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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aricò, A., Rodriguez, G.: A fast solver for linear systems with displacement structure. Internal Note. http://tex.unica.it/~arico/publications.html (2009)
Bini, D., Pan, V.Y.: Polynomial and matrix computations. In: Progress in Theoretical Computer Science, vol. 1. Birkhäuser Boston, Boston (1994)
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)
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)
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)
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)
Frigo, M., Leiserson, C.E., Prokop, H., Ramachandran, S.: Cache-oblivious algorithms. Annual IEEE Symposium on Foundations of Computer Science 0, 285 (1999)
Gerasoulis, A.: A fast algorithm for the multiplication of generalized Hilbert matrices with vectors. Math. Comp. 50(181), 179–188 (1988)
Gohberg, I., Kailath, T., Olshevsky, V.: Fast Gaussian elimination with partial pivoting for matrices with displacement structure. Math. Comp. 64(212), 1557–1576 (1995)
Gu, M.: Stable and efficient algorithms for structured systems of linear equations. SIAM J. Matrix Anal. Appl. 19(2), 279–306 (1998)
Hansen, P.C., Chan, T.F.: Fortran subroutines for general Toeplitz systems. ACM Trans. Math. Softw. 18(3), 256–273 (1992)
Heinig, G., Rost, K.: Algebraic methods for Toeplitz-like matrices and operators. In: Operator Theory: Advances and Applications, vol. 13. Birkhäuser, Basel (1984)
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)
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)
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)
Kailath, T., Sayed, A.H.: Displacement structure: theory and applications. SIAM Rev. 37(3), 297–386 (1995)
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)
Rodriguez, G.: Fast solution of Toeplitz- and Cauchy-like least-squares problems. SIAM J. Matrix Anal. Appl. 28(3), 724–748 (2006)
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)
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)
Walker, J.S.: Fast Fourier transforms. Studies in Advanced Mathematics, 2nd edn. CRC, Boca Raton (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Poloni, F. A note on the O(n)-storage implementation of the GKO algorithm and its adaptation to Trummer-like matrices. Numer Algor 55, 115–139 (2010). https://doi.org/10.1007/s11075-010-9361-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-010-9361-5