What Happens If Your First Guess Using Newton's Method Is Exact?

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Discussion Overview

The discussion revolves around the implications of using Newton's method for finding roots of a function when the initial guess is exactly the root. Participants explore the behavior of subsequent approximations and seek clarification on the mathematical principles involved.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant proposes a scenario where the first guess in Newton's method is the exact root of the function and questions what happens to subsequent approximations.
  • Another participant references the formula for Newton's method and asks what occurs when the function value at the guess is zero.
  • A subsequent reply suggests that if the function value is zero, the next approximation remains the same as the current guess.
  • Further clarification is sought regarding the implications of this simplification and whether it leads to failure in obtaining further approximations.

Areas of Agreement / Disagreement

Participants appear to agree that if the first guess is the exact root, the next approximation remains unchanged. However, there is some uncertainty regarding the implications of this outcome and whether it constitutes a failure in the method.

Contextual Notes

Participants have not fully resolved the implications of the simplification when the function value is zero, nor have they provided specific examples to illustrate their points.

Who May Find This Useful

This discussion may be of interest to those studying numerical methods, particularly in the context of root-finding algorithms and their behaviors under specific conditions.

oreon
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Hi everybody, I have a kinda theory to explain but I need a little help to find out the explanation of it. It is,

suppose your first guess using Newton's method to find a root is lucky and you guess the exact root of f(x). (Not an approximation,but exact) What happens to your second approximation of the root and later approximations? Justfiy the answer with calculus and specific examples.

do I fail when I do the second approximation of the root and for the other approximations? and how does that happen?

I got liitle bit confuse about this.
 
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Just look at the formula used in Newton's method:

x_(n + 1) = x_n - f(x_n) / f'(x_n).

What happens if f(x_n) = 0? How can x_(n + 1) be simplified then?
 
so you are saying that, if it is equal to zero, then it can not be simplified.

Am I right sir?
 
What it says is then that x(n+1)=x(n)
 
Oh I got it. Thank you for your helps...
 

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