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

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 rank-deficient matrix $A$ from an initial vector. Theoretical convergence properties of the algorithm are analyzed, and an accuracy bound is derived for the computed approximate null vector. By integrating a deflation technique with a specialized termination criterion, LSQRNV is extended to LSQRNS, which computes an orthonormal basis for the null space $\mathcal{N}(A)$ and explicitly determines the nullity of $A$. A rigorous accuracy bound is established for the resulting 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 computation.