On the Construction of Semi-Iterative Methods

Given a nonsingular linear system $A{\bf x} = {\bf b}$, a splitting $A = M - N$ leads to the one-step iteration (1) ${\bf x}_m = T{\bf x}_{m - 1} = {\bf c}$ with $T: = M^{ - 1} N$ and $c: = M^{ - 1} {\bf b}$. We investigate semiiterative methods with respect to (1) under the assumption that the eigenvalues of T are contained in some compact set $U \subset \mathbb{C}$, $1 \notin U$. Using results about “maximal convergence” of polynomials and “uniformly distributed” nodes from Approximation Theory we describe semiiterative methods which are asymptotically optimal with respect to U. Under mildly restrictive assumptions on U we construct semiiterative methods which are asymptotically optimal with respect to U and which can be interpreted as one-step nonstationary methods.