Third-order convergence theorem by using majorizing function for a modified Newton method in Banach space

Abstract Following the ideas of Frontini and Sormani, we present a modified Newton method in Banach space which is used to solve the nonlinear operator equation. We establish the Newton–Kantorovich convergence theorem for the modified Newton method with third-order convergence in Banach space by using majorizing function. We also get the error estimate. Finally, two examples are provided to show the application of our theorem.

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

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

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

[4]  M. A. Salanova,et al.  Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method , 2000 .

[5]  Huang Zhengda,et al.  A note on the Kantorovich theorem for Newton iteration , 1993 .

[6]  Sergio Amat,et al.  A fast Chebyshev's method for quadratic equations , 2004, Appl. Math. Comput..

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

[8]  Ioannis K. Argyros.  On the comparison of a weak variant of the Newton-Kantorovich and Miranda theorems , 2004 .

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

[10]  Ioannis K. Argyros,et al.  An improved error analysis for Newton-like methods under generalized conditions , 2003 .

[11]  M. A. Hernández.  A modification of the classical Kantorovich conditions for Newton's method , 2001 .

[12]  Ioannis K. Argyros,et al.  On the Newton-Kantorovich hypothesis for solving equations , 2004 .

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

[14]  Ioannis K. Argyros.  On Newton's method under mild differentiability conditions and applications , 1999, Appl. Math. Comput..

[15]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[16]  M. Frontini,et al.  Third-order methods from quadrature formulae for solving systems of nonlinear equations , 2004, Appl. Math. Comput..

[17]  I. Argyros.  On a theorem of L.V. Kantorovich concerning Newton's method , 2003 .

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

[20]  Ioannis K. Argyros,et al.  Results on the Chebyshev method in banach spaces , 1993 .