Filtered Rayleigh-Ritz and the Prony-Ritz equivalence class.

Recent work [Phys. Rev. Lett. 134, 241901 (2025)0031-900710.1103/pcvc-734h; Phys. Rev. D 112, 054506 (2025)10.1103/zjzt-rv86; Phys. Rev. D 112, 014514 (2025)2470-001010.1103/fp74-q35q] has shown that the (block) Lanczos algorithm can be used to extract approximate energy spectra and matrix elements from (matrices of) correlation functions in quantum field theory, and identified exact coincidences between Lanczos analysis methods and others [Eur. Phys. J. A 61, 26 (2025)1434-601X10.1140/epja/s10050-025-01495-8; Phys. Rev. D 112, 074506 (2025)10.1103/x3kb-8zw8]. In this work, we note another coincidence: the Lanczos algorithm is equivalent to the well-known Rayleigh-Ritz method applied to Krylov subspaces. Rayleigh-Ritz provides optimal eigenvalue approximations within Hermitian subspaces, including those arising in the decompositions of non-Hermitian operators describing noisy data. We explore the relation between Lanczos and Prony's method, their block generalizations, generalized pencil of functions (GPOF), and methods based on the generalized eigenvalue problem (GEVP), and find they all fall into a larger "Prony-Ritz equivalence class," identified as all methods which solve a finite-dimensional spectrum exactly given sufficient correlation function (matrix) data. This equivalence allows simpler and more numerically stable implementations of (block) Lanczos analyses.