Approximating the pth root by composite rational functions
A landmark result from rational approximation theory states that x 1/p on [0, 1] can be approximated by a type-(n, n) rational function with root-exponential accuracy. Motivated by the recursive optimality property of Zolotarev functions (for the square root and sign functions), we investigate appro...
| Main Authors: | , |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
Elsevier
2021
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| _version_ | 1826307349206269952 |
|---|---|
| author | Gawlik, ES Nakatsukasa, Y |
| author_facet | Gawlik, ES Nakatsukasa, Y |
| author_sort | Gawlik, ES |
| collection | OXFORD |
| description | A landmark result from rational approximation theory states that x
1/p on [0, 1] can be approximated
by a type-(n, n) rational function with root-exponential accuracy. Motivated by the recursive optimality
property of Zolotarev functions (for the square root and sign functions), we investigate approximating
x
1/p by composite rational functions of the form rk
(x,rk−1(x,rk−2(· · · (x,r1(x, 1))))). While this class
of rational functions ceases to contain the minimax (best) approximant for p ≥ 3, we show that it
achieves approximately pth-root exponential convergence with respect to the degree. Moreover, crucially,
the convergence is doubly exponential with respect to the number of degrees of freedom, suggesting that
composite rational functions are able to approximate x
1/p
and related functions (such as |x| and the
sector function) with exceptional efficiency |
| first_indexed | 2024-03-07T07:00:57Z |
| format | Journal article |
| id | oxford-uuid:ffb991bf-8eeb-442d-b905-c6159bd72de6 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:00:57Z |
| publishDate | 2021 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:ffb991bf-8eeb-442d-b905-c6159bd72de62022-03-27T13:47:16ZApproximating the pth root by composite rational functionsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:ffb991bf-8eeb-442d-b905-c6159bd72de6EnglishSymplectic ElementsElsevier2021Gawlik, ESNakatsukasa, YA landmark result from rational approximation theory states that x 1/p on [0, 1] can be approximated by a type-(n, n) rational function with root-exponential accuracy. Motivated by the recursive optimality property of Zolotarev functions (for the square root and sign functions), we investigate approximating x 1/p by composite rational functions of the form rk (x,rk−1(x,rk−2(· · · (x,r1(x, 1))))). While this class of rational functions ceases to contain the minimax (best) approximant for p ≥ 3, we show that it achieves approximately pth-root exponential convergence with respect to the degree. Moreover, crucially, the convergence is doubly exponential with respect to the number of degrees of freedom, suggesting that composite rational functions are able to approximate x 1/p and related functions (such as |x| and the sector function) with exceptional efficiency |
| spellingShingle | Gawlik, ES Nakatsukasa, Y Approximating the pth root by composite rational functions |
| title | Approximating the pth root by composite rational functions |
| title_full | Approximating the pth root by composite rational functions |
| title_fullStr | Approximating the pth root by composite rational functions |
| title_full_unstemmed | Approximating the pth root by composite rational functions |
| title_short | Approximating the pth root by composite rational functions |
| title_sort | approximating the pth root by composite rational functions |
| work_keys_str_mv | AT gawlikes approximatingthepthrootbycompositerationalfunctions AT nakatsukasay approximatingthepthrootbycompositerationalfunctions |