  We present a thorough numerical analysis of the relaxational dynamics of the
Sherrington-Kirkpatrick spherical model with an additive non-disordered
perturbation for large but finite sizes $N$. In the thermodynamic limit and at
low temperatures, the perturbation is responsible for a phase transition from a
spin glass to a ferromagnetic phase. We show that finite size effects induce
the appearance of a distinctive slow regime in the relaxation dynamics, the
extension of which depends on the size of the system and also on the strength
of the non-disordered perturbation. The long time dynamics is characterized by
the two largest eigenvalues of a spike random matrix which defines the model,
and particularly by the statistics of the gap between them. We characterize the
finite size statistics of the two largest eignevalues of the spike random
matrices in the different regimes, sub-critical, critical and super-critical,
confirming some known results and anticipating others, even in the less studied
critical regime. We also numerically characterize the finite size statistics of
the gap, which we hope may encourage analytical work which is lacking. Finally,
we compute the finite size scaling of the long time relaxation of the energy,
showing the existence of power laws with exponents that depend on the strenght
of the non-disordered perturbation, in a way which is governed by the finite
size statistics of the gap.
