We present a general numerical method for investigating prescribed Ricci curvature problems on toric Kähler 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 Lü-Page-Pope quasi-Einstein metrics on CP²♯\overline{CP}² (in both cases the metrics are known explicitly). We also find numerical approximations to the Wang-Zhu soliton on CP²♯ 2\overline{CP}² (here the metric is not known explicitly). Finally, a substantial numerical investigation of the quasi-Einstein equation on CP²♯ 2\overline{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.

A good deal of the work was conducted whilst SH was visiting TM at California State University Fullerton. SH would like to thank the department for their hospitality. This trip was funded by a CSUF Startup grant.