Newton–Kantorovich Convergence Theorem of a Modified Newton’s Method Under the Gamma-Condition in a Banach Space

A Newton–Kantorovich convergence theorem of a modified Newton’s method having third order convergence is established under the gamma-condition in a Banach space to solve nonlinear equations. It is assumed that the nonlinear operator is twice Fréchet differentiable and satisfies the gamma-condition. We also present the error estimate to demonstrate the efficiency of our approach. A comparison of our numerical results with those obtained by other Newton–Kantorovich convergence theorems shows high accuracy of our results.

[1]  Chong Li,et al.  Smale's α-theory for inexact Newton methods under the γ-condition☆ , 2010 .

[2]  Jose M. Gutikez.  A new semilocal convergence theorem for Newton's method , 1997 .

[3]  Miguel Ángel Hernández,et al.  An acceleration of Newton's method: Super-Halley method , 2001, Appl. Math. Comput..

[4]  Herbert H. H. Homeier.  On Newton-type methods with cubic convergence , 2005 .

[5]  Chong Li,et al.  Convergence of Newton's Method and Uniqueness of the Solution of Equations in Banach Spaces II , 2003 .

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

[7]  Yueqing Zhao,et al.  Convergence analysis for a deformed Newton's method with third-order in Banach space under γ-condition , 2009, Int. J. Comput. Math..

[8]  Qingbiao Wu,et al.  Third-order convergence theorem by using majorizing function for a modified Newton method in Banach space , 2006, Appl. Math. Comput..

[9]  Zhou Yuren,et al.  About Newton method , 2000 .

[10]  Yitian Li,et al.  A modification of Newton method with third-order convergence , 2006, Appl. Math. Comput..

[11]  I. Argyros,et al.  A generalized Kantorovich theorem on the solvability of nonlinear equations , 2009 .

[12]  Xinghua Wang.  Convergence on the iteration of Halley family in weak conditions , 1997 .

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

[14]  S. Smale.  Newton’s Method Estimates from Data at One Point , 1986 .

[15]  Xinghua Wang,et al.  Convergence of Newton's method and uniqueness of the solution of equations in Banach space , 2000 .

[16]  Ioannis K. Argyros,et al.  On the midpoint method for solving equations , 2010, Appl. Math. Comput..