动力学系统精细算法的逼近机理与误差上界

Approximate mechanism and upper error limit of precise computation for dynamics system

发现以下三者的协同作用是实现精细算法高精度、高效率的内在机理和根本原因：１）｜ｘ｜＜∞，指数矩阵ｅＨｘ的Ｍａｃｌａｕｒｉｎ级数展开式绝对收敛；２）初始Ｍａｃｌａｕｒｉｎ级数展开式中的有效展开项总数能够通过递推算法以指数方式扩展；３）新增有效展开项的系数能够通过递推算法以指数或拟指数方式逼近其真值。此外，本文还给出了精细算法的截项误差递推公式和相关的误差上界，发现随着保留项数Ｍ或递推阶数Ｎ的增大，精细算法的逼近误差上界以指数方式减小。

The present paper discovers that the inherent mechanism and fundamental cause of realizing the high approximating precision and computing efficiency by the precision computation method for dynamics system lies in the cooperation of the following three factors:1) |x|<∞,the expansion of exponential matrix e Hx in Maclaurin series is absolutely convergent. 2) the active expansion item of exponential matrix e Hx in Maclaurin series increase exponentially as recurrent course. 3) the coefficients of newly adding active expansion items will approximate their actual values exponentially or quasi exponentially as recurrent course. Besides, the recursion formula of the error from cut items and the related error upper limit of the precision computation method is presented, and the following rule is discovered that the approximate error upper limit of the precision computation method decreases exponentially as increasing of reserved item M or recursion order N.