If the numerical range of a matrix is contained in the right half-plane, the GMRES algorithm for solving linear systems will make progress at every iteration. In his Ph.D. dissertation, Howard Elman derived a bound that guarantees convergence. When all eigenvalues are in the right half-plane but the numerical range contains the origin, GMRES need not make progress at every step, and Elman's bound does not apply. By solving a Lyapunov equation, one can construct an inner product in which the numerical range is contained in the right half-plane. One can bound GMRES (run in the standard Euclidean norm) by applying Elman's bound in this new inner product, at the cost of a multiplicative constant that characterizes the distortion caused by the change of inner product. Using Lyapunov inverse iteration, one can build a family of suitable inner products, trading off the location of the numerical range with the size of constant. This approach complements techniques recently proposed by Greenbaum and colleagues for eliminating the origin from the numerical range for GMRES convergence analysis.

翻译：若矩阵的数值范围包含于右半平面，则在求解线性系统时，GMRES算法每次迭代均能取得进展。Howard Elman在其博士论文中推导出一个保证收敛性的界。当所有特征值位于右半平面但数值范围包含原点时，GMRES未必能步步推进，此时埃尔曼界不再适用。通过求解李雅普诺夫方程，可以构造一个内积，使得在该内积下数值范围包含于右半平面。通过在新内积中应用埃尔曼界，并引入一个表征内积变换导致的畸变程度的乘法常数，即可对（标准欧氏范数下运行的）GMRES进行界定。利用李雅普诺夫逆迭代，可以构建一族合适的内积，在数值范围位置与常数大小之间进行权衡。该方法对Greenbaum等人近期提出的、通过消除数值范围中的原点以分析GMRES收敛性的技术形成了补充。