Skip to main content
SpringerLink
Log in

The accurate solution of certain continuous problems using only single precision arithmetic

  • Part II Numerical Mathematics
  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

A typical approach for finding the approximate solution of a continuous problem is through discretization with meshsizeh such that the truncation error goes to zero withh. The discretization problem is solved in floating point arithmetic. Rounding-errors spoil the theoretical convergence and the error may even tend to infinity.

In this paper we present algorithms of moderate cost which use only single precision and which compute the approximate solution of the integration and elliptic equation problems with full accuracy. These algorithms are based on the modified Gill-Møller algorithm for summation of very many terms, iterative refinement of a linear system with a special algorithm for the computation of residuals in single precision and on a property of floating point subtraction of nearby numbers.

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. O. Axelsson and I. Gustafsson,A preconditioned conjugate gradient method for finite element equations, which is stable for rounding errors. Inf. Proc. 80, pp. 723–728.

  2. I. Babuška,Numerical stability in problems of linear algebra. SIAM J. Num. Anal., 1972, 9, pp. 53–77.

    Google Scholar 

  3. M. Jankowski, A. Smoktunowicz and H. Woźniakowski,A note on floating-point summation of very many terms. Journal of Information Processing and Cybernetics-EIK, 19 (1983) 9, 435–440.

    Google Scholar 

  4. M. Jankowski, and H. Woźniakowski,Iterative refinement implies numerical stability. BIT, 17 (1977), pp. 303–311.

    Google Scholar 

  5. A. Kielbasiński,Summation algorithm with corrections and some of its applications, Math. Stos., 1973, I, pp. 22–41 (in Polish).

    Google Scholar 

  6. Z. Mejran,Algorithm for solving linear algebraic systems arising from the difference method for Poisson's equation. Mat. Stos., 1978, XII, pp. 35–41 (in Polish).

    Google Scholar 

  7. O. Møller,Quasi-double-precision in floating-point addition. BIT, 5 (1965), pp. 37–50, 251–255.

    Google Scholar 

  8. P. Pohl,Iterative improvement without double precision in a boundary value problem. BIT, 14 (1974), pp. 361–365.

    Google Scholar 

  9. G. Ramos,Roundoff error analysis of the fast Fourier transform. Tech. Rep. STAN-CS-70-146, 1970, Stanford Universiy.

  10. A. A. Samarskij,Theory of the Difference Schemes. Moscow 1977 (in Russian).

  11. J. H. Wilkinson,Rounding Errors in Algebraic Processes. Prentice-Hall, Englewood Cliffs, 1963.

    Google Scholar 

  12. H. Woźniakowski,Numerical stability of the Chebyshev method for the solution of large linear systems. Numer. Math., 1977, 28, pp. 191–209.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Informatics, University of Warsaw, 00-901, Warsaw, Poland

    M. Jankowski & H. Woźniakowski

  2. Department of Computer Science, Columbia University, 10027, New York, N.Y., U.S.A.

    M. Jankowski & H. Woźniakowski

Authors
  1. M. Jankowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Woźniakowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

On leave of absence from Institute of Informatics, University of Warsaw, 00-901 Warsaw, Poland.

This research was supported in part by the National Science Foundation under Grant MCS-7823676.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jankowski, M., Woźniakowski, H. The accurate solution of certain continuous problems using only single precision arithmetic. BIT 25, 635–651 (1985). https://doi.org/10.1007/BF01936142

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01936142

Keywords

Search

Navigation