Lebesgue Constant Minimizing Shape Preserving Barycentric Rational Interpolation

Interpolation is an important tool of mathematical. Polynomial interpolation although the structure is simple and easy to construct,when the number of the interpolation function is higher may be product Runge phenomenon. Rational interpolation convergence rate is faster than polynomial interpolation, but it possible has problems like the deficit not exist and the effect of approximation may be not so good. The barycentric rational interpolation compared with other interpolation not only relaxed requirements on the number of the interpolation functions but also no poles, no unattainable, specially good numerical stability. Different interpolation weights can be get distinct interpolation function, so how to choose optimal weights becomes a key question. In this paper, based on Lebesgue constant minimizing barycentric rational interpolation, we studied the shape control of barycentric rational interpolation and bivariate barycentric rational interpolation. Generally speaking, take the Lebesgue constant minimizing as the objective function, take the weights as the only decision variable of the optimization model and take some constraint conditions to satisfy interpolation function has no unattained points, no poles and has a unique solution.In this paper, we will add some constraint conditions of shape control on the above optimization model to construct shape-preserving barycentric rational interpolation optimization model. The optimization algorithm of bivariate barycentric rational interpolation is constructed based on univariate barycentric rational interpolation, and give specific optimization model. Finally, a lot of numerical examples are given to show the feasibility and the effectiveness of the new method.