Understanding Newton Method for Solving Equations

In summary, the Newton method for finding roots of an equation may not always work due to certain conditions that need to be met, such as the derivative of the function being less than 1 in the neighborhood of the root and the initial guess being inside that neighborhood. If these conditions are not met, the method may result in an incorrect or infinite solution. It is best to try and see what happens with different initial guesses, but it is important to also consider the behavior of the function and potential problem areas to avoid incorrect solutions. The Wikipedia page for Newton's Method provides helpful examples of where the method may not work.
  • #1
sara_87
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Homework Statement



I read that the Newton method for finding roots of an equation doesn't always work.
How do I know whether it works or not for a given equation?


Homework Equations





The Attempt at a Solution


For an example, consider:

f(x) = 1/(x2-1)+1/(x2-4) - x-1

When I try to solve this using programming, I get 2.9642 for some initial guesses but if I put the initial guess as, say, 5 I get that the root is -inf

I don't understand why.
And also for other examples I either get x=0 after 0 steps (in the Newton method) or I get x=inf.

Thank you
 
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  • #2
You need to have

[tex] \abs {f'(x)} <= 1 [/tex] in the neighborhood of your guess and the solution. A local extreme between the 2 could also cause troubles.
 
  • #3
I don't understand. What do you mean 'A local extreme between the 2 could also cause troubles' ?

Thank you
 
  • #4
At a local extremum (maximum or minimum) the derivative is 0. Since Newton's method involves dividing by 0, it one of your iterations happens to give you that value, there's an obvious problem!
 
  • #5
Oh so if the value gives infinity, this means that there is a problem. How do I avoid this problem.
In general, for a given function, how do I know whether I can implement the Newton Method?
 
  • #6
As integral said, you need to know that f'(x)< 1 in some neighborhood of of the root, that f'(x) is NOT 0 in that neighborhood, and that your starting value is inside that neighborhood. Frankly, most of the time, the best thing to do is to "try and see what happens".
 
  • #7
You would have to look at a graph of the function to see where your initial guess will take you with Newton's Method. In your case, at your initial guess x1=5, the slope of the tangent line of your function is -13/352800 and just below the x-axis. This will make x2 far to the left in negative x values and each subsequent approximation will further diverge to negative infinity.

http://en.wikipedia.org/wiki/Newton%27s_method" might help you out with some good examples of where you can run into trouble with Newton's Method.
 
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Related to Understanding Newton Method for Solving Equations

What is Newton's method for solving equations?

Newton's method is an algorithm used to find the roots or solutions of a given equation. It is based on the idea of using an initial guess for the root and then refining that guess through a series of iterations until a satisfactory solution is obtained.

How does Newton's method work?

Newton's method involves finding the tangent line to the curve of the given equation at a chosen point and then finding the x-intercept of that tangent line. This x-intercept will be a closer approximation to the actual root of the equation. This process is repeated until the desired level of accuracy is achieved.

When is Newton's method most useful?

Newton's method is most useful when finding the roots of a differentiable equation, meaning an equation that has a continuous and smooth graph. It is also effective when the initial guess is close to the actual root and when the equation is difficult to solve using other methods.

What are the limitations of Newton's method?

Newton's method may not converge or give an accurate solution if the initial guess is too far from the actual root. It also may not work if there are multiple roots or if the equation has a repeated root. In addition, it may involve complex calculations and requires knowledge of the derivative of the equation.

Can Newton's method be used for solving any type of equation?

No, Newton's method is not suitable for all types of equations. It is most effective for finding the roots of single-variable equations, but it can also be extended to some multi-variable equations. It is not suitable for equations that do not have a continuous and smooth graph or for equations with multiple roots.

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