New preconditioners for systems of linear equations with Toeplitz structure
The construction of preconditioners for solving systems of linear equations \(T x = b\), where \(T\) is a block Toeplitz matrix with Toeplitz blocks (BTTB matrix) is discussed. At first, level-2 circulant preconditioners based on generalized Jackson kernels are considered. This type of preconditioners are block circulant matrices with circulant blocks. The presented preconditioners are an extension of circulant preconditioners proposed by \textit{R. H. Chan} et al. [SIAM J. Numer. Anal. 38, No. 3, 876--896 (2000; Zbl 0978.65035)] and [Numer. Math. 92, No. 1, 17--40 (2002; Zbl 1007.65033)]. A second kind of preconditioners is derived from a splitting of BTTB matrices. The proposed BTTB preconditioners are applied to solve BTTB least squares problems. For illustrating the efficiency of the presented preconditioners, three numerical examples are given. The new preconditioners are compared with preconditioners known from the literature.
block circulant matrix with circulant blocks
block Toeplitz matrix with Toeplitz blocks
preconditioned conjugate gradient method
scientific article; zbMATH DE number 6418666
scientific article