Shape Preserving Interpolation Using C2 Rational Cubic Spline

Abstract

This paper discusses the construction of new $C^{\mathrm{2}}$ rational cubic spline interpolant with cubic numerator and quadratic denominator. The idea has been extended to shape preserving interpolation for positive data using the constructed rational cubic spline interpolation. The rational cubic spline has three parameters ${\mathrm{\alpha }}_i$, ${\mathrm{\beta }}_i$, and ${\gamma }_i$. The sufficient conditions for the positivity are derived on one parameter ${\mathrm{\gamma }}_i$ while the other two parameters ${\mathrm{\alpha }}_i$ and ${\mathrm{\beta }}_i$ are free parameters that can be used to change the final shape of the resulting interpolating curves. This will enable the user to produce many varieties of the positive interpolating curves. Cubic spline interpolation with $C^{\mathrm{2}}$ continuity is not able to preserve the shape of the positive data. Notably our scheme is easy to use and does not require knots insertion and $C^{\mathrm{2}}$ continuity can be achieved by solving tridiagonal systems of linear equations for the unknown first derivatives $d_i$, $i=\mathrm{1},\dots ,n-\mathrm{1}$. Comparisons with existing schemes also have been done in detail. From all presented numerical results the new $C^{\mathrm{2}}$ rational cubic spline gives very smooth interpolating curves compared to some established rational cubic schemes. An error analysis when the function to be interpolated is $f\left(t\right)\in C^{\mathrm{3}}\left[t_{\mathrm{0}},t_n\right]$ is also investigated in detail.