Methods For Rational Spline Interpolation Based On Cauchy-Vandermonde System

This paper mainly studies the existence, the expression, the computation and the error estimate for rational interpolating splines with prescribed poles ( also called as CV rational interpolating splines)over the space R_3,2²(Δ,U₄).Firstly, the rational interpolation formulas represented by Cauchy-Vandermonde system are introducted. An alternate proof about its existence and uniqueness and its explicit represention are given, especially the rational interpolation formulas for simple konts and double konts, and the algorithm complexity in case of double konts is improved from O(n²) to O(n).Secondly, a rational spline space R_3,2²(Δ,U₄) with prescribed poles is defined, over which the existence and uniqueness of rational interpolating splines with prescribed poles are proved. Both expressions of CV rational interpolating splines are derived from the definition of the space R_3,2²(Δ,U₄) and its error bounds are estimated according to the band of the second derivation, from which it follows that its convergence and its derivative one are proved.Thirdly, The effects on the CV rational interpolating splines from the perturbation of the two boundary conditions are analyzed. From this the error bounds of first and second derivatives of CV rational interpolating spline are given.Lastly , the CV rational interpolating splines above are extended to the case of two variables. Their existence and uniqueness are proved for two usually boundary conditions as well. The CV rational interpolating splines of two variables are represented as tensor product from of the case of simple variable. The error bounds of first and second partial derivatives for the two-dimension rational splines produced by the error of boundary conditions are estimated.The new method for rational interpolating splines proposed in this paper is a generalization of three-degree polynomial ones, avoids the diffculty solving no-linear equation system in the case of ordinary rational interpolation, and the parameterεadds the freedom for rational interpolation,thus as an interpolating tool it is flexible,effective and can describe the singularity of function.