Circulant Preconditioners for Toeplitz Matrices with Positive continuous Generating Functions
Copyright policy )Circulant Preconditioners for Toeplitz Matrices with Positive continuous Generating Functions
We consider the solution of n-by-n Toeplitz systems A n x = b {A_n}x = b by the preconditioned conjugate gradient method. The preconditioner C n {C_n} is the circulant matrix that minimizes ‖ B n − A n ‖ F {\left \| {{B_n} - {A_n}} \right \|_F} over all circulant matrices B n {B_n} . We show that if the generating function f is a positive 2 π 2\pi -periodic continuous function, then the spectrum of the preconditioned system C n − 1 A n C_n^{ - 1}{A_n} will be clustered around one. In particular, if the preconditioned conjugate gradient method is applied to solve the preconditioned system, the convergence rate is superlinear.
Iterative numerical methods for linear systems, Toeplitz systems, preconditioned conjugate gradient method, Numerical computation of matrix norms, conditioning, scaling, circulant preconditioner matrices, superlinear convergence, positive continuous generating functions
Iterative numerical methods for linear systems, Toeplitz systems, preconditioned conjugate gradient method, Numerical computation of matrix norms, conditioning, scaling, circulant preconditioner matrices, superlinear convergence, positive continuous generating functions
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