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Doctoral Thesis
DOI
10.11606/T.45.2016.tde-22082016-192357
Document
Author
Full name
André Pierro de Camargo
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2015
Supervisor
Committee
Mascarenhas, Walter Figueiredo (President)
Barros, Saulo Rabello Maciel de
Cuminato, José Alberto
Pierro, Alvaro Rodolfo de
Ranga, Alagacone Sri
Title in Portuguese
Estabilidade numérica de fórmulas baricêntricas para interpolação
Keywords in Portuguese
Estabilidade numérica
Fórmula baricêntrica
Interpolação
Abstract in Portuguese
O problema de reconstruir uma função f a partir de um número finito de valores conhecidos f(x0), f(x1), ..., f(xn) aparece com frequência em modelagem matemática. Em geral, não é possível determinar f completamente a partir de f(x0), f(x1), ..., f(xn), mas, em muitos casos de interesse, podemos encontrar aproximações razoáveis para f usando interpolação, que consiste em determinar uma função (um polinômio, ou uma função racional ou trigonométrica, etc) g que satisfaça g(xi) = f(xi); i = 0, 1, ..., n: Na prática, a função interpoladora g é avaliada em precisão finita e o valor final computado de g(x) pode diferir do valor exato g(x) devido a erros de arredondamento. Essa diferença pode, inclusive, ultrapassar o erro de interpolação E(x) = f(x) - g(x) em várias ordens de magnitude, comprometendo todo o processo de aproximação. A estabilidade numérica de um algoritmo reflete sua sensibilidade em relação a erros de arredondamento. Neste trabalho apresentamos uma análise detalhada da estabilidade numérica de alguns algoritmos utilizados no cálculo de interpoladores polinomiais ou racionais que podem ser postos na forma baricêntrica. Os principais resultados deste trabalho também estão disponíveis em língua inglesa nos artigos - Mascarenhas, W e Camargo, A. P., On the backward stability of the second barycentric formula for interpolation, Dolomites research notes on approximation v. 7 (2014) pp. 1-12. - Camargo, A. P., On the numerical stability of Floater-Hormann's rational interpolant, Numerical Algorithms, DOI 10.1007/s11075-015-0037-z. - Camargo, A. P., Erratum: On the numerical stability of Floater-Hormann's rational interpolant", Numerical Algorithms, DOI 10.1007/s11075-015-0071-x. - Camargo, A. P. e Mascarenhas, W., The stability of extended Floater-Hormann interpolants, Numerische Mathematik, submetido. arXiv:1409.2808v5
Title in English
Numerical stability of barycentric formulae for interpolation.
Keywords in English
Barycentric formulae
Interpolation
Numerical stability
Abstract in English
The problem of reconstructing a function f from a finite set of known values f(x0), f(x1), ..., f(xn) appears frequently in mathematical modeling. It is not possible, in general, to completely determine f from f(x0), f(x1), ..., f(xn) but, in several cases of interest, it is possible to find reasonable approximations for f by interpolation, which consists in finding a suitable function (a polynomial function, a rational or trigonometric function, etc.) g such that g(xi) = f(xi); i = 0, 1, ..., n: In practice, the interpolating function g is evaluated in finite precision and the final computed value of g(x) may differ from the exact value g(x) due to rounding. In fact, such difference can even exceed the interpolation error E(x) = f(x)-g(x) in several orders of magnitude, compromising the entire approximation process. The numerical stability of an algorithm reflect is sensibility with respect to rounding. In this work we present a detailed analysis of the numerical stability of some algorithms used to evaluate polynomial or rational interpolants which can be put in the barycentric format. The main results of this work are also available in english in the papers - Mascarenhas, W e Camargo, A. P., On the backward stability of the second barycentric formula for interpolation, Dolomites research notes on approximation v. 7 (2014) pp. 1-12. - Camargo, A. P., On the numerical stability of Floater-Hormann's rational interpolant, Numerical Algorithms, DOI 10.1007/s11075-015-0037-z. - Camargo, A. P., Erratum: On the numerical stability of Floater-Hormann's rational interpolant", Numerical Algorithms, DOI 10.1007/s11075-015-0071-x. - Camargo, A. P. e Mascarenhas, W., The stability of extended Floater-Hormann interpolants, Numerische Mathematik, submetido. arXiv:1409.2808v5
 
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Publishing Date
2016-09-09
 
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