Search for the roots of a function adjacent to imaginary numbers

Question

I am trying to find the root of a function, which may be just before it starts to have only imaginary values. (In particular, this is the intersection of the line and the semicircle). Obviously, neither Brent nor the halving method will work; also there will be no Newton method. Is there any less obvious one that will?

Answer 1

Instead of trying to solve the equation

f (x) == 0

instead you can try to solve

abs (f (x)) == 0.

For example, you can use halving to find the lows. In cases like the ones you mention, it may even be useful to solve

abs (f (x)) ** 2 == 0,

.

Answer 2

- ? , laguerre, http://mathworld.wolfram.com/LaguerresMethod.html