IGMRES method for linear systems

The Index Generalized Minimal RESidual (IGMRES) algorithm is designed to compute the Drazin-inverse solution of a linear system of equations $Ax=b$, where $A$ is an arbitrary square matrix with index $\gamma$. If $\gamma=0$, then the this method method coincide with Generalized Minimal RESidual (GMRES) method. Also, the {$k^{th}$} ideal index generalized minimal residual polynomial of $A$ is introduced and the roots of these polynomials are studied. Moreover, by numerical results the convergence rate of these methods are compared by two examples.