scholarly journals On Some Efficient Techniques for Solving Systems of Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Janak Raj Sharma ◽  
Puneet Gupta
Keyword(s):  
Nonlinear Equations ◽  
Computational Efficiency ◽  
Newton Iteration ◽  
Systems Of Nonlinear Equations ◽  
Order Method ◽  
Third Order ◽  
Newton Step ◽  
The Third ◽  
Large Systems ◽  
Theoretical Results

We present iterative methods of convergence order three, five, and six for solving systems of nonlinear equations. Third-order method is composed of two steps, namely, Newton iteration as the first step and weighted-Newton iteration as the second step. Fifth and sixth-order methods are composed of three steps of which the first two steps are same as that of the third-order method whereas the third is again a weighted-Newton step. Computational efficiency in its general form is discussed and a comparison between the efficiencies of proposed techniques with existing ones is made. The performance is tested through numerical examples. Moreover, theoretical results concerning order of convergence and computational efficiency are verified in the examples. It is shown that the present methods have an edge over similar existing methods, particularly when applied to large systems of equations.

scholarly journals Modified Potra-Pták Multi-step Schemes with Accelerated Order of Convergence for Solving Systems of Nonlinear Equations

2018 ◽  
Vol 24 (1) ◽  
pp. 3
Author(s):  
Himani Arora ◽  
Juan Torregrosa ◽  
Alicia Cordero
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Sixth Order ◽  
Systems Of Nonlinear Equations ◽  
Order Method ◽  
Third Order ◽  
Boundary Problems ◽  
Newton Step ◽  
The Family ◽  
Efficiency And Reliability

In this study, an iterative scheme of sixth order of convergence for solving systems of nonlinear equations is presented. The scheme is composed of three steps, of which the first two steps are that of third order Potra-Pták method and last is weighted-Newton step. Furthermore, we generalize our work to derive a family of multi-step iterative methods with order of convergence 3 r + 6 , r = 0 , 1 , 2 , … . The sixth order method is the special case of this multi-step scheme for r = 0 . The family gives a four-step ninth order method for r = 1 . As much higher order methods are not used in practice, so we study sixth and ninth order methods in detail. Numerical examples are included to confirm theoretical results and to compare the methods with some existing ones. Different numerical tests, containing academical functions and systems resulting from the discretization of boundary problems, are introduced to show the efficiency and reliability of the proposed methods.

scholarly journals An efficient derivative free iterative method for solving systems of nonlinear equations

2013 ◽  
Vol 7 (2) ◽  
pp. 390-403 ◽  
Author(s):  
Janak Sharma ◽  
Himani Arora
Keyword(s):  
Nonlinear Equations ◽  
New Method ◽  
Systems Of Nonlinear Equations ◽  
First Order ◽  
Numerical Tests ◽  
Several Variables ◽  
Derivative Free ◽  
Derivative Free Method ◽  
Large Systems ◽  
Theoretical Results

We present a derivative free method of fourth order convergence for solving systems of nonlinear equations. The method consists of two steps of which first step is the well-known Traub's method. First-order divided difference operator for functions of several variables and direct computation by Taylor's expansion are used to prove the local convergence order. Computational efficiency of new method in its general form is discussed and is compared with existing methods of similar nature. It is proved that for large systems the new method is more efficient. Some numerical tests are performed to compare proposed method with existing methods and to confirm the theoretical results.

scholarly journals An Efficient Family of Traub-Steffensen-Type Methods for Solving Systems of Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Janak Raj Sharma ◽  
Puneet Gupta
Keyword(s):  
Nonlinear Equations ◽  
Systems Of Nonlinear Equations ◽  
Numerical Tests ◽  
New Family ◽  
Several Variables ◽  
Derivative Free ◽  
The Family ◽  
Large Systems ◽  
Steffensen Method ◽  
Theoretical Results

Based on Traub-Steffensen method, we present a derivative free three-step family of sixth-order methods for solving systems of nonlinear equations. The local convergence order of the family is determined using first-order divided difference operator for functions of several variables and direct computation by Taylor's expansion. Computational efficiency is discussed, and a comparison between the efficiencies of the proposed techniques with the existing ones is made. Numerical tests are performed to compare the methods of the proposed family with the existing methods and to confirm the theoretical results. It is shown that the new family is especially efficient in solving large systems.

scholarly journals Third-Order Newton-Type Methods Combined with Vector Extrapolation for Solving Nonlinear Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Wen Zhou ◽  
Jisheng Kou
Keyword(s):  
Nonlinear Systems ◽  
Nonlinear Equations ◽  
Semilocal Convergence ◽  
Numerical Results ◽  
The Other ◽  
Systems Of Nonlinear Equations ◽  
Order Method ◽  
Third Order ◽  
Composite Method ◽  
Vector Extrapolation

We present a third-order method for solving the systems of nonlinear equations. This method is a Newton-type scheme with the vector extrapolation. We establish the local and semilocal convergence of this method. Numerical results show that the composite method is more robust and efficient than a number of Newton-type methods with the other vector extrapolations.

