光滑函数实根计算的渐进显式公式

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 =φ_i(s) such that A_i(s_j)= C(φ_i(s_j)). This paper proposes an explicit formula based on the reparameterized function φ_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.