Computation of the zeros of Laguerre–Sobolev polynomials

Computation of the zeros of Laguerre–Sobolev polynomials | 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 Computation of the zeros of Laguerre–Sobolev polynomials T. Laudadio, N. Mastronardi, F. Marcellan, N. Van Buggenhout, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9167775/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 8 You are reading this latest preprint version Abstract A new algorithm for computing all the zeros of Laguerre–Sobolev orthogonal polynomials, based on the Ehrlich–Aberth method, is described in this work. The Ehrlich–Aberth method is a Newton–like method, requiring, at each iteration, the evaluation of the polynomial and its derivative in the computed approximations of the zeros. The Laguerre–Sobolev polynomials are related to the classical Laguerre orthogonal polynomials by a four term recurrence relation, that allows to evaluate the former polynomials and their derivatives in a point. This relation can be then exploited in the Ehrlich–Aberth method. Laguerre–Sobolev polynomials exhibit a behavior similar to that of Laguerre polynomials: their values grow rapidly as their degrees increase, and overflow occurs in floating point arithmetic if their degree exceeds 170. In order to avoid overflow, novel recurrence relations are proposed to simultaneously compute the ratio between the Laguerre–Sobolev polynomials and the corresponding derivatives in a point. The proposed algorithm turns out to be very efficient and accurate, with O(n 2 ) computational complexity and O(n) memory, where n is the degree of the polynomial. AMS Classification: 33C45 , 33C47 , 65H04 Laguerre-Sobolev orthogonal polynomials zeros of polynomials Ehrlich-Aberth method Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 19 May, 2026 Reviewers agreed at journal 24 Apr, 2026 Reviews received at journal 23 Apr, 2026 Reviewers agreed at journal 06 Apr, 2026 Reviewers invited by journal 01 Apr, 2026 Editor assigned by journal 01 Apr, 2026 Submission checks completed at journal 01 Apr, 2026 First submitted to journal 19 Mar, 2026 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. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9167775","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":616414892,"identity":"9d018768-5fc3-4750-92ab-42479eadaccc","order_by":0,"name":"T. Laudadio","email":"","orcid":"","institution":"Consiglio Nazionale delle Ricerche","correspondingAuthor":false,"prefix":"","firstName":"T.","middleName":"","lastName":"Laudadio","suffix":""},{"id":616414894,"identity":"1499ec32-e25e-4e20-b014-a5c8099db9e9","order_by":1,"name":"N. 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The Ehrlich–Aberth method is a Newton–like method, requiring, at each iteration, the evaluation of the polynomial and its derivative in the computed approximations of the zeros.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe Laguerre–Sobolev polynomials are related to the classical Laguerre orthogonal polynomials by a four term recurrence relation, that allows to evaluate the former polynomials and their derivatives in a point. This relation can be then exploited in the Ehrlich–Aberth method.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eLaguerre–Sobolev polynomials exhibit a behavior similar to that of Laguerre polynomials: their values grow rapidly as their degrees increase, and overflow occurs in floating point arithmetic if their degree exceeds 170.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIn order to avoid overflow, novel recurrence relations are proposed to simultaneously compute the ratio between the Laguerre–Sobolev polynomials and the corresponding derivatives in a point. The proposed algorithm turns out to be very efficient and accurate, with O(n\u003csup\u003e2\u003c/sup\u003e) computational complexity and O(n) memory, where n is the degree of the polynomial.\u003c/p\u003e\n\u003cp\u003eAMS Classification: 33C45 , 33C47 , 65H04\u003c/p\u003e","manuscriptTitle":"Computation of the zeros of Laguerre–Sobolev polynomials","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-04-06 05:04:23","doi":"10.21203/rs.3.rs-9167775/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"editorInvitedReview","content":"","date":"2026-05-19T14:49:36+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"264375101114651312630577061143053712043","date":"2026-04-24T23:26:16+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-04-23T19:46:47+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"132715465433337549136328800589284032308","date":"2026-04-06T12:34:57+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-04-01T09:15:12+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-04-01T09:13:19+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-04-01T06:42:38+00:00","index":"","fulltext":""},{"type":"submitted","content":"Numerical Algorithms","date":"2026-03-19T09:26:42+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"numerical-algorithms","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"numa","sideBox":"Learn more about [Numerical Algorithms](http://link.springer.com/journal/11075)","snPcode":"11075","submissionUrl":"https://submission.nature.com/new-submission/11075/3","title":"Numerical Algorithms","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"8f039eb3-f7ce-4fbf-912f-d416dbc2b577","owner":[],"postedDate":"April 6th, 2026","published":true,"recentEditorialEvents":[{"type":"editorInvitedReview","content":"","date":"2026-05-19T14:49:36+00:00","index":16,"fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-04-06T05:04:23+00:00","versionOfRecord":[],"versionCreatedAt":"2026-04-06 05:04:23","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9167775","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9167775","identity":"rs-9167775","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}