Some remarks on Fitzpatrick and Flynn's Gr¨ obner

Applied Mathematics and Computation, 2018

We modify Cuyt and Verdonk's approach to multivariate Newton-Padé approximations. Explicit formulas are given for coefficients which can be computed once for given system of nodes, and the linear system of equations to find multivariate Newton-Padé approximants can be written using simple formulas with those coefficients and the Newton series of interpolated function.

BIT, 1986

It is proved that higher order interpolatory Pad~-type approximants in two variables do not exist. Let f be a formal double power series i=O j=O and let c be the linear functional such that c(xiy j) = cij, i, j = O, 1 .... The Pad6-type approximants of f are defined as follows [1, p. 190]. Let V be an arbitrary polynomial kl k2 V(x, y) = ~, Z biix'Yi i=O j=O and let W be

Journal of Computational and Applied Mathematics, 1990

We describe the minimal vector Padt approximation problem, which consists in finding PadC approximants with a common denominator for a number of series. These approximants are minimal in the sense that for a given order of approximation and a given discrepancy in numerator and denominator degrees, the degree of the rational approximant is minimal. Properties and solution methods are derived from an associated minimal partial realization problem.

Some proposals are made to give a general definition of matrix Padé approximants. Depending on the normalization of the denominator we define type I (constant term is the unit matrix) or type II ( by conditions on the leading coefficient) approximants. Existence and uniqueness are considered, determinant expressions are given and relation among type I/II and left/right approximants are considered. Also some ideas about the computation are included.

While the concept of Pad e approximant is essentially several centuries old, its multivariate version dates only from the early seventies. In the last century many univariate convergence results were proven, describing the approxima- tion power for several function classes. It is not our aim to give a general review of the univariate case, but to discuss only these theorems that have a multivariate counterpart. The rst section summarizes the theorems under discussion, in a univariate framework. The second and third section discuss the multivariate versions of these theorems, for dierent approaches to the multivariate Pad e approximation problem. 1 Convergence of univariate Pad´ e approximants. Given a function f(z), through its series expansion at a certain point in the complex plane, the Pad e approximant (n=m)f of degree n in the numerator and m in the

Revue européenne des éléments finis, 2004

Journal of advances in mathematics and computer science, 2022

We review Taylor approximation (TA), Padé approximation (PA), Restrictive Taylor approximation (RTA) and Restrictive Padé approximation (RPA). After comparing these four approximation methods with two other modified approximation methods: Modified Restrictive Taylor approximation (MRTA) and Modified Restrictive Padé approximation (MRPA), we give test examples to illustrate how the modified approximations could be used. The mathematical principles behind all these approximations could be applied for the development of new computing methods.

2011

In this paper, Numerical solution of Partial Diferential-Algebraic Equations(PDAEs) is considered by Multivariate Pade Approximations. We applied these method to one example. First Partial Diferential-Algebraic Equation(PDAE) has been converted to power series by two-dimensional differential transformation,Then the numerical solution of equation was put into Multivariate Pade series form. Thus we obtained numerical solution of Partial Diferential-Algebraic Equation(PDAE).

Journal of Computational and Applied Mathematics, 1980

Recently McCabe and Murphy have considered the two-point Pad6 approximants to a function for which (formal) power series expansions at the origin and at infinity are given. In this paper these approximations are slightly modified and determinant representations for them are given. The existence of various three-term recursion relations for the numerators and denominators of these approximants is shown. Based on these, a new continued fraction representation for these approximants is obtained and also an efficient recursive method is proposed for the determination of the coefficients of all the approximants that obtain from a given number of terms of the power series.

2011

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. Acknowledgement: The authors thank the Alexander von Humboldt Foundation for support in the framework of the Institutes partnership project and the Feodor-Lynen programme. The second author is supported in part by Grant MTM2006-13000-C03-03 of the D.G.I. of Spain.

Journal of Computational and Applied Mathematics, 1990

Pad& and Pad&type approximants are usually defined by replacing the function (1-xt)-' by its Hermite (that is confluent) interpolation polynomial and then applying the functional c defined by c(n') = c, where the c;'s are the coefficients of the series to be approximated. In this paper the functional d which, applied to (1-xt)-', gives the same Pad6 or Pad&type approximant as before is studied. It can be considered as the dual of the interpolation operator applied to the functional c.

Journal of Computational and Applied Mathematics, 2013

This introductory paper describes the main topics of this special issue, dedicated to Leonardo Traversoni, known at international level as the promoter of the conference series "Multivariate Approximation: Theory and Applications", to celebrate his 60th birthday.

Numerische Mathematik, 2002

We define the multivariate Padé-Bergman approximants (also called A 2 ρ 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 A 2 ρ 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).

Journal of Nonlinear Mathematical Physics, 2003

In this paper, we compare the degrees and the orders of approximation of vector and matrix Padé approximants for series with matrix coefficients. It is shown that, in this respect, vector Padé approximants have better properties. Then, matrix-vector Padé approximants are defined and constructed. Finally, matrix Padé approximants are related to the method of moments.