Abstract
Givenn+1 distinct points and arbitrary order derivative information at these points, a parallel algorithm to compute the coefficients of the corresponding Hermite interpolating polynomial inO (logn) parallel arithmetic operations usingO (n 2) processors is presented. The algorithm relies on a novel closed formula that yields the expansion of the generalized divided differences in terms of the given function and derivative values. We show that each one of the coefficients in this expansion and the required linear combinations can be evaluated efficiently.
The particular cases where up to first and second order derivative information is available are treated in detail. The proof of the general case, where arbitrarily high order derivative information is available, involves algebraic arguments that make use of the theory of symmetric, functions.
Zusammenfassung
Gegeben seienn+1 verschiedene Punkte sowie die Werte von Ableitungen beliebiger Ordnung in diesen Punkten. Für die Berechnung der Koeffizienten des zugehörigen Hermiteschen Interpolationspolynoms wird ein paralleler Algorithmus vorgestellt, derO (logn) parallele arithmetische Operationen aufO (n 2) Prozessoren benötigt. Der Algorithmus basiert auf einer neuartigen geschlossenen Darstellung der verallgemeinerten Differenzenquotienten durch die gegebenen Funktions- und Ableitungswerte. Wir zeigen, daß sowohl die Koeffizienten in dieser Darstellung als auch die benötigten Linearkombinationen effizient berechnet werden können.
Detailliert behandelt werden die Spezialfälle, daß die Ableitungen bis zur ersten bzw. zweiten Ordnung bekannt sind. Für den Beweis des allgemeinen Falles, wo Ableitungswerte beliebiger höherer Ordnung verfügbar sind, wird ein algebraischer Zugang gewählt, bei dem die Theorie symmetrischer Funktionen herangezogen wird.
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Supported in part by NSF Grant No. DCR-8603722.
Supported in part by the National Science Foundation under Grants Nos. NSF DCR84-10110 and NSF DCR 85-09770, the U. S. Department of Energy under Grant No. DOE DE-FG02-85ER25001, the Air Force Office of Scientific Research Grant No. AFOSR-85-0221 and an IMB Donation.
Supported by Lawrence Livermore National Laboratory Contract No. LLNL-7526225.
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Eğecioğlu, Ö., Gallopoulos, E. & Koç, Ç.K. Parallel Hermite interpolation: An algebraic approach. Computing 42, 291–307 (1989). https://doi.org/10.1007/BF02243225
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DOI: https://doi.org/10.1007/BF02243225