Bivariate Composite Barycentric Rational Interpolation

The interpolation is that according to the given values of function table to construct a simple continuously function such that it has the same function values of at all the given points exactly. Polynomial interpolants are used in many areas of numerical analysis. But the polynomial interpolant does not converge uniformly were given by Runge. When the interpolated function with poles,rational interpolation is more flexible than polynomial interpolants. However, it is difficult to avoid poles and unattainable points, control the position of the poles. Compared with Thiele-type continued fraction rational interpolant,barycentric rational interpolation which has small calculation quantity, good numerical stability.Besides, poles and unattainable points are prevented when chooing the appropriate weighets. In this paper, the new method of bivariate composite barycentric blending rational interpolation in rectangular domain is studied and the new method of bivariate composite barycentric rational interpolation in rectangular domain is studied. Firstly, bivariate Newton interpolation polynomial,bivariate blending rational interpolation and bivariate barycentric rational interpolation are constructed in a small rectangular domain respectively. Then, bivariate composite barycentric blending rational interpolation and bivariate composite barycentric rational interpolation are constructed by means of composite barycentric rational interpolation and some interpolation properties are proved,such as bivariate composite barycentric blending rational interpolant and bivariate composite barycentric rational interpolation have no poles and unattainable points.Finally,unary composite barycentric rational interpolation with prescribed poles is studied in this paper.Many numerical examples are given to show the effectiveness of the new method.