目的 多项式求实根问题有着广泛的应用。改进传统的裁剪方法,在多项式重根的情形下,保持计算稳定性的同时显著地提高相应的收敛阶。方法 提出了基于R³空间内的3次裁剪方法。该方法继承了传统裁剪求根方法的优点,充分利用了Bernstein基函数较好的计算稳定性,同时给出简单方法判别重根的存在性,从而使得重根的情形可以转化为单根的情形。结果 与已有的基于R¹和R²空间的3次裁剪方法相比,本文方法可以具有更好的逼近效果。单根情形下,本文方法与基于R²空间的3次裁剪方法同时具有5次收敛阶,略高于基于R¹空间3次裁剪方法的4次收敛阶;m(≥2)重根情形下,本文方法理论上可具有5次收敛阶,明显优于已有的基于R¹和R²空间的3次裁剪方法的4/m或5/m收敛阶。基于R¹,R²和R³空间的3次裁剪方法的计算时间复杂度大致相当,均为O(n²)。结论 本文方法可以快速判定重根的情形,同时具有更高的收敛阶和更好的逼近效果。

Objective The root-finding problem has a wide range of application in computer-aided design and computer graphics. This study attempts to improve conventional clipping. For a multiple root case, good computational stability is maintained and a higher convergence rate is achieved. Method This study presents a new clipping method based on R³ space. The proposed method utilizes the good computational stability of Bernstein basis functions, provides a simple method to evaluate the existence of multiple roots, and converts a multiple root case into a simple root case. Result Compared with conventional cubic clipping methods based on R¹ and R² spaces, the proposed method achieves a better approximation effect. The proposed method can also achieve the convergence rate of 5 for a multiple root case of multiplicity m, which is better than the convergence rates 4/m and 5/m of previous cubic clipping methods. The computational complexity of different cubic clipping methods are close. Conclusion The new method can evaluate the existence of multiple roots and can achieve a higher convergence rate and a better approximation effect for a multiple root case.