A Class Of Rational Splines And Its Shape-preserving Interpolation

In the paper a new method for rational spline interpolation of order ( n, 2 )~k( k = 1,2) with control parameter and prescribed poles is presented and its existence, representation, computation and error bounds are studied. Based on this rational splines, a class of shape-preserving interpolation of order 3 and 4 is discussed . Main contents are as follows:Firstly, the rational spline interpolation of order ( n, 2 )~k( k = 1,2) with control parameter and prescribed poles are constructed. Its existence and uniqueness are prove. The rational spline interpolating function is represented in two forms.Secondly, main attention is paid to studies for the the rational spline interpolation of degree 4.Its error is estimated. Several functions are used as numerical test with a computer. The curves for exact functions and their rational interpolating functions are plotted and their maximum error are computed, which shows the interpolation method presented in this paper is better than that of cubic polynomial splines, especially for the functions with prescribed poles. Especially, it reduces to the case of polynomial interpolating splines of degree 3 when q_i = 0.Thirdly, for two usual boundary conditions, the effect on the rational interpolation spline when they obtain a perturbation is analysed. The error bounds for first and second derivations for rational interpolation spline functions are given. The bounds are symmetrical, optimal in case of uniform knots.Finally, a method for a shape- preserving interpolation is developed by using the rational spline interpolating functions of degree 3 and 4. The sufficient conditions are derived for the rational spline interpolating functions to be monotonic and convex or concave in each interpolating interval. From this ,for the given data having the property of monotonicity or convexity its rational shape- preserving interpolating function of de- gree 3 or 4 can generated by choosing suitable control parameters in the knots.Examples show the effective of this method of shape- preserving interpolation .