Three Optimization Models For Weighted Multisplitting Preconditioners

In this paper, we study preconditioning techniques for large linear systems, and put forward a kind of methods to decide weighted multisplitting preconditioners. The main work of us is as follows:We use multisplitting matrices with weighting parameters as the preconditioner of A. By approaching to the identity matrix, the optimal weighting parameters are determined, and the scale of approaching is defined by F-norm,2-norm and oo-norm, respectively. Based on these three minimization models, three algorithms are presented and the convergence theories are established. In order to illustrate the general weighting parameters are better than nonnegative weighting parameters, we discuss the boundary of condition number when matrices are combined by nonnegative parameters. Then we find the condition number of convex combining matrices is not necessary smaller than any one of the original matrix. More over, by the relationship of matrix norm and spectral radius, we give the requirements of being nonsingular for weighted multisplit-ting matrix. They are very important to compute the condition number in numerical examples. Finally, numerical examples show that the multisplitting preconditioner with optimal weighting parameters, which are determined from minimizing F-norm,2-norm and∞-orm models, can improve the condition number of A effectively. The multisplitting preconditioners with weighting parameters, which are determined from F-norm and2-norm models, can improve the condition number of A better than any one of multisplitting preconditioners only. But the weighted multisplitting preconditioner which is determined from oo-norm model is not better than some one of multisplit-ting preconditioners. That’s because we add nonnegative constraint to the∞-norm model. It verifies general weighting parameters are better than nonnegative weighting parameters, too.