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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Brand, L.: Differential and Difference Equations. New York: Wiley 1966.
Bulirsch, R., Stoer, J.: Darstellung von Funktionen in Rechenautomaten (Sauer, R., Szabò, I. eds.), Mathematische Hilfsmittel des Ingenieurs, Teil III. Berlin-Heidelberg-New York: Springer 1968.
Clenshaw, C. W.: A note on the summation of Chebyshev series. MTAC,9, 118–120 (1955).
Elliott, D.: Error analysis of an algorithm for summing certain finite series. J. Australian Math. Soc.8, 213–221 (1968).
Forsythe, G. E.: Generation and use of orthogonal polynomials for data fitting with a digital computer. J. SIAM5, 74–88 (1957).
Gautschi, W.: Computational aspects of three-term recurrence relations. SIAM Rev.9, 24–82 (1967).
Goertzel, G.: An algorithm for the evaluation of finite trigonometric series. Amer. Math. Monthly65, 34–35 (1958).
Langel, R.: An algorithm for certain double sums of polynomial series. Math. Comp.21, 230–232 (1967).
Luke, Y. L.: The Special Functions and their Approximations I/II. New York: Academic Press 1969.
MacRobert, T. M.: Spherical Harmonics. Oxford-New York-Braunschweig: Pergamon Press 1967.
Miller, J. C. P.: Bessel functions, Part II (Math. Tables X). Cambridge University Press 1952.
Oliver, J.: Relative Error Propagation in the Recursive Solution of Linear Recurrence Relations. Numer. Math.9, 323–340 (1967).
Olver, F. W. J.: Numerical solution of second-order linear difference equations. J. Res. N.B.S.71B, 111–129 (1967).
Reinsch, Chr.: A Note on Trigonometric Interpolation. (Unpublished manuscript.)
Smith, F. J.: An Algorithm for Summing Orthogonal Polynomial Series and their Derivatives with Application to Curve-Fitting and Interpolation. Math. Comp.19, 33–36 (1965).
Stoer, J.: Einführung in die Numerische Mathematik I. (Heidelberger Taschenbuch Band 105.) Berlin-Heidelberg-New York: Springer 1972.
Tait, R.: Error Analysis of Recurrence Equations. Math. Comp.21, 629–638 (1967).
Wilkinson, J. H.: The Algebraic Eigenvalue Problem. Oxford: Clarendon Press 1965.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Deufihard, P. On algorithms for the summation of certain special functions. Computing 17, 37–48 (1976). https://doi.org/10.1007/BF02252258
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02252258