positive-definiteness of a huge sparse matrix

I want to determine the positive-definiteness of a huge sparse matrix (N^2,N^2) with nonzeros on the 9 off-diagnols, where N is the larger the better. I am using eigs(A,1,'SA') because of the memory issues, however, for N=260, each iteraction requires 1 hour. I really want to find out fast method to deal this problem.

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