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 γ. if γ = 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 con vergence rate of these methods are compared by two examples.