Stabilization Algorithm Of Numerical Differentiation And Its Realization

Numerical differentiation is a typical ill-posed problem.The essential difficulty of solving this problem lies in the instability of its solution,that is,a small perturbation in the measurement data may cause huge error in its derivative by direct computation.Therefore,how to construct stable differential algorithms has always been the focus of numerical differentiation.In this paper,two kinds of stabilization algorithms for solving numerical differential problem are considered:the finite difference method and the mollification method.The construction,error analysis and numerical implementation of both method are given in this paper.The finite difference method is based on the concept of algebra precision,where the method of undetermined coefficients are used to construct the arbitrary order finite difference scheme.Moreover,the the remainder of difference scheme is analyzed,and the choice strategy of the difference step is also given.In the mollification method,the general selection method of the mollification kernel function is given.The error analysis of the approximate derivatives and the choice strategy of the regularization parameters are given under the ? norm and L~2 norm respectively.Based on this,the specific selection of the mollification kernel function is presented.Finally,the numerical efficiency of the above both method is demonstrated by numerical experiments.