一种基于LSQR的大规模零空间计算算法
An LSQR-based algorithm for large-scale null space computations
AI总结：
提出基于LSQR的算法LSQRNV计算大型亏秩矩阵零向量，分析其收敛性并给出精度界。将其扩展为LSQRNS计算零空间正交基并确定零度，实验验证了算法对大规模零空间计算的有效性。
摘要：
计算大规模矩阵的零空间和零向量是数值线性代数和科学计算中的基本任务。本文提出一种基于LSQR的算法LSQRNV，从初始向量计算大型亏秩矩阵\(A\)的零向量。分析了算法的理论收敛性质，并为计算得到的近似零向量推导了精度界。通过将收缩技术与专门的终止准则相结合，将LSQRNV扩展为LSQRNS，它计算零空间\(\mathcal{N}(A)\)的正交基并明确确定\(A\)的零度。为所得的近似数值零空间建立了严格的精度界。此外，通过适当的参数设置，LSQRNV有效地确定大型矩阵在数值上是亏秩还是具有满列秩。数值实验证实了理论结果，证明了LSQRNS在大规模零空间计算中的稳健性、效率和有效性。
英文摘要：
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.
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