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：我们研究了特征向量经验谱分布接近于具有良好正交多项式参考测度的问题中Lanczos算法的稳定性。我们给出了一个向后稳定性结果，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，该结果可提升为向前稳定性结果。我们的分析表明，即使在有限精度算术下执行计算，许多大型随机矩阵模型上的Lanczos算法事实上是向前稳定的，因此几乎具有确定性。由于Lanczos算法通常并非向前稳定，这再次说明随机矩阵远非"任意旧矩阵"，因此在使用它们测试数值算法时必须谨慎。