Finding Roots of (Large) Determinants

Question

I have a matrix of the form $$\mathbb{A}_{i,j}(\omega) = \sum_n \frac{\sin(nx_i)\sin(nx_j)}{-\omega^2+C(n)}+\sum_{r_x}\sum_{r_y}\frac{\sin(r_xx_i)\sin(r_xx_j)\sin(r_y\alpha)}{-\omega^2+D(r_x,r_y)}.$$ In truth, the matrix formulation is much more complex that this, but I have tried to make things simpler by introducing $C(n)$ and $D(r_x,r_y)$ which are functions of $n$ and $r_x, r_y$, respectively. The goal of all of this is to find the roots of $\det(\mathbb{A})$ (ie. what values of $\omega$ make $\det(\mathbb{A})=0)$. I see no way to do this analytically, especially since $\mathbb{A}$ is oftentimes large. Right now, I am finding roots computationally, by sweeping through $\omega$. This is not very efficient, nor accurate and even with fine grid spacing of $\omega$, roots are skipped in very basic cases. I am familiar with the basic root algorithms, (Newton's Method, bisection methods, etc...). However, I don't know if those methods would work given that $\det(\mathbb{A})$ is not only transcendental in $\omega$, but it is not feasible to find an analytic expression for $\det(\mathbb{A})$. Are there any root finding algorithms that are applicable in this situation? Am I missing some simplification that I can make here?