We study the GMRES algorithm applied to linear systems of equations involving a scaled and shifted $N\times N$ matrix whose entries are independent complex Gaussians. When the right hand side of this linear system is independent of this random matrix, the $N\to\infty$ behavior of the GMRES residual error can be determined exactly. To handle cases where the right hand side depends on the random matrix, we study the pseudospectra and numerical range of Ginibre matrices and prove a restricted version of Crouzeix's conjecture.

翻译：我们研究将GMRES算法应用于涉及一缩放平移$N\times N$矩阵（其元素为独立复高斯随机变量）的线性方程组。当该线性系统的右端项独立于该随机矩阵时，GMRES残差误差的$N\to\infty$行为可以精确确定。为处理右端项依赖于随机矩阵的情形，我们研究Ginibre矩阵的伪谱和数值范围，并证明了Crouzeix猜想的一个受限版本。