Algebraic approximations to special Kahler metrics

I will begin by defining the space of algebraic metrics in a particular Kahler class and recalling the Tian-Ruan-Zelditch result saying that they are dense in the space of all Kahler metrics in this class.  I will then discuss the relationship between some special algebraic metrics called 'balanced metrics' and distinguished Kahler metrics (Extremal metrics, cscK, Kahler-Ricci solitons...). Finally I will talk about some numerical algorithms due to Simon Donaldson for finding explicit examples of these balanced metrics (possibly with some pictures).