Enhance Stability of Successive Over-Relaxation Method and Orthogonalized Symmetry Successive Over-Relaxation in a Larger Range of Relaxation Parameter

The successive over-relaxation method and its symmetric extension to the symmetric successive over-relaxation method inherit the advantages of direct method and iterative method; they are simple iterative algorithms to solve the linear systems. We derive the equivalent forms of successive over-relaxation method and symmetric successive over-relaxation method in terms of residual vector and descent vector. Then a new orthogonalized technique is developed to stabilize the successive over-relaxation and symmetric successive over-relaxation methods. For the orthogonalized successive over-relaxation method, the range of relaxation parameter can be extended, even with a negative value. Based on the maximal projection technique, the sub-optimal value of the relaxation parameter for the orthogonalized successive over-relaxation method is derived to enhance its accuracy; the golden section search algorithm is used to find the minimal point of a derived merit function. The orthogonalized successive over-relaxation and orthogonalized symmetric successive over-relaxation methods show absolute convergence. According to the new form of successive over-relaxation method, a new approach of the accelerated over-relaxation method can be achieved by multiplying the descent vector of the successive over-relaxation method by a stabilization factor. Numerical examples confirm that the orthogonalized successive over-relaxation and orthogonalized symmetric successive over-relaxation methods outperform the successive over-relaxation and symmetric successive over-relaxation methods.