Advances in High Precision Partial Derivative Approximation for Black Box Functions of n Variables

Advances in High Precision Partial Derivative Approximation for Black Box Functions of n Variables | 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 Advances in High Precision Partial Derivative Approximation for Black Box Functions of n Variables Koya Yamamoto, John L. Junkins This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6926796/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 20 Apr, 2026 Read the published version in The Journal of the Astronautical Sciences → Version 1 posted You are reading this latest preprint version Abstract We review and extend the complex step approach for approximating partial derivatives of smooth vector functions with n independent variables. This method, which leverages complex arithmetic, offers a highly accurate alternative to traditional finite difference techniques, and highly efficient alternative to operator overloading required techniques such as Automatic differentiation. Certain aspects of the multivariable second and higher order derivative approximations are novel. We demonstrate that for 64-bit arithmetic, 15-digit precision is generally obtained for the first partials. 10 to 13 digit precision is routinely obtained for the second partials, and 5 to 8 digit precision is obtained for the third-order partials. The formulation presented does not require operator overloading and minimum derivation or coding are required. We provide several illustrative examples that demonstrate the utility, accuracy, efficiency, and generality of this approach, including an astrodynamic application. Complex step differentiation Higher order derivative approximation State transition tensors Sensitivity analysis Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 20 Apr, 2026 Read the published version in The Journal of the Astronautical Sciences → 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-6926796","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":477495679,"identity":"b0c5f15f-058c-4e19-aca2-d34ae5368961","order_by":0,"name":"Koya Yamamoto","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAsUlEQVRIiWNgGAWjYDACHhBRgcwhTssZA1K1MLaRosXgzOFjEj/n/ZHXnZHA+OBtGzFazralSfZuMzDcdiOB2XAuUVrO85hJ8G4zYARqYZPmJVaL5N85BvZALey/idNytsdMmrfBIBFkCzNRWiTPHEu2ljlmnLztzMNmyTnniNDCdyb54M03NXK2244nH/zwpowILQoHGFgkIEzGBiLUA4F8AwPzB+KUjoJRMApGwYgFAA51OEPHutQjAAAAAElFTkSuQmCC","orcid":"","institution":"Texas A\u0026M University","correspondingAuthor":true,"prefix":"","firstName":"Koya","middleName":"","lastName":"Yamamoto","suffix":""},{"id":477495682,"identity":"91e325f3-ae70-4487-a3f8-572f16590827","order_by":1,"name":"John L. Junkins","email":"","orcid":"","institution":"Texas A\u0026M University","correspondingAuthor":false,"prefix":"","firstName":"John","middleName":"L.","lastName":"Junkins","suffix":""}],"badges":[],"createdAt":"2025-06-19 02:38:31","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6926796/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6926796/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s40295-026-00585-z","type":"published","date":"2026-04-20T15:58:54+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":107928113,"identity":"f5745716-7669-46a0-a973-8c34cb690029","added_by":"auto","created_at":"2026-04-27 16:08:08","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2754967,"visible":true,"origin":"","legend":"","description":"","filename":"CXSpringertemplateV2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6926796/v1_covered_9e31b347-b524-4551-9603-9f2f2332b676.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Advances in High Precision Partial Derivative Approximation for Black Box Functions of n Variables","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"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":"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":"Complex step differentiation, Higher order derivative approximation, State transition tensors, Sensitivity analysis","lastPublishedDoi":"10.21203/rs.3.rs-6926796/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6926796/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We review and extend the complex step approach for approximating partial derivatives of smooth vector functions with n independent variables. This method, which leverages complex arithmetic, offers a highly accurate alternative to traditional finite difference techniques, and highly efficient alternative to operator overloading required techniques such as Automatic differentiation. Certain aspects of the multivariable second and higher order derivative approximations are novel. We demonstrate that for 64-bit arithmetic, 15-digit precision is generally obtained for the first partials. 10 to 13 digit precision is routinely obtained for the second partials, and 5 to 8 digit precision is obtained for the third-order partials. The formulation presented does not require operator overloading and minimum derivation or coding are required. We provide several illustrative examples that demonstrate the utility, accuracy, efficiency, and generality of this approach, including an astrodynamic application.","manuscriptTitle":"Advances in High Precision Partial Derivative Approximation for Black Box Functions of n Variables","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-06-30 07:55:07","doi":"10.21203/rs.3.rs-6926796/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":"2c7cf426-b1a4-4d5b-9795-e079b43f9845","owner":[],"postedDate":"June 30th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-04-27T16:05:44+00:00","versionOfRecord":{"articleIdentity":"rs-6926796","link":"https://doi.org/10.1007/s40295-026-00585-z","journal":{"identity":"the-journal-of-the-astronautical-sciences","isVorOnly":false,"title":"The Journal of the Astronautical Sciences"},"publishedOn":"2026-04-20 15:58:54","publishedOnDateReadable":"April 20th, 2026"},"versionCreatedAt":"2025-06-30 07:55:07","video":"","vorDoi":"10.1007/s40295-026-00585-z","vorDoiUrl":"https://doi.org/10.1007/s40295-026-00585-z","workflowStages":[]},"version":"v1","identity":"rs-6926796","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6926796","identity":"rs-6926796","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}