Computing accurate splines of degree greater than three is still a challenging task in today's applications. In this type of interpolation, high-order derivatives are needed on the given mesh. As these derivatives are rarely known and are often not easy to approximate accurately, high-degree splines are difficult to obtain using standard approaches. In Beaudoin (1998), Beaudoin and Beauchemin (2003), and Pepin et al. (2019), a new method to compute spline approximations of low or high degree from equidistant interpolation nodes based on the discrete Fourier transform is analyzed. The accuracy of this method greatly depends on the accuracy of the boundary conditions. An algorithm for the computation of the boundary conditions can be found in Beaudoin (1998), and Beaudoin and Beauchemin (2003). However, this algorithm lacks robustness since the approximation of the boundary conditions is strongly dependant on the choice of $\theta$ arbitrary parameters, $\theta$ being the degree of the spline. The goal of this paper is therefore to propose two new robust algorithms, independent of arbitrary parameters, for the computation of the boundary conditions in order to obtain accurate splines of any degree. Numerical results will be presented to show the efficiency of these new approaches.

翻译：在当前的应用中，计算次数大于三的高精度样条仍然是一项具有挑战性的任务。在这种插值方法中，需要在给定网格上获得高阶导数。由于这些导数很少已知，且通常难以精确逼近，因此使用标准方法难以获得高阶样条。在Beaudoin (1998)、Beaudoin and Beauchemin (2003) 以及Pepin等人 (2019) 的研究中，分析了一种基于离散傅里叶变换从等距插值节点计算低阶或高阶样条逼近的新方法。该方法的精度在很大程度上取决于边界条件的精度。Beaudoin (1998) 和 Beaudoin and Beauchemin (2003) 中给出了一个计算边界条件的算法。然而，该算法缺乏鲁棒性，因为边界条件的逼近强烈依赖于θ任意参数的选择，其中θ是样条的次数。因此，本文的目标是提出两种新的、独立于任意参数的鲁棒算法，用于计算边界条件，以获得任意次数的精确样条。数值结果将展示这些新方法的有效性。