A new multidimensional continued fraction algorithm

It has been believed that the continued fraction expansion of $(\alpha,\beta)$ $(1,\alpha,\beta$ is a ${\mathbb{Q}}$-basis of a real cubic field$)$ obtained by the modified Jacobi-Perron algorithm is periodic. We conducted a numerical experiment (cf. Table B, Figure 1 and Figure 2) from which we conjecture the non-periodicity of the expansion of $(\langle\sqrt[3]{3}\rangle, \langle\sqrt[3]{9}\rangle)$ ( $\langle x\rangle$ denoting the fractional part of $x$). We present a new algorithm which is something like the modified Jacobi-Perron algorithm, and give some experimental results with this new algorithm. From our experiments, we can expect that the expansion of $(\alpha,\beta)$ with our algorithm always becomes periodic for any real cubic field. We also consider real quartic fields.