A widely known result of Howard Elman, and its improvements due to Gerhard Starke, Michael Eiermann and Oliver Ernst, gives a bound on the (worst-case) GMRES residual norm using quantities related to the field of values of the given matrix and of its inverse. In this note we give a simple and direct proof that these bounds also hold for the ideal GMRES approximation. Our work was motivated by a question of Otto Strnad, a student at the Charles University in Prague.

翻译：霍华德·埃尔曼(Howard Elman)的一个众所周知的结果及其因格哈德·斯塔克(Gerhard Starke)、迈克尔·艾尔曼(Michael Eiermann)和奥利弗·恩斯特(Oliver Ernst)而带来的改进,在使用与特定矩阵及其反面价值领域有关的数量时,对GMRES残留规范(最差的情况)作出了约束。在本说明中,我们简单和直接地证明这些界限也符合GMRES的理想近距离。我们的工作受到布拉格查尔斯大学学生Otto Strnad问题的启发。