On Newton-type methods with cubic convergence
- 作者：Homeier ; H.H.H.
- 关键词：41A25 ; 65D99 ; Rootfinding ; Newton method ; Newton-type method ; Newton theorem ; Inverse function ; Iterative methods ; Nonlinear equations
- 刊名：Journal of Computational and Applied Mathematics
- 出版年：2005
- 期刊代码：72_03770427
- 类别：cp
- 出版时间：April 15, 2005
- 卷：176
- 期：2
- 页码：425-432
- 文件大小：173 K
摘要
Recently, there has been some progress on Newton-type methods with cubic convergence that do not require the computation of second derivatives. Weerakoon and Fernando (Appl. Math. Lett. 13 (2000) 87) derived the Newton method and a cubically convergent variant by rectangular and trapezoidal approximations to Newton's theorem, while Frontini and Sormani (J. Comput. Appl. Math. 156 (2003) 345; 140 (2003) 419 derived further cubically convergent variants by using different approximations to Newton's theorem. Homeier (J. Comput. Appl. Math. 157 (2003) 227; 169 (2004) 161) independently derived one of the latter variants and extended it to the multivariate case. Here, we show that one can modify the Werrakoon–Fernando approach by using Newton's theorem for the inverse function and derive a new class of cubically convergent Newton-type methods.