Quadratic Interpolating Spline With Variable Parameters

Rational interpolation plays an important part in approximation theory. It hasa pleasant result not only when the functions have poles, but also when they haven't.So rational interpolation has become more and more important in approximationtheory today.In this paper, two quadratic interpolating functions have been developed, whichhave variable parameters. This kind of rational interpolating functions are more flex-ible. The shape of the curve can be changed from varying the variable parameters.Then a rational interpolating spline has been developed according to the rationalinterpolating functions mentioned before. This rational spline has one parameteron each subinterval. The common rational interpolating splines are fixed when theinitial data are given, and the shape of the curve are also fixed. However, for rationalspline which has variable parameters, the shape of the curve can be changed fromvarying the variable parameters on corresponding subintervals. These are the mainresults of the paper:1. A quadratic rational interpolating function g(x) has been developed, whichdepends on the two function values and one derivative value of the end-pointof the given interval. It has two variable variable parameters u,v. In thispaper, it has been proved that:(1) The interpolating function g(x) is monotonic preserving, when restrainingthe parameter v in proper range.(2) The rational interpolating function is monotonic about each variable pa-rameter, which makes it easier to change the shape of the curve throughvarying the variable parameters.Moreover, an error estimation illustrates that this interpolation scheme is sta- ble. Compare with Hermite polynomial interpolation and Lagrange polyno-mial interpolation, this rational interpolating function has better approximateresults.2. A quadratic interpolating function g(x) has been developed, which depends onfunction values and derivative values of both end-points of the given interval.It has only one variable parameter u. In this paper, it has been proved that:(1) For all u, g(x) is monotonic preserving.(2) g(x) is monotonic about the variable parameter u.(3) |f(x) -g(x)| = O(h); For strictly monotone function f(x), the errorbound will be O(h~3); When some initial conditions can be satisfied, theerror bound can be O(h~4) through constraining the variable parameteru.3. A rational interpolating spline has been developed according to the rationalinterpolating functions mentioned before. It has one parameter on each subin-terval.