Before revision:  The Meataxe algorithm , a popular one known for computing such a decomposition runs in (n^{6(d+1)  on a generalized matrix reduction technique that runs in O( n ^{2\omega+1}) time for d=2 and in O(n^{d(2\omega +1)) time for d>2 where \omega<2.
After revision:  The Meataxe algorithm , a popular one known for computing such a decomposition runs in We present an efficient algorithm to compute the unique decomposition of a finitely presented persistence module M defined over the multiparameter \mathbb{Z on a generalized matrix reduction technique that runs in O( n ^{2\omega+1}) time for d=2 and in O(n^{d(2\omega +1)) time for d>2 where \omega<2.