Rational minimax approximation via adaptive barycentric representations
Computing rational minimax approximations can be very challenging when there are singularities on or near the interval of approximation - precisely the case where rational functions outperform polynomials by a landslide. We show that far more robust algorithms than previously available can be develo...
| Main Authors: | , , , |
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| Format: | Journal article |
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Society for Industrial and Applied Mathematics
2018
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| _version_ | 1826289689738346496 |
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| author | Filip, S Nakatsukasa, Y Trefethen, L Beckermann, B |
| author_facet | Filip, S Nakatsukasa, Y Trefethen, L Beckermann, B |
| author_sort | Filip, S |
| collection | OXFORD |
| description | Computing rational minimax approximations can be very challenging when there are singularities on or near the interval of approximation - precisely the case where rational functions outperform polynomials by a landslide. We show that far more robust algorithms than previously available can be developed by making use of rational barycentric representations whose support points are chosen in an adaptive fashion as the approximant is computed. Three variants of this barycentric strategy are all shown to be powerful: (1) a classical Remez algorithm, (2) a "AAA-Lawson" method of iteratively reweighted least-squares, and (3) a differential correction algorithm. Our preferred combination, implemented in the Chebfun MINIMAX code, is to use (2) in an initial phase and then switch to (1) for generically quadratic convergence. By such methods we can calculate approximations up to type (80, 80) of $|x|$ on $[-1, 1]$ in standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan, and Carpenter required 200-digit extended precision. |
| first_indexed | 2024-03-07T02:32:43Z |
| format | Journal article |
| id | oxford-uuid:a7cc7d2a-b2c7-4dbc-a10c-4583a5f0bda3 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T02:32:43Z |
| publishDate | 2018 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:a7cc7d2a-b2c7-4dbc-a10c-4583a5f0bda32022-03-27T02:56:53ZRational minimax approximation via adaptive barycentric representationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:a7cc7d2a-b2c7-4dbc-a10c-4583a5f0bda3Symplectic Elements at OxfordSociety for Industrial and Applied Mathematics2018Filip, SNakatsukasa, YTrefethen, LBeckermann, BComputing rational minimax approximations can be very challenging when there are singularities on or near the interval of approximation - precisely the case where rational functions outperform polynomials by a landslide. We show that far more robust algorithms than previously available can be developed by making use of rational barycentric representations whose support points are chosen in an adaptive fashion as the approximant is computed. Three variants of this barycentric strategy are all shown to be powerful: (1) a classical Remez algorithm, (2) a "AAA-Lawson" method of iteratively reweighted least-squares, and (3) a differential correction algorithm. Our preferred combination, implemented in the Chebfun MINIMAX code, is to use (2) in an initial phase and then switch to (1) for generically quadratic convergence. By such methods we can calculate approximations up to type (80, 80) of $|x|$ on $[-1, 1]$ in standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan, and Carpenter required 200-digit extended precision. |
| spellingShingle | Filip, S Nakatsukasa, Y Trefethen, L Beckermann, B Rational minimax approximation via adaptive barycentric representations |
| title | Rational minimax approximation via adaptive barycentric representations |
| title_full | Rational minimax approximation via adaptive barycentric representations |
| title_fullStr | Rational minimax approximation via adaptive barycentric representations |
| title_full_unstemmed | Rational minimax approximation via adaptive barycentric representations |
| title_short | Rational minimax approximation via adaptive barycentric representations |
| title_sort | rational minimax approximation via adaptive barycentric representations |
| work_keys_str_mv | AT filips rationalminimaxapproximationviaadaptivebarycentricrepresentations AT nakatsukasay rationalminimaxapproximationviaadaptivebarycentricrepresentations AT trefethenl rationalminimaxapproximationviaadaptivebarycentricrepresentations AT beckermannb rationalminimaxapproximationviaadaptivebarycentricrepresentations |