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Claire58 · Jul 5, 2010, 09:07 AM

Why does the newton's method not always work well. Also what is a specific function example I could use to support my explanation?
My professor wants us to read the subject on our own and then do a project on it! I'm bad enough at math as it is. Please help me understand what this is all about.

galactus · Jul 5, 2010, 02:00 PM

There can be some problems with Newton's method, but overall, it works well. Especially, for solving hard to solve equations, such as

$x^{3}+x-1=0$

But, some reasons it does not work is suppsoe we have $f'(x_{n})=0$ for some n, then Newton involves a division by zero, making it impossible to generate $x_{n+1}$ .

But, this is to be expected because the tangent line to y=f(x) is parallel to the x-axis where $f'(x_{n})=0$ , and so the tangent line does not cross the x axis to generate the next approximation.

Sometimes the value of $x_{n}$ produced by Newton do not converge to a solution. Take for example,

$x^{\frac{1}{3}}=0$ which has x=0 as its only solution, and try to approximate by using Newton with a starting value of $x_{0}=1$ .
If we do this, then we quickly see that the values do not converge to 0 and Newton does not work.

Here is a graph. Note how the lines do not converge around 0.

At the bottom is the graph for one that does converge, $x^{3}+x-1=0$

Note how the tangent lines are grouping around the solution, which is about 59/86