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This paper provides two new fourth-order force gradient symplectic intrgrators,each of which is obtained from a symmetric product of two identied optimal third-order force gradient symplectic algorithms reported in the literature. They are both greatly superior to the fourth-order non-gradient symplectic method of Forest and Ruth in the accuracy of either energy on chaotic perturbed Kepler problems or the energy eigenvalues for one-dimensional Schrö,dinger equations. So are they to the known optimalfourth-order force gradient symplectic scheme.
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