The Research On Several Problems Of Michalik Continued Fraction

The method of interpolation,as one of the powerful numerical methods in scientific engineering calculation,refers to use the coordinates of some discrete points to construct a function with continuous definition,so that the function is completely consistent with the corresponding value of the interpolated function at a set point.As the basis of the whole numerical approximation,polynomial interpolation is easy to calculate with simple form and has been widely used in equation for root extraction,the process of numerical solution of differential and integral equation.Because the oscillatory phenomenon of higher-order polynomial interpolation limits its development,it is very important to study rational interpolation.Although the form of rational interpolation is more complex than that of polynomial,it has higher approximation accuracy and significant advantages in the speed of approximation.As an important member of rational interpolation,Thiele continued fraction has always played an indispensable role.However,due to the pole of the Thiele continued fraction,and the branched continued fraction algorithm based on Thiele continued fraction is suitable for rectangular grid data,which can not effectively deal with the scattered data interpolation.In view of this,Michalik continued fraction interpolation method and its related problems are studied in this paper.The main contents of this paper include that the bivariate Michalik continued fraction interpolation algorithm,the bivariate Michalik continued fraction interpolation with prescribed poles,a comparative study on two kinds of bivariate rational interpolations based on scattered data,Michalik barycentric blending rational interpolation and so on.The main contents of this paper are summarized as follows:On the basis of Michalik continued fraction theory,the bivariate form of Michalik continued fraction is studied by constructing the bivariate coefficient recursive algorithm on the scattered point set.At the same time,three recursive formulas and characteristic theorems are given.Used to discuss the recursive relationship between the various orders of rational functions and the number of estimated numerators and denominators.In order to solve the problem of unattainable points in traditional continued fraction interpolation,the interpolation nodes are rearranged sequentially and the auxiliary functions are designed.The correction processing method of bivariate Michalik continuous with unattainable points is proposed.lt fully demonstrates that Michalik continued fraction is suited for multivariate function and the method is used for expanding function into continued fraction.The bivariate Michalik continued interpolation method is simple and easy to program,the numerical example shows that the bivariate Michalik interpolation constructed in the text has a good approximation effect;Under the condition that the pole information of the interpolation function is given,the basis function with the pole is introduced,and the original function value is multiplied by a certain number,the bivariate Michalik continued fraction is used to establish the poleless interpolation function,and finally the division is performed.This method maintains the original multiplicity while maintaining the original pole position.The bivariate rational interpolations with prescribed poles constructed by progressive diagonal rational interpolation and non-tensor product interpolation algorithm are further disscussed in the text,numerical example analyzes and compares the approximation effect of two interpolation algorithms;By mixing the traditional barycentric rational interpolation with Michalik's continued fraction and selecting the reasonable weight function,two kinds of interpolation algorithms are mixed at each interpolation node to obtain a unified and bivariate Michalik barycentric blending rational interpolation.With the advantages of no poles and unattainable points.The numerical experiments show that the interpolation function obtained a good approximation effect after blending.Figure[16]Table[6]Reference[48]...