  The need to compute small con-eigenvalues and the associated con-eigenvectors
of positive-definite Cauchy matrices naturally arises when constructing
rational approximations with a (near) optimally small $L^{\infty}$ error.
Specifically, given a rational function with $n$ poles in the unit disk, a
rational approximation with $m\ll n$ poles in the unit disk may be obtained
from the $m$th con-eigenvector of an $n\times n$ Cauchy matrix, where the
associated con-eigenvalue $\lambda_{m}>0$ gives the approximation error in the
$L^{\infty}$ norm. Unfortunately, standard algorithms do not accurately compute
small con-eigenvalues (and the associated con-eigenvectors) and, in particular,
yield few or no correct digits for con-eigenvalues smaller than the machine
roundoff. We develop a fast and accurate algorithm for computing
con-eigenvalues and con-eigenvectors of positive-definite Cauchy matrices,
yielding even the tiniest con-eigenvalues with high relative accuracy. The
algorithm computes the $m$th con-eigenvalue in $\mathcal{O}(m^{2}n)$ operations
and, since the con-eigenvalues of positive-definite Cauchy matrices decay
exponentially fast, we obtain (near) optimal rational approximations in
$\mathcal{O}(n(\log\delta^{-1})^{2})$ operations, where $\delta$ is the
approximation error in the $L^{\infty}$ norm. We derive error bounds
demonstrating high relative accuracy of the computed con-eigenvalues and the
high accuracy of the unit con-eigenvectors. We also provide examples of using
the algorithm to compute (near) optimal rational approximations of functions
with singularities and sharp transitions, where approximation errors close to
machine precision are obtained. Finally, we present numerical tests on random
(complex-valued) Cauchy matrices to show that the algorithm computes all the
con-eigenvalues and con-eigenvectors with nearly full precision.
