Explanation of Newton's Method

In summary, the conversation discusses the concept of Newton's method for finding roots and addresses the question of what happens when the first guess is the exact root. It is explained that in this case, the subsequent approximations will also be the exact root. This is justified using calculus and the formula for Newton's method. The conversation also touches on the concept of simplifying the formula when f(x_n) is equal to 0.
  • #1
oreon
10
0
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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  • #2
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?
 
  • #3
so you are saying that, if it is equal to zero, then it can not be simplified.

Am I right sir?
 
  • #4
What it says is then that x(n+1)=x(n)
 
  • #5
Oh I got it. Thank you for your helps...
 

1. What is Newton's Method?

Newton's Method is a numerical method for finding the roots of a given equation. It is named after Sir Isaac Newton, who first described the method in the 17th century.

2. How does Newton's Method work?

Newton's Method works by using an initial guess to iteratively find a better approximation of the root of an equation. It involves calculating the slope of the curve at the initial guess and using that slope to determine a new guess that is closer to the actual root.

3. What is the purpose of using Newton's Method?

The purpose of using Newton's Method is to find the roots of a given equation. This can be useful in various mathematical and scientific applications, such as optimization problems and solving differential equations.

4. What are the limitations of Newton's Method?

Newton's Method may not always converge to the actual root of an equation. It may also fail to find a root if the initial guess is too far from the actual root or if the equation has multiple roots. Additionally, it can be computationally expensive for complex equations.

5. Are there any alternatives to Newton's Method?

Yes, there are other numerical methods for finding roots of equations, such as the Bisection Method and the Secant Method. These methods may have different convergence rates and may be more suitable for certain types of equations.

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