Computing algebraic numbers of bounded height

We describe an algorithm which, given a number field K and a bound B, finds all the elements of K having relative height at most B. Two lists of numbers are computed: one consisting of elements x is an element of K for which it is known with certainty that H-K(x) <= B, and one containing elements x such that vertical bar H-K(x)-B vertical bar < theta for a tolerance theta chosen by the user. We show that every element of K whose height is at most B must appear in one of the two lists.