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I am looking mainly for implementations but also for theoretical algorithms to compute gaps between smallest positive eigenvalues of symmetric, singular matrix or real numbers. To be precise, I want to find index n of the eigenvalue such that between $\lambda_n$ and $\lambda_{n+1}$ there is the largest gap (I assume that eigenvalues are sorted). The matrix I am dealing with is huge, so I am allowed to use only sparse representation. I cannot store all the zero entries due to memory constraints.
Can you recommend any software which can compute it to me and uses sparse matrix representations? Maybe you know some papers with algorithms solving this problem. Thanks in advance!

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this sounds like a great question for the scientific computing stackexchange –  guest Feb 25 at 22:40
    
I never heard of software specifically for computing gaps, but computing $k$ minimal consecutive eigenvalues is well-known how to do for sparse matrices, see e.g. caam.rice.edu/software/ARPACK (ARPACK has interfaces to Python and Matlab, so you don't have to program in Fortran to use it) –  Dima Pasechnik Feb 25 at 22:49
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The title of your post "largest gap between smallest nonzero eigenvalues of sparse matrix" is somewhat contradictory and doesn't agree with the actual statement in the body of the question. Do you care whether $\lambda_{n}$ and $\lambda_{n+1}$ are both nonzero? Do you care how small they are? –  Brian Borchers Feb 26 at 2:54
    
Thanks Dima, I will definitely have a look at ARPACK! Brian -- you are right, the title is vague. I want both \lambda_n and \lambda_{n+1} to be nonzero. –  user47459 Feb 26 at 14:22
    
It seems like short of getting all the nonzero eigenvalues there may be no way to zoom into the "largest-gap".... –  Suvrit Feb 26 at 15:56

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