Rational cubic clipping with linear complexity for computing roots of polynomials

Abstract

Many problems in computer aided geometric design and computer graphics can be turned into a root-finding problem of polynomial equations. Among various clipping methods, the ones based on the Bernstein–Bézier form have good numerical stability. One of such clipping methods is the k-clipping method, where k=2,3 and often called a cubic clipping method when k=3. It utilizes O(n2) time to find two polynomials of degree k bounding the given polynomial f(t) of degree n, and achieves a convergence rate of k+1 for a single root. The roots of the bounding polynomials of degree k are then used for bounding the roots of f(t). This paper presents a rational cubic clipping method for finding two bounding cubics within O(n) time, which can achieve a higher convergence rate 5 than that of 4 of the previous cubic clipping method. When the bounding cubics are obtained, the remaining operations are the same as those of previous cubic clipping method. Numerical examples show the efficiency and the convergence rate of the new method.