A generalization of the simple continued fraction algorithm

In this paper we present a generalization of the continued fraction algorithm, based on a geometric and matrix-theoretic approach. We first give a geometric representation in the plane R 2 {R^2} , of the simple continued fraction algorithm, described in terms of geometric and arithmetic properties of 2 × 2 2 × 2 matrices with nonnegative integer entries and determinant 1. The algorithm of this paper is then derived as a natural generalization of the situation in R 2 {R^2} . We describe a computational procedure for our algorithm, and give several examples.