The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.

翻译：普通差分方程式的数值解决方案可以被解释为附近修改方程式的精确解决方案。 通过分析修改方程式来调查数字解决方案的行为被称为后向错误分析。如果原始和修改方程式共享结构属性,那么精确和大致解决方案共享几何特征,例如保存数量的存在。Conjugate sympecticic 方法保留了修改的中位形式,在应用到汉密尔顿系统时可以使用修改的汉密尔顿语。我们展示了如何使用混合版本的变式和间位技术来计算经修改的中位和汉密尔顿结构。与其他方法不同,我们的后方差错分析方法并不依赖ansatz,而是系统地计算结构,条件是该方法的变式公式是已知的。该技术以带有矩阵系数的对称线多步法为示例。