For the iterative solution of linear systems of algebraic equations Ax = b(1), with and det(A) ≠ 0, numerous methods exist. Although a classification of them seems not to be possible one may note that the first step for the construction of an iterative method usually begins with a splitting of A in (1). Thus A is written as A = M − N, where det(M) ≠ 0 and M is easily inverted, so that (1) is equivalent to x = Tx + c (2), T ≔ M−1N, c ≔ M−1. The discussion will be restricted to the so-called linear (stationary) iterative methods, although some nonstationary ones will be mentioned, and emphasis will be given to those which fall into the categories of extrapolation, relaxation and similar ones. For this it will be assumed that the spectrum σ(T) of T in (2) is contained in a well-defined compact region R, whose complement with respect to the complex plane is simply connected, and 1 ∉ R. Under these assumptions and for each specific class of methods described each time an attempt will be made to present the ‘optimum’ one. The optimum is the one out of the class of methods for which the sequence of vectors yielded converges asymptotically to the unique solution of (1) as fast as possible.
