Paige-style finite precision stability analysis for block Lanczos

Establish a Paige-style finite precision stability result for the block Lanczos algorithm by proving that a perturbed version of the exact block three-term recurrence AQ_k = Q_k T_k + E_{k-1} Q_k e_1^T holds when the algorithm is executed in floating point arithmetic. Derive explicit bounds on the perturbation and related quantities, analogous to the classical stability guarantees known for standard (single-vector) Lanczos.

Background

The monograph highlights that, unlike the well-understood finite precision behavior of the standard Lanczos algorithm, the block Lanczos method lacks a Paige-style analysis guaranteeing perturbed recurrence relations and quantifying stability. This gap is notable given widespread practical use of block methods for matrix functions.

Providing such a result would unify theory between single-vector and block variants, offer rigorous guarantees for block Krylov subspace computations, and help guide robust implementations that handle deflation, rank deficiencies, and loss of orthogonality in finite precision arithmetic.