Why Does Newton Raphson Method Fail for Some Functions?

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In summary, the Newton Raphson method may fail for some functions if the derivative is equal to 0, if the initial value is taken at a maximum or minimum point, if the initial value is taken at a point of inflection, if the initial value is taken near a maximum and minimum point, or if the initial value is taken near a point of inflection. The failure can be explained by the iteration formula, which becomes undefined in these cases. Additionally, if the derivative is not continuous at the root, convergence may also fail. Further explanations and analyses of the method can be found in Kellison's book on Numerical Analysis and on the website http://www.karlscalculus.org/NRbox.html.
  • #1
relinquished™
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Hello. I've been approached with a problem of explaining why Newton Raphson method fails for some functions. I came across a book in Numerical Analysis (Kellison's book) that the method may fail if

1.) f'(x)=0
2.) The initial value is taken at a maximum or minumum point,
3.) The initial value is taken at a point of inflection,
4.) The initial value is taken near a maximum point and a minimum point,
5.) The initial value is taken near a point of inflection.

Now I can explain (1). Newton's Method will fail since the iteration is given by

[itex]
x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
[/itex]

Therefore making the whole thing undefined.

As for (2), since minima and maxima have f'(x)=0, they will fail for the same reasons as (1)

As for (3), ... I'm totally clueless. I don't know how a point of inflection (f''(x)=0) could be related to the Iteration that uses f'

As for (4), I also know that choosing near a minima or maxima might make the method "oscillate", but what I'm looking for are more concrete answers; something that can be related to f'(x) or something that resembles a proof.

As for (5), I have no idea why it may fail for an initial value near a point of inflection..

All help is appreciated,

reli~
 
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  • #2
I'm still in the early stages of calc myself but...

for 5 I try to explain it to myself like this: imagine you're using Newton on a function that behaves a bit like x^2 - C as it heads for the x-axis but with a couple of inflexion points near the area you choose as your guess...If for your first guess the concavity of the curve is up then the tangent at the curve will be steeper than the secant between the root you want and the value of your first guess...so as you follow the tangent you are moving in the same direction as the secant but not as fast (along the x-axis)...such that your guess takes you closer to the root, future guesses under favourable conditions such as this would get you closer still.

But now suppose that you're unlucky for the second guess. The function has changed it's concavity between the x value of our first guess and the x value of our second guess. Now it's the secant that is steeper than the tangent (unless the tangent changes it's direction), for any point on this tangent you will be above the curve...and you will hit the x-axis at the other side of your root or further away from it...from this point onwards you might find yourself attracted to a different root or worse still sent on a wild goose chase

http://img132.imageshack.us/img132/24/Newtonspy8.gif [Broken]
 
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  • #3
Also if the derivative is not continuous at the root, then convergence may fail to occur.

Indeed, let f(0) = 0 and f(x) = x + x^2\sin(2/x) elsewhere.
 
  • #4
For 4/, when the initial value x0 is near the minima, or maxima, then, it may be possible that: [tex]f'(x_0) \approx 0[/tex]. Hence, it will make x1 considerably large: [tex]x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}[/tex]
Not really sure about 3/, and 5/...
 
  • #5
http://www.karlscalculus.org/NRbox.html [Broken] makes a pretty good analysis of the method. There you will find answers to your questions.
 
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1. Why does Newton Raphson method sometimes fail to find a root?

There are a few reasons why Newton Raphson method may fail to find a root. One possible reason is that the function may have multiple roots, and the initial guess may be too far from the desired root. Another reason could be that the function is not differentiable at the root, causing the algorithm to break down.

2. How can I avoid the failure of Newton Raphson method?

To avoid the failure of Newton Raphson method, it is important to carefully choose the initial guess. It should be as close to the desired root as possible and should be in a region where the function is differentiable. Additionally, using a different root-finding method, such as bisection method, can also be a good alternative.

3. Can the failure of Newton Raphson method be detected?

Yes, the failure of Newton Raphson method can be detected by analyzing the behavior of the algorithm. If the algorithm fails, it will either converge to a non-root or will diverge to infinity. In these cases, the algorithm should be stopped and the results should be carefully examined.

4. Is Newton Raphson method the most efficient root-finding method?

No, Newton Raphson method is not always the most efficient root-finding method. While it can converge quickly for well-behaved functions, it may fail for more complex or difficult functions. Other methods, such as secant method or Brent's method, may be more efficient in these cases.

5. Are there any alternatives to Newton Raphson method?

Yes, there are many alternatives to Newton Raphson method for finding roots of a function. Some popular methods include bisection method, secant method, and Brent's method. The choice of method may depend on the properties of the function and the desired accuracy of the solution.

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