Third-order iterative methods under Kantorovich conditions

Applied Mathematics and Computation, 2005

Using the iteration formulas of second order [Mamta, V. Kanwar, V.K. Kukreja, Sukhjit Singh, On a class of quadratically convergent iteration formulae, Appl. Math. Comput. 2004, in press] for solving single variable nonlinear equations, two classes of third-order multipoint methods without using second derivative are derived. The main advantage of these classes is that they do not fail if the derivative of the function is either zero or very small in the vicinity of the required root. Further, a new family of Secantlike method with guaranteed super linear convergence is obtained by discrete modifications and their comparison with respect to the existing classical methods is given.

International Journal of Computational Methods, 2009

The aim of this paper is to discuss the convergence of a third order method for solving nonlinear equations F (x) = 0 in Banach spaces by using recurrence relations. The 21 convergence of the method is established under the assumption that the second Fréchet derivative of F satisfies a condition that is milder than Lipschitz/Hölder continuity con-23 dition. A family of recurrence relations based on two parameters depending on F is also derived. An existence-uniqueness theorem is also given that establish convergence of the 25 method and a priori error bounds. A numerical example is worked out to show that the method is successful even in cases where Lipschitz/Hölder continuity condition fails. (2005); Wu and Zhao ] has been carried out to provide 1 improvements in these methods, their applications and convergence. Third order one point iterative methods ; ; 3 Wu and Zhao ; ] are used in many applications. They can also be used in stiff systems ], where a quick convergence is required. But 5 a very restrictive condition of one point iteration of order N is that they depend explicitly on the first N − 1 derivatives of F . All these higher order derivatives 7 are very difficult to compute. On the other hand, multi point third order iterative methods [Ye and Li (2006); Wu and Zhao (2006); Ortega and Rheinboldt (1970); 9 Hernández (2001); Hernández (2000)] use information at a number of points have also gained importance recently.

Applied Mathematics and Computation, 2007

In this paper, we suggest and analyze a new two-step iterative method for solving nonlinear equations, which is called the modified Householder method without second derivatives for nonlinear equation. We also prove that the modified method has cubic convergence. Several examples are given to illustrate the efficiency and the performance of the new method. New method can be considered as an alternative to the present cubic convergent methods for solving nonlinear equations.

Applied Mathematics and Computation, 2015

A new predictor-corrector iterative procedure, that combines Newton's method as predictor scheme and a fifth-order iterative method as a corrector, is designed for solving nonlinear equations in Banach spaces. We analyze the local order of convergence and the regions of accessibility of the new method comparing it with Newton's method, both theoretical and numerically.

Stud. Univ. Babes- …, 2007

In this short note we give certain comments and improvements of some three-step iterative methods recently considered by N.A. Mir and T.

Applied Mathematics and Computation, 2006

In this paper, we suggest and analyze a new three-step iterative method for solving nonlinear equations. We show that this new iterative method has third-order convergence. Several numerical examples are given to illustrate the efficiency and performance of this new method. New method can be viewed as an improvement of the previously known iterative methods.

Journal of Mathematical Chemistry, 2020

This paper deal with the study of local convergence of fourth and fifth order iterative method for solving nonlinear equations in Banach spaces. Only the premise that the first order Fréchet derivative fulfills the Lipschitz continuity condition is needed. Under these conditions, a convergence theorem is established to study the existence and uniqueness regions for the solution for each method. The efficacy of our convergence study is shown solving various numerical examples as a nonlinear integral equation and calculating the radius of the convergence balls. We compare the radii of convergence balls and observe that by our approach, we get much larger balls as existing ones. In addition, we also include the real and complex dynamic study of one of the methods applied to a generic polynomial of order two.

Applied Mathematics and Computation, 2007

In this paper, we suggest and analyze a new three-step iterative method for solving nonlinear equations involving only first derive of the function using a new decomposition technique which is due to Noor [M. Aslam Noor, Numerical Analysis and Optimization, Lecture Notes,

Bulletin of The Australian Mathematical Society, 1997

A family of third-order iterative processes (that includes Chebyshev and Halley's methods) is studied in Banach spaces. Results on convergence and uniqueness of solution are given, as well as error estimates. This study allows us to compare the most famous third-order iterative processes.

2011

Recently, Parida and Gupta [J. Comp. Appl. Math. 206 (2007), 873-877] used Rall's recurrence relations approach (from 1961) to approximate roots of nonlinear equations, by developing several methods, the latest of which is free of second derivative and it is of third order. In this paper, we use an idea of Kou and Li [Appl. Math. Comp. 187 (2007), 1027-1032] and modify the approach of Parida and Gupta, obtaining yet another third-order method to approximate a solution of a nonlinear equation in a Banach space. We give several applications to our method.

Computers & Mathematics with Applications, 2011

Gupta [J. Comp. Appl. Math. 206 (2007) 873-877] used Rall's recurrence relations approach (from 1961) to approximate roots of nonlinear equations, by developing several methods, the latest of which is free of second derivative and it is of third order. In this paper, we use an idea of Kou and Li [Appl. Math. Comp. 187 (2007), 1027-1032] and modify the approach of Parida and Gupta, obtaining yet another third-order method to approximate a solution of a nonlinear equation in a Banach space. We give several applications to our method.

Nonlinear Analysis-theory Methods & Applications, 2008

Applied Mathematics and Computation, 2007

a three-step iterative method with fourth-order for solving nonlinear equations has been proposed. In this paper, we show that the iterative method given by the authors has quadratic convergence, not the fourth-order one as claimed in their work.

Applied Mathematics and Computation, 2013

In this paper we introduce a new third-order iterative method for solving nonlinear equations. This method is converging on larger intervals than some other similar known methods. A comparison between the new method and other third-order methods is presented by using the basins of attraction for the real roots for some test problems. The numerical results are also presented.

Journal of Computational and Applied Mathematics, 2007

The convergence of iterative methods for solving nonlinear operator equations in Banach spaces is established from the convergence of majorizing sequences. An alternative approach is developed to establish this convergence by using recurrence relations. For example, the recurrence relations are used in establishing the convergence of Newton's method [L.B. Rall, Computational Solution of Nonlinear Operator Equations, Robert E. Krieger, New York, 1979] and the third order methods such as Halley's, Chebyshev's and super Halley's [V. Candela, A. Marquina, Recurrence relations for rational cubic methods I: the Halley method, Computing 44 (1990) 169-184; V. Candela, A. Marquina, Recurrence relations for rational cubic methods II: the Halley method, Computing 45 (1990) 355-367; J.A. Ezquerro, M.A. Hernández, Recurrence relations for Chebyshev-type methods, Appl. Math. Optim. 41 (2000) 227-236; J.M. Gutiérrez, M.A. Hernández, Third-order iterative methods for operators with bounded second derivative, J. Comput. Appl. Math. 82 (1997) 171-183; J.M. Gutiérrez, M.A. Hernández, Recurrence relations for the Super-Halley method, Comput. Math. Appl. 7(36) (1998) 1-8; M.A. Hernández, Chebyshev's approximation algorithms and applications, Comput. Math. Appl. 41 (2001) 433-445 [10]].