A collection of algorithms for computing
with functions defined on the unit disk or the surface of
the unit two-sphere is presented. Central to these
algorithms is a new scheme for approximating functions to
essentially machine precision that combines a
structure-preserving iterative variant of Gaussian
elimination together with the double Fourier sphere
method. The scheme produces low rank approximations of
functions on the disk and sphere, ameliorates oversampling
issues near the origin of the disk and poles of the
sphere, converges geometrically for sufficiently analytic
functions, and allows for stable differentiation. The low
rank representation makes operations such as function
evaluation, differentiation, and integration particularly
efficient. A demonstration of the algorithms using the new
Diskfun and Spherefun features of Chebfun will also be
given. The audience is encouraged to download the software
from http://www.chebfun.org and experiment with the
methods. This is joint work with Prof. Alex Townsend
and Heather Wilber (both at Cornell University).
|