Numerical Properties of Different Root-Finding Algorithms Obtained for Approximating Continuous Newton’s 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.

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

Jurnal Matematika Integratif ◽

10.24198/jmi.v13i2.11785 ◽

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

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

Algorithms ◽

10.3390/a13040078 ◽

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.

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On Global Aspects of Real Newton’s Method in Synthesis of Linkages

Volume 2: 30th Annual Mechanisms and Robotics Conference, Parts A and B ◽

10.1115/detc2006-99087 ◽

2006 ◽

Author(s):

Jin Xie ◽

Kaiyin Yan ◽

Yong Chen

Keyword(s):

Dynamical System ◽

Newton’S Method ◽

Newton's Method ◽

Global Analysis ◽

Basins Of Attraction ◽

Random Motion ◽

Iterative Root ◽

Degree Polynomial ◽

Root Finding ◽

Deterministic Dynamical System

Nonlinear equations arise from the synthesis of linkages. Newton’s method is one of the most accessible and easiest to implement of the iterative root-finding algorithms for these equations. As a discrete deterministic dynamical system, Newton’s method contains subsystems which have highly random motion. In a so-called chaotic zone, there is a rapid interchange between the basins of attraction for each root of the equation. Choosing initial points from such chaotic zone, one can obtain a certain number of roots or possible all of them under the Newton’s method. In this paper, how to locate the chaotic zones is addressed following the global analysis of real Newton’s method. It is show that there exist four chaotic zones for a general 4th degree polynomial. As an example, the equation derived from exact synthesis for five positions is solved.

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Newton’s method as a dynamical system: Efficient root finding of polynomials and the Riemann 𝜁 function

Holomorphic Dynamics and Renormalization ◽

10.1090/fic/053/10 ◽

2008 ◽

pp. 213-224

Author(s):

Dierk Schleicher

Keyword(s):

Dynamical System ◽

Newton’S Method ◽

Newton's Method ◽

Root Finding

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

Jurnal Matematika Integratif ◽

10.24198/jmi.v13.n2.11785.73-81 ◽

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

Download Full-text