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Notes and References

The modern treatment of iterative methods dates back to the relaxation method of Southwell [189]. This was the precursor to the SOR method, though the order in which approximations to the unknowns were relaxed varied during the computation. Specifically, the next unknown was chosen based upon estimates of the location of the largest error in the current approximation. Because of this, Southwell's relaxation method was considered impractical for automated computing. It is interesting to note that the introduction of multiple-instruction, multiple data-stream (MIMD) parallel computers has rekindled an interest in so-called asynchronous, or chaotic iterative methods (see Chazan and Miranker [52], Baudet [29], and Elkin [88]), which are closely related to Southwell's original relaxation method. In chaotic methods, the order of relaxation is unconstrained, thereby eliminating costly synchronization of the processors, though the effect on convergence is difficult to predict.

The notion of accelerating the convergence of an iterative method by extrapolation predates the development of SOR. Indeed, Southwell used overrelaxation to accelerate the convergence of his original relaxation method. More recently, the ad hoc SOR method, in which a different relaxation factor is used for updating each variable, has given impressive results for some classes of problems (see Ehrlich [80]).

The three main references for the theory of stationary iterative methods are Varga [206], Young [212] and Hageman and Young [119]. The reader is referred to these books (and the references therein) for further details concerning the methods described in this section.