A generalization of the simple continued fraction algorithm

Abstract:

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 ${R^2}$, of the simple continued fraction algorithm, described in terms of geometric and arithmetic properties of $2 \times 2$ matrices with nonnegative integer entries and determinant 1. The algorithm of this paper is then derived as a natural generalization of the situation in ${R^2}$. We describe a computational procedure for our algorithm, and give several examples.