Numerical Algorithms

, Volume 55, Issue 1, pp 115–139 | Cite as

A note on the O(n)-storage implementation of the GKO algorithm and its adaptation to Trummer-like matrices

Original Paper

Abstract

We propose a new O(n)-space implementation of the GKO-Cauchy algorithm for the solution of linear systems where the coefficient matrix is Cauchy-like. Moreover, this new algorithm makes a more efficient use of the processor cache memory; for matrices of size larger than n ≈ 500–1,000, it outperforms the customary GKO algorithm. We present an applicative case of Cauchy-like matrices with non-reconstructible main diagonal. In this special instance, the O(n) space algorithms can be adapted nicely to provide an efficient implementation of basic linear algebra operations in terms of the low displacement-rank generators.

Keywords

Cauchy-like matrix GKO algorithm Displacement structure Structured linear system Toeplitz matrix 

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References

  1. 1.
    Aricò, A., Rodriguez, G.: A fast solver for linear systems with displacement structure. Internal Note. http://tex.unica.it/~arico/publications.html (2009)
  2. 2.
    Bini, D., Pan, V.Y.: Polynomial and matrix computations. In: Progress in Theoretical Computer Science, vol. 1. Birkhäuser Boston, Boston (1994)Google Scholar
  3. 3.
    Bini, D.A., Iannazzo, B, Poloni, F.: A fast Newton’s method for a nonsymmetric algebraic Riccati equation. SIAM J. Matrix Anal. Appl. 30(1), 276–290 (2008)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bini, D.A., Meini, B., Poloni, F.: Fast solution of a certain Riccati equation through Cauchy-like matrices. Electron. Trans. Numer. Anal. 33(1), 84–104 (2008–2009)MATHMathSciNetGoogle Scholar
  5. 5.
    Boros, T., Kailath, T., Olshevsky, V.: Pivoting and backward stability of fast algorithms for solving Cauchy linear equations. Linear Algebra Appl. 343/344, 63–99 (2002, Special issue on structured and infinite systems of linear equations)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Cybenko, G.: The numerical stability of the Levinson–Durbin algorithm for Toeplitz systems of equations. SIAM J. Sci. Statist. Comput. 1(3), 303–319 (1980)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Frigo, M., Leiserson, C.E., Prokop, H., Ramachandran, S.: Cache-oblivious algorithms. Annual IEEE Symposium on Foundations of Computer Science 0, 285 (1999)Google Scholar
  8. 8.
    Gerasoulis, A.: A fast algorithm for the multiplication of generalized Hilbert matrices with vectors. Math. Comp. 50(181), 179–188 (1988)MATHMathSciNetGoogle Scholar
  9. 9.
    Gohberg, I., Kailath, T., Olshevsky, V.: Fast Gaussian elimination with partial pivoting for matrices with displacement structure. Math. Comp. 64(212), 1557–1576 (1995)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gu, M.: Stable and efficient algorithms for structured systems of linear equations. SIAM J. Matrix Anal. Appl. 19(2), 279–306 (1998)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hansen, P.C., Chan, T.F.: Fortran subroutines for general Toeplitz systems. ACM Trans. Math. Softw. 18(3), 256–273 (1992)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Heinig, G., Rost, K.: Algebraic methods for Toeplitz-like matrices and operators. In: Operator Theory: Advances and Applications, vol. 13. Birkhäuser, Basel (1984)Google Scholar
  13. 13.
    Hennessy, J.L., Patterson, D.A.: Computer Architecture: A Quantitative Approach (The Morgan Kaufmann Series in Computer Architecture and Design). Morgan Kaufmann, San Mateo (2002)Google Scholar
  14. 14.
    Kailath, T., Chun, J.: Generalized displacement structure for block-Toeplitz, Toeplitz-block, and Toeplitz-derived matrices. SIAM J. Matrix Anal. Appl. 15(1), 114–128 (1994)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kailath, T., Olshevsky, V.: Diagonal pivoting for partially reconstructible Cauchy-like matrices, with applications to Toeplitz-like linear equations and to boundary rational matrix interpolation problems. In: Proceedings of the Fifth Conference of the International Linear Algebra Society (Atlanta, GA, 1995), vol. 254, pp. 251–302 (1997)Google Scholar
  16. 16.
    Kailath, T., Sayed, A.H.: Displacement structure: theory and applications. SIAM Rev. 37(3), 297–386 (1995)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Olshevsky, V.: Pivoting for structured matrices and rational tangential interpolation. In: Fast algorithms for structured matrices: theory and applications (South Hadley, MA, 2001), vol. 323, pp. 1–73. Contemp. Math. Amer. Math. Soc., Providence (2003)Google Scholar
  18. 18.
    Rodriguez, G.: Fast solution of Toeplitz- and Cauchy-like least-squares problems. SIAM J. Matrix Anal. Appl. 28(3), 724–748 (2006)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Sweet, D.R.: Error analysis of a fast partial pivoting method for structured matrices. In: Proceedings SPIE. Advanced Signal Processing Algorithms SPIE, vol. 2563, pp. 266–280, (1995)Google Scholar
  20. 20.
    Van Barel, M., Heinig, G., Kravanja, P.: A stabilized superfast solver for nonsymmetric Toeplitz systems. SIAM J. Matrix Anal. Appl. 23(2), 494–510 (2001)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Walker, J.S.: Fast Fourier transforms. Studies in Advanced Mathematics, 2nd edn. CRC, Boca Raton (1996)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Scuola Normale SuperiorePisaItaly

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