(untitled) (Advances in Applied Mathematics and Mechanics, 2020)
Two adaptive techniques for choosing relaxation factor, namely, Minimal Residual Relaxation (MRR) and Orthogonal Projection Relaxation (OPR), on basic iterative methods for solving linear systems are proposed. Unlike classic relaxation, in which the optimal relaxation factor is generally difficult to find, in these proposed techniques, non-stationary relaxation factor based on minimal residual or orthogonal projection method is calculated adaptively in each relaxation step with acceptable cost for Jacobi, Gauss-Seidel or symmetric Gauss-Seidel iterative methods. In order to avoid the "stagnation" of the successive locally optimal relaxations, a recipe of inserting several basic iterations between every two adjacent relaxations is suggested and the resulting MRR(m)/OPR(m) strategy is more stable and efficient (here m denotes the number of basic iterations inserted). To solve linear systems with multiple right-hand sides efficiently, block-form relaxation strategies are proposed based on the MRR(m) and OPR(m). Numerical experiments show that the presented MRR(m)/OPR(m) algorithm is more robust and effective than classic relaxation methods. It is also showed that the proposed block relaxation strategies can efficiently accelerate the solution of systems with multiple right-hand sides in terms of total solution time as well as number of iterations.