Preconditioners for ill { conditioned Toeplitz matrices
نویسندگان
چکیده
This paper is concerned with the solution of systems of linear equations ANx = b, where fANg N2N denotes a sequence of positive deenite Hermitian ill{conditioned Toeplitz matrices arising from a (real{valued) nonnegative generating function f 2 C2 with zeros. We construct positive deenite Hermitian preconditioners MN such that the eigenvalues of M ?1 N AN are clustered at 1 and the corresponding PCG{method requires only O(N log N) arithmetical operations to achieve a prescribed precision. We sketch how our preconditioning technique can be extended to symmetric Toeplitz systems, doubly symmetric block Toeplitz systems with Toeplitz blocks and non{ Hermitian Toeplitz systems. Numerical tests connrm the theoretical expectations.
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