Using ω-circulant matrices for the preconditioning of Toeplitz systems

Toeplitz systems can be solved efficiently by using iterative methods such as the conjugate gradient algorithm. If a suitable preconditioner is used, the overall cost of the method is O(n logn) arithmetic operations. Circulant matrices are frequently employed for the preconditioning of Toeplitz systems. They can be chosen as preconditioners themselves, or they can be used for the computation of approximate inverses. In this article, we take the larger class of ω-circulant matrices instead of the well-known circulants to extend preconditioners of both types. This extension yields an additional free parameter ω which can be chosen in a way that speeds up convergence of the conjugate gradient method. The additional computational effort arising from the use of ω-circulant instead of circulant matrices is low.