On interpolation by rational functions with prescribed poles with applications to multivariate interpolation

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Autoren

  • G. Mühlbach

Organisationseinheiten

Forschungs-netzwerk anzeigen

Details

OriginalspracheEnglisch
Seiten (von - bis)203-216
Seitenumfang14
FachzeitschriftJournal of Computational and Applied Mathematics
Jahrgang32
Ausgabenummer1-2
PublikationsstatusVeröffentlicht - 26 Nov. 1990

Abstract

This paper is concerned with interpolation in the sense of Hermite by certain rational functions of one or several complex variables. In the univariate setting the interpolants are generalized polynomials of a Cauchy-Vandermonde space, whereas in the multivariate setting the interpolants are elements of suitable subspaces of tensor products of Cauchy-Vandermonde spaces. A Newton-type algorithm is given computing an interpolating univariate rational function with prescribed poles with no more than O(M2) arithmetical operations where M is the number of nodes. It is proved that the generalized divided differences are analytic functions of the nodes if the function to be interpolated is analytic. The algorithm will be extended to the multivariate setting. For subsets of grids possessing the rectangular property and for certain subspaces of a tensor product of Cauchy-Vandermonde spaces an algorithm computing an interpolating rational function of two variables is given whose complexity is O(m2n+n2m), where m and n are the numbers of interpolation points in the x- and y-direction, respectively.

ASJC Scopus Sachgebiete

Zitieren

On interpolation by rational functions with prescribed poles with applications to multivariate interpolation. / Mühlbach, G.
in: Journal of Computational and Applied Mathematics, Jahrgang 32, Nr. 1-2, 26.11.1990, S. 203-216.

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Download
@article{9157c846db9c473d96ee980f6c05a2d1,
title = "On interpolation by rational functions with prescribed poles with applications to multivariate interpolation",
abstract = "This paper is concerned with interpolation in the sense of Hermite by certain rational functions of one or several complex variables. In the univariate setting the interpolants are generalized polynomials of a Cauchy-Vandermonde space, whereas in the multivariate setting the interpolants are elements of suitable subspaces of tensor products of Cauchy-Vandermonde spaces. A Newton-type algorithm is given computing an interpolating univariate rational function with prescribed poles with no more than O(M2) arithmetical operations where M is the number of nodes. It is proved that the generalized divided differences are analytic functions of the nodes if the function to be interpolated is analytic. The algorithm will be extended to the multivariate setting. For subsets of grids possessing the rectangular property and for certain subspaces of a tensor product of Cauchy-Vandermonde spaces an algorithm computing an interpolating rational function of two variables is given whose complexity is O(m2n+n2m), where m and n are the numbers of interpolation points in the x- and y-direction, respectively.",
keywords = "Interpolation, multivariate rational interpolation, prescribed poles, rational functions",
author = "G. M{\"u}hlbach",
year = "1990",
month = nov,
day = "26",
doi = "10.1016/0377-0427(90)90431-X",
language = "English",
volume = "32",
pages = "203--216",
journal = "Journal of Computational and Applied Mathematics",
issn = "0377-0427",
publisher = "Elsevier",
number = "1-2",

}

Download

TY - JOUR

T1 - On interpolation by rational functions with prescribed poles with applications to multivariate interpolation

AU - Mühlbach, G.

PY - 1990/11/26

Y1 - 1990/11/26

N2 - This paper is concerned with interpolation in the sense of Hermite by certain rational functions of one or several complex variables. In the univariate setting the interpolants are generalized polynomials of a Cauchy-Vandermonde space, whereas in the multivariate setting the interpolants are elements of suitable subspaces of tensor products of Cauchy-Vandermonde spaces. A Newton-type algorithm is given computing an interpolating univariate rational function with prescribed poles with no more than O(M2) arithmetical operations where M is the number of nodes. It is proved that the generalized divided differences are analytic functions of the nodes if the function to be interpolated is analytic. The algorithm will be extended to the multivariate setting. For subsets of grids possessing the rectangular property and for certain subspaces of a tensor product of Cauchy-Vandermonde spaces an algorithm computing an interpolating rational function of two variables is given whose complexity is O(m2n+n2m), where m and n are the numbers of interpolation points in the x- and y-direction, respectively.

AB - This paper is concerned with interpolation in the sense of Hermite by certain rational functions of one or several complex variables. In the univariate setting the interpolants are generalized polynomials of a Cauchy-Vandermonde space, whereas in the multivariate setting the interpolants are elements of suitable subspaces of tensor products of Cauchy-Vandermonde spaces. A Newton-type algorithm is given computing an interpolating univariate rational function with prescribed poles with no more than O(M2) arithmetical operations where M is the number of nodes. It is proved that the generalized divided differences are analytic functions of the nodes if the function to be interpolated is analytic. The algorithm will be extended to the multivariate setting. For subsets of grids possessing the rectangular property and for certain subspaces of a tensor product of Cauchy-Vandermonde spaces an algorithm computing an interpolating rational function of two variables is given whose complexity is O(m2n+n2m), where m and n are the numbers of interpolation points in the x- and y-direction, respectively.

KW - Interpolation

KW - multivariate rational interpolation

KW - prescribed poles

KW - rational functions

UR - http://www.scopus.com/inward/record.url?scp=38249017511&partnerID=8YFLogxK

U2 - 10.1016/0377-0427(90)90431-X

DO - 10.1016/0377-0427(90)90431-X

M3 - Article

AN - SCOPUS:38249017511

VL - 32

SP - 203

EP - 216

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1-2

ER -