A numerical comparison of two minimal residual methods for linear polynomials in unitary matrices

Two minimal residual methods for solving linear systems of the form $(\alpha U+\beta I)x=b$, where $U$ is a unitary matrix, are compared numerically. The first method uses conventional Krylov subspaces, while the second involves generalized Krylov subspaces. Experiments favor the second method if $|\alpha|>|\beta|$. Moreover, the greater the ratio $|\alpha|/|\beta|$, the higher the superiority of the second method.