An LSQR-based algorithm for large-scale null space computations

Abstract

Computing the null space and null vectors of large-scale matrices is a fundamental task in numerical linear algebra and scientific computing. In this paper, an LSQR-based algorithm, termed LSQRNV, is proposed to compute a null vector of a large-scale rank-deficient matrix $A$ from an initial vector. The theoretical convergence properties of the algorithm are analyzed, demonstrating that it converges to a numerical null vector of $A$ at a rate dictated by its numerical condition number, and a rigorous accuracy bound is derived for the resulting approximation. By integrating a deflation technique with a tailored termination criterion, LSQRNV is extended to LSQRNS, which computes an orthonormal basis for the numerical null space of $A$ and explicitly determines its nullity. The aforementioned accuracy bound is rigorously generalized to the computed approximate numerical null space. Furthermore, with appropriate parameter settings, LSQRNV efficiently determines whether a large matrix is numerically rank-deficient or has full column rank. Numerical experiments corroborate the theoretical results, demonstrating the robustness, efficiency, and effectiveness of LSQRNS for large-scale null-space computations.