A Class of Higher-Order Newton-Like Methods for Systems of Nonlinear Equations

2021 ◽  
pp. 2150059
Author(s):  
Janak Raj Sharma ◽  
Sunil Kumar ◽  
Ioannis K. Argyros
Keyword(s):  
Nonlinear Equations ◽  
Inverse Operator ◽  
General Setting ◽  
Basins Of Attraction ◽  
Function Evaluation ◽  
Systems Of Nonlinear Equations ◽  
Third Order ◽  
Additional Function ◽  
Numerical Experimentation ◽  
The Cost

In this paper, a class of efficient iterative methods with increasing order of convergence for solving systems of nonlinear equations is developed and analyzed. The methodology uses well-known third-order Potra–Pták iteration in the first step and Newton-like iterations in the subsequent steps. Novelty of the methods is the increase in convergence order by an amount three per step at the cost of only one additional function evaluation. In addition, the algorithm uses a single inverse operator in each iteration, which makes it computationally more efficient and attractive. Local convergence is studied in the more general setting of a Banach space under suitable assumptions. Theoretical results of convergence and computational efficiency are verified through numerical experimentation. Comparison of numerical results indicates that the developed algorithms outperform the other similar algorithms available in the literature, particularly when applied to solve the large systems of equations. The basins of attraction of some of the existing methods along with the proposed method are given to exhibit their performance.

scholarly journals An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior

2019 ◽  
Vol 17 (1) ◽  
pp. 1567-1598
Author(s):  
Tianbao Liu ◽  
Xiwen Qin ◽  
Qiuyue Li
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Second Derivative ◽  
Third Order ◽  
Numerical Tests ◽  
The Family ◽  
Theoretical Results ◽  
High Computational Efficiency

Abstract In this paper, we derive and analyze a new one-parameter family of modified Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padé approximant. The convergence analysis of the family is also considered, and the methods have convergence order three. Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modified Cauchy methods is obtained by using weight function. We also perform some numerical tests and the comparison with existing optimal fourth-order methods to show the high computational efficiency of the proposed scheme, which confirm our theoretical results. The basins of attraction of this optimal fourth-order family and existing fourth-order methods are presented and compared to illustrate some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, from the fractal graphics, with the increase of the value m of the series in iterative methods, the chaotic behaviors of the methods become more and more complex, which also reflected in some existing fourth-order methods.

Conversion between Cartesian and geodetic coordinates on a rotational ellipsoid by solving a system of nonlinear equations

2011 ◽  
Vol 60 (2) ◽  
pp. 145-159 ◽  
Author(s):  
Marcin Ligas ◽  
Piotr Banasik
Keyword(s):  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
System Of Nonlinear Equations ◽  
Order Convergence ◽  
Ellipsoidal Height ◽  
Third Order ◽  
The Third ◽  
Geodetic Coordinates ◽  
Modified Newton’S Method

Conversion between Cartesian and geodetic coordinates on a rotational ellipsoid by solving a system of nonlinear equationsA new method to transform from Cartesian to geodetic coordinates is presented. It is based on the solution of a system of nonlinear equations with respect to the coordinates of the point projected onto the ellipsoid along the normal. Newton's method and a modification of Newton's method were applied to give third-order convergence. The method developed was compared to some well known iterative techniques. All methods were tested on three ellipsoidal height ranges: namely, (-10 - 10 km) (terrestrial), (20 - 1000 km), and (1000 - 36000 km) (satellite). One iteration of the presented method, implemented with the third-order convergence modified Newton's method, is necessary to obtain a satisfactory level of accuracy for the geodetic latitude (σφ < 0.0004") and height (σh< 10-6km, i.e. less than a millimetre) for all the heights tested. The method is slightly slower than the method of Fukushima (2006) and Fukushima's (1999) fast implementation of Bowring's (1976) method.

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