{"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/a-practical-framework-for-infinite","title":"A practical framework for infinite-dimensional linear algebra","arxiv_id":"1409.5529","date":"2014-09-19","proceeding":null,"authors":["Sheehan Olver","Alex Townsend"],"abstract":"We describe a framework for solving a broad class of infinite-dimensional linear equations, consisting of almost banded operators, which can be used to resepresent linear ordinary differential equations with general boundary conditions. The framework contains a data structure on which row operations can be performed, allowing for the solution of linear equations by the adaptive QR approach. The algorithm achieves $O(n^{\\rm opt})$ complexity, where $n^{\\rm opt}$ is the number of degrees of freedom required to achieve a desired accuracy, which is determined adaptively. In addition, special tensor product equations, such as partial differential equations on rectangles, can be solved by truncating the operator in the $y$-direction with $n_y$ degrees of freedom and using a generalized Schur decomposition to upper triangularize, before applying the adaptive QR approach to the $x$-direction, requiring $O(n_y^2 n_x^{\\rm opt})$ operations. The framework is implemented in the ApproxFun package written in the Julia programming language, which achieves highly competitive computational costs by exploiting unique features of Julia.","url_abs":"https://arxiv.org/abs/1409.5529v1","url_pdf":"https://arxiv.org/pdf/1409.5529v1.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":"a-practical-framework-for-infinite","repo_url":"https://github.com/dlfivefifty/ApproxFun.jl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null},{"paper_slug":"a-practical-framework-for-infinite","repo_url":"https://github.com/wormell/Poltergeist.jl","is_official":0,"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}