Shape-preserving piecewise rational interpolant with quartic numerator and quadratic denominator

An explicit representation of a piecewise rational interpolant with quartic numerator and quadratic denominator is presented. For positivity data, monotone data and convex data, the shape-preserving properties of the interpolant are given. The interpolant is C2C2 continuous spline with a shape parameter wiwi on each subinterval. The values of wiwi to guarantee shape preservation are estimated. A convergence analysis establishes an error bound in terms of wiwi and shows that the interpolant is O(h2)O(h2) or O(h3)O(h3) accurate. Several examples are supplied to support the practical value of the given interpolation method.