Note to the mixed-type splitting iterative method for the positive real linear system

摘要

In [5] a new iterative method is given for the linear system of equations Au=b , where A is large, sparse and nonsymmetrical and A(T)+A is symmetric and positive definite (SPD) or equivalently A is positive real. The new iterative method is based on a mixed-type splitting of the matrix A and is called the mixed-type splitting iterative method. The iterative method contains an auxiliary matrix D-1 that is restricted to be symmetric. In this note, the auxiliary matrix is allowed to be more general and it is shown that by proper choice of D-1 , the new iterative method is still convergent. It is also shown that by special choice of D-1, the new iterative method becomes the well-known (point) accelerated overrelaxation (AOR) [1] method. Hence, it is shown that the (point) AOR method applied to the positive real system is convergent under the proper choice of the overrelaxation parameters omega and tau.