The behavior of symmetric Krylov subspace methods for solving M x= (M - [gamma] I)v

We analyze the behavior of Krylov subspace methods for the solution of the symmetric system Mx = (M - gammaI)upsilon when gamma is close to some of the extreme eigenvalues of M. We show that a stagnation phase may occur if the structure of the right-hand side is not taken into account, and we analyze the occurrence and persistence of such stagnation. A natural alternative strategy is proposed and we show that the new approach provides a better approximation, with the same number of matrix-vector multiplications. Numerical experiments are also included.