I have a square matrix $F$ whose elements depend non-linearly on a complex parameter $s$. I would like to know the values of $s$ such that $\det(F)=0$, i.e., those $s$ that make $F$ singular. I would like to know if there is an efficient way of approaching this problem.
This problem is encountered when I want to know the natural frequencies of a linear time-invariant circuit . The same form is also arrived at when we try to solve the propagation constant of a waveguide by using the Method of Moment (MoM, which is the name usually called by people in microwave engineering society) . In , the above condition is used as the definition for the natural frequency, and no hint was provided for actually computing them for a given circuit. The Newton method was suggested in . However, since the differentiation of a matrix determinant w.r.t. a parameter is usually not available analytically, the finite-difference approximation was suggested in  to replace the Jacobian in Newton's method.
In my case, the matrix $F$ may be very large. Since computing matrix determinant is $O(n^3)$ (by $LU$ decomposition), the Newton method with finite difference approximation seems a little bit costly, keeping in mind that we need to evaluate the finite difference along both the real and imaginary axes, at each iteration.
Any comments are very appreciated.
 L. O. Chua, C A. Desoer, and E. S. Kuh, Linear and Nonlinear Circuits, McGraw-Hill, 1987.
 R. F. Harrington, Field Computation by Moment Methods, Macmillan, 1968.