Optimal, quasi-optimal and superlinear band-Toeplitz preconditioners for asymptotically ill-conditioned positive definite Toeplitz systems

In this paper we are concerned with the solution of n x n Hermitian Toeplitz systems with nonnegative generating functions f. The preconditioned conjugate gradient (PCG) method with the well-known circulant preconditioners fails in the case where f has zeros. In this paper we consider as preconditioners band-Toeplitz matrices generated by trigonometric polynomials g of fixed degree l. We use different strategies of approximation of f to devise a polynomial g which has some analytical properties of f, is easily computable and is such that the corresponding preconditioned system has a condition number bounded by a constant independent of n. For each strategy we analyze the cost per iteration and the number of iterations required for the convergence within a preassigned accuracy. We obtain different estimates of l for which the total cost of the proposed PCG methods is optimal and the related rates of convergence are superlinear. Finally, for the most economical strategy, we perform various numerical experiments which fully confirm the effectiveness of approximation theory tools in the solution of this kind of linear algebra problems.

[1]  S. Capizzano,et al.  Preconditioning Strategies for Hermitian Toeplitz Systems with Nondefinite Generating Functions , 1996, SIAM J. Matrix Anal. Appl..

[2]  O. Axelsson,et al.  On the rate of convergence of the preconditioned conjugate gradient method , 1986 .

[3]  O. Axelsson,et al.  Finite element solution of boundary value problemes - theory and computation , 2001, Classics in applied mathematics.

[4]  Fabio Di Benedetto,et al.  Analysis of Preconditioning Techniques for Ill-Conditioned Toeplitz Matrices , 1995, SIAM J. Sci. Comput..

[5]  Raymond H. Chan,et al.  Multigrid Method for Ill-Conditioned Symmetric Toeplitz Systems , 1998, SIAM J. Sci. Comput..

[6]  Stefano Serra,et al.  On the extreme eigenvalues of hermitian (block) toeplitz matrices , 1998 .

[7]  P. Tang.  A fast algorithm for linear complex Chebyshev approximations , 1988 .

[8]  I. Gohberg,et al.  Convolution Equations and Projection Methods for Their Solution , 1974 .

[9]  R. Chan.  Toeplitz Preconditioners for Toeplitz Systems with Nonnegative Generating Functions , 1991 .

[10]  G. Strang,et al.  Toeplitz equations by conjugate gradients with circulant preconditioner , 1989 .

[11]  Stephen J. Wright,et al.  Parallel Algorithms for Banded Linear Systems , 1991, SIAM J. Sci. Comput..

[12]  Alan V. Oppenheim,et al.  Applications of digital signal processing , 1978 .

[13]  Giuseppe Fiorentino,et al.  C. G. preconditioning for Toeplitz matrices , 1993 .

[14]  Dario Bini,et al.  SPECTRAL AND COMPUTATIONAL PROPERTIES OF BAND SYMMETRIC TOEPLITZ MATRICES , 1983 .

[15]  Raymond H. Chan,et al.  Fast Band-Toeplitz Preconditioners for Hermitian Toeplitz Systems , 1994, SIAM J. Sci. Comput..

[16]  D. Jackson,et al.  The theory of approximation , 1982 .

[17]  Raymond H. Chan,et al.  Toeplitz-Circulant Preconditioners for Toeplitz Systems and their Applications to Queueing Networks with Batch Arrivals , 1996, SIAM J. Sci. Comput..

[18]  N. Ahmed,et al.  Discrete Cosine Transform , 1996 .

[19]  C. Loan.  Computational Frameworks for the Fast Fourier Transform , 1992 .

[20]  U. Grenander,et al.  Toeplitz Forms And Their Applications , 1958 .

[21]  Gene H. Golub,et al.  Matrix computations , 1983 .

[22]  M. J. D. Powell,et al.  On the Maximum Errors of Polynomial Approximations Defined by Interpolation and by Least Squares Criteria , 1967, Comput. J..

[23]  Paola Favati,et al.  On a Matrix Algebra Related to the Discrete Hartley Transform , 1993, SIAM J. Matrix Anal. Appl..

[24]  Stefano Serra,et al.  On the extreme spectral properties of Toeplitz matrices generated byL1 functions with several minima/maxima , 1996 .

[25]  Stefano Serra,et al.  Multigrid methods for toeplitz matrices , 1991 .

[26]  Dario Andrea Bini,et al.  Matrix structures in parallel matrix computations , 1988 .

[27]  Stefano Serra-Capizzano.  Conditioning and solution of Hermitian (block) Toeplitz systems by means of preconditioned conjugate gradient methods , 1995, Optics & Photonics.

[28]  Stefano Serra,et al.  Preconditioning strategies for asymptotically ill-conditioned block Toeplitz systems , 1994 .

[29]  T. Chan.  An Optimal Circulant Preconditioner for Toeplitz Systems , 1988 .

[30]  G. Meinardus.  Approximation of Functions: Theory and Numerical Methods , 1967 .

[31]  O. Axelsson,et al.  On the eigenvalue distribution of a class of preconditioning methods , 1986 .