Usage of the convergence test of the residual norm in the Tsuno-Nodera version of the GMRES algorithm

Tsuno and Nodera proposed a new variant of the GMRES(m) algorithm. Their algorithm is referred to as the GMRES(≥ m_max) algorithm and performs the restart process adaptively, considering the distribution of the zeros of the residual polynomial. However, unless the zeros of the residual polynomial are distributed uniformly, mass is always chosen and their algorithm becomes almost the same as the GMRES(m) algorithm with m = m_max. In this paper, we include a convergence test for the residual norm in the GMRES(≥ m_max) algorithm and propose a new restarting technique based on two criteria. Even if the distribution of zeros does not become uniform, the restart can be performed by using the convergence test of the residual norm. Numerical examples simulated on a Compaq Beowulf computer demonstrate that the proposed technique accelerates the convergence of the GMRES(≥ m_max) algorithm.