Newtons method and convergence

In summary, the conversation discusses difficulties understanding the quadratic convergence of Newton's method. The speaker imagines approximations as the base of a right angle triangle, but cannot see how this leads to quadratic convergence. They ask for sources or explanations on this topic, specifically in regards to x2 - 2.
  • #1
nobahar
497
2
Hello!
In Newton's method, I am having difficulty understanding why the convergence is quadratic.
I can imagine the closer and closer approximations moving along the x-axis as the base of a right angle triangle, but I cannot see from here how I arrive at the convergence being quadratic.
Any one know of any sources that explain this in a straightforward manner? Or would anyone like to attempt to explain this for me?! It's really bugging me!
Thanks in advance.
 
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  • #2
Should it be obvious that for, say, x2 - 2, the rate of convergence of the error should be quadratic? The examples I find on the internet are far from obvious, even those that proport to be geometrical interpretations!
Any help would be really appreciated!
Thanks.
 

What is Newton's Method?

Newton's Method is an algorithm used to find the roots of a differentiable function, also known as the solutions to the equation f(x) = 0. It uses an iterative process to approximate the root by continuously refining the estimate.

How does Newton's Method work?

Newton's Method starts with an initial guess, x0, and then uses the derivative of the function, f'(x), to find the slope of the tangent line at that point. The next estimate, x1, is then found by using the equation x1 = x0 - f(x0)/f'(x0). This process is repeated until the desired level of accuracy is achieved.

What is the convergence of Newton's Method?

The convergence of Newton's Method refers to the rate at which the algorithm approaches the actual root. In general, the method has quadratic convergence, meaning that the number of correct digits in the approximation doubles with each iteration.

Are there any limitations to Newton's Method?

Yes, there are several limitations to Newton's Method. The method may fail to converge if the initial guess is too far from the actual root or if the function has multiple roots. Additionally, the method may also fail if the function is not differentiable at the root.

How do I know if Newton's Method has converged?

Newton's Method is considered to have converged when the difference between two consecutive estimates, |xn - xn-1|, is smaller than a predetermined tolerance level. This tolerance level is typically set by the user and depends on the desired level of accuracy.

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