best real approximation to complex numbers

Question

I have a system of equations and its answers are complex, but I want real numbers. Is there any way to find the best real approximation to a complex number?

Answer 1

The best real approximation to a complex number is its projection onto the real axis, that is, its real part.

Answer 2

You might be interested in Bairstow's method, which allows you to use real approximation techniques to locate the complex conjugate pairs of roots to real polynomials of arbitrary degree.

Effectively one applies a Newton method to find a real quadratic factor of the given real polynomial.

Perhaps a fuller description of your "system of equations" would be helpful.

Added: In response to the system of equations added as a comment (3 times), it seems strange that you ask this. As stated the equation for C is not related to the equations involving A and B, so the "best" real value of C s.t. $C^2$ is negative would be zero. There are of course two purely imaginary exact roots for C. The situation for A and B is not much more complicated. You have two possible exact real roots for $A = \pm \alpha ^ {1/2}$, and for each of these two exact roots for $B = \pm (\beta + 2a)^{1/2}$. One wonders if $A$ and $a$ in your equation for B are meant to be the same value. Otherwise you have quite forgotten to tell us anything about $a$. Assuming it is $A$, then taking the positive root for $A$ allows us to overcome the negative value for $\beta$ and get two exact real roots for $B$. If you take the negative real root for $A$, then obviously the exact roots for $B$ are both purely imaginary (so their closest real approximations would be zero).

I won't go into a lot more detail (all that is required is taking some square roots), but I await your clarification of my suspicion that the equation for C is not complete as given.