Generating sequences of rank 1 valuations

We consider a rank 1 valuation ν centered in a regular 3-dimensional local ring R. We assume that the residue field k of ν is contained in R. An algorithm for constructing a generating sequences for ν in R is provided. This is a generalization of our previous algorithm where only valuations of rational rank 1 were considered. We then give a comparison example of a rational rank 3 valuation centered in a ring of polynomials in three variables for which we explicitly compute a set of key polynomials and a generating sequence.