The kernel polynomial method (KPM) is a powerful numerical method for approximating spectral densities. Typical implementations of the KPM require an a prior estimate for an interval containing the support of the target spectral density, and while such estimates can be obtained by classical techniques, this incurs addition computational costs. We propose an spectrum adaptive KPM based on the Lanczos algorithm without reorthogonalization which allows the selection of KPM parameters to be deferred to after the expensive computation is finished. Theoretical results from numerical analysis are given to justify the suitability of the Lanczos algorithm for our approach, even in finite precision arithmetic. While conceptually simple, the paradigm of decoupling computation from approximation has a number of practical and pedagogical benefits which we highlight with numerical examples.

翻译：核多项式方法（KPM）是近似谱密度的强大数值方法。KPM的典型实现需要预先估计包含目标谱密度支撑的区间，虽然可以通过经典技术获得此类估计，但这会带来额外的计算成本。我们提出了一种基于无重正交化Lanczos算法的谱自适应KPM，该方法允许将KPM参数的选择推迟到高成本计算完成之后。数值分析的理论结果证明了Lanczos算法在我们方法中的适用性，即使在有限精度算术下也是如此。虽然概念上简单，但这种将计算与近似解耦的范式具有许多实践和教学上的优势，我们通过数值示例进行了强调。