Abstract
A slight extension of Clenshaw's summation technique is given together with a useful backward error analysis. Moreover, a class of algorithms for the evaluation of general double sums of doubly-indexed special functions is derived. An associated graph representation is introduced that permits easy classification of each algorithm in terms of certain stability aspects. As an example, details of a summation algorithm for spherical harmonics are worked out.
Zusammenfassung
Es wird eine Verallgemeinerung der Summation nach Clenshaw behandelt. Eine für die Anwendungen nützliche Rückwärtsanalyse wird angegeben. Darüberhinaus wird eine Klasse von Algorithmen zur Auswertung allgemeiner Doppelsummen über doppeltindizierte spezielle Funktionen hergeleitet. Eine zugeordnete Graphendarstellung wird eingeführt. Sie erlaubt eine einfache Klassifikation jedes einzelnen Algorithmus bezüglich gewisser Stabilitätsaspekte. Als Beispiel wird ein effizienter Algorithmus zur Summation von Kugelfunktionen angegeben.
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Deufihard, P. On algorithms for the summation of certain special functions. Computing 17, 37–48 (1976). https://doi.org/10.1007/BF02252258
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DOI: https://doi.org/10.1007/BF02252258