Multisplitting of a Symmetric Positive Definite Matrix

Parallel iterative methods are studied, and the focus is on linear algebraic systems whose matrix is symmetric and positive definite. The set of unknowns may be viewed as a union of subsets of unknowns (possibly with overlap). The parallel iteration matrix is then formed by a weighted sum of iteration matrices that are associated with splittings of the matrix corresponding to the blocks. When the blocks are from a matrix in dissection form, it can be shown under suitable conditions that the parallel algorithm is convergent. When the multisplitting version of successive over-relaxation (SOR) is used, the SOR parameter is required to be less than $\omega _0 < 2.0$. Calculations done on the Alliant FX/8 multiprocessing/vector computer indicate speedups of nine to ten.