scholarly journals Beyond Newton: A New Root-Finding Fixed-Point Iteration for Nonlinear Equations

Algorithms ◽  
2020 ◽  
Vol 13 (4) ◽  
pp. 78
Author(s):  
Ankush Aggarwal ◽  
Sanjay Pant
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Short Note ◽  
Basin Of Attraction ◽  
Computational Cost ◽  
Initial Guess ◽  
Mathematical Functions ◽  
New Class ◽  
Root Finding ◽  
Roots Of Equations

Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is Newton’s method. However, its convergence depends heavily on the initial guess, with poor choices often leading to slow convergence or even divergence. In this short note, we seek to enlarge the basin of attraction of the classical Newton’s method. The key idea is to develop a relatively simple multiplicative transform of the original equations, which leads to a reduction in nonlinearity, thereby alleviating the limitation of Newton’s method. Based on this idea, we derive a new class of iterative methods and rediscover Halley’s method as the limit case. We present the application of these methods to several mathematical functions (real, complex, and vector equations). Across all examples, our numerical experiments suggest that the new methods converge for a significantly wider range of initial guesses. For scalar equations, the increase in computational cost per iteration is minimal. For vector functions, more extensive analysis is needed to compare the increase in cost per iteration and the improvement in convergence of specific problems.

scholarly journals On the robust Newton’s method with the Mann iteration and the artistic patterns from its dynamics

2021 ◽  
Author(s):  
Krzysztof Gdawiec ◽  
Wiesław Kotarski ◽  
Agnieszka Lisowska
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Basins Of Attraction ◽  
Picard Iteration ◽  
Mann Iteration ◽  
Iteration Parameter ◽  
Root Finding ◽  
Base Method ◽  
Fabric Design ◽  
Root Finding Method

AbstractThere are two main aims of this paper. The first one is to show some improvement of the robust Newton’s method (RNM) introduced recently by Kalantari. The RNM is a generalisation of the well-known Newton’s root finding method. Since the base method is undefined at critical points, the RNM allows working also at such points. In this paper, we improve the RNM method by applying the Mann iteration instead of the standard Picard iteration. This leads to an essential decrease in the number of root finding steps without visible destroying the sharp boundaries among the basins of attractions presented in polynomiographs. Furthermore, we investigate visually the dynamics of the RNM with the Mann iteration together with the basins of attraction for varying Mann’s iteration parameter with the help of polynomiographs for several polynomials. The second aim of this paper is to present the intriguing polynomiographs obtained from the dynamics of the RNM with the Mann iteration under various sequences used in this iteration. The obtained polynomiographs differ considerably from the ones obtained with the RNM and are interesting from the artistic perspective. Moreover, they can easily find applications in wallpaper or fabric design.

Thomas Simpson and ‘Newton's method of approximation’: an enduring myth

1992 ◽  
Vol 25 (3) ◽  
pp. 347-354 ◽  
Author(s):  
Nick Kollerstrom
Keyword(s):  
Newton’S Method ◽  
Computer Use ◽  
Newton's Method ◽  
Time Warp ◽  
Newton Raphson ◽  
Roots Of Equations ◽  
Image Position ◽  
Raphson Method ◽  
As If

A resurgence of interest has occurred in ‘Newton's method of approximation’ for deriving the roots of equations, as its repetitive and mechanical character permits ready computer use. If x = α is an approximate root of the equation f(x) = 0, then the method will in most cases give a better approximation aswhere f′(x) is the derivative of the function into which α has been substituted. Older books sometimes called it ‘the Newton–Raphson method’, although the method was invented essentially in the above form by Thomas Simpson, who published his account of the method in 1740. However, as if through a time-warp, this invention has migrated back in time and is now matter-of-factly placed by historians in Newton's De analysi of 1669. This paper will describe the steps of this curious historical transposition, and speculate as to its cause.

