  We analyze the coupling between the ferroelectric and magnetic order
parameters in the magnetoelectric multiferroic BiFeO_3 using density functional
theory within the local spin density approximation and the LSDA+U method. We
show that weak ferromagnetism of the Dzyaloshinskii-Moriya type occurs in this
material, and we analyze the coupling between the resulting magnetization and
the structural distortions. We explore the possibility of
electric-field-induced magnetization reversal and show that, although it is
unlikely to be realized in BiFeO_3, it is not in general impossible. Finally we
outline the conditions that must be fulfilled to achieve switching of the
magnetization using an electric field.
