Abstract
The solutions of symmetric positive definite Toeplitz systems Ax = b are studied by the preconditioned conjugate gradient method. The preconditioner is the circulant matrix C that minimizes the Frobenius norm ‖C - A‖F [T. Chan, “An Optimal Circulant Preconditioner for Toeplitz Systems,” UCLA Department of Mathematics, CAM Report 87-06, June 1987]. The convergence rate of these iterative methods is known to depend on the distribution of the eigenvalues of C-1A. For Toeplitz matrix A with entries which are Fourier coefficients of a positive function in the Wiener class, this paper establishes the invertibility of C, finds the asymptotic behaviour of the eigenvalues of the preconditioned matrix C-1A as the dimension increases and proves that they are clustered around 1.
Original language | English |
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Pages (from-to) | 503-506 |
Number of pages | 4 |
Journal | SIAM Journal on Numerical Analysis |
Volume | 26 |
Issue number | 2 |
DOIs | |
Publication status | Published - Apr 1989 |
Externally published | Yes |
Keywords
- Toeplitz matrix
- circulant matrix
- preconditioned conjugate gradient method