[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81587-en":3,"doc-seo-81587-105":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},81587,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","The Matrix-Vector Complexity of Ax = b","Matrix–vector algorithms, especially Krylov subspace methods, are central to solving large linear systems of equations. This paper proves worst-case lower bounds on the number of matrix–vector products required to approximately solve a general system Ax=b to relative residual accuracy ε. For two-sided algorithms using randomization and access to both A and its transpose, Ω(κ log(1/ε)) products are necessary. For one-sided algorithms without transpose access, n products are necessary even when κ=1. Results include sharp constants, confirming Krylov optimality.","arXiv :2602 .04842v 3 [ cs .DS] 9 Jul 2026  \nThe matrix-vector complexity of Ax = b  \nMichał Derezi ´nski University of Michigan [derezin@umich. edu](derezin@umich. edu)  \nEthan N. Epperly UC Berkeley  \n[eepperly@berkeley. edu](eepperly@berkeley. edu)  \nRaphael A. Meyer  \nUC Berkeley & ICSI[ram900@berkeley. edu](ram900@berkeley. edu)  \nJuly 13, 2026  \nAbstract  \nMatrix–vector algorithms, particularly Krylov subspace methods, are widely viewed as the most effective algorithms for solving large systems of linear equations. This paper establishes lower boundson the worst-case number of matrix–vector products needed by such an algorithm to approximately solve a general linear system. The first main result is that, for any matrix–vector algorithm which is allowed the use of randomization and can perform products with both a matrix and its transpose,Ω (κlog(1/ε)) matrix–vector products are necessary to solve a linear system with condition number κ to accuracy ε, matching an upper bound for conjugate gradient on the normal equations. The second main result is that one-sided algorithms, which lack access to the transpose, must use n matrix–vector products to solve an n × n linear system, even when the problem is perfectly conditioned. Both main results include explicit constants that match known upper bounds up to a factor of four. These results rigorously demonstrate the limitations of matrix–vector algorithms and confirm the optimality of widely used Krylov subspace algorithms.  \n1 Introduction  \nSolving systems of linear algebraic equations is a fundamental problem in computer science and mathematics, and algorithms for this task are used in almost every area of modern computation, including machine learning [MRT18], scientific computing [Gre97], and optimization [BV04] . Given an invertible matrix A ∈ Rn×n and a vector b ∈ Rn , the objective is to find the vector x ∈ Rn for which Ax = b. The standard algorithms for this task are variants of Gaussian elimination, which run in O ( nω ) operations when implemented using fast matrix multiplication.1 Determining the optimal complexity for the linear system problem is a major open question [Spi24] .  \nIf we are willing to settle for an approximate solution to Ax = b, the design space for algorithms becomes larger. Significant attention has gone to the class of matrix–vector algorithms, which learn about A only through the matrix–vector product (matvec) primitives z 7→ Az and z 7→ A⊺ z. These methods are commonly understood to be among the only approaches for solving large linear systems [Gre97] . The most popular matrix–vector algorithms for solving linear systems are Krylov subspace methods, such asthe conjugate gradient and GMRES algorithms, which see wide use in practice. Given the importance of these methods, it is natural to ask:  \nHow many matvecs are neccessary and sufficient to approximately solve Ax = b?  \nTo specify this problem completely, we must determine an accuracy requirement for an approximate solution  . For this article, we seek a vector  with a small relative residual:  \n∥A − b∥2 ≤ ε∥b∥2 for specified ε > 0. (1.1)  \n1 Here, ω ≤ [2.371 . . . is](2.371 . . . is) the matrix multiplication exponent.  \nThroughout this paper, ∥·∥2 will denote the ℓ2 norm of a vector or the spectral norm of a matrix. See Section E for a discussion of other error metrics.  \n1.1 Background and research questions  \nMost work on matrix–vector complexity forthe Ax = b problem focuses on the case when Ais symmetric positive definite (SPD) . In this case, the standard Krylov subspace algorithms are conjugate gradient and MINRES [Gre97, Algs. 2 & 4]. The latter achieves the guarantee (1.1) using O( pκ log(1/ε)) matvecs. Here,κ : = cond(A)  : = ∥A∥2 ∥A−1∥2 = σmax (A)/σmin(A) is the condition number. Classical lower bounds confirm that Ω ( p κ log(1/ε)) matvecs are necessary for any deterministic algorithm to achieve this guarantee [NY83, Sec. 7.2] . However, with the rapidly increasing use of random","cbCaibDDkrLvbATf","https://ap.wps.com/l/cbCaibDDkrLvbATf","pdf",397561,3,1,30,"English","en",105,"# Introduction\n## Background and research questions","[{\"question\":\"What accuracy criterion does the paper use for approximating Ax=b?\",\"answer\":\"It requires a vector x such that the relative residual satisfies ||A x − b||2 ≤ ε ||b||2 for a specified ε \\u003e 0.\"},{\"question\":\"What is the main lower bound for two-sided randomized matrix–vector algorithms?\",\"answer\":\"For algorithms with randomization and the ability to multiply by both A and A⊺, the paper shows Ω(κ log(1/ε)) matrix–vector products are necessary in the worst case.\"},{\"question\":\"How does the complexity change for one-sided algorithms that cannot use the transpose?\",\"answer\":\"For one-sided algorithms lacking access to A⊺, the paper proves that n matrix–vector products are necessary to solve an n×n system even when the problem is perfectly conditioned (κ=1).\"}]",1784174536,76,{"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},"the-matrix-vector-complexity-of-ax-b","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/the-matrix-vector-complexity-of-ax-b/81587/",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 accuracy criterion does the paper use for approximating Ax=b?","Question",{"text":75,"@type":76},"It requires a vector x such that the relative residual satisfies ||A x − b||2 ≤ ε ||b||2 for a specified ε > 0.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the main lower bound for two-sided randomized matrix–vector algorithms?",{"text":80,"@type":76},"For algorithms with randomization and the ability to multiply by both A and A⊺, the paper shows Ω(κ log(1/ε)) matrix–vector products are necessary in the worst case.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the complexity change for one-sided algorithms that cannot use the transpose?",{"text":84,"@type":76},"For one-sided algorithms lacking access to A⊺, the paper proves that n matrix–vector products are necessary to solve an n×n system even when the problem is perfectly conditioned (κ=1).","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":22,"slug":121},"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]