The talk will have two themes. The first is to give an account of recent work of Chen, LeBrun and Weber, proving the existence of a new Einstein metric on a certain 4-manifold. The second will describe a general class of numerical solutions to Calabi's extremal metric equation. The two themes are related by the fact that the metric of Chen, LeBrun and Weber is conformal to an extremal metric, so explicit geometric information can be obtained.