{"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/fast-measure-modification-of-orthogonal","title":"Fast measure modification of orthogonal polynomials via matrices with displacement structure","arxiv_id":"2412.17663","date":"2024-12-23","proceeding":null,"authors":["Karim Gumerov","Samantha Rigg","Richard Mikael Slevinsky"],"abstract":"It is well known that matrices with low Hessenberg-structured displacement rank enjoy fast algorithms for certain matrix factorizations. We show how $n\\times n$ principal finite sections of the Gram matrix for the orthogonal polynomial measure modification problem has such a displacement structure, unlocking a collection of fast algorithms for computing connection coefficients (as the upper-triangular Cholesky factor) between a known orthogonal polynomial family and the modified family. In general, the ${\\cal O}(n^3)$ complexity is reduced to ${\\cal O}(n^2)$, and if the symmetric Gram matrix has upper and lower bandwidth b, then the ${\\cal O}(b^2n)$ complexity for a banded Cholesky factorization is reduced to ${\\cal O}(b n)$. In the case of modified Chebyshev polynomials, we show that the Gram matrix is a symmetric Toeplitz-plus-Hankel matrix, and if the modified Chebyshev moments decay algebraically, then a hierarchical off-diagonal low-rank structure is observed in the Gram matrix, enabling a further reduction in the complexity of an approximate Cholesky factorization powered by randomized numerical linear algebra.","url_abs":"https://arxiv.org/abs/2412.17663v1","url_pdf":"https://arxiv.org/pdf/2412.17663v1.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":"fast-measure-modification-of-orthogonal","repo_url":"https://github.com/JuliaApproximation/FastTransforms.jl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"fast-measure-modification-of-orthogonal","repo_url":"https://github.com/MikaelSlevinsky/FastTransforms.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}