Studies Of The Richardson-iterative Algorithm

Study the convergence of the Richardson scheme is important in theory and practicality. Using the singular value decomposition (SVD), we derive an iterative representation formula for the Richardson scheme and consequently establish the necessary and sufficient conditions for its convergence. In this article we study the relaxation parameter's selection in the Richardson-iterative algorithms, and study the convergence and convergence rate of the iterative algorithms when relaxation parameter is invariable. We establish the necessary and sufficient conditions for its convergence when relaxation parameters in an interval not invariable. Our results allow the relaxation parameters to be complex. Moreover, it is found that the Richardson scheme can converge within finite iterations when the relaxation parameters are chosen to be the inverses of the non-zero singular values. When the relaxation parameters increase monotonously from 1/λ_max to 2/λ_max and the step length is adjusted properly,the irerative convergence rate can be improved.