A family of piecewise rational quartic interpolants is given. Identified uniquely by the value of a tension parameter λᵢ, each interpolant of the family can be C² 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 λᵢ to guarantee shape preservation is given. A convergence analysis establishes an error bound in terms of λᵢ and shows that O(h³) accuracy is obtained for C² continuity. Several examples are supplied to support the practical value of the method.
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Xuli Han (2008) studied this question.
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