The field of values bound on ideal GMRES
  A widely known result of Elman, and its improvements due to Starke, Eiermann and Ernst, gives a bound on the worst-case GMRES residual norm using quantities related to the field of values of the given matrix and its inverse. We prove that these bounds also hold for the ideal GMRES approximation, and we derive and discuss some improvements of the bounds. 
J\"org Liesen, Petr Tich\'y
math.NA cs.NA