Cubic Tangent Method in \mathbbR^2 Space for Computing Real Roots of Polynomials Within an Interval

The root finding problem has wide applications in intersection, minimum distance computation, and so on.The cubic clipping method utilizes the stability of the computation by using Bernstein basis functions and avoids the unstability of the numerical solution.It also has the convergence rate of order 4.Different from previous clipping methods approximating f(t) in \mathbbR^1 space, this paper presents an improved cubic clipping method approximating (t, f(t)) in \mathbbR^2 space instead, which has a higher convergence rate of order 5 for a single root.It selects three points on the curve (t, f(t)), and constructs two cubic polynomial curves A_i(u), i=1, 2 which interpolate both three points and two of their three directional tangent vectors.By using a suitable reparameterization function φ(t), A_i(φ(t)) and f(t) have the approximation of order 5. Furthermore, when the length of the given interval is small enough, the two polynomials A_i(u) can be directly used as the two bounding polynomials, which avoids the corresponding time-consuming computation needed in the previous cubic clipping method.Examples show the better efficiency, approximation effect and convergence rate of the new method.