Abstract
A particular class of preconditioners for the conjugate gradient method and other iterative methods is proposed for the solution of linear systemsA n,mx=b, whereA n,m is ann×n positive definite block Toeplitz matrix withm×m Toeplitz blocks. In particular we propose a sparse preconditionerP n,m such that the condition number of the preconditioned matrix turns out to be less than a suitable constant independent of bothn andm, even if the condition number ofA n,m tends to ∞. This leads to iterative methods which require a number of steps independent ofm andn in order to reduce the error by a given factor.
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References
O. Axelsson and G. Lindskog,On the rate of convergence of the preconditioned conjugate gradient method, Numer. Math, 48 (1988), pp. 499–523.
D. Bini and M. Capovani,Spectral and computational properties of band symmetric Toeplitz matrices, Linear Algebra Appl., 52/53 (1983), pp. 99–126.
J. R. Bunch,Stability of methods for solving Toeplitz systems of equations, SIAM J. Sci. Stat. Comp., 6 (1985), pp. 349–364.
R. H. Chan,Toeplitz preconditioners for Toeplitz systems with nonnegative generating functions, IMA J. Numer. Anal., 11 (1991), pp. 333–345.
R. H. Chan and T. F. Chan,Circulant preconditioner for elliptic problems, to appear in Numer. Linear Algebra Applics.
R. H. Chan and X. Q. **,A family of block preconditioners for block systems, SIAM J. Sci. Stat. Comp., 13 (1992), pp. 1218–1235.
D. Coppersmith and S. Winograd,Matrix multiplication via arithmetic progressions, J. Symbolic Comp., 9 (1990), pp. 1–6.
P. Davis,Circulant Matrices, John Wiley and Sons, New York, 1979.
F. Di Benedetto,Preconditioning of block-Toeplitz matrices by sine transforms, to appear in SIAM J. Sci. Stat. Comp.
F. Di Benedetto, G. Fiorentino, and S. Serra,C.G. preconditioning for Toeplitz matrices, Comput. Math. Appl., 6 (1993), pp. 33–45.
F. W. Dorr,The direct solution of the discrete Poisson equation on a rectangle, SIAM Review, 12 (1970), pp. 248–263.
G. Fiorentino and S. Serra,Multigrid methods for Toeplitz matrices, Calcolo, 28 (1992), pp. 283–305.
G. Fiorentino and S. Serra,Multigrid methods for symmetric positive definite block Toeplitz matrices with nonnegative generating functions, to appear (1994).
G. H. Golub and C. F. Van Loan,Matrix Computations, The Johns Hopkins University Press, 1983.
U. Grenander and G. Szegö,Toeplitz Forms and Their Applications, Second edition, Chelsea, New York, 1984.
W. Hackbush,Multigrid Methods and Applications, Springer-Verlag, 1985.
M. R. Hestenes and E. Stiefel,Methods of conjugate gradients for solving linear systems, Nat. Bur. Standards, J. of Res., 49 (1952), pp. 409–436.
I. S. Iakhvidov,Hankel and Toeplitz forms: Algebraic Theory, Birkhäuser, Boston, 1982.
T. K. Ku and C. C. J. Kuo,On the spectrum of a family of preconditioned Toeplitz matrices, SIAM J. Sci. Stat. Comp., 13 (1992), pp. 948–966.
S. Serra,Preconditioning techniques for ill-conditioned block Toeplitz systems with nonnegative generating functions, tech. rep., University of Pisa, 1993.
G. Szegö,Orthogonal polynomials, American Mathematical Society Colloquium Publications, 23 (1939).
R. S. Varga,Matrix Iterative Analysis. Prentice-Hall, 1962.
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Serra, S. Preconditioning strategies for asymptotically ill-conditioned block Toeplitz systems. BIT 34, 579–594 (1994). https://doi.org/10.1007/BF01934269
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DOI: https://doi.org/10.1007/BF01934269