# Numerically find all zeros of multivariate function

How do I find all zeros of a multivariate function , i.e. f(x1,x2,x2,...xn)=0 numerically? I don't know exact analytic form of f , but can numerically compute f at every point on its domain.

Example: Given a matrix A=[i , a; 3+i , 9], I have to find all complex a such that A has an complex eigenvalue with arguement 37 degrees?

Here i is sqrt(-1).

$|\mathbf{A}-\lambda\mathbf{I}|=0 \rightarrow \left[ \begin{array}{cc} i-\lambda & a\\3+i&9-\lambda \end{array}\right]=0$
So you need to find the complex values with argument 37 degree: $\lambda=|\lambda| e^{i\angle 37^{\circ}}$. Plugging this:
$(i-|\lambda| e^{i\angle 37^{\circ}})(9-|\lambda| e^{i\angle 37^{\circ}})-a(3+i)=0$. Therefore:
$a=\frac{(i-|\lambda| e^{i\angle 37^{\circ}})(9-|\lambda| e^{i\angle 37^{\circ}})}{3+i}$