Finding roots of a function that are adjacent to imaginary numbers

Question

I'm trying to find a root of a function that may be immediately before it begins having only imaginary values. (Specifically, it's the intersection of a line and a half-circle.) Obviously neither Brent's nor the bisection method will work; neither will Newton's method. Is there a less-obvious one that will?

Answer 1

is it polynomial function? maybe you can use laguerre method, http://mathworld.wolfram.com/LaguerresMethod.html

Answer 2

Rather than trying to solve the equation

f(x) == 0

you could instead try to solve

abs(f(x)) == 0.

For example you could use bisection to find minima. In cases like the one you mention it may even be beneficial to solve

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

because this way you void some square roots.