Abstract
In this paper we propose a new algorithm to solve large Toeplitz systems. It consists of two steps. First, we embed the system of order N into a semi-infinite Toeplitz system and compute the first N components of its solution by an algorithm of complexity O(N log2 N). Then we check the accuracy of the approximate solution, by an a posteriori criterion, and update the inaccurate components by solving a small Toeplitz system. The numerical performance of the method is then compared with the conjugate gradient method, for 3 different preconditioners. It turns out that our method is compatible with the best PCG methods concerning the accuracy and superior concerning the execution time.
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References
F.L. Bauer. Ein direktes Iterations Verfahren zur Hurwitz-Zerlegung eines Polynoms. Arch. Elektr. Übertragung, 9:285–290, 1955.
F.L. Bauer. Beiträge zur Entwicklung numerischer Verfahren für programmgesteuerte Rechenanlagen, ii. Direkte Faktorisierung eines Polynoms. Sitz. Ber. Bayer. Akad. Wiss., pp. 163–203, 1956.
B.P. Bogert, M.J.R. Healy, and J.W. Tukey. The frequency analysis of time series for echoes: cepstrum pseudo-autocovariance, cross-cepstrum and saphe cracking. In: M. Rosenblatt (ed.), Proc. Symposium Time Series Analysis, John Wiley, New York, 1963, pp. 209–243.
A. Böttcher and B. Silbermann. Operator-valued Szegő-Widom theorems. In: E.L. Basor and I. Gohberg (Eds.), Toeplitz Operators and Related Topics, volume 71 of Operator Theory: Advances and Applications. Birkhäuser, Basel-Boston, 1994, pp. 33–53.
A. Calderón, F. Spitzer, and H. Widom. Inversion of Toeplitz matrices. Illinois J. Math., 3:490–498, 1959.
T.F. Chan. An optimal circulant preconditioner for Toeplitz systems. SIAM J. Sci. Stat. Comput., 9(4):766–771, 1988.
R.H. Chan and M.K. Ng. Conjugate Gradient Methods for Toeplitz Systems. SIAM Review, 38(3): 297–386, 1996.
R. Chan and G. Strang. Toeplitz equations by conjugate gradients with circulant preconditioner. SIAM J. Sci. Stat. Comput., 10(1):104–119, 1989.
I.C. Gohberg and I.A. Feldman. Convolution Equations and Projection Methods for their Solution, volume 41 of Transl. Math. Monographs. Amer. Math. Soc., Providence, RI, 1974.
I. Gohberg, S. Goldberg, and M.A. Kaashoek. Classes of Linear Operators, Vol. II, volume 63 of Operator Theory: Advances and Applications. Birkhäuser, Basel-Boston, 1993.
I. Gohberg and M.A. Kaashoek. Projection method for block Toeplitz operators with operator-valued symbols. In: E.L. Basor and I. Gohberg (Eds.), Toeplitz Operators and Related Topics, volume 71 of Operator Theory: Advances and Applications. Birkhauser, Basel-Boston, 1994, pp. 79–104.
T.N.T. Goodman, C.A. Micchelli, G. Rodriguez, and S. Seatzu. Spectral factorization of Laurent polynomials. Adv. Comput. Math., 7:429–454, 1997.
G. Heinig and K. Rost, Algebraic Methods for Toeplitz-like Matrices and Operators, volume 13 of Operator Theory: Advances and Applications. Birkhäuser, Basel-Boston, 1984.
T. Kailath and A.H. Sayed. Displacement structure: theory and applications. SIAM Review, 37(3): 297–386, 1996.
T. Kailath and A.H. Sayed. Fast reliable algorithms for matrices with structure. SIAM, Philadelphia, PA, 1999.
M.G. Krein. Integral equations on the half-line with kernel depending upon the difference of the arguments. Uspehi Mat. Nauk., 13(5):3–120, 1958. (Russian, translated in AMS Translations, 22:163–288, 1962).
The MathWorks Inc. Matlab ver. 6.5. Natick, MA, 2002.
C.V.M. van der Mee, M.Z. Nashed, and S. Seatzu. Sampling expansions and interpolation in unitarily translation invariant reproducing kernel Hilbert spaces. Adv. Comput. Math., 19(4):355–372, 2003.
C.V.M. van der Mee and S. Seatzu. A method for generating infinite positive definite self-adjoint testmatrices and Riesz bases. SIAM J. Matrix Anal. Appl. (to appear).
A.V. Oppenheim and R.W. Schafer. Discrete-Time Signal Processing. Prentice Hall Signal Processing Series. Prentice Hall, Englewood Cli.s, NJ, 1989.
M. Tismenetsky. A Decomposition of Toeplitz Matrices and Optimal Circulant Preconditioning. Linear Algebra Appl., 154-156:105–121, 1991.
E.E. Tyrtyshnikov. Optimal and superoptimal circulant preconditioners. SIAM J. Matrix Anal. Appl., 13:459–473, 1992.
G. Wilson. Factorization of the covariance generating function of a pure moving average process. SIAM J. Numer. Anal., 6(1):1–7, 1969.
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Dedicated to Israel Gohberg on the occasion of his 75th birthday
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Rodriguez, G., Seatzu, S., Theis, D. (2005). An Algorithm for Solving Toeplitz Systems by Embedding in Infinite Systems. In: Gohberg, I., et al. Recent Advances in Operator Theory and its Applications. Operator Theory: Advances and Applications, vol 160. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7398-9_19
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DOI: https://doi.org/10.1007/3-7643-7398-9_19
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