A Fast Implementation For GMRES Method Augmented With Error Approximations

As we all know, there are a lot of methods for solving nonsymmetric linear systems. Among these methods, the generalized minimum residual (GMRES) is the most popular choice. It first builds an orthonormal basis thanks to the Arnoldi process. Then a least-squares problem is solved by using Givens rotations.In this work, we first review the GMRES-Aya method, which is similar to the standard GMRES. However, the GMRES-Aya method does not use Givens rotations for solving a least-squares problem. Then we analyse the angle of the sequential and every other residual vectors, i.e., the sequential angles and the skip angles. We observe that the skip angles are much smaller. Generally speaking, small skip angles suggest that the slow convergence phase may be encountered in the iterative process. Therefore, enlightened by the article [3], we add some previous error approximations to the next approximate Krylov subspace, then we can observe that both the skip and sequential angles can be maintained at a relatively large degree, which implies that the convergence effect is better. Finally, we give some numerical examples to compare the new method (LGMRES-Aya) with the LGMRES method and the GMRES-Aya method. The numerical examples show that the new LGMRES-Aya method is more efficient than the LGMRES method and the GMRES-Aya method.