Under the preconditioned matrix P=I+R, the new USSOR iterative method was put forward. By the theory of matrix, it proved the convergence of the preconditioned USSOR iterative method when the coefficient matrix is a M-matrix or an H-matrix, gave a comparison theorem between the preconditioned USSOR iterative method and the classic USSOR iterative method when the coefficient matrix is a M-matrix, and disclosed precondition iterative method accelerates the convergence speed of the USSOR iterative method. Then a numerical example was used to demonstrate the validity.

Let [WTHX]A[WTBZ]=([WTBX]aij)∈Cn×n, if there exists α∈(0,1) which can make |aii|≥Rαi(A)S1-αi(A)  be right for i∈N={1,2,…,n}, then [WTHX]A[WTBZ] is called an Ostrowski diagonally dominant matrix. We extended the concept to generalized Ostrowski diagonally dominant matrix，and obtained a new criteria conditions for a matrix to be a nonsingular [WTHX]H[WTBZ]-matrix. The theory of Ostrowski diagonally dominant matrix and nonsingular [WTHX]H[WTBZ]-matrix was improved and completed. These conclusions provide strong basis for the research of relative fields, such as computational mathematics, matrix theory, control theory, mathematical economics, etc.

According to several iteration methods for solving large linear system, when coefficient matrix is of α-diagonal strictly dominance, a new upper bound for the spectral radius of the iterative matrices was presented. Parameter estimation for JOR method was discussed. Results are applicable not only for α-diagonal strictly dominance, but also for generalized α-diagonal strictly dominant matrices. The known conclusion was improved. Finally, two numerical examples were given for illustrating advantage of results.