Improvement of GMRES algorithm using various updated QR decompositions

Abstract

The GMRES is one of the iteration methods how to solve linear equations.Especially, it is effective for nonsingular, nonsymetric, sparse matrix.Full GMRES is a typical method as increase n of $Q_n$, $H_n$ one by one until $\|b-A\cdot x_n\|$ is sufficiently small. But, sizes of $Q_n$, $\widetilde{H_n}$ are changed and bigger than previous size. we need to preserve all data in each iteration. So, many memories are needed for saving data. the speed of calculation is increasingly slow. So we try to restarted GMRES that runs in Krylov space with fixed dimension. There is a Hessenberg matrix in Anoldi iteration. So, we studied various QR decompositions that can solve $\widetilde{H_n}\cdot y_n=B$. Since it is a Hessenberg form, we change the algorithm of standard QR decomposition. And also, we try to combine the QR decomposition algorithm in Anoldi process.