Greedy Kaczmarz Algorithm Using Optimal Intermediate Projection Technique for Coherent Linear Systems

Abstract

The Kaczmarz algorithm is a common iterative method for solving linear systems. As an effective variant of Kaczmarz algorithm, the greedy Kaczmarz algorithm utilizes the greedy selection strategy. The two-subspace projection method performs an optimal intermediate projection in each iteration. In this paper, we introduce a new greedy Kaczmarz method, which give full play to the advantages of the two improved Kaczmarz algorithms, so that the generated iterative sequence can exponentially converge to the optimal solution. The theoretical analysis reveals that our algorithm has a smaller convergence factor than the greedy Kaczmarz method. Experimental results confirm that our new algorithm is more effective than the greedy Kaczmarz method for coherent systems and the two-subspace projection method for appropriate scale systems. AMS subject classifications: 15A06, 65F10, 65F20, 65F25, 65F50