  This paper provides a theoretical and numerical comparison of classical first
order splitting methods for solving smooth convex optimization problems and
cocoercive equations. In a theoretical point of view, we compare convergence
rates of gradient descent, forward-backward, Peaceman-Rachford, and
Douglas-Rachford algorithms for minimizing the sum of two smooth convex
functions when one of them is strongly convex. A similar comparison is given in
the more general cocoercive setting under the presence of strong monotonicity
and we observe that the convergence rates in optimization are strictly better
than the corresponding rates for cocoercive equations for some algorithms. We
obtain improved rates with respect to the literature in several instances
exploiting the structure of our problems. In a numerical point of view, we
verify our theoretical results by implementing and comparing previous
algorithms in well established signal and image inverse problems involving
sparsity. We replace the widely used $\ell_1$ norm by the Huber loss and we
observe that fully proximal-based strategies have numerical and theoretical
advantages with respect to methods using gradient steps. In particular,
Peaceman-Rachford is the more performant algorithm in our examples.
