Volume 65, pp. 271-280, 2026.

Again about the numerical solution of systems of linear algebraic equations with symmetric nonsingular nondefinite matrices by the Lanczos method

Leonid Knizhnerman

Abstract

The Lanczos method for approximating solutions of systems of linear algebraic equations with symmetric nondegenerate (possibly nondefinite) matrices is investigated. We recall our 2002 and 1995 theorems, which contain Cauchy–Hadamard-type estimates in terms of bounded self-adjoint operators in Hilbert spaces and the $m$th root of the residual or the error norm at step $m$; those theorems assert that a “reasonably good” bound holds at least at every other step. Using a slight and easy modification of the old proof, we prove a non-asymptotic upper bound for matrices in terms of the individual minimal residuals; this new result retains the every-other-step formulation. A lower residual bound for the case of even discrete spectral measures is also obtained.

Full Text (PDF) [256 KB], BibTeX , DOI: 10.1553/etna_vol65s271

Key words

Lanczos method, residual, minimal residual method (MINRES), system of linear algebraic equations with symmetric nondefinite matrix, upper and lower residual bounds

AMS subject classifications

65F10

Links to the cited ETNA articles

[3]	Vol. 60 (2024), pp. 421-427 Tyler Chen and Gérard Meurant: Near-optimal convergence of the full orthogonalization method