Computation of polynomial and rational approximations in complex domains by the τ-method

Computation of polynomial and rational approximations in complex domains by the τ-method | 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 polynomial and rational approximations in complex domains by the τ-method Irina Georgieva, Clemens Hofreither This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4441013/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 25 Jul, 2024 Read the published version in Numerical Algorithms → Version 1 posted 10 You are reading this latest preprint version Abstract We investigate numerical methods for computation of polynomial and rational approximations of functions in complex domains based on Faber polynomials and the Lanczos τ-method. Our interest is motivated by applications in fractional partial differential equations. We give an overview of previous results related to the basis of Faber polynomials associated with a complex domain, Faber expansion, and the Lanczos τ-method. We also collect numerical algorithms for the computational realization of these concepts. Our main new contribution is a τ-method for rational approximation in complex domains which uses Faber polynomials in the perturbation term. We realize it via a novel hybrid symbolic-numeric algorithm which can be applied to arbitrary functions satisfying a suitable differential equation. We present some numerical examples, where we use sectors lying in the complex plane as our domains of interest. We compare results for the various polynomial and rational approximation techniques outlined above; in particular, we observe exponential convergence with respect to the rational degree for our proposed method. Lanczos τ-method polynomial approximation rational approximation complex approximation symbolic computation Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 25 Jul, 2024 Read the published version in Numerical Algorithms → Version 1 posted Editorial decision: Revision requested 01 Jul, 2024 Reviews received at journal 26 Jun, 2024 Reviews received at journal 24 Jun, 2024 Reviewers agreed at journal 30 May, 2024 Reviewers agreed at journal 30 May, 2024 Reviewers agreed at journal 30 May, 2024 Reviewers invited by journal 30 May, 2024 Editor assigned by journal 29 May, 2024 Submission checks completed at journal 28 May, 2024 First submitted to journal 18 May, 2024 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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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-4441013","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":308682936,"identity":"9c20949a-d9a6-4318-8cc7-6edacfa8ff80","order_by":0,"name":"Irina Georgieva","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAzUlEQVRIiWNgGAWjYBACNgbGBgaGCiBLAsgGggQitZxhYOAhWgsYMLaRooVP+nDbg5/ztsnbSzewPfjxhyFP3oGQw/gS2w17t9027JE5wG7Y28ZQbHiAkBYexjYJ3m23GXskEtgkeBsYEjc2EKFF8u+c2/YgLZJ//hCpRZq34XYiSIs0DxtD4nwCOiBaZI7dTu65kdhuLNsmUWxASIt8D/szyTc1t23bZyQfe/jmj02ePCGHIQFQnALTgMEB4rXA7CXBllEwCkbBKBgZAADm3Tr5rnVQWgAAAABJRU5ErkJggg==","orcid":"","institution":"Institute of Mathematics and Informatics, Bulgarian Academy of Sciences","correspondingAuthor":true,"prefix":"","firstName":"Irina","middleName":"","lastName":"Georgieva","suffix":""},{"id":308682937,"identity":"930027a0-48cc-4a08-8b96-ffaaabb83e6a","order_by":1,"name":"Clemens Hofreither","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Clemens","middleName":"","lastName":"Hofreither","suffix":""}],"badges":[],"createdAt":"2024-05-18 12:23:17","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4441013/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4441013/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11075-024-01897-7","type":"published","date":"2024-07-25T16:15:51+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":61596214,"identity":"6adadc46-ad6d-4ac6-906f-2fc1ef9c7d71","added_by":"auto","created_at":"2024-08-01 17:25:46","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":5729693,"visible":true,"origin":"","legend":"","description":"","filename":"RationalTau.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4441013/v1_covered_39e92772-9854-4906-8c01-76f2d38a5721.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eComputation of polynomial and rational approximations in complex domains by the τ-method\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"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},"keywords":"Lanczos τ-method, polynomial approximation, rational approximation, complex approximation, symbolic computation","lastPublishedDoi":"10.21203/rs.3.rs-4441013/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4441013/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe investigate numerical methods for computation of polynomial and rational approximations of functions in complex domains based on Faber polynomials and the Lanczos τ-method. 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