Cubic spline interpolation with optimal end conditions

Traditional end conditions for cubic spline interpolation consist of values, the first or the second derivatives of interpolated functions on the boundary interpolation knots. The not-a-knot end condition proposed by de Boor (1985) is a kind of end condition of cubic spline interpolation for the practical application without the requirements of the derivatives at the end knots. However, a significant disadvantage of such end condition is that there is a sharp decrease in the accuracy of the interpolation at boundary intervals. In this paper, by changing the locations of two spline knots in not-a-knot end condition, we propose the optimal arrangement of shifted spline knots for cubic spline interpolation. The proposed scheme leads to an approximately 3.4 times increasing in the accuracy of the interpolation compared to the de Boor’s not-a-knot end condition. Furthermore, we also present the optimal end conditions of cubic spline interpolation to approximate the first and the second derivatives of interpolated functions. The representative examples show the effectiveness and the superiority of the proposed method.