A Kind Of Lanczos Method For Higher Derivatives Of Approximate Functions

As a typical ill-posed problem in the sense of Hadamard,the numerical differential problem refers to the problem of calculation of derivative by using approximate or discrete data of unknown function.In order to overcome the ill-posedness of numerical differential,many special treatment methods are proposed,such as polishing method,difference method,integral derivation method,partial differential equation based method and general regularization theory.This dissertation mainly studies the integral approximation method of the higher derivative of the stable calculation approximate function based on the Lanczos method,which is also called Lanczos method because of its proposal.Using the orthogonality of Legendre polynomials,we present a class of high-precision integral method to approximate the higher order derivatives of approximate functions,that is,to construct a series of integral operatorsD_n^,_hmto approximate the higher-order derivatives of noise functions.In addition,these integral operators haveO((?))convergence rates,wheredis the noise level of the approximate function.Secondly,the regularization parameter selection problem of the proposed high precision integral method is studied,four posterior selection strategies of regularization parameter selection are proposed,and the convergence estimation of regularization solution is given.There are two strategies that need to know the noise level,while the other strategies do not.Finally,using the high-precision integral approximation method given in this dissertation,the numerical experiments of approximate calculation of higher order derivatives are given under the criterion of prior and posterior selection of regularization parameters.The numerical results show that the proposed high precision integral approximation method is feasible,and the proposed posterior regularization parameter is also feasible.