An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a method that requires less evaluations of the function that defines the ODE and its derivatives than the usual version. On the other hand, an efficient numerical solution of the equation that arises from the discretization by means of Newton's method is introduced for an implicit scheme of any order. Numerical experiments illustrate that the resulting algorithm is simpler to implement and has better performance than its exact counterpart.

翻译：提出了一种针对常微分方程组初值问题的高效近似隐式泰勒方法。该方法基于泰勒方法的近似公式，相比常规版本，减少了定义常微分方程的函数及其导数的评估次数。另一方面，针对任意阶隐式格式，引入了一种基于牛顿法的离散化方程高效数值解法。数值实验表明，所得算法相比精确版本更易于实现且性能更优。