We have determined from first principles the athermal elastic constant tensor of Mg2SiO4 forsterite with the plane wave pseudopotential method over a wide range of pressure (0-100 GPa) that encompasses the full range over which forsterite has been observed experimentally. The computed elastic constants are in excellent agreement with experimental data up to the maximum pressure of the experiments (16 GPa). We calculate the single-crystal elastic anistropy from the elastic constants. We find that the anisotropy is strong (azimuthal P- and S-wave anisotropy: 25% and 20%, respectively, polarization anisotropy: 15%), in agreement with experiment, and that it depends weakly on pressure over the range 0--25 GPa, in contrast to the behavior of other silicates and oxides.¿ 1997 American Geophysical Union