  We discuss continued fractions on real quadratic number fields of class
number 1. If the field has the property of being 2-stage euclidean, a
generalization of the euclidean algorithm can be used to compute these
continued fractions. Although it is conjectured that all real quadratic fields
of class number 1 are 2-stage euclidean, this property has been proven for only
a few of them. The main result of this paper is an algorithm that, given a real
quadratic field of class number 1, verifies this conjecture, and produces as
byproduct enough data to efficiently compute continued fraction expansions. If
the field was not 2-stage euclidean, then the algorithm would not terminate. As
an application, we enlarge the list of known 2-stage euclidean fields, by
proving that all real quadratic fields of class number 1 and discriminant less
than 8000 are 2-stage euclidean.
