  We present a general numerical method for investigating prescribed Ricci
curvature problems on toric K\"ahler manifolds. This method is applied to two
generalisations of Einstein metrics, namely Ricci solitons and quasi-Einstein
metrics. We begin by recovering the Koiso--Cao soliton and the L\"u--Page--Pope
quasi-Einstein metrics on $\mathbb{CP}^{2}\sharp\overline{\mathbb{CP}}^{2}$ (in
both cases the metrics are known explicitly). We also find numerical
approximations to the Wang--Zhu soliton on $\mathbb{CP}^{2}\sharp
2\overline{\mathbb{CP}}^{2}$ (here the metric is not known explicitly).
Finally, a substantial numerical investigation of the quasi-Einstein equation
on $\mathbb{CP}^{2}\sharp 2\overline{\mathbb{CP}}^{2}$ is conducted. In this
case it is an open problem as to whether such metrics exist on this manifold.
We find metrics that solve the quasi-Einstein equation to the same degree of
accuracy as the approximations to the Wang--Zhu soliton solve the Ricci soliton
equation.
