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Square rooting is as difficult as multiplication

Die Komplexität des Wurzelziehens

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Abstract

It is shown that multiplication of numbers and square rooting have the same complexity, i. e. from a program for multiplication one can construct a program for square rooting with the same asymptotic time complexity (1 step≦1 bit-operation) and vice versa. It follows from the Schönhage-Strassen algorithm that square rooting can be performed in 0 (n logn log logn) bit-operations.

Zusammenfassung

Es wird gezeigt, daß Multiplikation von Zahlen und Bestimmen der Quadratwurzel von gleicher Komplexität sind, d. h. aus einem Programm zur Multiplikation kann man eines zum Wurzelziehen konstruieren, das größenordnungsmäßig die gleiche Zeitkomplexität hat (1 Schritt ≦ 1 Bit-Operation) und umgekehrt. Mit dem Schönhage-Strassen-Algorithmus erhält man so einen 0 (n logn log logn)-Algorithmus zum Berechnen der Quadratwurzel.

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References

  1. Aho, Hopcroft, Ullman: The Design and Analysis of Computer Algorithms. Addison-Wesley 1974.

  2. Alt: Algorithms for Square Root Extraction. Report A 77-12, Fachbereich Angewandte Mathematik und Informatik, Saarbrücken, 1977.

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Authors and Affiliations

  1. Fachbereich 10 Angewandte Mathematik und Informatik, Universität des Saarlandes, D-6600, Saarbrücken, Federal Republic of Germany

    H. Alt

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  1. H. Alt
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Alt, H. Square rooting is as difficult as multiplication. Computing 21, 221–232 (1979). https://doi.org/10.1007/BF02253055

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  • DOI: https://doi.org/10.1007/BF02253055

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