A unifying approach to the construction of circulant preconditioners (original) (raw)

The main result is the "black dot algorithm" and its fast version for the construction of a new circulant preconditioner for Toeplitz matrices. This new preconditioner C is sought directly as a solution to one of possible settings of the approximation problem A ≈ C + R, where A is a given matrix and R should be a "low-rank" matrix. This very problem is a key to the analysis of superlinear convergence properties of already established circulant and other matrix-algebra preconditioners. In this regard, our new preconditioner is likely to be the best of all possible circulant preconditioners. Moreover, in contrast to several "functionbased" circulant preconditioners used for "bad" symbols, it is constructed entirely from the entries of a given matrix and performs equally as the best of the known or better than those for the same symbols.