finally, we briefly study a family of cubic HPF.