Taylor算法的改进

Taylor算法是提供高精度出发值的传统方法之一,但它需要计算多元函数f(x,y)的高阶导数,计算量太大,所以不够实用。本文在不改变原方法稳定性区域的条件下,用函数值的组合去代替高阶导数的计算,这就大大地减少了计算量,尤其在方程组的情形是这样;此外,通过选取适当的参数使局部截断误差达到极小从而使所得方法和相应的古典Runge—Kutta 方法有时完全相当,有时甚至还有略优的数值精度(它们有时略低于Taylor 算法的精度)。

The improvement of Taylor algorithm

Taylor algorithm is one of the classical methods which are capable of producing high- ly accurate starting values,however,in this case the higher derivatives of function f(x.y) in several variables ave evaluated,since the tremendous calculating cost are invo ved,then it is impracticable,In the paper,the excessively tedious computation of higher deriratives is replaced by the combination of some functions,the stability region of the method is not altered,then the computational effort is reduced significantly,especially for systems. In addition,by the choice of the approprite parameters it is possiable to get minimal value of the local truncatio nerror,these methods are well-matched with the corresponding classi- cal Runge-Kutta methods,they may have higher numerical accuracy.