Divided difference
DOI:
https://doi.org/10.5902/2179460X25033Abstract
Given a sequence of real numbers, not necessarily distinctive, τ = (τi)ni=1, it is said that a polynomial P interpolates, a function f in τ when, for each τi of τ, that, occurs m times, in results P(j-1)(τi) = f(j-1)(τi), j = 1,..., m where P(j-1) and f(j-1) respectively represent the derivative or order j-1 of P and f.
The kth divided difference of f becomes defined, in the points of τi,..., τi+K of τ, as being a leader coefficient of polynomial of degree of maximum k that interpolates f in τi,..., τi+K.
Having in mind the definition of kth divided difference, initially, in his paper, the uniqueness of polynomial of interpolation is studied. Furtheron, the definition and some properties of kth divided difference are presented, and finally it is demonstrated the if f is at a class od Ck in [a,b], where a = min {τj, ..., τj+k} and b = max { τj, ..., τj+k}, then previous definition of kth divided difference becomes equivalent to Int(0-1) int(0-t1)...int(0-tk-1) f(k) [tk(τi+k – τi+k-1) + ... + t1 (τi+1 – τi) + τi] dtk...dt1.
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References
PRENTER, P. M. Splines and Variational Methods. Wiley, Interscience Publication, 1975.
ISSACSON, E. and KELLER, H. B. Analysis of Numerical Methods. New York, John Wiley & Sons, 1966.
DE BOOR, C. A practical guide to Spline. New York, Springer-Verlag, 1978.
GAZZONI, A. T. F. Uma limitação para a interpolação de funções contínuas por Spline. Rio de Janeiro, 1979. (Tese de Mestrado-UFRJ).
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