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I have a physical problem that involves a dispertion relation between two parameters, the frequency $\omega$ and the propagation coefficient $\beta_{n}$, where $n$ is an integer index. The dispersion equation, whose solution will give us the relation $\omega=\omega(\beta_{n})$, is deduced from the system's matrix by requiring that its determinant vanish, $$ \det[ M ]=0, $$ where $M$ is a Hermitian matrix of order $2(2N+1)\times 2(2N+1)$, with $N$ being the highest allowed values of the index $n$ (basically the truncation of a sum over $-N\leq n\leq N$). The elements of the matrix $M$ include terms, such as Bessel and Hankel functions, that are functions of $\omega$ and $\beta_{n}$ (too long and convoluted to practically write here explicitly). Typically such a system is solved numerically for a given $\omega$ by forcing the determinant above to vanish and taking the corresponding roots as solutions of $\beta_{n}$.

For a given $\omega$, when I try to directly solve this determinant =0 equation, to find $\beta_{n}$, I run into numerical problems (for example in Mathematica) and so far finding the roots of this determinant directly seem to be hopeless. However, I am aware that a method like Arnoldi's method can find the eigenvalues of a given matrix quickly, and it works for my matrix $M$, if I test it with given $\omega$ and $\beta_{n}$ values.

However, I am not sure how to express my problem as an eigenvalue problem here, so as to take advantage of the Arnoldi method, which gives approximate eigenvalues of the matrix? How can I recast this problem so that the eigenvalues returned would give $\beta_{n}$? What is the relation between roots of $\det[M]$ and of the eigenvalues of $M$?

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