An iterative method for the positive real linear system

In this paper 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. The iterative method contains an auxiliary matrix D-1. It is shown that by proper choice of D-1 the new iterative method is convergent. It is also shown that by special choice of DI, the new iterative method becomes the well-known (point) successive overrelaxiation (SOR) [1] method. Hence, it is shown that the (point) SOR method applied to the positive real system is convergent if the overrelaxiation parameter omega is in (0, omega (U)). The upper bound omega (U) is also given in terms of the norm and smallest eigenvalue of related matrices (see Eq. (23)).