  When a linear system Ax = y is solved by means of iterative methods (mainly
CG and GMRES) and the convergence rate is slow, one may consider a
preconditioner P. The use of such preconditioner changes the spectrum of the
matrix defining the system and could result into a great acceleration of the
convergence rate. The construction of optimal rank preconditioners is strongly
related to the possibility of splitting A as A = P + R + E, where E is a small
perturbation and R is of low rank. In the present work we extend the black-dot
algorithm for the computation of such splitting for P circulant, to the case
where P is in L, for several known low-complexity matrix algebras L. The
algorithm so obtained is particularly efficient when A is Toeplitz plus Hankel
like. We finally discuss in detail the existence and the properties of the
decomposition A = P + R + E when A is Toeplitz, also extending to the
phi-circulant and Hartley-type cases some results previously known for P
circulant.
