A convergence theorem of SOR iteration method

For solving a system of linear equations of Ax=b,A into A=M-N is often splited,where M is non-singular.It is known that x(k+1)=M-1Nx(k)+M-1b(k=0,1,2,…) converges to the solution x=A-1b for each x(0) if and only if spectral radius ρ(M-1N)1,the matrix M-1 N is called an iterative matrix.It is easy to see estimates for bounds for ρ(M-1N)are of interest.Some iteration methods for solving linear system are studied,when coefficient matrix is doubly α-diagonal strictly dominance,and a convergence theorem and some corollaries are given.Obtained results are applicable for doubly α-diagonal strictly dominance matrix,and for generalized diagonal strictly dominance matrix also.The results not only is that problem of estimates of upper bounds of the spectral radius for some iteration matrices is solved but also convenient for apply.Finally,two numerical examples are given for illustrating advantage of results.