The structure of symplectic integrators up to fourth-order can be completely
and analytical understood when the factorization (split) coefficents are
related linearly but with a uniform nonlinear proportional factor. The analytic
form of these {\it extended-linear} symplectic integrators greatly simplified
proofs of their general properties and allowed easy construction of both
forward and non-forward fourth-order algorithms with arbitrary number of
operators. Most fourth-order forward integrators can now be derived
analytically from this extended-linear formulation without the use of symbolic
algebra.Comment: 12 pages, 2 figures, submitted to Phys. Rev. E, corrected typo
