Addendum to “a note on construction of circulant preconditioners from kernels”

摘要

A unified treatment of construction of block circulant preconditioners for block Toeplitz systems from the viewpoint of kernels has been given [Appl. Math. Comput. 83 (1997) 3]. It was shown that some well-known block circulant preconditioners can be derived from convoluting the generating functions of systems with some famous kernels. A convergence analysis was also given there. For solving a large class of block Toeplitz systems, a linear convergence rate is obtained by using the preconditioned conjugate gradient (PCG) method with block circulant preconditioner. In this addendum, by using a given convergence result [Linear Algebra Appl. 232 (1996) 1], a superlinear convergence rate is obtained. Numerical results are given to illustrate the rate of convergence.