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.

翻译：埃尔曼(Elman)及其因斯塔克(Starke ) 、 艾尔曼(Eiermann)和安永斯特(Ernst)的改善而广为人知的结果(GMRES ), 利用与特定矩阵价值领域及其反向相关的数量,对GMRES最坏的残余规范做出了约束。 我们证明这些界限也符合GMRES的理想近似值,我们得出并讨论这些界限的一些改进。