Computing

, Volume 45, Issue 2, pp 95–117 | Cite as

Systolic computation of interpolating polynomials

  • P. R. Cappello
  • E. Gallopoulos
  • Ç. K. Koç
Article

Abstract

Several time-optimal and spacetime-optimal systolic arrays are presented for computing a process dependence graph corresponding to the Aitken algorithm. It is shown that these arrays also can be used to compute the generalized divided differences, i.e., the coefficients of the Hermite interpolating polynomial. Multivariate generalized divided differences are shown to be efficiently computed on a 2-dimensional systolic array. The techniques also are applied to the Neville algorithm, producing similar results.

AMS Subject Classifications

65D05 68Q80 68N99 

Key words

Newton interpolation Hermite interpolation Aitken's algorithm Neville's algorithm systolic array 

Systolische Berechnung von interpolierenden Polynomen

Zusammenfassung

Zur Berechnung eines zum Aitken-Algorithmus gehörigen Prozeßabhängigkeitsgraphen werden einige Zeit-optimale und Raum-Zeit-optimale systolische Felder vorgestellt. Es wird gezeigt, daß man diese Felder auch zur Berechnung verallgemeinerter dividierter Differenzen verwenden kann, wie sie als Koeffizienten des Hermiteschen Interpolationspolynoms auftreten. Die effiziente Berechnung multivariater verallgemeinerter dividierter Differenzen auf einem zweidimensionalen systolischen Feld wird gezeigt. Für den Neville-Algorithmus ergeben die Techniken ähnliche Ergebnisse.

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References

  1. [1]
    Annaratone, A. M., Arnould, E., Gross, T., Kung, H.-T., Lam, M., Menzilcioglu, O., Webb, J., The WARP Computer: Architecture, Implementation, and Performance, IEEE Trans. on Computers, C-36, No. 12, pp. 1523–1538, December 1987.Google Scholar
  2. [2]
    Berezin, I. S., Zhidkov, N. P., Computing Methods, Vol. 1, Addison-Wesley, 1965.Google Scholar
  3. [3]
    Cappello, P. R., Steiglitz, K., Unifying VLSI Array Designs with Linear Transformations of Space-Time, in Advances in Computer Research, edited by F. P. Preparata, Vol. 2, pp. 23–65, JAI Press, 1984.Google Scholar
  4. [4]
    Fortes, J. A. B., Moldovan D. I., Parallelism detected and algorithm transformation techniques useful for VLSI architecture design, J. Parallel Distrib. Comput.,2, pp. 277–301, August 1985.Google Scholar
  5. [5]
    Foulser, D. E., Schreiber, R., “The Saxpy Matrix-1: a General-Purpose Systolic Computer,” IEEE Computer,20, No. 7, pp. 35–43, July 1987.Google Scholar
  6. [6]
    Garey, M. R., Johnson, D. S., Computers and Intractability, Freeman, 1979.Google Scholar
  7. [7]
    Gasca, M., Maeztu, J. I., On Lagrange and Hermite Interpolation inR k,Numerische Mathematik,39, No. 1, pp. 1–14, 1982.Google Scholar
  8. [8]
    Guenther, R. B., Roetman, E. L., Some observations on interpolation in higher dimensions, Mathematics of Computation,24, No. 111, pp. 517–527, July 1970.Google Scholar
  9. [9]
    Hildebrand, F. B., Introduction to Numerical Analysis, McGraw-Hill, 1956.Google Scholar
  10. [10]
    IMS T800 transputer, Rpt. 72 TRN 117 01, INMOS Ltd., Almondsbury, Bristol, UK, November 1986.Google Scholar
  11. [11]
    Krogh, F., Efficient algorithms for polynomial interpolation and divided differences, Mathematics of Computation,24, No. 109, pp. 185–190, January 1970.Google Scholar
  12. [12]
    McKeown, G. P., Iterated interpolation using a systolic array, ACM Transactions on Mathematical Software,12, No. 2, pp. 162–170, June 1986.Google Scholar
  13. [13]
    Miranker, W. L., Winkler, A., Spacetime representations of computational structures, Computing,32, pp. 93–114, 1984.Google Scholar
  14. [14]
    Moldovan, D. I., On the Analysis and Synthesis of VLSI Algorithms, IEEE Transactions on Computers, C-31, pp. 1121–1126, November 1982.Google Scholar
  15. [15]
    Moldovan, D. I., On the Design of Algorithms for VLSI Systolic Arrays”, Proc. IEEE,71, No. 1, pp. 113–120, January 1983.Google Scholar
  16. [16]
    Quinton, P., Automatic synthesis of systolic arrays from uniform recurrent equations, Proc. 11th Ann. Symp. on Computer Architecture, pp. 208–214, 1984.Google Scholar
  17. [17]
    Rao, S. K., Regular Iterative Algorithms and Their Implementation on Processor Arrays, Ph.D. dissertation, Stanford University, October, 1985.Google Scholar
  18. [18]
    Salzer, H. E., Some New Divided Difference Algorithms, inOn Numerical Approximation, edited by Langer, R. E., pp. 61–98, The University of Wisconsin Press, 1956.Google Scholar
  19. [19]
    Salzer, H. E., Divided differences for functions of two variables for irregularly spaced arguments, Numerische Mathematik,6, No. 2, pp. 68–77, 1964.Google Scholar
  20. [20]
    Schumaker, L. L., Fitting Surface to Scattered Data, in Approximation Theory, Vol. II, edited by G. G. Lorentz, C. K. Chui, and L. L. Schumaker, pp. 203–268, Academic Press, 1976.Google Scholar
  21. [21]
    Thacher, Jr., H. C., Derivation of interpolation formulas in several independent variables, Annals of New York Academy of Sciences,86, No. 3, pp. 758–775, May 1960.Google Scholar
  22. [22]
    Tsao, N. K., R. Prior, On Multipoint Numerical Interpolation, ACM Transactions on Mathematical Software, Vol. 4, No. 1, pp. 51–56, March 1978.Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • P. R. Cappello
    • 1
  • E. Gallopoulos
    • 2
  • Ç. K. Koç
    • 3
  1. 1.Department of Computer ScienceUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Center for Super computing Research and DevelopmentUniversity of Illinois at Urbana ChampaignUrbanaUSA
  3. 3.Department of Electrical EngineeningUniversity of HoustonHoustonUSA

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