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The Kernel Polynomial Method is a popular alternative to SLQ, particularly in physics; see e.g. Weiße et al., 2006. As with SLQ, the KPM approximates ϕ(x;A)\phi(x;\vec{A}) by approximating the Chebyshev moments. However, the resulting approximation is now a smooth function (as opposed to a sum of Dirac delta functions). Similar theoretical convergence guarantees are available Chen et al., 2025Chen, 2024.

References
  1. Weiße, A., Wellein, G., Alvermann, A., & Fehske, H. (2006). The kernel polynomial method. Reviews of Modern Physics, 78(1), 275–306. 10.1103/revmodphys.78.275
  2. Chen, T., Trogdon, T., & Ubaru, S. (2025). Randomized Matrix-Free Quadrature: Unified and Uniform Bounds for Stochastic Lanczos Quadrature and the Kernel Polynomial Method. SIAM Journal on Scientific Computing, 47(3), A1733–A1757. 10.1137/23m1600414
  3. Chen, T. (2024). The Lanczos algorithm for matrix functions: a handbook for scientists. https://arxiv.org/abs/2410.11090
Randomized Numerical Linear Algebra with Examples
Stochastic Lanczos Quadrature
Randomized Numerical Linear Algebra with Examples
Sampling-Based Methods