Name
Abstract | ||
|---|---|---|
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) an "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 vertical bar x vertical bar on [-1,1] in standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan, and Carpenter [Math. USSR Sb., 74 (1993), pp. 271-290] required 200-digit extended precision. |
Year | DOI | Venue |
|---|---|---|
2018 | 10.1137/17M1132409 | SIAM JOURNAL ON SCIENTIFIC COMPUTING |
Keywords | Field | DocType |
barycentric formula,rational minimax approximation,Remez algorithm,differential correction algorithm,AAA algorithm,Lawson algorithm | Mathematical optimization,Minimax,Polynomial,Mathematical analysis,Floating point,Minimax approximation algorithm,Rate of convergence,Rational function,Remez algorithm,Mathematics,Barycentric coordinate system | Journal |
Volume | Issue | ISSN |
40 | 4 | 1064-8275 |
Citations | PageRank | References |
1 | 0.36 | 13 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Silviu-Ioan Filip | 1 | 2 | 1.08 |
| Yuji Nakatsukasa | 2 | 97 | 17.74 |
| Lloyd N. Trefethen | 3 | 1024 | 203.66 |
| Bernhard Beckermann | 4 | 376 | 38.48 |