Coalgebra offers a general framework for modelling different types of state-based systems. Our aim is to present generic algorithms to decide behavioural equivalence for coalgebras which generalize partition refinement. The underlying idea of the algorithms is to work on the final chain and to factor out redundant information. If the algorithm terminates, the result of the construction is a representative of the given coalgebra that is not necessarily unique and that allows to precisely answer questions about behavioural equivalence. We apply the algorithm to weighted automata over semirings in order to obtain a procedure for checking language equivalence for a large number of semirings.