We study the stability of the Lanczos algorithm run on problems whose eigenvector empirical spectral distribution is near to a reference measure with well-behaved orthogonal polynomials. We give a backwards stability result which can be upgraded to a forward stability result when the reference measure has a density supported on a single interval with square root behavior at the endpoints. Our analysis implies the Lanczos algorithm run on many large random matrix models is fact forward stable, and hence nearly deterministic, even when computations are carried out in finite precision arithmetic. Since the Lanczos algorithm is not forward stable in general, this provides yet another example of the fact that random matrices are far from ``any old matrix'', and care must be taken when using them to test numerical algorithms.

翻译：我们研究了在特征向量经验谱分布接近具有良好正交多项式性质的参考测度的问题上运行Lanczos算法的稳定性。我们给出了一个后向稳定性结果，当参考测度的密度支撑在单个区间上且端点处具有平方根行为时，该结果可提升为前向稳定性结果。我们的分析表明，即使计算在有限精度算术下进行，在许多大型随机矩阵模型上运行的Lanczos算法实际上也是前向稳定的，因此几乎是确定性的。由于Lanczos算法通常并非前向稳定，这为随机矩阵远非"任何旧矩阵"这一事实提供了又一个例证，并提醒在使用随机矩阵测试数值算法时必须谨慎。