[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82734-en":3,"doc-seo-82734-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},82734,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","An LSQR-Based Algorithm for Large-Scale Null Space Computations","Computing null spaces and null vectors for large, rank-deficient matrices is central to numerical linear algebra and scientific computing. This paper introduces an LSQR-based method (LSQRNV) that computes a numerical null vector from an initial vector, with convergence controlled by the numerical condition number and a rigorous accuracy bound for the approximation. A deflation strategy and termination rule extend it to LSQRNS, yielding an orthonormal basis, the nullity, and generalized accuracy guarantees. Proper parameters also enable reliable numerical rank-deficiency tests, supported by robust experiments.","arXiv :2607 .03341v2 [math .NA] 8 Jul 2026  \nAN LSQR-BASED ALGORITHM FOR LARGE-SCALE NULL SPACE  \nCOMPUTATIONS∗  \nJINZHI HUANG†  \nAbstract. 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 computesan 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.  \nKey words. null vector, null space, nullity, rank, LSQR algorithm, convergence, deflation MSC codes. 65F10, 15A03, 15A06  \n1. Introduction. Null space and null vectors are intrinsic properties of a matrix, and they play important roles in a variety of applications, such as determining the rank and range space of a matrix [11, 22], conducting linear discriminant analysis [15, 24], developing torque-controlled and autonomous robots [6 , 19], mitigating catastrophic forgetting in continual learning [25, 26], and solving linear inverse problems in image restoration [14, 23] .  \nFor a general matrix A ∈ RM ×N , Coleman and Pothen [3 , 4] propose a pair of two-stage algorithms that utilize bipartite matching to compute the fundamental and triangular bases for its null space N(A) . Gilbert and Heath [8] develop several QR factorization-based algorithms to construct sparse bases for N(A) . However, the exorbitant computational and storage demands of these algorithms render them prohibitive for large-scale matrices. In response, various solvers tailored to specific matrix structures have emerged. For instance, Berry et al. [1] propose an algorithm based on Gaussian elimination and orthogonal factorization for banded matrices. Gotsmanand Toledo [12] employ sparse LU factorization with partial pivoting for rectangular matrices with small nullity. Foster and Davis [7] employ rank-revealing sparse QR factorization for matrices with very small or large nullity, whereas Park and Nakatsukasa [21] design sketch-and-solve methods for tall and skinny matrices. Crucially, all these methods are based on certain factorizations of A or relevant matrices, restricting their practical applicability primarily to matrices with particular structures. For a general large-scale sparse matrix A, Kressner and Shao [18] propose a randomized small-block Lanczos method that computes an orthonormal basis for N(A) by applying the block Lanczos algorithm [10] to ATA. However, working with the cross-product matrix ATA inherently squares the singular values of A. In the presence of moderately small sin-  \n∗ Submitted to the editors DATE.  \nFunding: This work was supported by the Youth Fund of the National Science Foundation of China (Grant No. 12301485) and the Jiangsu Province Youth Science and Technology Talent Support Program (Grant No. JSTJ-2025-828) .  \n†School of Mathematical Sciences, Soochow University, 215006 Suzhou, China (jzhuang21@suda. [edu.cn](edu.cn)).  \n2 JINZHI HUANG  \ngular values of A, this squaring effect clusters the eigenvalues of ATA densely near the origin, which not only degrades the convergence rate but also obscures the num","cbCaicvXQMT2zJrH","https://ap.wps.com/l/cbCaicvXQMT2zJrH","pdf",6035280,2,1,22,"English","en",105,"# Introduction\n## Motivation and prior approaches\n## Proposed LSQR-based framework","[{\"question\":\"What does LSQRNV compute in this paper?\",\"answer\":\"LSQRNV computes a null vector of a large-scale rank-deficient matrix A from an initial vector.\"},{\"question\":\"How is the convergence behavior of LSQRNV characterized?\",\"answer\":\"The convergence is analyzed theoretically and shown to approach a numerical null vector at a rate determined by the numerical condition number, accompanied by a rigorous accuracy bound.\"},{\"question\":\"What additional capability does LSQRNS provide compared with LSQRNV?\",\"answer\":\"LSQRNS integrates deflation and a tailored termination criterion to produce an orthonormal basis for the numerical null space and to explicitly determine the nullity, with accuracy bounds generalized to the computed basis.\"}]",1784182569,55,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"an-lsqr-based-algorithm-for-large-scale-null-space-computations","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/an-lsqr-based-algorithm-for-large-scale-null-space-computations/82734/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does LSQRNV compute in this paper?","Question",{"text":75,"@type":76},"LSQRNV computes a null vector of a large-scale rank-deficient matrix A from an initial vector.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the convergence behavior of LSQRNV characterized?",{"text":80,"@type":76},"The convergence is analyzed theoretically and shown to approach a numerical null vector at a rate determined by the numerical condition number, accompanied by a rigorous accuracy bound.",{"name":82,"@type":73,"acceptedAnswer":83},"What additional capability does LSQRNS provide compared with LSQRNV?",{"text":84,"@type":76},"LSQRNS integrates deflation and a tailored termination criterion to produce an orthonormal basis for the numerical null space and to explicitly determine the nullity, with accuracy bounds generalized to the computed 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