Convergence properties of the symmetric and unsymmetric successive overrelaxation methods and related methods

Abstract

The paper is concerned with variants of the successive overrelaxation method (SOR method) for solving the linear system A u = b Au = b . Necessary and sufficient conditions are given for the convergence of the symmetric and unsymmetric SOR methods when A A is symmetric. The modified SOR, symmetric SOR, and unsymmetric SOR methods are also considered for systems of the form D 1 u 1 − C U u 2 = b 1 , − C L u 1 + D 2 u 2 = b 2 {D_1}{u_1} - {C_U}{u_2} = {b_1}, - {C_L}{u_1} + {D_2}{u_2} = {b_2} where D 1 {D_1} and D 2 {D_2} are square diagonal matrices. Different values of the relaxation factor are used on each set of equations. It is shown that if the matrix corresponding to the Jacobi method of iteration has real eigenvalues and has spectral radius μ ¯ > 1 \bar \mu > 1 , then the spectral radius of the matrix G G associated with any of the methods is not less than that of the ordinary SOR method with ω = 2 ( 1 + ( 1 − μ ¯ 2 ) 1 / 2 ) − 1 \omega = 2{(1 + {(1 - {\bar \mu ^2})^{1/2}})^{ - 1}} . Moreover, if the eigenvalues of G G are real then no improvement is possible by the use of semi-iterative methods.