Abstract
A good preconditioner is extremely important in order for the conjugate gradients method to converge quickly. In the case of Toeplitz matrices, a number of recent studies were made to relate approximation of functions to good preconditioners. In this paper, we present a new result relating the quality of the Toeplitz preconditionerC for the Toeplitz matrixT to the Chebyshev norm ∥(f− g)/f∥∞, wheref and g are the generating functions forT andC, respectively. In particular, the construction of band-Toeplitz preconditioners becomes a linear minimax approximation problem. The case whenf has zeros (but is nonnegative) is especially interesting and the corresponding approximation problem becomes constrained. We show how the Remez algorithm can be modified to handle the constraints. Numerical experiments confirming the theoretical results are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Ammar and W. Gragg, Superfast solution of real positive definite Toeplitz systems, SIAM J. Matrix Appl. 9 (1988) 61–76.
J. Bunch, Stability of methods for solving Toeplitz systems of equations, SIAM J. Sci. Statist. Comp. 6 (1985) 349–364.
R. Chan and M. Yeung, Circulant preconditioners for Toeplitz matrices with positive continuous generating functions, Math. Comp. 58 (1992) 233–240.
R. Chan and K. Ng, Fast iterative solvers for Toeplitz-plus-band systems, SIAM J. Sci. Statist. Comp., to appear. Also as Research Report, Dept. of Math. HKU, #91-11.
R. Chan and G. Strang, Toeplitz equations by conjugate gradients with circulant preconditioner, SIAM J. Sci. Statist. Comp. 10 (1989) 104–119.
T. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Statist. Comp. 9 (1988) 766–771.
V.F. Demjanov and V.N. Malozemov,Introduction to Minimax (Wiley, New York, 1974).
I. Gohberg and I. Fel'dman,Convolution Equations and Projection Methods for Their Solutions, vol. 41, Transaction of Mathematical Monographs (American Mathematical Society, Rhode Island, 1974).
G. Golub and C. Van Loan,Matrix Computations, 2nd ed. (The Johns Hopkins University Press, Maryland, 1989).
U. Grenander and G. Szegö,Toeplitz Form and Its Applications, 2nd ed. (Chelsea, New York, 1984).
P. Laurent and C. Carasso, An algorithm of successive minimization in convex programming, R.A.I.R.O. Anal. Numer. 12 (1978) 377–400.
A. Oppenheim,Applications of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1978).
G. Strang, A proposal for Toeplitz matrix calculations, Stud. Appl. Math. 74 (1986) 171–176.
P.T.P. Tang, A fast algorithm for linear complex Chebyshev approximation, Math. Comp. 51 (1988) 721–739.
W. Trench, An algorithm for the inversion of finite Toeplitz matrices, SIAM J. Appl. Math. 12 (1964) 515–522.
S.J. Wright, Parallel algorithms for banded linear systems, SIAM J. Sci. Stat. Comp. 12 (1991) 824–842.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chan, R.H., Tang, P.T.P. Constrained minimax approximation and optimal preconditioned for Toeplitz matrices. Numer Algor 5, 353–364 (1993). https://doi.org/10.1007/BF02109196
Issue Date:
DOI: https://doi.org/10.1007/BF02109196