It has been believed that the continued fraction expansion of
(
α
,
β
)
(\alpha ,\beta )
(
1
,
α
,
β
(1,\alpha ,\beta
is a
Q
{\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
(
⟨
3
3
⟩
,
⟨
9
3
⟩
)
(\langle \sqrt [3]{3}\rangle , \langle \sqrt [3]{9}\rangle )
(
⟨
x
⟩
\langle x\rangle
denoting the fractional part of
x
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
(
α
,
β
)
(\alpha ,\beta )
with our algorithm always becomes periodic for any real cubic field. We also consider real quartic fields.