I have a matrix $a(\kappa)$ from which I am trying to determine $\kappa$ by using the equation $det(a(\kappa)) = 0$. The matrices I deal with are on the order of 100 X 100 to 500 X 500. Originally I was trying to do this symbolically which of course went no where fast, and so I posted the problem here: Find Determinant/or Row Reduce parameter dependent matrix.
As a result I am now essentially using a singular value decomposition to find $\kappa$. The code is
f[x_?NumericQ] := Last[SingularValueList[a /. \[Kappa] -> x, Tolerance -> 0]]
Then I create a plot for a range of $\kappa$ values which should contain the physically correct $\kappa$. I am getting values of $\kappa$ which fit my expectations, and seem to be correct. The correctness comes form the fact that 1) I am reproducing someone else's work, and 2) $\kappa$ is essentially an eigenvalue and I can check the eigenfunctions.
Right now I can get $\kappa$ to about 4 digits of accuracy that is, if I increase the size of my matrix, my $\kappa$ will converge to about 4 digits buy beyond that the value of the 5th digit is anyone's guess. Are there any tricks to increase the accuracy of the results?