scholarly journals On iterative techniques for estimating all roots of nonlinear equation and its system with application in differential equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mudassir Shams ◽  
Naila Rafiq ◽  
Nasreen Kausar ◽  
Praveen Agarwal ◽  
Choonkil Park ◽  
...  
Keyword(s):  
Nonlinear Equation ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Basin Of Attraction ◽  
Numerical Test ◽  
Computational Cost ◽  
System Of Nonlinear Equations ◽  
Single Root ◽  
Root Finding

AbstractIn this article, we construct a family of iterative methods for finding a single root of nonlinear equation and then generalize this family of iterative methods for determining all roots of nonlinear equations simultaneously. Further we extend this family of root estimating methods for solving a system of nonlinear equations. Convergence analysis shows that the order of convergence is 3 in case of the single root finding method as well as for the system of nonlinear equations and is 5 for simultaneous determination of all distinct and multiple roots of a nonlinear equation. The computational cost, basin of attraction, efficiency, log of residual and numerical test examples show that the newly constructed methods are more efficient as compared to the existing methods in literature.

scholarly journals Converging Newton’s Method With An Inflection Point of A Function

2017 ◽  
Vol 13 (2) ◽  
pp. 73
Author(s):  
Ridwan Pandiya ◽  
Ismail Bin Mohd
Keyword(s):  
Iterative Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Inflection Point ◽  
Initial Point ◽  
One Dimensional ◽  
Root Finding ◽  
Inflection Points ◽  
Starting Point ◽  
The Relationship

For long periods of time, mathematics researchers struggled in obtaining the appropriate starting point when implementing root finding methods, and one of the most famous and applicable is Newton’s method. This iterative method produces sequence that converges to a desired solution with the assumption that the starting point is close enough to a solution. The word “close enough” indicates that we actually do not have any idea how close the initial point needed so that this point can bring into a convergent iteration. This paper comes to answer that question through analyzing the relationship between inflection points of one-dimensional non-linear function with the convergence of Newton’s method. Our purpose is to illustrate that the neighborhood of an inflection point of a function never fails to bring the Newton’s method convergent to a desired solution

Extending the convergence domain of Newton’s method for twice Fréchet differentiable operators

2016 ◽  
Vol 14 (02) ◽  
pp. 303-319
Author(s):  
Ioannis K. Argyros ◽  
Á. Alberto Magreñán
Keyword(s):  
Convergence Analysis ◽  
Newton’S Method ◽  
Newton's Method ◽  
Computational Cost ◽  
Semilocal Convergence ◽  
Convergence Criteria ◽  
Fréchet Differentiable ◽  
Local Convergence Analysis ◽  
Appl Math ◽  
Semilocal Convergence Theorem

We present a semi-local convergence analysis of Newton’s method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Using center-Lipschitz condition on the first and the second Fréchet derivatives, we provide under the same computational cost a new and more precise convergence analysis than in earlier studies by Huang [A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993) 211–217] and Gutiérrez [A new semilocal convergence theorem for Newton’s method, J. Comput. Appl. Math. 79 (1997) 131–145]. Numerical examples where the old convergence criteria cannot apply to solve nonlinear equations but the new convergence criteria are satisfied are also presented at the concluding section of this paper.

scholarly journals Extending the applicability of Newton’s method using Wang’s– Smale’s α–theory

2017 ◽  
Vol 33 (1) ◽  
pp. 27-33
Author(s):  
IOANNIS K. ARGYROS ◽  
◽  
SANTHOSH GEORGE ◽  
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Computational Cost ◽  
Semilocal Convergence ◽  
Numerical Examples ◽  
Convergence Results

We improve semilocal convergence results for Newton’s method by defining a more precise domain where the Newton iterate lies than in earlier studies using the Smale’s α– theory. These improvements are obtained under the same computational cost. Numerical examples are also presented in this study to show that the earlier results cannot apply but the new results can apply to solve equations.

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