日時： 2017年9月20日(水) 14:30～15:30
場所： 東京大学工学部 14号館 5階 534号室
講演者： 中務 佑治 (University of Oxford)
講演題目: Best and near-best rational approximation
via adaptive barycentric representation
Rational approximation can outperform polynomial approximation by a
landslide when there are singularities in or near the domain of
approximation. However, its use has been limited relative to
polynomials, primarily due to ill-conditioning (in addition to spurious
poles). In this work we show that the conditioning, and the overall
utility of rational functions, can be improved dramatically by an
adaptive barycentric representation for rational functions, wherein one
chooses a basis depending not only on the domain but also the function
to be approximated.
We first introduce a new algorithm, called AAA (triple A, standing for
“adaptive Antoulas-Anderson”) for rational approximation on a real or
complex set of points. Even on a disk or interval the algorithm may
outperform existing methods, and on more complicated domains it is
especially competitive. The core ideas are (1) representation of the
rational approximant in barycentric form with interpolation at certain
support points and (2) greedy selection of the support points (which
determines the basis) to avoid exponential instabilities.
We next consider computing rational minimax approximations (best
rational approximant) on an real interval. We show that far more robust
algorithms than previously available can be developed using the
barycentric representation. Our improved rational Remez algorithm
calculates 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.
Based on joint work with S. Filip, O. Sete, and L. N. Trefethen (Oxford)
and B. Beckermann (Lille).