Calculation of machine precision second order derivatives using dual-complex numbers

Calculation of machine precision second order derivatives using dual-complex numbers | 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 Calculation of machine precision second order derivatives using dual-complex numbers H. Millwater, M. Balcer, N. Millwater This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3929470/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 It is well known that both complex and dual numbers can be employed to obtain machine precision first-order derivatives; however, neither, on their own, can compute machine precision 2nd order derivatives. To address this limitation, it is demonstrated in this paper that combined dual-complex numbers can be used to compute machine precision 1st and 2nd order derivatives. The dual-complex approach is simpler than utilizing multicomplex or hyper-dual numbers as existing dual libraries can be used as is or easily augmented to accept complex numbers, and the complexity of developing, integrating, and deploying multicomplex or hyper-dual libraries is avoided. The efficacy of this approach is demonstrated for both univariant and multivariate functions with examples from the Python, Julia, and Mathematica languages. sensitivity methods Complex Taylor Series Expansion dual numbers automatic differentiation numerical methods 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. 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-3929470","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":271179922,"identity":"69aa7d4f-d95c-4392-88af-e9ec009055a7","order_by":0,"name":"H. Millwater","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA90lEQVRIiWNgGAWjYBACAyjNw8cOYchBBZgJa2EDqTnAwGDMwEakFgaYlsQGQlrM2c8++8zDsE2GjZnH8POHinvpG+63P/zAUGGd2IBDi2VPuvFsHobbQIfxGEscOFOcu+EYkMFwJh2nFoMDaczMEC28GyQOtiWAtDBIMLYdxq3l/DO4ls0/Dv5LSDc4xv74B+M/PFpuIGzZJnGwISHB4BiDmQRjAz4tz5gZ5xiAtPB/szhzLMFw5rEcM4uEY+nGuB2WxszwpuK2PT97W/KNipoEeb7Dxx/f+FBjLYtLCwgw8RigCyXgUQ4CjD8IKBgFo2AUjIIRDgAmZlKUzM5h+gAAAABJRU5ErkJggg==","orcid":"","institution":"The University of Texas at San Antonio","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"H.","middleName":"","lastName":"Millwater","suffix":""},{"id":271179923,"identity":"2c706ba5-c6b5-4009-8a6f-c0a0e068b36f","order_by":1,"name":"M. Balcer","email":"","orcid":"","institution":"Los Alamos National Laboratory","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"M.","middleName":"","lastName":"Balcer","suffix":""},{"id":271179924,"identity":"8ca203b5-bb5f-425c-9f95-1aabdccfe318","order_by":2,"name":"N. Millwater","email":"","orcid":"","institution":"Southwest Research Institute","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"N.","middleName":"","lastName":"Millwater","suffix":""}],"badges":[],"createdAt":"2024-02-05 01:59:09","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3929470/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3929470/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":50827457,"identity":"e8576d14-d62b-42a2-ba11-20d776531ffb","added_by":"auto","created_at":"2024-02-08 01:22:32","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":559910,"visible":true,"origin":"","legend":"","description":"","filename":"complexdual2ndOrderSubmission3.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3929470/v1_covered_8f2bce48-4a60-4c18-a34f-e7af2334fe48.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Calculation of machine precision second order derivatives using dual-complex numbers","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"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},"keywords":"sensitivity methods, Complex Taylor Series Expansion, dual numbers, automatic differentiation, numerical methods","lastPublishedDoi":"10.21203/rs.3.rs-3929470/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3929470/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIt is well known that both complex and dual numbers can be employed to obtain machine precision first-order derivatives; however, neither, on their own, can compute machine precision 2nd order derivatives. To address this limitation, it is demonstrated in this paper that combined dual-complex numbers can be used to compute machine precision 1st and 2nd order derivatives. The dual-complex approach is simpler than utilizing multicomplex or hyper-dual numbers as existing dual libraries can be used as is or easily augmented to accept complex numbers, and the complexity of developing, integrating, and deploying multicomplex or hyper-dual libraries is avoided. The efficacy of this approach is demonstrated for both univariant and multivariate functions with examples from the Python, Julia, and Mathematica languages.\u003c/p\u003e","manuscriptTitle":"Calculation of machine precision second order derivatives using dual-complex numbers","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-02-07 03:45:57","doi":"10.21203/rs.3.rs-3929470/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":"401297be-a743-4820-be90-44486a78d3d4","owner":[],"postedDate":"February 7th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-02-08T01:14:19+00:00","versionOfRecord":[],"versionCreatedAt":"2024-02-07 03:45:57","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3929470","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3929470","identity":"rs-3929470","version":["v1"]},"buildId":"cBFmMYwuxLRRLfASyISRj","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}