In this paper, we present a new one-parameter fourth-order family of iterative methods for solvin... more In this paper, we present a new one-parameter fourth-order family of iterative methods for solving nonlinear equations. The new family requires two evaluations of the given function and one of its first derivative per iteration. The well-known Traub-Ostrowski's fourth-order method is shown to be part of the family. Several numerical examples are given to illustrate the efficiency and performance of the presented methods.
Journal of Computational and Applied Mathematics, 2001
Let f:X→Rk be a Lipschitz continuous function on a compact subset X⊂Rd. Subdivision algorithms ar... more Let f:X→Rk be a Lipschitz continuous function on a compact subset X⊂Rd. Subdivision algorithms are described that can be used to find all solutions of the equation f(x)=0 that lie in X. Convergence is shown and numerical examples are presented. Modifications of the basic algorithm which speed convergence are given for the case of nondegenerate zeros of a vector field.
Journal of Computational and Applied Mathematics, 2008
In this paper we consider constructing some higher-order modifications of Newton's method for sol... more In this paper we consider constructing some higher-order modifications of Newton's method for solving nonlinear equations which increase the order of convergence of existing iterative methods by one or two or three units. This construction can be applied to any iteration formula, and per iteration the resulting methods add only one additional function evaluation to increase the order. Some illustrative examples are provided and several numerical results are given to show the performance of the presented methods.
In this paper, the homotopy perturbation method and a modified homotopy perturbation method are u... more In this paper, the homotopy perturbation method and a modified homotopy perturbation method are used for analytical treatment of the wave equation and some nonlinear diffusion equations, respectively. Some examples are given to illustrate that a suitable choice of an initial solution can lead to the exact solution, this revealing the reliability and effectiveness of the method.
In this paper, we present two new families of iterative methods for multiple roots of nonlinear e... more In this paper, we present two new families of iterative methods for multiple roots of nonlinear equations. One of the families require one-function and two-derivative evaluation per step, and the other family requires two-function and one-derivative evaluation. It is shown that both are third-order convergent for multiple roots. Numerical examples suggest that each family member can be competitive to other third-order methods and Newton's method for multiple roots. In fact the second family is even better than the first.
In this paper, we construct some modifications of Newton's method for solving nonlinear equations... more In this paper, we construct some modifications of Newton's method for solving nonlinear equations, which is based on the method of undetermined coefficients. It is shown by way of illustration that the method of undetermined coefficients is a promising tool for developing new methods, and reveals its wide applicability by obtaining some existing methods as special cases. Two new sixth-order methods are developed. Numerical examples are given to support that the methods thus obtained can compete with other iterative methods.
Gupta [J. Comp. Appl. Math. 206 (2007) 873-877] used Rall's recurrence relations approach (from 1... more 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.
ABSTRACT The two-point boundary value problems occur in a wide variety of problems in engineering... more ABSTRACT The two-point boundary value problems occur in a wide variety of problems in engineering and science. In this paper, we implement the homotopy perturbation method for solving the linear and nonlinear two-point boundary value problems. The main aim of this paper is to compare the performance of the homotopy perturbation method with extended Adomian decomposition method and shooting method. As a result, for the same number of terms, the homotopy perturbation method yields relatively more accurate results with rapid convergence than other methods. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations.
In this paper we present a new efficient sixth-order scheme for nonlinear equations. The method i... more In this paper we present a new efficient sixth-order scheme for nonlinear equations. The method is compared to several members of the family of methods developed by [B. Neta, A sixth-order family of methods for nonlinear equations, Int. J. Comput. Math. 7 (1979) 157-161]. It is shown that the new method is an improvement over this well known scheme.
There are many methods for the solution of a nonlinear algebraic equation. The methods are classi... more There are many methods for the solution of a nonlinear algebraic equation. The methods are classified by the order, informational efficiency and efficiency index. Here we consider other criteria, namely the basin of attraction of the method and its dependence on the order. We discuss several methods of various orders and present the basin of attraction for several examples. It can be seen that not all higher order methods were created equal. Newton's, Halley's, Murakami's and Neta-Johnson's methods are consistently better than the others. In two of the examples Neta's 16th order scheme was also as good.
In this paper, we present some new variants of super-Halley method for solving nonlinear equation... more In this paper, we present some new variants of super-Halley method for solving nonlinear equations. These methods are free from second derivatives and require one function and two first derivative evaluations per iteration. Analysis of convergence shows that the methods are fourth-order. Several numerical examples are given to illustrate the performance of the presented methods.
Newton's method Iterative methods Nonlinear equations Order of convergence Method of undetermined... more Newton's method Iterative methods Nonlinear equations Order of convergence Method of undetermined coefficients Root-finding methods a b s t r a c t In this paper we present two new schemes, one is third-order and the other is fourth-order. These are improvements of second-order methods for solving nonlinear equations and are based on the method of undetermined coefficients. We show that the fourth-order method is more efficient than the fifth-order method due to Kou et al. [J. Kou, Y. Li, X. Wang, Some modifications of Newton's method with fifth-order covergence, J. Comput. Appl. Math., 209 (2007) 146-152]. Numerical examples are given to support that the methods thus obtained can compete with other iterative methods.
