Sequential and parallel synchronous alternating iterative methods

The so-called parallel multisplitting nonstationary iterative Model A was introduced by Bru, Elsner, and Neumann [Linear Algebra and its Applications 103:175-192 (1988)] for solving a nonsingular linear system $A\mbox{\mathversion{bold} $x$ } =\mbox{\mathversion{bold} $b$ }$ using a weak nonnegative multisplitting of the first type. In this paper new results are introduced when $A$ is a monotone matrix using a weak nonnegative multisplitting of the second type and when $A$ is a symmetric positive definite matrix using a $P$-regular multisplitting. Also, nonstationary alternating iterative methods are studied. Finally, combining Model A and alternating iterative methods, two new models of parallel multisplitting nonstationary iterations are introduced. When matrix $A$ is monotone and the multisplittings are weak nonnegative of the first or of the second type, both models lead to convergent schemes. Also, when matrix $A$ is symmetric positive definite and the multisplittings are $P$-regular, the schemes are also convergent.