We analyze randomized matrix-free quadrature algorithms for spectrum and spectral sum approximation. The algorithms studied include the kernel polynomial method and stochastic Lanczos quadrature, two widely used methods for these tasks. Our analysis of spectrum approximation unifies and simplifies several one-off analyses for these algorithms which have appeared over the past decade. In addition, we derive bounds for spectral sum approximation which guarantee that, with high probability, the algorithms are simultaneously accurate on all bounded analytic functions. Finally, we provide comprehensive and complimentary numerical examples. These examples illustrate some of the qualitative similarities and differences between the algorithms, as well as relative drawbacks and benefits to their use on different types of problems.

翻译：本文分析了用于谱与谱和逼近的随机化无矩阵求积算法。所研究的算法包括核多项式方法与随机Lanczos求积——这是两种广泛应用于此类任务的算法。我们对谱逼近的分析统一并简化了过去十年间针对这些算法出现的若干独立分析。此外，我们推导了谱和逼近的界，这些界保证了算法在所有有界解析函数上具有高概率的同步精度。最后，我们提供了全面且互补的数值算例。这些算例展示了算法之间某些定性的相似性与差异性，以及在处理不同类型问题时各自相对的优缺点。