  We propose a methodology for studying the performance of common splitting
methods through semidefinite programming. We prove tightness of the methodology
and demonstrate its value by presenting two applications of it. First, we use
the methodology as a tool for computer-assisted proofs to prove tight
analytical contraction factors for Douglas--Rachford splitting that are likely
too complicated for a human to find bare-handed. Second, we use the methodology
as an algorithmic tool to computationally select the optimal splitting method
parameters by solving a series of semidefinite programs.
