My question is as to which is the best method to find extremal eigenvalues of a real symmetric matrix? Currently I am using Lanczos Iterations followed by Bisection Method. Does anyone have a better suggestion?
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When you say extremal, do you also mean the smallest eigenvalue? In the latter case you might benefit from looking at: this question on MO
If you are looking for specific software implementations I suggest looking here. Download the source and look what authors of the software are using. Since these are widely used libraries, they are the fastest for the widest range of the problems. Any faster algorithm would definitely be data specific, meaning that for some range of problems it will be faster, but it will be slower for others. Note that this is a generic response, but the so is the question.