Some Research On Continued Fraction Interpolation Method

At present,in the fields of science,technology and nature,there are still a large number of nonlinear problems that need to be solved urgently.The parameter forms in approximation problems are generally various forms of parameters.In theory,these non-linear problems are still the focus of research in the field of science and technology.Interpolation,as one of the most powerful numerical algorithms in computational science,plays a vital role in solving nonlinear problems.Its role is to use the coordinates of discrete points to construct a function with a continuous definition,making this function exactly the same as the value of the function being interpolated at a given point.Among them,polynomial interpolation is the first interpolation method,which is widely used in the process of solving equations,differential and integral equations.Its advantages are simple form,easy to calculate,and it is the basic form of interpolation approximation algorithm.However,the higher order polynomial interpolation has oscillation and limitation,so the research on rational interpolation becomes the key point to understand the non-linear problem.Although rational interpolation algorithm is much more complex than polynomial interpolation in form,it has higher approximation accuracy and obvious advantages in the speed of approximation.Rational function interpolation has become one of the research hotspots of nonlinear problem-solving methods due to its advantages of good flexibility and high accuracy of approximation,and is widely used in approximation and scientific research.There are many ways to construct rational interpolation function,and continued fraction interpolation algorithm,as an important part of rational interpolation,has always been an indispensable role.The continued fraction interpolation itself has strong recursion,which is convenient to calculate the coefficients needed in rational function interpolation.Based on the theory of continued fraction interpolation,which was studied before,and combined with the current research status,this paper makes a deep research on the continued fraction interpolation method and its extended algorithm,which mainly includes three modules:the binary vector bifurcated continued fraction algorithm of pre given poles,the binary vector continued fraction interpolation algorithm of pre given poles,and the extended continued fraction interpolation algorithm of keeping asymptote,etc.The main contents of this paper are summarized as follows:First of all,on the basis of the general theory of continued fraction interpolation,the vector continued fraction interpolation with pre given poles is extended.By giving the pole information of the interpolation lattice,the elementary transformation of the row and column of the lattice is carried out,the corresponding element substitution is calculated,a new continued fraction is constructed,and a numerical example is given to show the effectiveness of the new method.Secondly,by constructing the inherited binary coefficient algorithm on the scattered point set,the binary vector form of continued fraction interpolation is studied,and a binary vector continued fraction interpolation algorithm with pre given poles is given.According to the given pole information of the interpolated function,a factor in the denominator polynomial of the interpolating function is constructed.Then,by multiplying the vector value of each corresponding interpolating node by a certain number,it becomes a binary vector interpolation problem without a given pole.Through the Samelson inverse of the vector,a binary non tensor product vector continued fraction interpolation is constructed,and then divided With a definite function,a binary vector continued fraction interpolation with pre given poles is obtained.This method has pre given poles and the original multiplicity remains unchanged.At the same time,by giving three recurrence formulas and characteristic theorems,the recurrence relations among the order of numerator,denominator and rational function are derived.By giving the function with horizontal or oblique asymptote,an extended continued fraction algorithm for keeping asymptote is given,which makes the continued fraction interpolation more accurate and tends to asymptote.The method is to add a function node on the basis of the original node number,use the limit obtained by the asymptote to calculate the function value of the node,so that the given interpolation node can be more accurate,use the new interpolation node and the original function node to construct a new continued fraction interpolation function,so as to be able to approach the asymptote well.Through numerical examples,the approximation effects of traditional continued fraction interpolation and new continued fraction interpolation are analyzed,and their errors are compared,which shows the effectiveness of extended involute preserving continued fraction algorithm.Figure[10]table[6]reference[48]...