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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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By "extermal eigenvalues" do you mean the largest and smallest (signed) real eigenvalues? – hardmath Jan 16 '11 at 20:35

you might benefit from using ARPACK, or if you have really large matrices, by using SLEPc

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

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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.

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BLAS does basic linear algebra; not eigenvalues. – user1709 Jan 16 '11 at 20:33
The place to look for is, I suggest starting with LAPACK, but ARPACK (as already mentioned above) is also worth a shot – Lagerbaer Jan 16 '11 at 23:17
@Slowsolver, yes, but any software which does eigenvalues will use some kind of BLAS, and the link contains the list of all such software. – mpiktas Jan 17 '11 at 7:06

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