Newton-Pade Approximation Research Articles

On Baker’s Patchwork Conjecture for Diagonal Padé Approximants

We prove that for entire functions f of finite order, there is a sequence of integers $$\mathcal {S}$$ such that as $$n\rightarrow \infty $$ through S, $$\begin{aligned} \min \left\{ \left| f-\left[ n/n\right] \right| \left( z\right) ,\left| f-\left[ n-1/n-1\right] \right| \left( z\right) \right\} ^{1/n}\rightarrow 0 \end{aligned}$$ uniformly for z in compact subsets of the plane. More generally this holds for sequences of Newton–Pade approximants and for functions whose errors of approximation by rational functions of type $$\left( n,n\right) $$ decay sufficiently fast. This establishes George Baker’s patchwork conjecture for large classes of entire functions.

Formal Orthogonal Polynomials and Newton–Padé Approximants

We proved in [8] that the denominators of Newton–Pade approximants for a formal Newton series are formal orthogonal with respect to linear functionals. The same functional is used along an antidiagonal of the Newton–Pade denominator table. The two linear functionals, corresponding to two adjacent antidiagonals, are linked with a very simple relation. Recurrence relations between denominators are given along an antidiagonal or two adjacent antidiagonals in the normal and non-normal case. The same recurrence relations are also satisfied by the Newton–Pade numerators, which implies another formal orthogonality.

Multivariate rational approximants for multiclass closed queuing networks

Closed Markovian networks of queues with multiclass customers and having a product form equilibrium state probability distribution are useful in the performance evaluation and design of computer and telecommunication systems. Therefore, the efficient computation of the normalizing function, the key element of the solution in product form, has attracted considerable effort. We consider a network that consists of one infinite-server (IS) station and one processor-sharing (PS) or FCFS single-server station. We use multivariate Newton-Pade approximants computed from data for small numbers of customers in each class, to estimate the normalizing function for a larger population in the network. The effectiveness and tremendous gain in computing time of this procedure are illustrated through various numerical experiments.

Singular rules for a multivariate quotient-difference algorithm

Using the multivariateqdg-algorithm developed in [5], it is possible to compute the partial numerators and denominators of a continued fraction representation associated with a descending staircase in a table of multivariate rational interpolants, more precisely, multivariate Newton-Pade approximants. The algorithm is only applicable if every three successive elements on the staircase are different. If a singularity occurs in the defining system of equations for the multivariate rational interpolant then singular rules must be developed. For the univariate Newton-Pade approximant this was done in [3] by Claessens and Wuytack. The idea to perturb the initial staircase and walk around the block structure in the table in order to avoid the singularity, is explored now in a multivariate setting. Another approach would be to use block bordering methods in combination with reverse bordering [4] in order to solve the rank deficient linear system of interpolation conditions (Newton-Pade approximation conditions) recursively. Since this last technique can also be used for scattered multivariate data exhibiting near-singularity, we describe the second approach in a separate paper [7]. Here we deal only with partially grid-structured data (satisfying the so-called rectangle rule or inclusion property).

The multipoint Pad� table and general recurrences for rational interpolation

We first review briefly the Newton-Pade approximation problem and the analogous problem with additional interpolation conditions at infinity, which we call multipoint Pade approximation problem. General recurrence formulas for the Newton-Pade table combine either two pairs of Newton-Pade forms or one such pair and a pair of multipoint Pade forms. We show that, likewise, certain general recurrences for the multipoint Pade table compose two pairs of multipoint Pade forms to get a new pair of multipoint Pade forms. We also discuss the possibility of superfast, i.e.,O(n log2n) algorithms for certain rational interpolation problems.

The structure of the singular solution table of the M-Padé approximation problem

A general interpolation problem is defined which contains as special cases the M-Padé approximation problem as well as classical interpolation problems, e.g., Newton-Padé approximation. To study the structure of the solution table a new concept—the minimal solution—is introduced; statements on the local and global structure of the solution table are obtained. As a special case the well-known block structure of the Padé table is derived. Finally, a characterization of the solution set is given.

General order Newton-Pad� approximants for multivariate functions

Pade approximants are a frequently used tool for the solution of mathematical problems. One of the main drawbacks of their use for multivariate functions is the calculation of the derivatives off(x 1, ...,x p ). Therefore multivariate Newton-Pade approximants are introduced; their computation will only use the value off at some points. In Sect. 1 we shall repeat the univariate Newton-Pade approximation problem which is a rational Hermite interpolation problem. In Sect. 2 we sketch some problems that can arise when dealing with multivariate interpolation. In Sect. 3 we define multivariate divided differences and prove some lemmas that will be useful tools for the introduction of multivariate Newton-Pade approximants in Sect. 4. A numerical example is given in Sect. 5, together with the proof that forp=1 the classical Newton-Pade approximants for a univariate function are obtained.

Bigradients, Hankel determinants and the Newton-Padé table

In [9], Warner introduced generalized bigradients in the study of the Newton-Pade table. In this paper we introduce generalized Hankel determinants and derive, using the framework of Newton-Pade approximation, a relationship between these generalized Hankel determinants and generalized bigradients. This generalizes a determinantal identity obtained by Householder and Stewart [7, p. 136].