Shape Preserving Interpolation UsingC2Rational Cubic Spline

Abstract

This paper discusses the construction of newC2rational 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 parametersαi,βi, andγi. The sufficient conditions for the positivity are derived on one parameterγiwhile the other two parametersαiandβiare 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 withC2continuity is not able to preserve the shape of the positive data. Notably our scheme is easy to use and does not require knots insertion andC2continuity can be achieved by solving tridiagonal systems of linear equations for the unknown first derivativesdi,i=1,…,n-1. Comparisons with existing schemes also have been done in detail. From all presented numerical results the newC2rational cubic spline gives very smooth interpolating curves compared to some established rational cubic schemes. An error analysis when the function to be interpolated isft∈C3t0,tnis also investigated in detail.