{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/compact-finite-differences-and-cubic-splines","title":"Compact Finite Differences and Cubic Splines","arxiv_id":"1805.07659","date":"2018-05-19","proceeding":null,"authors":["Robert M. Corless"],"abstract":"In this paper I uncover and explain---using contour integrals and residues---a connection between cubic splines and a popular compact finite difference formula. The connection is that on a uniform mesh the simplest Pad\\'e scheme for generating fourth-order accurate compact finite differences gives \\textsl{exactly} the derivatives at the interior nodes needed to guarantee twice-continuous differentiability for cubic splines. %I found this connection surprising, because the two problems being solved are different. I also introduce an apparently new spline-like interpolant that I call a compact cubic interpolant; this is similar to one introduced in 1972 by Swartz and Varga, but has higher order accuracy at the edges. I argue that for mildly nonuniform meshes the compact cubic approach offers some potential advantages, and even for uniform meshes offers a simple way to treat the edge conditions, relieving the user of the burden of deciding to use one of the three standard options: free (natural), complete (clamped), or \"not-a-knot\" conditions. Finally, I establish that the matrices defining the compact cubic splines (equivalently, the fourth-order compact finite difference formul\\ae) are positive definite, and in fact totally nonnegative, if all mesh widths are the same sign.","url_abs":"http://arxiv.org/abs/1805.07659v3","url_pdf":"http://arxiv.org/pdf/1805.07659v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"compact-finite-differences-and-cubic-splines","repo_url":"https://github.com/rcorless/CompactCubicSpline","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}