Black Dots: Newton's Method and a Simple One-Dimensional Fractal

2001 ◽  
Vol 94 (9) ◽  
pp. 734-737
Author(s):  
Tony J. Fisher
Keyword(s):  
Differentiable Function ◽  
Newton’S Method ◽  
Newton's Method ◽  
Simple Complex ◽  
One Dimensional ◽  
Fractal Patterns ◽  
Fractal Images ◽  
Complex Valued

Students in a standard calculus course learn Newton's method for finding the root of a differentiable function. Although they may often see a diagram that visually demonstrates how this method works, it often soon becomes yet another algorithm to memorize or to program into a calculator. In addition, students are sometimes told that using Newton's method on simple complex-valued functions can lead to beautiful fractal patterns. However, the connection between the sequence of steps that they have learned and the corresponding fractal images is fuzzy at best. This article describes a calculator exercise that can help students develop a better visual and numeric feel for Newton's method and discover how Newton's method can lead to a simple, one-dimensional fractal.

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

On Newton's Method and Rational Approximations to Quadratic Irrationals

2004 ◽  
Vol 47 (1) ◽  
pp. 12-16
Author(s):  
Edward B. Burger
Keyword(s):  
Differentiable Function ◽  
Newton’S Method ◽  
Newton's Method ◽  
Continued Fraction ◽  
Analogous Result ◽  
Golden Ratio ◽  
Fibonacci Number ◽  
Rational Approximations ◽  
Initial Value ◽  
Continued Fraction Expansions

AbstractIn 1988 Rieger exhibited a differentiable function having a zero at the golden ratio (−1 + )/2 for which when Newton's method for approximating roots is applied with an initial value x0 = 0, all approximates are so-called “best rational approximates”—in this case, of the form F2n/F2n+1, where Fn denotes the n-th Fibonacci number. Recently this observation was extended by Komatsu to the class of all quadratic irrationals whose continued fraction expansions have period length 2. Here we generalize these observations by producing an analogous result for all quadratic irrationals and thus provide an explanation for these phenomena.

On a newton-type method under weak conditions with dynamics

2020 ◽  
pp. 2150145
Author(s):  
Manoj Kumar Singh ◽  
Arvind K. Singh
Keyword(s):  
Iterative Methods ◽  
Newton’S Method ◽  
Newton's Method ◽  
Second Order ◽  
Numerical Results ◽  
Order Derivative ◽  
Algebraic Equations ◽  
Type Method ◽  
Nonlinear Algebraic Equations ◽  
Fractal Patterns

In this paper, we present new cubically convergent Newton-type iterative methods with dynamics for solving nonlinear algebraic equations under weak conditions. The proposed methods are free from second-order derivative and work well when [Formula: see text]. Numerical results show that the proposed method performs better when Newton’s method fails or diverges and competes well with same order existing method. Fractal patterns of different methods also support the numerical results and explain the compactness regarding the convergence, divergence, and stability of the methods to different roots.

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

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