Symplectic integrators from composite operator factorizations

I derive fourth order symplectic integrators by factorizing the exponential of two operators in terms of an additional higher order composite operator with positive coefficients. One algorithm requires only one evaluation of the force and one evaluation of the force and its gradient. When applied to Kepler's problem, the energy error function associated with these algorithms are approximately 10 to 80 times smaller than the fourth order Forest-Ruth, Candy-Rozmus integrator.