Miller-Stable s-Step Conjugate Gradient and Conjugate Residual Methods

Miller-Stable s-Step Conjugate Gradient and Conjugate Residual Methods | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Miller-Stable s -Step Conjugate Gradient and Conjugate Residual Methods Stephen Thomas This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8619286/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Communication avoiding s -step Krylov methods generate s basis vectors per outer iteration to reduce global synchronization on parallel architectures. The projected Gram system arising at each outer iteration must be solved accurately, yet direct factorization becomes numerically unstable when the Gram matrix is ill-conditioned. This paper establishes that the direction of Gauss-Seidel iteration for the Gram system affects numerical stability through the Miller-Gautschi theory of three-term recurrence relations. Forward Gauss-Seidel selects the minimal solution of the underlying recurrence while Backward Gauss-Seidel can amplify the dominant solution, leading to error growth. Numerical experiments demonstrate that Forward Gauss-Seidel requires two to three times fewer outer iterations than Backward Gauss-Seidel for block sizes s ≥ q 15. The paper also establishes that Conjugate Residuals is more robust than Conjugate Gradients under inexact Gram solves because CR minimizes a local residual norm objective at each step while CG enforces a global A -conjugacy constraint that accumulates errors across iterations. The combination of Forward Gauss-Seidel with s -step CR provides a communication avoiding method with reliability approaching that of standard CG while reducing synchronization by a factor of s . Numerical experiments on Poisson problems and matrices from the SuiteSparse collection confirm the theoretical predictions and demonstrate practical effectiveness. Mathematics Subject Classification (2020): 65F10, 65F15, 65G50, 65Y05 s-step methods communication avoiding Krylov methods conjugate gradients con jugate residuals Gauss-Seidel iteration Miller algorithm three-term recurrence finite precision arithmetic Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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The projected Gram system arising at each outer iteration must be solved accurately, yet direct factorization becomes numerically unstable when the Gram matrix is ill-conditioned. This paper establishes that the direction of Gauss-Seidel iteration for the Gram system affects numerical stability through the Miller-Gautschi theory of three-term recurrence relations. Forward Gauss-Seidel selects the minimal solution of the underlying recurrence while Backward Gauss-Seidel can amplify the dominant solution, leading to error growth. Numerical experiments demonstrate that Forward Gauss-Seidel requires two to three times fewer outer iterations than Backward Gauss-Seidel for block sizes \u003cem\u003es\u003c/em\u003e ≥ \u003cem\u003eq\u003c/em\u003e 15. The paper also establishes that Conjugate Residuals is more robust than Conjugate Gradients under inexact Gram solves because CR minimizes a local residual norm objective at each step while CG enforces a global \u003cem\u003eA\u003c/em\u003e-conjugacy constraint that accumulates errors across iterations. The combination of Forward Gauss-Seidel with \u003cem\u003es\u003c/em\u003e-step CR provides a communication avoiding method with reliability approaching that of standard CG while reducing synchronization by a factor of \u003cem\u003es\u003c/em\u003e. Numerical experiments on Poisson problems and matrices from the SuiteSparse collection confirm the theoretical predictions and demonstrate practical effectiveness.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMathematics Subject Classification (2020):\u003c/strong\u003e 65F10, 65F15, 65G50, 65Y05\u003c/p\u003e","manuscriptTitle":"Miller-Stable s-Step Conjugate Gradient and Conjugate Residual Methods","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-04 06:48:14","doi":"10.21203/rs.3.rs-8619286/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"cff982c9-95b1-4976-bf44-6830a43b1135","owner":[],"postedDate":"February 4th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-04-08T10:05:44+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-04 06:48:14","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8619286","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8619286","identity":"rs-8619286","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}