In this paper, we present a new third-order modification of Newton's method for multiple roots, w... more In this paper, we present a new third-order modification of Newton's method for multiple roots, which is based on existing third-order multiple root-finding methods. Numerical examples show that the new method is competitive to other methods for multiple roots.
In this paper, we present a new one-parameter fourth-order family of iterative methods for solvin... more In this paper, we present a new one-parameter fourth-order family of iterative methods for solving nonlinear equations. The new family requires two evaluations of the given function and one of its first derivative per iteration. The well-known Traub-Ostrowski's fourth-order method is shown to be part of the family. Several numerical examples are given to illustrate the efficiency and performance of the presented methods.
In this paper, we present a new fifth-order method for solving nonlinear equations. Per iteration... more In this paper, we present a new fifth-order method for solving nonlinear equations. Per iteration the new method requires two function and two first derivative evaluations. It is shown that the new method is fifth-order convergent. Several numerical examples are given to illustrate the performance of the presented method.
In this paper, we construct some fourth-order modifications of Newton's method for solving nonlin... more In this paper, we construct some fourth-order modifications of Newton's method for solving nonlinear equations. Any two existing fourth-order methods can be effectively used to give rise to new fourth-order methods. Per iteration the new methods require two evaluations of the function and one of its first-derivative. Numerical examples are given to show the performance of the presented methods.
In this paper, we present some sixth-order class of modified Ostrowski's methods for solving nonl... more In this paper, we present some sixth-order class of modified Ostrowski's methods for solving nonlinear equations. Per iteration each class member requires three function and one first derivative evaluations, and is shown to be at least sixthorder convergent. Several numerical examples are given to illustrate the performance of some of the presented methods.
In this paper new fourth order optimal root-finding methods for solving nonlinear equations are p... more In this paper new fourth order optimal root-finding methods for solving nonlinear equations are proposed. The classical Jarratt's family of fourth-order methods are obtained as special cases. We then present results which describe the conjugacy classes and dynamics of the presented optimal method for complex polynomials of degree two and three. The basins of attraction of existing optimal methods and our method are presented and compared to illustrate their performance.
In this paper, we present a new one-parameter fourth-order family of iterative methods for solvin... more In this paper, we present a new one-parameter fourth-order family of iterative methods for solving nonlinear equations. The new family requires two evaluations of the given function and one of its first derivative per iteration. The well-known Traub-Ostrowski's fourth-order method is shown to be part of the family. Several numerical examples are given to illustrate the efficiency and performance of the presented methods.
Journal of Computational and Applied Mathematics, 2001
Let f:X→Rk be a Lipschitz continuous function on a compact subset X⊂Rd. Subdivision algorithms ar... more Let f:X→Rk be a Lipschitz continuous function on a compact subset X⊂Rd. Subdivision algorithms are described that can be used to find all solutions of the equation f(x)=0 that lie in X. Convergence is shown and numerical examples are presented. Modifications of the basic algorithm which speed convergence are given for the case of nondegenerate zeros of a vector field.
Journal of Computational and Applied Mathematics, 2008
In this paper we consider constructing some higher-order modifications of Newton's method for sol... more In this paper we consider constructing some higher-order modifications of Newton's method for solving nonlinear equations which increase the order of convergence of existing iterative methods by one or two or three units. This construction can be applied to any iteration formula, and per iteration the resulting methods add only one additional function evaluation to increase the order. Some illustrative examples are provided and several numerical results are given to show the performance of the presented methods.
In this paper, the homotopy perturbation method and a modified homotopy perturbation method are u... more In this paper, the homotopy perturbation method and a modified homotopy perturbation method are used for analytical treatment of the wave equation and some nonlinear diffusion equations, respectively. Some examples are given to illustrate that a suitable choice of an initial solution can lead to the exact solution, this revealing the reliability and effectiveness of the method.
In this paper, we present two new families of iterative methods for multiple roots of nonlinear e... more In this paper, we present two new families of iterative methods for multiple roots of nonlinear equations. One of the families require one-function and two-derivative evaluation per step, and the other family requires two-function and one-derivative evaluation. It is shown that both are third-order convergent for multiple roots. Numerical examples suggest that each family member can be competitive to other third-order methods and Newton's method for multiple roots. In fact the second family is even better than the first.
In this paper, we construct some modifications of Newton's method for solving nonlinear equations... more In this paper, we construct some modifications of Newton's method for solving nonlinear equations, which is based on the method of undetermined coefficients. It is shown by way of illustration that the method of undetermined coefficients is a promising tool for developing new methods, and reveals its wide applicability by obtaining some existing methods as special cases. Two new sixth-order methods are developed. Numerical examples are given to support that the methods thus obtained can compete with other iterative methods.
