AAA rational approximation on a continuum
AAA rational approximation has normally been carried out on a discrete set, typically hundreds or thousands of points in a real interval or complex domain. Here we introduce a continuum AAA algorithm that discretizes a domain adaptively as it goes. This enables fast computation of high-accuracy rati...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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Society for Industrial and Applied Mathematics
2024
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| _version_ | 1797113007727181824 |
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| author | Driscoll, T Nakatsukasa, Y Trefethen, LN |
| author_facet | Driscoll, T Nakatsukasa, Y Trefethen, LN |
| author_sort | Driscoll, T |
| collection | OXFORD |
| description | AAA rational approximation has normally been carried out on a discrete set, typically hundreds or thousands of points in a real interval or complex domain. Here we introduce a continuum AAA algorithm that discretizes a domain adaptively as it goes. This enables fast computation of high-accuracy rational approximations on domains such as the unit interval, the unit circle, and the imaginary axis, even in some cases where resolution of singularities requires exponentially clustered sample points, support points, and poles. Prototype MATLAB (or Octave) and Julia codes aaax, aaaz, and aaai are provided for these three special domains; the latter two are equivalent by a Möbius transformation. Execution is very fast since the matrices whose SVDs are computed have only three times as many rows as columns. The codes include a AAA-Lawson option for improvement of a AAA approximant to minimax, so long as the accuracy is well above machine precision. The result returned is pole-free in the approximation domain. |
| first_indexed | 2024-03-07T08:00:12Z |
| format | Journal article |
| id | oxford-uuid:405ca260-401d-40db-898c-13b67907339f |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-04-09T03:57:38Z |
| publishDate | 2024 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:405ca260-401d-40db-898c-13b67907339f2024-03-28T09:48:36ZAAA rational approximation on a continuumJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:405ca260-401d-40db-898c-13b67907339fEnglishSymplectic ElementsSociety for Industrial and Applied Mathematics2024Driscoll, TNakatsukasa, YTrefethen, LNAAA rational approximation has normally been carried out on a discrete set, typically hundreds or thousands of points in a real interval or complex domain. Here we introduce a continuum AAA algorithm that discretizes a domain adaptively as it goes. This enables fast computation of high-accuracy rational approximations on domains such as the unit interval, the unit circle, and the imaginary axis, even in some cases where resolution of singularities requires exponentially clustered sample points, support points, and poles. Prototype MATLAB (or Octave) and Julia codes aaax, aaaz, and aaai are provided for these three special domains; the latter two are equivalent by a Möbius transformation. Execution is very fast since the matrices whose SVDs are computed have only three times as many rows as columns. The codes include a AAA-Lawson option for improvement of a AAA approximant to minimax, so long as the accuracy is well above machine precision. The result returned is pole-free in the approximation domain. |
| spellingShingle | Driscoll, T Nakatsukasa, Y Trefethen, LN AAA rational approximation on a continuum |
| title | AAA rational approximation on a continuum |
| title_full | AAA rational approximation on a continuum |
| title_fullStr | AAA rational approximation on a continuum |
| title_full_unstemmed | AAA rational approximation on a continuum |
| title_short | AAA rational approximation on a continuum |
| title_sort | aaa rational approximation on a continuum |
| work_keys_str_mv | AT driscollt aaarationalapproximationonacontinuum AT nakatsukasay aaarationalapproximationonacontinuum AT trefethenln aaarationalapproximationonacontinuum |