Modified parallel multisplitting iterative methods for non-Hermitian positive definite systems

In this paper we present three modified parallel multisplitting iterative methods for solving non-Hermitian positive definite systems Ax = b . The first is a direct generalization of the standard parallel multisplitting iterative method for solving this class of systems. The other two are the iterative methods obtained by optimizing the weighting matrices based on the sparsity of the coefficient matrix A. In our multisplitting there is only one that is required to be convergent (in a standard method all the splittings must be convergent), which not only decreases the difficulty of constructing the multisplitting of the coefficient matrix A, but also releases the constraints to the weighting matrices (unlike the standard methods, they are not necessarily be known or given in advance). We then prove the convergence and derive the convergent rates of the algorithms by making use of the standard quadratic optimization technique. Finally, our numerical computations indicate that the methods derived are feasible and efficient. Communicated by Charles A. Micchelli. This work is supported by NSF of China (11071184) and NSF of Shanxi Province (2010011006). C.-L. Wang (B) Department of Mathematics, Taiyuan Normal University, Taiyuan 030012, Shanxi Province, People’s Republic of China e-mail: [email protected] G.-Y. Meng Department of Computer Science, Xinzhou Normal University, Xinzhou 034000, Shanxi Province, People’s Republic of China X.-R. Yong Department of Mathematical Sciences, University of Puerto Rico at Mayaguez, Mayaguez, PR 00681-9018, USA