Intuition for small spectral inflation n_k in Paige’s finite precision Lanczos analysis

Ascertain a conceptual and preferably rigorous explanation for why the inflation parameter n_k in Paige’s finite precision analysis of the Lanczos algorithm, which bounds the spectrum of the computed tridiagonal T_k within [λ_min(A) − n_k, λ_max(A) + n_k], is typically small in practice. Characterize the dependence of n_k on properties of the input (A, b), the iteration count k, and the machine precision εmach to clarify the practical behavior observed in finite precision computations.

Background

Paige’s foundational analysis of the Lanczos algorithm in finite precision arithmetic establishes that the tridiagonal matrix T_k produced by the algorithm has its spectrum contained within a slightly expanded interval around the true spectrum of A: A(T_k) ⊂ [λ_min(A) − n_k, λ_max(A) + n_k], with n_k expected to be small. While the bounds are quantitative, the monograph notes a gap in intuitive understanding of why n_k should be small across practical problems.

Clarifying the mechanisms or conditions that ensure small n_k—e.g., properties of spectral distributions, conditioning of moment-to-tridiagonal maps, or iteration-local error propagation—would improve practical confidence in Lanczos-based methods and provide guidance for algorithmic design and parameter selection in finite precision settings.