  We propose a variation of the forward--backward splitting method for solving
structured monotone inclusions. Our method integrates past iterates and two
deviation vectors into the update equations. These deviation vectors bring
flexibility to the algorithm and can be chosen arbitrarily as long as they
together satisfy a norm condition. We present special cases where the deviation
vectors, selected as predetermined linear combinations of previous iterates,
always meet the norm condition. Notably, we introduce an algorithm employing a
scalar parameter to interpolate between the conventional forward--backward
splitting scheme and an accelerated O(1/n^2)-convergent forward--backward
method that encompasses both the accelerated proximal point method and the
Halpern iteration as special cases. The existing methods correspond to the two
extremes of the allowed scalar parameter range. By choosing the interpolation
scalar near the midpoint of the permissible range, our algorithm significantly
outperforms these previously known methods when addressing a basic monotone
inclusion problem stemming from minimax optimization.
