Advertisement

Kronecker products of matrices and their implementation on shuffle/exchange-type processor networks

  • Otto Lange
Architectural Aspects (Session 5.1)
Part of the Lecture Notes in Computer Science book series (LNCS, volume 237)

Abstract

In generalized spectral analysis an orthogonal transformation provides a set of spectral coefficients by a vector-matrix-multiplication. If the pnxpn-transformation matrix Hn is a Kronecker product of n pxp-matrices Mr, r=0,1,...,n−1, then — as has been shown by Good — there exists a factorization \(\mathop \prod \limits_{r = 0}^{n - 1} \)Gr of the transformation matrix Hn in such a way that the vector-matrix-multiplication can be reduced from 0(N2) to 0(NlogN) operations, where N=pn. The specific structure of the matrix factors Gr, r=0,1...,n−1, permits an implementation of the multiplication on processor networks, in case of p=2 on permutation networks of the Shuffle/Exchange-type.

Keywords

Kronecker Product Orthogonal Transformation Spectral Coefficient Processor Element Processor Network 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. /AK70/.
    Andrews, H. C., Kane, J.: Kronecker Matrices, Computer Implementation and Generalized Spectra J. ACM, Vol. 17, No. 2, April 1970.Google Scholar
  2. /AN70/.
    Andrews, H. C.: Computer Techniques in Image Processing Academic Press, New York/London, 1970.Google Scholar
  3. /GO58/.
    Good, I. J.: The Interaction Algorithm and Practical Fourier Analysis J. Roy. Statist. Soc. London, B 20, 1958.Google Scholar
  4. /LA70/.
    Lawrie, D. H.: Access and Alignment of Data in an Array Processor IEEE Transactions on Computers, Vol. C-25, No. 12, Dec. 1976.Google Scholar
  5. /PA80/.
    Parker, D. S.: Notes on Shuffle/Exchange-Type Switching Networks IEEE Transactions on Computers, Vol. C-29, No. 3, March 1980.Google Scholar
  6. /PE68/.
    Pease, M. C.: An Adaption of the Fast Fourier Transform for Parallel Processing J. ACM, Vol. 15, April 1968.Google Scholar
  7. /WA23/.
    Walsh, J. L.: A Closed Set of Normal Orthogonal Functions Ann. J. Math. 55, 1923.Google Scholar
  8. /WG68/.
    Whelchel, J. E., Guinn, D. F.: The Fast Fourier Hadamard Transform and its Use in Signal Representation and Classification The Electronic and Aerospace Systems Convention Record (EASCON), IEEE New York, 1968.Google Scholar
  9. /ST71/.
    Stone, H. S.: Parallel Processing with the Perfect Shuffle IEEE Transaction on Computers, Vol. C-20, No. 2, Feb. 1971.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Otto Lange
    • 1
  1. 1.Allgemeine Elektrotechnik und DatenverarbeitungssystemeRWTH AachenGermany

Personalised recommendations