Abstract
Three algorithms for computing the coefficients of translated polynomials are discussed and compared from the point of view of complexity. The two classical translation algorithms based on explicit application of the Taylor expansion theorem and the Ruffini-Horner method, respectively, have complexityO (n 2). A third algorithm based on the fast Fourier transform is shown to have complexityO (n logn). However, when the cost of arithmetic operations is explicitly taken into consideration, the Ruffini-Horner algorithm is one order of magnitude better than the one based on the Taylor expansion and competes quite well with the algorithm based on the fast Fourier transform.
Zusammenfassung
Wir vergleichen die Komplexität von 3 Algorithmen zur Berechnung der Polynomkoeffizienten an beliebigen Anschlußstellen. Die beiden klassischen Algorithmen beruhen auf der expliziten Anwendung der Taylor-Formel und der Methode von Ruffini-Horner. Beide haben KomplexitätO (n 2). Ein dritter Algorithmus verwendet die schnelle Fourier-Transformation und hat die KomplexitätO (n logn). Wenn man aber die unterschiedlichen Kosten der verschiedenen arithmetischen Operationen genauer in Betracht zieht, dann ist der Algorithmus von Ruffini-Horner um eine Größenordnung besser als die Taylor-Entwicklung und mit dem Algorithmus, der auf schneller Fourier-Transformation beruht, durchaus vergleichbar.
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Akritas, A.G., Danielopoulos, S.D. On the complexity of algorithms for the translation of polynomials. Computing 24, 51–60 (1980). https://doi.org/10.1007/BF02242791
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DOI: https://doi.org/10.1007/BF02242791