Circulant preconditioners with unbounded inverses: Why non-optimal preconditioners may possess a better quality than optimal ones

• Main Author: Tyrtyshnikov, E.E.
• Format: Conference Proceeding
• Language: English
• Published: United States 31.12.1994
• Subjects: CONVERGENCE; ITERATIVE METHODS; MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; MATRICES
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/0024-3795(93)00092-E

There exist several preconditioning strategies for systems of linear equations with Toeplitz coefficient matrices. The most popular of them are based on the Strang circulants and the Chan optimal circulants. Let A-n be an n-by-n Toeplitz matrix. Then the Strang preconditioner S-n copies the central n/2 diagonals of A-n while other diagonals are determined by the circulant properties of S-n. The Chan circulant C-n coincides with the minimizer of the deviation A-n - C-n in the sense of the matrix Frobenius norm. At the first glance the Chan circulant should provide a faster convergence rate since it exploits more information on the coefficient matrix. The preconditioning quality is heavily dependent on clusterization of the preconditioned eigenvalues. According to recent results by R. Chan it is known that both considered circulants possess the clustering property if the coefficient Toeplitz matrices A-n are generated by a function which first belongs to the Wiener class and second is separated from zero. Both circulants provide approximately the same clustering rate, and therefore both should possess the same preconditioning quality. However, the most interesting case is the one when the generating function may take the zero value, and hence the circulants have unbounded in n inverses. In these cases the Strang preconditioners may appear to be singular and we recommend to use the so called improved Strang preconditioners (in which a zero eigenvalue of the Strang circulant is replaced by some positive value).