Range functions are a fundamental tool for certified computations in geometric modeling, computer graphics, and robotics, but traditional range functions have only quadratic convergence order ($m=2$). For ``superior'' convergence order (i.e., $m>2$), we exploit the Cornelius--Lohner framework in order to introduce new bivariate range functions based on Taylor, Lagrange, and Hermite interpolation. In particular, we focus on practical range functions with cubic and quartic convergence order. We implemented them in Julia and provide experimental validation of their performance in terms of efficiency and efficacy.

翻译：区间函数是几何建模、计算机图形学及机器人学中用于验证计算的一种基础工具，但传统区间函数仅具有二次收敛阶 ($m=2$)。为获得“高阶”收敛性（即 $m>2$），我们基于 Cornelius-Lohner 框架，利用泰勒、拉格朗日和埃尔米特插值方法引入新的双变量区间函数。特别地，我们聚焦于具有三次和四次收敛阶的实用化区间函数，并在 Julia 语言中实现，通过实验验证了其在效率与有效性方面的性能表现。