Parallel multisplitting methods with optimal weighting matrices for linear systems

In this paper, the parallel multisplitting iterative methods with optimal weighting matrices are presented to solve a linear system of equations in which the coefficient matrix is a symmetric positive definite matrix. The zero pattern in weighting matrices is determined by preset set, the non-zero entries of weighting matrices in overlap multisplitting methods are determined optimally by finding the optimal point in hyperplane or convex combination of m points generated by parallel multisplitting iterations. Several schemes of finding the optimal weighting matrices are given. Especially, the nonnegative assumption in the weighting matrices is eliminated. The convergence properties are discussed for these parallel multisplitting methods. Finally, our numerical examples show that these parallel multisplitting methods with the optimal weighting matrices are effective.