Using Newton's method to optimize the square-modulus of a complex-valued function of a real variable?

Question

Assuming I have $F(x): \mathbb{R} \to \mathbb{C}$ and I can estimate a first and second derivative at a point x. I'd like to use Newton's method to find maxima of $|F(x)|^2$, how can I do this? In particular I'm confused by the fact the derivatives are complex-valued and how to turn those into an update for x.

Answer 1

You can interpret your functions as real-valued functions.

Express $F$ as

$$F(x)=f(x)+ig(x)$$

where $f,g:\mathbb{R}\to\mathbb{R}$ (I don’t know the form of your $F$ but this is in principle possible). Then we have that

$$|F(x)|^2=f(x)^2+g(x)^2$$

and you can apply Newton’s method as usual.