标题： 改进多项式界并加速GMRES：通过求解矩形上的min-max问题以及通过去除低秩部分 显示英文标题

标题： Improved Polynomial Bounds and Acceleration of GMRES by Solving a min-max Problem on Rectangles, and by Deflating

摘要： 对于左预处理、右预处理和分裂预处理的广义极小残量法 (GMRES)，考虑了多项式收敛界。这些界包含了加权 GMRES 和降阶 GMRES 的情形，适用于线性方程组 Ax = b。特别是，当 A 是正定矩阵时的情况被研究。众所周知的多项式界被推广到所考虑的情形，并简化为在复平面上的矩形上求解一个极小极大问题。讨论并比较了几种方法。通过使用特定的降阶空间和预条件子，可以改进新的界，从而加速 GMRES 的收敛。数值例子验证了所得结果。 显示英文摘要

摘要： Polynomial convergence bounds are considered for left, right, and split preconditioned GMRES. They include the cases of Weighted and Deflated GMRES for a linear system Ax = b. In particular, the case of positive definite A is considered. The well-known polynomial bounds are generalized to the cases considered, and then reduced to solving a min-max problem on rectangles on the complex plane. Several approaches are considered and compared. The new bounds can be improved by using specific deflation spaces and preconditioners. This in turn accelerates the convergence of GMRES. Numerical examples illustrate the results obtained.