Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods (Q1358158)

Collecting the various results in the literature on which kinds of splitting of a matrix of a system of linear equations are connected with certain iteration matrices, the authors derive a necessary and sufficient condition for the existence of a splitting and a non-uniqueness result in the singular case. They then consider alternating iterations with different splittings. The previously established results are used to deduce results on the existence of splittings for the combined iteration and the combined convergence properties. Particular attention is paid to monotone and positive (semi)definite matrices. A final comparison result shows that under certain conditions the combined iteration converges at least as fast as each single iteration.