What starting value to use for Newton method?

Question

I have a function:

$x^3-$$6x^2$$+8$

for which I need to find the root(s).

The Newton Method states that I must have a number $x_n$ to have as a starting value before evaluating, I have seen many people use $x_n$=$1$ as their starting number, and even $-1$, how do I know which one to use? especially for functions with multiple zeroes.

Answer 1

Generally speaking, given a function for which you want to find a root using Newton's method, you have to have some prior knowledge of the rough location of a zero, and the behavior of the function near that zero, in order to pick a good starting value of Newton's method (for example as in the answer of @ClaudeLeibovici).

If you don't have such prior knowledge, finding a choice of the starting value which "works" is a difficult and, sometimes, indeterminate problem. You can see this quite clearly in the paper of Curry, Garnett and Sullivan entitled "On the iteration of a rational function: computer experiments with Newton's method", in which they study the behavior of (complex valued) Newton's method for polynomials of the form $z^3 + (A-1) z - A$.

Answer 2

For almost all functions, almost any choice of initial number will work, so you can relax. If a choice doesn't work, then just pick another and start over. For a specific problem, try to sketch out a rough graph. In the case of your function, you have

$$f'(x) = 3x^2-12x$$

which has roots $x=0$ and $x=4$. Also

$$f''(x) = 6x-12$$

which has root $x=2$.

Now, $f(0) = 8$ and $f(4)=$ ...negative $20$-ish. So there's root between the two extrema. It makes the inflection point a good starting number. So my advice would be to use $x_0 = 2.$

Answer 3

If you have a single root, by Darboux theorem (with a few restrictions) select $x_0$ such that $$f(x_0)\times f''(x_0) >0$$ At least, there will not be any overshoot of the solution