Two Kaczmarz Algorithms Using Optimal Intermediate Projection Technique

In recent years,the research that Kaczmarz type algorithms solve large-scale linear systems has received extensive attention.In such algorithms,the use of greedy selection strategy is a common improvement idea,where both the theory and experiments show significant acceleration effect.In addition,the two-subspace projection method is also an effective improved algorithm,in which the optimal intermediate projection technique plays an important role.In order to organically combine the two improved techniques,this paper proposes an algorithm called greedy Kaczmarz algorithm using optimal intermediate projection technique.Through theoretical analysis,it can be seen that the iterative sequence generated by the algorithm can converge to the optimal solution,and the convergence factor of the algorithm is smaller than that of the greedy Kaczmarz method.Experimental results show that the algorithm is superior to the greedy Kaczmarz method when solving coherent linear systems.Meanwhile,the new algorithm is more effective than the two-subspace projection method when solving some moderately coherent linear systems.In addition,some greedy randomized algorithms which showed good results in both theory and experiments have been proposed successively.Among them,the relaxation stochastic greedy Kaczmarz algorithm is a representative algorithm.In this paper,we propose another new by applying the optimal intermediate projection technique to the relaxation stochastic greedy Kaczmarz algorithm.The effectiveness of the new algorithm be proved from the corresponding theoretical convergence analysis and numerical experiments given in this paper.