Multivariate Frobenius-Padé approximants: properties and algorithms
type: Item
instance of: scholarly article
title: Multivariate Frobenius-Padé approximants: properties and algorithms
zbMATH Open document ID: 1127.41005
author: Ana C. Matos
published in: Journal of Computational and Applied Mathematics
publication date: 2007-04-13T00:00:00Z
review text: The definition of the two variable Frobenius-Padé approximant using formal power series expansions in orthogonal polynomials studied in this paper was introduced in [\textit{A. C. Matos} and \textit{J. Van Iseghem}, J. Comput. Appl. Math. 176, No. 2, 231--258 (2005; Zbl 1067.65018)]. Consider the two variable function \[ f(x,y)=\sum_{i=0}^{\infty}\,\sum_{j=0}^{\infty}\,c_{ij}P_i(x)P_j(y), \] with \(\{P_i\}\) a system of orthogonal polynomials on \([a,b]\) with respect to a weight \(w\). Given two non-negative integers \(m,n\in \mathbb N\), find the polynomials \(P(x,y),\,Q(x,y)\) \[ P(x,y)=\sum_{(i,j)\in N}\,a_{ij}P_i(x)P_j(y), \;Q(x,y)=\sum_{(i,j)\in D}\,b_{ij}P_i(x)P_j(y), \] satisfying \[ f(x,y)Q(x,y)-P(x,y)=\sum_{(i,j)\in\mathbb N^2\setminus E}\,d_{ij}P_i(x)P_j(y),\tag \(*\) \] where \(D\subset \mathbb N^2\) has \(m\) elements, \(E\subset \mathbb N^2\) is the set of \(n+m-1\) indices of null terms in \((\ast)\), \(N\subset E\) is a set of \(n\) elements, \(H=E\setminus N\) has \(m-1\) elements. Just as in the case for the one variable Hermite-Padé approximant, the solution is found in two steps -- solve the homogeneous linear system for the \(b_{ij}\), -- calculate with these the numerator coefficients \(a_{ij}\). Then, with the aid of the ordinary three term recurrence relation for the orthogonal polynomials, the author introduces algorithms to calculate the values of a sequence of approximants at a given point. Moreover, for different choices of \(D,\,E\) and \(N\), the author studies -- using the Chebyshev polynomials \(T_n\) -- the problem of solving the homogeneous system for the denominator coefficients using the Hessenberg displacement rank structure of the associated block-matrix. Finally, the so called `mixed' Frobenius-Padé approximant approach is considered. Let \[ f(x,y)=\sum_{j=0}^{\infty}\,\sum_{i=0}^{\infty}\,f_{ij}P_i(y)P_j(x)= \sum_{j=0}^{\infty}\,f_j(y)P_j(x), \] then 1. Consider \[ f_y(x)=\sum_{j=0}^{\infty}\,f_j(y)P_j(x) \] with \(y\) a param…
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 65F05
Mathematics Subject Classification ID: 65F30
zbMATH DE Number: 5142317
zbMATH Keywords: multivariate Padé approximants
zbMATH Keywords: Chebyshev series
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: orthogonal expansions
zbMATH Keywords: rational approximation
zbMATH Keywords: displacement rank structure
reviewed by: Marcel G. de Bruin
MaRDI profile type: Publication
full work available at URL: j.cam.2006.03.007
OpenAlex ID: W2128238196
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DOI: 10.1016/J.CAM.2006.03.007
description: scientific article; zbMATH DE number 5142317