论文信息 - On Newton-type methods with cubic convergence

On Newton-type methods with cubic convergence

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. © 2004 Elsevier B.V. All rights reserved.

H. H. H. Homeiera | H. Homeiera

[1] Bruce W. Char,et al. Maple V Language Reference Manual , 1993, Springer US.

[2] Bruce W. Char,et al. Maple V Library Reference Manual , 1992, Springer New York.

[3] Herbert H. H. Homeier. A modified Newton method with cubic convergence: the multivariate case , 2004 .

[4] J. M. Gutiérrez,et al. Geometric constructions of iterative functions to solve nonlinear equations , 2003 .

[5] Herbert H. H. Homeier. A modified Newton method for rootfinding with cubic convergence , 2003 .

[6] M. Frontini,et al. Some variant of Newton's method with third-order convergence , 2003, Appl. Math. Comput..

[7] Sunethra Weerakoon,et al. A variant of Newton's method with accelerated third-order convergence , 2000, Appl. Math. Lett..

[8] M. Frontini,et al. Modified Newton's method with third-order convergence and multiple roots , 2003 .