【作者】 付莹；

【导师】 王晶昕；

【作者基本信息】 辽宁师范大学 ， 基础数学， 2006， 硕士

【摘要】 数值微分就是用离散方法近似地求出函数在某点的导数值，关于数值微分已有许多求解方法，但这些方法都有各自的局限性，并且关于高阶导数近似逼近的方法研究相对较少。本文在分析和概括数值微分基本思想和基本方法的基础上，提出了两种求解高阶数值微分的新方法，并给出了高阶导数近似逼近的误差估计。 方法一是利用Tikhonov正则化方法，通过构造一个五次样条函数研究了不等距分布下的高阶数值微分问题。由于在讨论中的采样取值是随机的，使得该方法具有较大的实用性。方法二是在等距分布的条件下，基于三次样条插值函数的三弯距算法，通过利用节点处弯距的加权和近似计算区间中点处的二阶导数。该方法的优点在于简单，易懂，并且具有比用三次样条插值函数求二阶数值微分更高阶的计算精度。更多还原

【Abstract】 Numerical differentiation is that derivative value of a function at a certain point is approximately solved in discrete method.There has been a lot of solutions to numerical differentiation.However,they have their limitations of their own.Moreover,there are relatively few researches on derivative of higher order approximation.Based on the analysis and summary of basic ideas and methods,this thesis proposes two new estimation of error of higher order approximation.Solution 1 is to research the question of higher order numerical differentiation under non-uniform distribution by using Tikhonov regularization method and constructing a five order spline funtion.Because the values of in discussion are random,this solution is of great practicability. Another solution is in addition of uniform distribution ,and based on the three-bending-moment algorithm of the cubic spline interpolation function.By making use of the bending moment weighted sum at the knots and compute the approximation of the second order derivative at the mid-point of interval.This solutions’advantages is that it is simple,easy to understand,and more higher order computational accuracy than getting two order numerical differentiation by using cubic spline interpolation function.更多还原