I am processing my professional work on acoustic logging.
one problem met is as follows:

there is a N by N square complex matrix, (N maybe 4, 8, 12). each entry of the matrix is a known function of one complex variable. the determinant of the sqaure matrix must be set to zero.

I have read NR book, but almost the methods are real-number related instead of complex number. further, the methods used for real-number matrix are bisection, Newtown-Ralphson, secant methods..... all of method are root-finding method for polynomials.

can anyone give any advice about the problem?

Thanks!