Barycentric Rational Interpolation Based On Padé-type Approximation

As an important aspect of interpolation,nonlinear approximation is widely used in engineering computation and image processing.Essentially,it is to construct a function with continuous definition from the values of the known discrete points,so that it is exactly the same as the value of the approximated function at the given point.As the basis of the whole numerical approximation,polynomial interpolation has the advantages of simple structure and easy calculation,but the function constructed by higher-order interpolation may have Runge phenomenon and has limitations.When there are more interpolation nodes,compared with polynomial interpolation,rational function interpolation converges faster and has better approximation effect,but there are some inevitable shortcomings,such as Thiele type rational interpolation may have deficit quotient does not exist,cannot avoid poles and unreachable points and other problems.The rational function constructed by barycentric rational interpolation under certain conditions not only satisfies the conditions of interpolation but also avoids poles.At the same time,it also has the advantages of small computation and good numerical stability.On the other hand,Padé approximation is a very effective rational approximation method,which is characterized by being able to reflect as much information as possible about the function being approximated,especially the information about the position of the pole,but the position of the pole is not easy to control.The basic idea of the Padé-type approximation proposed later is to select the pole of the rational fraction in advance,and then determine its numerator and denominator.So that it can produce a better approximation effect.On this basis,this paper further studies the composite barycentric rational interpolation method based on Padé-type approximation,and proves that it satisfies the interpolation conditions and has no poles in the interpolation interval.At the same time,it studies the conformal method of composite barycentric rational interpolation based on Padé-type approximation,which mainly studies the problem of harmonic concavity and concavity.The minimum sum of the squares of the errors is taken as the objective function,and the conformal condition,no poles and interpolation weight is not zero are taken as the constraint conditions.The optimization model is created and the optimal weight is solved.The experimental results show that the conformal method is feasible.In addition,a composite barycenter interpolation method based on Padé approximation is proposed.Finally,the composite barycentric rational interpolation method based on Padé-type approximation is extended to binary form,and the new interpolation method is applied to definite integral,and several numerical examples are given to verify the effectiveness of the new method.Figure [18] Table [3] Reference [55]...