A Type of C~2 Piecewise Rational Interpolation

A family of piecewise rational quintic interpolation is presented. Each interpolation of the family, which is identified uniquely by the value of a parameter αi, is of C2 continuity without solving a system of consistency equations for the derivative values at the knots, and can be expressed by the basis functions. Interpolant is of O(hr) accuracy when f(x)?Cr[a,b], and the errors have only a small floating for a big change of the parameter αi, it means the interpolation is stable for the parameter. The interpolation can preserve the shape properties of the given data, such as monotonicity and convexity, and a proper choice of parameter αi is given.