An investigation of restarted GMRES method by using flexible starting vectors

We discuss a variant of restarted GMRES method that allows changes of the restarting vector at each cycle of iterations. The merit of the variant is that previously generated information can be utilized to select a new starting vector, such that the occurrence of stagnation be mitigated or the convergence be accelerated. The more appealing utilization of the new method is in conjunction with a harmonic Ritz vector as the starting vector, which is discussed in detail. Numerical experiments are carried out to demonstrate that the proposed procedure can effectively mitigate the occurrence of stagnation due to the presence of small eigenvalues in modulus.