11,489 research outputs found
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 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 on in
standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan,
and Carpenter required 200-digit extended precision.Comment: 29 pages, 11 figure
On the connectedness of planar self-affine sets
In this paper, we consider the connectedness of planar self-affine set
arising from an integral expanding matrix with
characteristic polynomial and a digit set
. The necessary and sufficient conditions only
depending on are given for the to be connected.
Moreover, we also consider the case that is non-consecutively
collinear.Comment: 18 pages; 18 figure
On the speed of convergence of Newton's method for complex polynomials
We investigate Newton's method for complex polynomials of arbitrary degree
, normalized so that all their roots are in the unit disk. For each degree
, we give an explicit set of
points with the following universal property: for every normalized polynomial
of degree there are starting points in whose Newton
iterations find all the roots with a low number of iterations: if the roots are
uniformly and independently distributed, we show that with probability at least
the number of iterations for these starting points to reach all
roots with precision is . This is an improvement of an earlier result in
\cite{Schleicher}, where the number of iterations is shown to be in the worst case (allowing multiple roots)
and for
well-separated (so-called -separated) roots.
Our result is almost optimal for this kind of starting points in the sense
that the number of iterations can never be smaller than for fixed
.Comment: 13 pages, 1 figure, to appear in AMS Mathematics of Computatio
Tchebychev Polynomial Approximations for Order Boundary Value Problems
Higher order boundary value problems (BVPs) play an important role modeling
various scientific and engineering problems. In this article we develop an
efficient numerical scheme for linear order BVPs. First we convert the
higher order BVP to a first order BVP. Then we use Tchebychev orthogonal
polynomials to approximate the solution of the BVP as a weighted sum of
polynomials. We collocate at Tchebychev clustered grid points to generate a
system of equations to approximate the weights for the polynomials. The
excellency of the numerical scheme is illustrated through some examples.Comment: 21 pages, 10 figure
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