An Auto-Tuning Technique of the Weighted Jacobi-Type Iteration Used for Preconditioners of Krylov Subspace Methods

The Jacobi iteration is often used for preconditioners with high parallel efficiency of Krylov subspace methods to solve very large linear systems. However, these preconditioners do not always show great improvement of the convergence rate, because of the strict convergence condition and the poor convergence property of the Jacobi iteration. In order to resolve this difficulty, we recently introduced the weighted Jacobi-type iteration which has a weight parameter and a scaling diagonal matrix, and proposed the optimization technique for its weight parameter. As its efficient development, in this paper, we propose an auto-tuning technique not only for the weight parameter but also for the scaling diagonal matrix of the weighted Jacobi-type iteration used for preconditioners. The numerical experiments indicate that our auto-tuning technique is well played to solve very large linear systems.