We study randomized matrix-free quadrature algorithms for spectrum and spectral sum approximation. The algorithms studied are characterized by the use of a Krylov subspace method to approximate independent and identically distributed samples of $\mathbf{v}^{\mathsf{H}} f(\mathbf{A}) \mathbf{v}$, where $\mathbf{v}$ is an isotropic random vector, $\mathbf{A}$ is a Hermitian matrix, and $f(\mathbf{A})$ is a matrix function. This class of algorithms includes the kernel polynomial method and stochastic Lanczos quadrature, two widely used methods for approximating spectra and spectral sums. Our analysis, discussion, and numerical examples provide a unified framework for understanding randomized matrix-free quadrature algorithms and sheds light on the commonalities and tradeoffs between them. Moreover, this framework provides new insights into the practical implementation and use of these algorithms, particularly with regards to parameter selection in the kernel polynomial method.

翻译：我们研究的是光谱和光谱和近似光谱的无矩阵二次方程算法。所研究的算法的特征是使用Krylov 子空间方法来估计独立和同样分布的 $\ mathbf{v ⁇ mathsf{H} f(mathbf{A}})\ mathbf{f{v}$, 其中$\mathbf{v}$是一个异热带随机矢量, $\mathbf{A}$是Hermitian 矩阵, $f(mathbf{A}) 是一个矩阵函数。这一算法类别包括内核多球多角度法和随机的兰氏二次方形样本,这是两种广泛使用的相近光谱光谱和光谱等方法。 我们的分析、讨论和数字示例提供了一个统一框架, 用以理解随机的无基矩阵四方算算法, 以及它们之间的共性和交替性。 此外, 这个框架为这些算法的实际实施和使用提供了新的洞察, 以及这些矩阵的参数选择方法。