Numerical Differentiation And Its Application

Numerical differentiation is a classical ill-posed problem in the sense of Hada-mard. The small errors in the measurement may cause huge errors in the numerical results. This problem has been treated by several methods. In this paper, we discuss the numerical differentiation problem by Tikhonov regularization. The regularized solutions based on Spline (unctions are constructed in the one-dimensional case and the regularized solutions based on Green function are constructed in the two-dimensional case. By using a simple strategy of choosing the regularization parameter, which is based on the conditional stability of the differentiation problem, we show that our algorithm can be realized easily and fast.We discuss the existence and uniqueness of the regularized solutions both for the one-dimensional case and two-dimensional case. The error estimates are also given. For one-dimensional case, the high-order numerical derivatives are also discussed. For two-dimensional cases, the first and second order derivatives are discussed. For the case that the dimension is greater than 2. the same argument given in this paper still works.The properties of the regularized solution are analyzed. It is shown that the regularized solutions will blow up in the neighborhood of the discontinuous points. This local,property can be used to locate the discontinuous points for non-smooth functions numerically. We show this by several examples.We give numerical examples and applications for both the one-dimensional and two-dimensional cases.