Many problems in computer aided geometric design and computer graphics can be turned into a root-finding problem of a polynomial equation. Among various solutions, clipping methods based on the Bernstein–Bézier form usually have good numerical stability. A traditional clipping method using polynomials of degree r can achieve a convergence rate of r + 1 for a single root. It utilizes two polynomials of degree r to bound the given polynomial f ( t ) of degree n, where r = 2 , 3 , and the roots of the bounding polynomials are used for clipping off the subintervals containing no roots of f ( t ) . This paper presents a rational cubic clipping method for finding the roots of a polynomial f ( t ) within an interval. The bounding rational cubics can achieve an approximation order of 7 and the corresponding convergence rate for finding a single root is also 7. In addition, differently from the traditional cubic clipping method solving the two bounding polynomials in O ( n 2 ) , the new method directly constructs the two rational cubics in O ( n ) which can be used for bounding f ( t ) in many cases. Some examples are provided to show the efficiency, the approximation effect and the convergence rate of the new method. 
A rational cubic clipping method for computing real roots of a polynomial