  We develop new algorithms for approximating extremal toric K\"ahler metrics.
We focus on an extremal metric on
$\mathbb{CP}^{2}\sharp2\overline{\mathbb{CP}}^{2}$, which is conformal to an
Einstein metric (the Chen-LeBrun-Weber metric). We compare our approximation to
one given by Bunch and Donaldson and compute various geometric quantities. In
particular, we demonstrate a small eigenvalue of the scalar Laplacian of the
Einstein metric which gives a numerical evidence that the Einstein metric is
conformally unstable under the Ricci flow.
