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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 dont 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~

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 dont 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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