Newton's Method and Intervals

Question

I'm trying to find the root of $x^4-2x^3+5x^2-6=0$ in the interval $[1,2]$.

For $x_1$, can I use $1$ or $2$ since they're in the interval, or do I have to choose a number between $1$ and $2$?

It depends on the specifics of the problem. An example of an issue that could occur would be if you have a function which is positive on $[1,1.5)$, zero at $1.5$, and has a maximum at $1.25$. Then if you start at $1$, most likely you will be sent to the left, away from the root at $1.5$ that you actually want to find. — Ian, Dec 07 '15 at 19:44
A concrete example of this issue would be $-(x-1.3)^2+0.04$. (It might be helpful to you to actually run Newton's method on this example to see what happens.) — Ian, Dec 07 '15 at 19:49
If you don't know anything, then 1.5 is the safest bet, but the problem can still occur in that case. If possible, knowing the intervals of increasing and decreasing of your function, or at least some bounds on them, is very helpful. — Ian, Dec 07 '15 at 19:51
using $x_1=1$ will be fine, the main thing you need to avoid is points where $f'(x)=0$ and your function has zero derivative only at $x=0$ — WW1, Dec 07 '15 at 19:56
Ah I did not see that there is no linear term. That means you can solve $f'(x)=0$ easily. The only possible issue is then if the method sends you to the left of zero. But this is impossible because the second derivative is always positive. — Ian, Dec 07 '15 at 21:25
@CharmaineDR: using Newton's Method with $x_0 = 1$, $x_0 = 2$ or $x_0 = 1.5$ all converge to the correct root of $x^* = 1.2175621547507618$. — Moo, Dec 08 '15 at 02:28

score 0 · Answer 1 · answered Dec 08 '20 at 06:25

To summarize what's been said in the comments, any value in the interval $[1,2]$ will give you the right root in this example. In general, though, you have to be a little more careful. See this question and its answers for more detail on what can go wrong if you make a bad guess.

Newton's Method and Intervals

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