# Methods of solving roots involving matrix determinant

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 [1]. 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) [2]. In [1], 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 [2]. 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 [2] 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.

[1] L. O. Chua, C A. Desoer, and E. S. Kuh, Linear and Nonlinear Circuits, McGraw-Hill, 1987.

[2] R. F. Harrington, Field Computation by Moment Methods, Macmillan, 1968.