Advertisement

Fast Computation of a Restricted Subset of Eigenpairs of a Varying Hermitian Matrix

  • Pierre Comon
Part of the NATO ASI Series book series (volume 70)

Abstract

Standard algorithms require an order of dm 2 flops to compute d eigenpairs of a m × m semi-definite positive hermitian matrix, R. A larger amount of computations (of same order) is required to find d left singular pairs of a square matrix, D, such that R = DD H ; this is known to be numerically more stable. Note that D is not unique, the best choice in the sequel turns out not to be the Cholesky factor of R. The algorithm proposed in this contribution will compute d left singular pairs of D in a faster way, such that it requires an order of αmd 2 flops. This algorithm takes advantage of the fact that the matrix considered is nothing else but a matrix whose d eigenpairs are known, perturbed by a rank-one modification. It is pointed out in the sequel why the technique of Bunch et al [1] is not relevant for O(md 2) algorithms, neither for square-root implementations.

Keywords

Cholesky Factor Ritz Vector Subspace Iteration Positive Definite Hermitian Matrix Singular Triplet 
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. [1]
    J.R. Bunch, C.P. Nielsen and D.C. Sorenson, “Rank One Modification of the Symmetric Eigenproblem”, Numerische Math., vol. 31, 1978, 3148.Google Scholar
  2. [2]
    P. Comon, “Adaptive Computation of a Few Extreme Eigenpairs of a Positive Definite Hermitian Matrix”, Proceedings of Eusipco 88, Sept. 5–8, Grenoble, France.Google Scholar
  3. [3]
    P. Comon and G.H. Golub, “Tracking of a Few Extreme Singular Values and Vectors in Signal Processing”, to be submitted.Google Scholar
  4. [4]
    J.J. Dongara, C.B. Moler, J.R. Bunch and G.W. Stewart, LINPACK User’s Guide, SIAM, Philadelphia, 1979.CrossRefGoogle Scholar
  5. [5]
    G.H. Golub and W. Kahan, “Calculating the Singular Values and Pseudo Inverse of a Matrix”, SIAM Journal Num. Anal., ser. B, vol. 2, no 2, 1965, 205–224.MathSciNetGoogle Scholar
  6. [6]
    G.H. Golub, F.T. Luk and M.L. Overton, “A Block Lanczos Method for Computing the Singular Values and Corresponding Singular Vectors of a Matrix”, ACM Trans. on Mathematical Software, vol 7, no 2, june 1981, 149–169.Google Scholar
  7. [7]
    G.H. Golub and C. Reinsch, “Singular Value Decomposition and Least Squares Solutions”, Numerische Math., 1970, vol. 14, 403–420, or Handbook of Automatic Computation, 1971, Wilkinson and Reinsch editors, Springer Verlag.Google Scholar
  8. [8]
    G.H. Golub and C.F. Van Loan, Matrix Computations, 1983, Hopkins.Google Scholar
  9. [9]
    J. Karhunen, “Adaptive Algorithm for Estimating Eigenvectors of Correlation Type Matrices”, IEEE International Conference on ASSP 8, San Diego, 1984.Google Scholar
  10. [10]
    E. Oja and J. Karhunen, “On Stochastic Approximation of Eigenvectors and Eigenvalues of the Expectation of a Random Matrix”, Journal of Mathematical Analysis and Applications, vol. 106, no 1, Feb 1985, 69–84.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    V.U. Reddy, B. Egardt and T. Kailath, “Least Squares Type Algorithm for Adaptive Implementation of Pisarenko’s Harmonic Retrieval Method”, IEEE Trans. on ASSP, vol 30, no 3, june 1982, 399–405.CrossRefGoogle Scholar
  12. [12]
    D.C. Sorenson, “Updating techniques in Parallel Computation”, this volume.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Pierre Comon
    • 1
  1. 1.Thomson Sintra, ASMCagnes sur merFrance

Personalised recommendations