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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:

enter image description here

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.

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What form does $A_{\lambda}$ have? Chances are that we can suggest a much better way of approaching this problem, but you haven't provided sufficient information about how the problem arose. – Brian Borchers Feb 8 at 5:05
@BrianBorchers The form of $A$ is $\phi_{\omega}(\|x_i-y_j\|)$ where $\phi_\omega$ is a bessel function, so $A_\lambda$ it is not sparse. – Beni Bogosel Feb 8 at 6:58

1 Answer

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This is an exceptionally bad algorithm to find eigenvalues, at the very least if you're just looking for the eigenvalues of a linear operator or matrix. There are many better algorithms to do what you are trying to do.

While it is true that the mathematical definition of an eigenvalue is that $A_\lambda$ is singular, i.e. that $\det A_\lambda=0$ in the finite dimensional case, this is not a practical definition to find eigenvalues in actual applications.

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Do you have any suggestions of better algorithms? – Victor Liu Feb 8 at 1:35
Are you simply trying to find the eigenvalues of a matrix? – Wolfgang Bangerth Feb 8 at 3:05
I'm not looking for the eigenvalues of a matrix, but of the Laplace operator on a set. You may say that it is a bad algorithm, but there are articles on the subject which say otherwise. – Beni Bogosel Feb 8 at 6:59
To find eigenvalues of operators I would look at the Lanczos and Arnoldi algorithms. – Wolfgang Bangerth Feb 8 at 13:10
It would seem like he is solving a nonlinear matrix eigenvalue problem. How does one apply Lanczos or Arnoldi to such problems? – Victor Liu Feb 9 at 4:10
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