  The Berlekamp--Massey--Sakata algorithm and the Scalar-FGLM algorithm both
compute the ideal of relations of a multidimensional linear recurrent
sequence.Whenever quering a single sequence element is prohibitive, the
bottleneck of these algorithms becomes the computation of all the needed
sequence terms. As such, having adaptive variants of these algorithms, reducing
the number of sequence queries, becomes mandatory.A native adaptive variant of
the Scalar-FGLM algorithm was presented by its authors, the so-called Adaptive
Scalar-FGLM algorithm.In this paper, our first contribution is to make the
Berlekamp--Massey--Sakata algorithm more efficient by making it adaptive to
avoid some useless relation test-ings. This variant allows us to divide by four
in dimension 2 and by seven in dimension 3 the number of basic operations
performed on some sequence family.Then, we compare the two adaptive algorithms.
We show that their behaviors differ in a way that it is not possible to tweak
one of the algorithms in order to mimic exactly the behavior of the other. We
detail precisely the differences and the similarities of both algorithms and
conclude that in general the Adaptive Scalar-FGLM algorithm needs fewer queries
and performs fewer basic operations than the Adaptive Berlekamp--Massey--Sakata
algorithm.We also show that these variants are always more efficient than the
original algorithms.
