A symmetric product of two optimal third-order force gradient symplectic algorithms
两个三阶最优化力梯度辛积分器的对称组合

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 Schr dinger equations. So are they to the known optimalfourth-order force gradient symplectic scheme.