Skip to main content
SpringerLink
Log in

Spigot Algorithm and Root Computing

  • Published:
Reliable Computing

Abstract

The word spigot indicates that the "digits" (generally in a widened meaning) of the result number are extracted successively from left to right (as if pumped through a spigot) by using only integer arithmetic, as opposed to the iterative approach, where the result number as the whole is improved after each iteration step by using (high-precision) floating-point arithmetic. The approach of spigot computing as used in papers by S. Kamal Abdaly (Comm. ACM 13(1970)), S. Rabinowitz and S. Wagon (American Mathematical Monthly 102(3) (1995)), A. H. J. Sale (Comput. J. 11(1968)) is now systematized and correctness is formally proved; the way for achieving an arbitrary accuracy is shown.

Then a method for computing roots of arbitrary rational numbers is developed. If a root is not rational, spigot approach is used to compute its decimal approximation with an arbitrary given accuracy; if the root is rational, its numerator and denominator are computed exactly. This method for root computing is absolutely reliable: it is both formally proved and tested by numerical examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bailey, D., Borwein, P., and Plouffe, S.: On the Rapid Computation of Various Polylogarithmic Constants, Math. Comp. 66 (1997), pp. 903–913.

    Google Scholar 

  2. Gosper, R. W.: Acceleration of Series, Memo no. 304, M.I.T. Artificial Intelligence Laboratory, Cambridge, 1974.

  3. Kamal Abdaly, S.: Algorithm 393-Special Series Summation with Arbitrary Precision, Comm. ACM 13 (1970), p. 570.

    Google Scholar 

  4. Knopp, K.: Theorie und Anwendung der unendlichen Reihen, Verlag von Julius Springer, 1964.

  5. Knuth, D. E.: The Art of Computer Programming, Vol. 2, Seminumerical Algorithms, Third Edition, Addison-Wesley Reading, 1997.

  6. Rabinowitz, S. and Wagon, S.: A Spigot Algorithm for the Digits of π, American Mathematical Monthly 102 (3) (1995), pp. 195–203.

    Google Scholar 

  7. Reichel, H.: Formale Spezifikation und Verifikation, Vorlesungsscript, TU Dresden Fakultät Informatik Institut für theoretische Informatik, 1993.

  8. Sale, A. H. J.: The Calculation of e to Many Significant Digits, Comput. J. 11 (1968), pp. 229–230.

    Google Scholar 

  9. Stoschek, E. P.: Abenteuer Algorithmus, Teil 2, Dresden University Press, 1997.

  10. Stoschek, E. P.: Internet Workshop on Design of Algorithms; Module 9, http://wwwtcs.inf.tu-dresden.de/DresdenAlgorithmicsChannel/modul9/task9.htm.

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Institute of Theoretical Computer Science, Chair of Theory of Algorithms and Automates, Dresden University of Technology, Dresden, Germany

    Dang-Khoa Do

Authors
  1. Dang-Khoa Do
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Do, DK. Spigot Algorithm and Root Computing. Reliable Computing 7, 247–273 (2001). https://doi.org/10.1023/A:1011403105587

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1011403105587

Keywords

Search

Navigation