Accurate C² Rational Interpolants in Tension

A family of rational interpolants is defined for a set of real data points (xᵢ ,fᵢ ),1 i n , with x₁ < x₂ < ⋯ < xₙ. Each interpolant of the family is a C² spline with a cubic numerator and a quadratic denominator which is identified uniquely by the value of a positive tension parameter C₀. The solution constructed is nonparametric. A value of C₀ close to zero will produce piecewise linear behaviour while a large value of C₀ will give approximately the cubic spline solution. The method used arises from a study of other rational functional forms for effecting monotonic or convex interpolation. A convergence analysis establishes an error bound in terms of C₀, and shows that O(h⁴ ) accuracy is obtained when exact boundary derivatives are prescribed. Several examples are supplied to support the practical value of the method.