Circulant preconditioners from B-splines

In this paper, we propose a new family of circulant preconditioners for solving Toeplitz systems, They are based on B-splines. The R. Chan and T. Chan preconditioners can be constructed from the first and the second order B-splines. Numerical results show that preconditioners from higher-order B-splines perform much better than well-known ones even in the cases where the Toeplitz matrices are ill-conditioned. Like that of the other circulant preconditioners, the construction of B-spline preconditioners requires only the entries of the given Toeplitz matrix and does not require an apriori knowledge of its generating function. Thus they are most suitable for applications where the generating function of the given Toeplitz matrix is not known explicitly.