光滑函数实根计算的渐进显式公式

求根问题在计算机图形学、机器人技术、地磁导航等领域应用广泛。基于重新参数化方法（reparamaterization-based method，RBM），给出了用于计算给定光滑函数在某区间内唯一实根的渐进式显式公式。给定光滑函数f（t），用有理多项式A_i（s）对曲线C（t）=（t，f（t））进行插值，得到重新参数化函数t =?_i（s），使得A_i（s_j）=C（?_i（s_j））。提出了基于重新参数化函数?_i（s）的显式公式用于渐进式逼近f（t）对应的实根，在n个函数计算的成本下，收敛阶可达到3·2^n-2，其中n≥3。与类牛顿法相比，本文方法提高了计算稳定性，且收敛速度更快、计算效率更高。与裁剪法相比，本文方法不需要求解包围多项式，且可用于非多项式函数计算，计算效率更高。数值实例表明，每增加一个插值点，逼近阶可提高一倍，且可获得较传统裁剪法更高的计算效率。

Explicit formulae for progressively computing a real root of the smooth function

The issue of finding the roots of equations has wide applications in computer graphics, robotics, and geomagnetic navigation. Based on the reparameterization technique, this paper presents an explicit formula for computing the real root of a smooth function within a certain interval. Given a smooth function f (t), by using the rational polynomial A_i(s) to interpolate the curve C(t)=(t,f (t)), it deduces a reparameterization function t =?_i(s) such that A_i(s_j)= C(?_i(s_j)). This paper proposes an explicit formula based on the reparameterized function ?_i(s) to progressively approximate the root of f (t), which can achieve the convergence order of 3·2^n-2 at the cost of n functions, where n≥3. Compared with the Newton-like method, it improves the computational stability, and can achieve faster convergence rate and better efficiency. Compared with the clipping methods, it does not need to solve the bounding polynomial, and is applicable for solving non-polynomial functions. Numerical examples show that each time one more interpolation point is added, the approximation order will be doubled, hence much better computational efficiency than traditional clipping methods.