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 of Stieltjes series

Using Nuttall's compact formula for the [n, n − 1] Pad'e approximant, the authors show that there is a natural connection between the Padé approximants of a series of Stieltjes and orthogonal polynomials. In particular, we obtain the precise error formulas. The [n, n − 1] Padé approximant in this case is just a Gaussian quadrature of the Stieltjes integral. Hence, analysis of the error is now possible and under very mild conditions it is shown that the [n, n + j], j − 1, Padé approximants converge to the Stieltjes integral.     

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.     

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.     

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.     

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 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.     

Boundary layer equations in the problem of axially symmetric jet flow

The problem of stationary axially symmetric jet flour of a viscous incompressible fluid from a round tube into a free space is considered. The mathematical model of this flow in the Mises variables and for large Reynolds numbers is reduced to the boundary-value problem for a degenerate integro-differential equation of the second order. The existence and uniqueness of bounded solutions to this problem are proved, and stabilization as a time-like variable increases is shown. Bibliography: 7 titles. To Vsevolod Alekseevich Solonnikov on the occasion of his 75th birthday Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 48-63.     

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.     

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.     

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     

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.     

Global existence of shock front solution to axially symmetric piston problem in compressible flow

In this paper we study the global existence of BV solution to two-dimensional piston problem in fluid dynamics. Different from previous results on related problems we remove the restriction on the strength of the leading shock and require the velocity of the piston is rather fast or the density is quite small instead. The main tool in our proof is Glimm Scheme with some improvement. To define the Glimm functional we derive more precise estimates for the interaction of elementary waves, particularly in the region near the leading shock. The paper is partially supported by National Natural Science Foundation of China 10531020, the National Basic Research Program of China 2006CB805902 and the Doctorial Foundation of National Educational Ministry 20050246001.     

Global existence of shock front solution to axially symmetric piston problem in compressible flow

In this paper we study the global existence of BV solution to two-dimensional piston problem in fluid dynamics. Different from previous results on related problems we remove the restriction on the strength of the leading shock and require the velocity of the piston is rather fast or the density is quite small instead. The main tool in our proof is Glimm Scheme with some improvement. To define the Glimm functional we derive more precise estimates for the interaction of elementary waves, particularly in the region near the leading shock.     

Theoretical and numerical considerations about Padé approximants for the matrix logarithm

We show that for a vast class of matrix Lie groups, which includes the orthogonal and the symplectic, diagonal Padé approximants of log((1+x)/(1−x)) are structure preserving. The conditioning of these approximants is analyzed. We also present a new algorithm for the Briggs–Padé method, based on a strategy for reducing the number of square roots in the inverse scaling and squaring procedure.     

Discrete non-local boundary conditions for split-step Padé approximations of the one-way Helmholtz equation

This paper deals with the efficient numerical solution of the two-dimensional one-way Helmholtz equation posed on an unbounded domain. In this case, one has to introduce artificial boundary conditions to confine the computational domain. The main topic of this work is the construction of the so-called discrete transparent boundary conditions for state-of-the-art parabolic equation methods, namely a split-step discretization of the high-order parabolic approximation and the split-step Padé algorithm of Collins. Finally, several numerical examples arising in optics and underwater acoustics illustrate the efficiency and accuracy of our approach.     

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.