# Algorithms to compute largest gap between smallest nonzero eigenvalues of sparse symmetric matrix

I am looking mainly for c/c++ 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 $\lambda_{n+1}$ (both are nonzero) 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 c/c++ software which can compute it to me and uses sparse matrix representations? I have tried to install fortran ARPACK, but with no success. Maybe you know some papers with algorithms solving this problem. Thanks in advance!

• Hi. Why did you fail to install fortran Arpack? which operating system are you using (install it in ubuntu is very easy, for example)? have you tried with Arpack++? Actually I dont think there is any algorithm to get the largest gap, so it seems that you have to calculate all of them, or maybe you can use some information that you have about your matrix (where is the matrix coming from?). – sebas Feb 26 '14 at 22:40
• When you say largest gap do you mean the maximum $\lambda_n - \lambda_{n+1}$? – meawoppl Mar 2 '14 at 19:24