[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85205-en":3,"doc-seo-85205-105":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},85205,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","Sketch-and-Restart: Randomized Sketching in Quadrature-Based Restarting for Matrix Functions","Develops a sketch-and-restart framework for computing the matrix function action f(A)b with A large, sparse, and non-Hermitian. The approach unifies quadrature-based restarting with Arnoldi-like decompositions from sketched or truncated Arnoldi processes, yielding restarted algorithms with either fixed or adaptively chosen Krylov subspace size. It introduces a new sketched harmonic Arnoldi variant and proves convergence for restarted methods for Stieltjes functions when A is positive real, validated by experiments on savings from sketching, reduced storage, and thick restarting acceleration.","arXiv :2607 . 10354v1 [math .NA] 11 Jul 2026  \nSKETCH-AND-RESTART: RANDOMIZED SKETCHING IN  \nQUADRATURE-BASED RESTARTING FOR MATRIX FUNCTIONS  \nSTEFAN G¨UTTEL∗ , JINGYU LIU†, AND LAURI NYMAN‡  \nAbstract. We develop a sketch-and-restart framework for computing the action of a matrix function on a vector, f (A)b, where A is large, sparse, and non-Hermitian. The framework combines quadrature-based restarting with Arnoldi-like decompositions generated by sketched or truncated Arnoldi processes. Within this framework, we develop two classes of restarted algorithms. The first uses a fixed Krylov subspace dimension and is based either on the sketched Arnoldi process or on anew sketched harmonic Arnoldi process proposed in this work. The second class chooses the Krylov subspace dimension adaptively by running the truncated Arnoldi process until the condition number of the generated basis, estimated from its sketch, exceeds a prescribed threshold. We also establish the convergence of the restarted sketched harmonic Arnoldi method for Stieltjes functions under the assumption that A is positive real. Numerical experiments demonstrate the effectiveness of the proposed framework, including the computational savings achieved through sketching, the storage reduction enabled by adaptive truncation, and the acceleration obtained from thick restarting.  \nKey words. matrix function, Krylov method, sketching, restarting  \nAMS subject classifications. 65F60, 65F50, 68W20  \n1. Introduction. This paper considers the computation of f (A)b, where A ∈ CN ×N is a matrix, b ∈ CN is a vector, and f is a suitable scalar function. Such computations arise in many areas of scientific computing.  \nIf A is small and dense, one can first compute f (A) explicitly using algorithms such as those described in [14], and then apply it to b. For large matrices, however, this approach is infeasible because of its computational and memory costs. Krylov subspace methods are therefore commonly used to approximate f(A)b, as they require only matrix-vector products with A, which can often be computed efficiently when A is sparse. Their main limitation is the need to store and orthogonalize the Krylov basis. For non-Hermitian matrices, the Arnoldi process requires O (Nm2 ) operations to generate m orthonormal basis vectors and O (Nm) storage for the basis, and these costs can become substantial when m is large.  \nMany approaches have been proposed to address these limitations. One important class is based on restarting, in which only a fixed number of basis vectors need to be stored [1, 8] . The quadrature-based restarting method of [11] uses an integral representation of f, together with interpolation polynomials, to derive an integral representation of the error. Approximating this error representation by quadrature allows the process to be restarted without storing all previously generated basis vectors. However, restarted methods may converge slowly, and this issue can be alleviated by thick restarting strategies [4,9,16] .  \nAnother class of approaches is based on randomized sketching [5,13] . These methods approximate f (A)b using a nonorthonormal basis of a Krylov subspace, often generated by the sketched Arnoldi process [2] or the truncated Arnoldi process [17] . The sketched Arnoldi process computes the inner products required for orthogonaliza-  \n∗ Department of Mathematics, The University of Manchester, Oxford Road, Manchester, M139PL, United Kingdom, [stefan.guettel@manchester.ac.uk](stefan.guettel@manchester.ac.uk)  \n†School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai, 200433, China, [jyliu22@m.fudan.edu.cn](jyliu22@m.fudan.edu.cn)  \n‡Department of Mathematics, The University of Manchester, Oxford Road, Manchester, M139PL, United Kingdom, [lauri.nyman@manchester.ac.uk](lauri.nyman@manchester.ac.uk)  \n2 S. G¨UTTEL, J. LIU, AND L. NYMAN  \ntion in the sketched space, thereby reducing the orthogonalization cost by nearly one half. The trunca","cbCaipB2MpnTYdA4","https://ap.wps.com/l/cbCaipB2MpnTYdA4","pdf",520578,2,1,23,"English","en",105,"# Introduction\n# Sketch-and-Restart Framework\n# Quadrature-Based Restarting Extension\n# Proposed Algorithms\n# Convergence Analysis","[{\"question\":\"What problem does the sketch-and-restart framework address?\",\"answer\":\"It targets efficient computation of f(A)b for large, sparse, non-Hermitian matrices, avoiding the high cost of storing and orthogonalizing large Krylov bases.\"},{\"question\":\"How do quadrature-based restarting and randomized sketching work together here?\",\"answer\":\"The method combines quadrature-based error representations with Arnoldi-like decompositions generated by sketched or truncated Arnoldi processes, producing a common restart mechanism across related Krylov approximations.\"},{\"question\":\"What are the two classes of restarted algorithms proposed?\",\"answer\":\"One class uses a fixed Krylov subspace dimension via sketched Arnoldi or a new sketched harmonic Arnoldi process; the second class adapts the Krylov dimension by running truncated Arnoldi until a sketch-estimated basis condition number exceeds a threshold.\"}]",1784201738,58,{"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},"sketch-and-restart-randomized-sketching-in-quadrature-based-restarting-for-matrix-functions","",{"@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/sketch-and-restart-randomized-sketching-in-quadrature-based-restarting-for-matrix-functions/85205/",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","2026-07-23","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 problem does the sketch-and-restart framework address?","Question",{"text":75,"@type":76},"It targets efficient computation of f(A)b for large, sparse, non-Hermitian matrices, avoiding the high cost of storing and orthogonalizing large Krylov bases.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do quadrature-based restarting and randomized sketching work together here?",{"text":80,"@type":76},"The method combines quadrature-based error representations with Arnoldi-like decompositions generated by sketched or truncated Arnoldi processes, producing a common restart mechanism across related Krylov approximations.",{"name":82,"@type":73,"acceptedAnswer":83},"What are the two classes of restarted algorithms proposed?",{"text":84,"@type":76},"One class uses a fixed Krylov subspace dimension via sketched Arnoldi or a new sketched harmonic Arnoldi process; 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