Abstract
A family of piecewise rational quartic interpolants is given. Identified uniquely by the value of a tension parameter $\lambda_i$, each interpolant of the family can be $C^2$ spline without solving a linear or nonlinear system of consistency equations for the derivative values at the knots. The interpolant can preserve the local convexity/concavity properties of the given data. A proper choice of $\lambda_i$ to guarantee shape preservation is given. A convergence analysis establishes an error bound in terms of $\lambda_i$ and shows that $O(h^3)$ accuracy is obtained for $C^2$ continuity. Several examples are supplied to support the practical value of the method.
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