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 , of the simple continued fraction algorithm, described in terms of geometric and arithmetic properties of 2 x 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 Ä2.We describe a computational procedure for our algorithm, and give several examples.In this paper we present a generalization of the continued fraction algorithm, based on a geometric and matrix-theoretic approach.There are several ways in which the simple continued fraction algorithm (CFA) may be represented geometrically in R2.We summarize briefly, in Section 3 of this paper, one such interpretation, which is described entirely in terms of geometric and arithmetic properties of 2 x 2 matrices with nonnegative entries and determinant 1.Our approach displays, in an easy and natural manner, all the main properties of continued fractions: the coordinates of the continued fraction for a positive real number x, its convergents, and the convergence of the algorithm.The question of periodicity is not so simple, and we do not discuss it here.In Section 1 we give a simple computational procedure for our algorithm, and some examples of its use.In Section 4 we show how this algorithm is a natural generalization to R3 of the geometric and matrix-theoretic situation in R2 given in Section 3.In Section 5 we show that our algorithm is always convergent.Geometrically, the idea of the proof is very simple.The actual details are elementary but rather tedious.Basically, our algorithm bears a strong resemblance to the Jacobi-Perron algorithm in its matrix-theoretic aspects, and we hope in another paper to investigate the Jacobi-Perron algorithm itself from the point of view of this paper.We have not yet worked out a theoretical basis for comparing our algorithm to the Jacobi-Perron algorithm, but the examples of Section 2 show some interesting differences.For instance, for ($7, ffi), our algorithm became periodic after 13 steps, while the Jacobi-Perron algorithm takes only 3 steps.On the other hand, our rational approximations at comparable stages are superior in accuracy.