Gupta [J. Comp. Appl. Math. 206 (2007) 873-877] used Rall's recurrence relations approach (from 1... more 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.
ABSTRACT The two-point boundary value problems occur in a wide variety of problems in engineering... more ABSTRACT The two-point boundary value problems occur in a wide variety of problems in engineering and science. In this paper, we implement the homotopy perturbation method for solving the linear and nonlinear two-point boundary value problems. The main aim of this paper is to compare the performance of the homotopy perturbation method with extended Adomian decomposition method and shooting method. As a result, for the same number of terms, the homotopy perturbation method yields relatively more accurate results with rapid convergence than other methods. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations.
In this paper we present a new efficient sixth-order scheme for nonlinear equations. The method i... more In this paper we present a new efficient sixth-order scheme for nonlinear equations. The method is compared to several members of the family of methods developed by [B. Neta, A sixth-order family of methods for nonlinear equations, Int. J. Comput. Math. 7 (1979) 157-161]. It is shown that the new method is an improvement over this well known scheme.
There are many methods for the solution of a nonlinear algebraic equation. The methods are classi... more There are many methods for the solution of a nonlinear algebraic equation. The methods are classified by the order, informational efficiency and efficiency index. Here we consider other criteria, namely the basin of attraction of the method and its dependence on the order. We discuss several methods of various orders and present the basin of attraction for several examples. It can be seen that not all higher order methods were created equal. Newton's, Halley's, Murakami's and Neta-Johnson's methods are consistently better than the others. In two of the examples Neta's 16th order scheme was also as good.
In this paper, we present some new variants of super-Halley method for solving nonlinear equation... more In this paper, we present some new variants of super-Halley method for solving nonlinear equations. These methods are free from second derivatives and require one function and two first derivative evaluations per iteration. Analysis of convergence shows that the methods are fourth-order. Several numerical examples are given to illustrate the performance of the presented methods.
Newton's method Iterative methods Nonlinear equations Order of convergence Method of undetermined... more Newton's method Iterative methods Nonlinear equations Order of convergence Method of undetermined coefficients Root-finding methods a b s t r a c t In this paper we present two new schemes, one is third-order and the other is fourth-order. These are improvements of second-order methods for solving nonlinear equations and are based on the method of undetermined coefficients. We show that the fourth-order method is more efficient than the fifth-order method due to Kou et al. [J. Kou, Y. Li, X. Wang, Some modifications of Newton's method with fifth-order covergence, J. Comput. Appl. Math., 209 (2007) 146-152]. Numerical examples are given to support that the methods thus obtained can compete with other iterative methods.
In this paper, we present a new third-order modification of Newton's method for multiple roots, w... more In this paper, we present a new third-order modification of Newton's method for multiple roots, which is based on existing third-order multiple root-finding methods. Numerical examples show that the new method is competitive to other methods for multiple roots.
In this paper, we present a new one-parameter fourth-order family of iterative methods for solvin... more In this paper, we present a new one-parameter fourth-order family of iterative methods for solving nonlinear equations. The new family requires two evaluations of the given function and one of its first derivative per iteration. The well-known Traub-Ostrowski's fourth-order method is shown to be part of the family. Several numerical examples are given to illustrate the efficiency and performance of the presented methods.
In this paper, we present a new fifth-order method for solving nonlinear equations. Per iteration... more In this paper, we present a new fifth-order method for solving nonlinear equations. Per iteration the new method requires two function and two first derivative evaluations. It is shown that the new method is fifth-order convergent. Several numerical examples are given to illustrate the performance of the presented method.
In this paper, we construct some fourth-order modifications of Newton's method for solving nonlin... more In this paper, we construct some fourth-order modifications of Newton's method for solving nonlinear equations. Any two existing fourth-order methods can be effectively used to give rise to new fourth-order methods. Per iteration the new methods require two evaluations of the function and one of its first-derivative. Numerical examples are given to show the performance of the presented methods.
In this paper, we present some sixth-order class of modified Ostrowski's methods for solving nonl... more In this paper, we present some sixth-order class of modified Ostrowski's methods for solving nonlinear equations. Per iteration each class member requires three function and one first derivative evaluations, and is shown to be at least sixthorder convergent. Several numerical examples are given to illustrate the performance of some of the presented methods.
In this paper new fourth order optimal root-finding methods for solving nonlinear equations are p... more In this paper new fourth order optimal root-finding methods for solving nonlinear equations are proposed. The classical Jarratt's family of fourth-order methods are obtained as special cases. We then present results which describe the conjugacy classes and dynamics of the presented optimal method for complex polynomials of degree two and three. The basins of attraction of existing optimal methods and our method are presented and compared to illustrate their performance.
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