  The idea of computing Matveev complexity by using Heegaard decompositions has
been recently developed by two different approaches: the first one for closed
3-manifolds via crystallization theory, yielding the notion of Gem-Matveev
complexity; the other one for compact orientable 3-manifolds via generalized
Heegaard diagrams, yielding the notion of modified Heegaard complexity. In this
paper we extend to the non-orientable case the definition of modified Heegaard
complexity and prove that for closed 3-manifolds Gem-Matveev complexity and
modified Heegaard complexity coincide. Hence, they turn out to be useful
different tools to compute the same upper bound for Matveev complexity.
