I have a numerical problem in which I need to find the values $\lambda$ for which the determinant of a matrix $A_\lambda$ is zero. (The solutions $\lambda$ will give the eigenvalues of an operator...)
For that I learnt that it is better to look for the singularities of the function $\lambda \mapsto \log(|det(A_\lambda)|)$. An example of the plot of this kind of function on $[0,10]$ is given in the following figure:
I have two questions:
1) What algorithm is best for finding such a singularity if it is unique in a certain interval $[a,b]$?
2) How should I approach the problem of finding all singularities in an interval $[a,b]$?
[I'm not sure how to choose the right tags for this question, so please feel free to add the right ones.]
Perhaps an answer for 1) could be the golden ratio search: http://en.wikipedia.org/wiki/Golden_section_search , for the case when there is a single minimum in the interval.
Still, I have no idea how to generalize this to the case where there are multiple intervals to be chosen.