关键词: 数值微分 不适定性 离散正则化方法 正则参数 偏差原理

Abstract:

Numerical differentiation is a typical inverse problem. It is known to be ill-posed in the sense of Hadamard that small perturbations in the function   to be differentiated may lead to large errors in the computed derivative. For the first time Tikhonov＇ s regularizing method is used to discuss some th heoretical and technical problems of applying the discrete regularization method to do the problem of numerical differentiation, among which, the analys  is of convergence and stability of discrete regularization solution, the selection of regularization parameters with and without knowing the error level   of the input data, the development of the stable algorithms, are concerned and presented, and the numerical tests with comparison are also Given. A the   oretical analysis and numeri- cal test shows that using the discrete regularization method given in this paper on numerical differentiation possesses th   e advantages of wide application, high accuracy and good numerical stability.

Key words: numerical differentiation, ill-posed, discrete regularization method, regularization parameter, dis crepancy principle