ABSTRACT
The use of circulant preconditioners is therefore good from the point of view of computational costs. In the real symmetric case, a sufficient condition is a cluster in the spectrum of the preconditioned system. Under appropriate hypotheses, some useful theorems guarantee that the underlying preconditioners work. Providing preconditioners from the kernel comes directly from classical Fourier analysis and needs the convolution of two functions. Another approach based on circulant matrices is the so-called superoptimal preconditioner. The word superoptimal was chosen because the former was born to be a competitor for T. Chan's optimal preconditioner. The simple existence of clusters of eigenvalues for the underlying sequence of preconditioned systems cannot be enough for fast convergence. In many cases, the complex entries of the matrix are localized, for example, on the main diagonal, thus many preconditioning strategies applied directly to the complex linear system will have the undesirable effect of spreading nonreal entries in the preconditioning matrix.
