I have a problem of root finding. Format of the function can be arbitrary whereas I want to solve the following one:
$$f(x)= a+e^{-x^2}(b+cx+dx^2)$$ where a,b,c and d are given parameters, not limited to real numbers. I have read about the root-finding algorithm of polynomial, but still have no idea how to solve functions like I wrote here. Besides the initial guess, how do we determine the value of next iteration based on the former one, and how to find out the conjugate pairs since complex roots are wanted ?
I am not aware of the status of the art in the area of finding solutions of equations in the complex domain. So take my answer for what it is, that is to say not much.
What I used to do in the old times was first to avoid calculations in the complex domain since they are very expensive with respect to CPU time. So, I used to do everything in real.
Let us consider the case of $$a+e^{-z^2}(b+cz+dz^2)=0$$ in which all coefficients are assumed to be complex. Set $z=x+i y$, replace and expand. Isolate the real and imaginary parts and set them equal to $0$. This means that you end with two equations $$R(x,y)=0$$ $$I(x,y)=0$$ that you need to solve simultaneously for $x$ and $y$. Assuming that we know a "reasonable" starting point, we can use Newton-Raphson and write $$R(x,y)=R(x_n,y_n)+\frac{dR(x,y)}{dx}(x_{n+1}-x_n)+\frac{dR(x,y)}{dy}(y_{n+1}-y_n)=0$$ $$I(x,y)=I(x_0,y_0)+\frac{dI(x,y)}{dx}(x_{n+1}-x_n)+\frac{dI(x,y)}{dy}(y_{n+1}-y_n)=0$$ the partial derivatives being computed at $(x_n,y_n)$. Solving the system gives you $(x_{n+1},y_{n+1})$ and you continue.
Another kind of approaches is minimization of the norm, say $$\Phi(x,y)=R^2(x,y)+I^2(x,y)$$ In this area, there are some very advanced methods and hopefully you could get, at least, at a point $\Phi(x,y)=0$ which will correspond to one of the possible solutions. Techniques involving global minimization are supposed to provide all minima.