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 in 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。当参考测度的密度支撑在具有端点平方根行为的单个区间上时，我们给出了一个可升级为前向稳定性结果的后向稳定性结论。我们的分析表明，即使在有限精度算术下进行计算，许多大型随机矩阵模型上运行的Lanczos算法实际上是前向稳定的，因此几乎具有确定性。由于Lanczos算法在一般情况下并非前向稳定，这进一步证明了随机矩阵远非"普通矩阵"，因此在用其测试数值算法时必须谨慎。