  In this work we apply the recently introduced framework of degenerate
preconditioned proximal point algorithms to the hybrid proximal extragradient
(HPE) method for maximal monotone inclusions. The latter is a method that
allows inexact proximal (or resolvent) steps where the error is controlled by a
relative-error criterion. Recently the HPE framework has been extended to the
Douglas-Rachford method by Eckstein and Yao. In this paper we further extend
the applicability of the HPE framework to splitting methods. To that end we use
the framework of degenerate preconditioners that allows to write a large class
of splitting methods as preconditioned proximal point algorithms. In this way
we modify many splitting methods such that one or more of the resolvents can be
computed inexactly with an error that is controlled by an adaptive criterion.
Further, we illustrate the algorithmic framework in the case of
Chambolle-Pock's primal dual hybrid gradient method and the Davis-Yin's forward
Douglas-Rachford method. In both cases, the inexact computation of the
resolvent shows clear advantages in computing time and accuracy.
