The behavior of symmetric Krylov subspace methods for solving Mx=(M−γI)v Original Research Article

We analyze the behavior of Krylov subspace methods for the solution of the symmetric system Mx=(M−γI)v when γ 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.