So as the title would suggest I'm currently reading about Newton's method for finding roots. I'm having trouble understanding the reasoning for a function without a root.
It reads as following:
"Consider the function $f(x) =1+x^2$. Clearly f has no real roots though it does have complexroots $x\pm i$.The Newton method formula for f is:
$x_{n+1} = x_{n} – \frac{1+x^2}{2x_{n}}=\frac{x^2-1}{2x_{n}}$"
What is happening here?
Many thanks to whomever might expand this a little for me!
Best Answer
You have to choose complex starting values, otherwise the method cannot converge to complex roots.
With the correct iteration formula $$x_{n+1}=x_n - \frac{f(x)}{f'(x_n)} = x_n - \frac{x_n^2+1}{2x_n} = \frac{2x_n^2-x_n^2 -1}{2x_ n}=\frac{x_n^2 - 1}{2x_n}$$ and a complex starting value you get e.g.
and for the other root