Skip to main content
SpringerLink
Log in

Accuracy and speed of real and complex interpolation

Genauigkeit und Geschwindigkeit der reellen und komplexen Interpolation

  • Published:
Computing Aims and scope Submit manuscript

Abstract

Because complex arithmetic takes only a little more time than real arithmetic on modern computers it is advisable to take a closer look at some of the known mathematical tools and compare the advantages in complex computations.

This paper compares real and complex interpolation from the point of view of the application for larger problems. It shows that real interpolation breaks down very early whereas complex interpolation is superior by many orders of magnitude with regard to time as well as accuracy. It is shown theoretically that complex interpolation on the unit circle is the best that can be achieved.

Zusammenfassung

Die komplexe Arithmetik braucht nur ein wenig mehr Computerzeit als die reelle und es ist empfehlenswert, manche bekannte mathematische Verfahren und ihre Vorteile bei der Verwendung der komplexen Rechnungen zu untersuchen und zu vergleichen.

Dieser Beitrag untersucht die reelle und komplexe Interpolation von dem Standpunkt der Anwendung für größere Probleme. Er stellt klar, daß bei der Erhöhung des Polynomgrades die reelle Interpolation frühzeitig zusammenbricht, während die komplexe Interpolation um viele Größenordnungen in der Zeit und Genauigkeit der reellen überlegen ist. Es wird theoretisch bewiesen, daß die komplexe Interpolation an dem Einheitskreis die besterreichbare ist.

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. Berezin, I. S., and N. P. Zhidkov: Computing Methods. Pergamon Press. 1965.

  2. Timan, A. F.: Theory of Approximation of Functions of a Real Variable. Pergamon Press. 1963.

  3. Ralston, A.: A First Course in Numerical Analysis. McGraw-Hill. 1965.

  4. Cooley, J. W., P. A. W. Lewis, and P. D. Welch: The Finite Fourier Transform, IEEE Transactions on Audio- and Electroacoustics. Vol. AU-17, pp. 77–85, June 1969.

    Google Scholar 

  5. Walsh, J. L.: Interpolation and Approximation by Rational Functions in the Complex Domain. American Mathematical Society, 2nd edition, 1956.

  6. Cooley, J. W., and J. W. Tukey: An Algorithm for the Machine Calculation of Complex Fourier Series. Math. of Computation19, 297–301 (1965).

    Google Scholar 

  7. Cooley, J. W., P. A. W. Lewis, and P. D. Welch: The Fast Fourier Transform Algorithm and its Applications. IBM Corp., Research Paper RC-1743, 1967.

  8. Wilkinson, J. H.: The Algebraic Eigenvalue Problem. Oxford: Clarendon Press. 1965.

    Google Scholar 

  9. Singhal, K., and J. Vlach: Interpolation using the Fast Fourier Transform. Proc. IEEE60, 1558 (1972).

    Google Scholar 

  10. Cooley, J. W., P. A. W. Lewis, and P. D. Welch: The Fast Fourier Transform Algorithm: Programming Considerations in the Calculation of Sine, Cosine and Laplace Transforms. J. Sound and Vib.12, 315–337 (1970).

    Google Scholar 

  11. Gentleman, W. M., and G. Sande: Fast Fourier Transforms — for Fun and Profit. Fall Joint Computer Conf., 1966, AFIPS Proc., Vol. 29.

  12. Kaneko, T., and B. Liu: Accumulation of Round-off Error in Fast Fourier Transforms. J. Assoc. Comp. Mach.17, 637 (1970).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Electrical Engineering Department, University of Waterloo, Waterloo, Ontario, Canada

    K. Singhal &  J. Vlach

Authors
  1. K. Singhal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Vlach
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

With 3 Figures

This work was supported by the National Research Council under grant A 7398.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Singhal, K., Vlach, J. Accuracy and speed of real and complex interpolation. Computing 11, 147–158 (1973). https://doi.org/10.1007/BF02252904

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02252904

Keywords

Search

Navigation