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How can I efficiently check if a tridiagonal system with 1's in diagonal is ill-conditioned or not ? The common way is to get the ratio of largest and smallest singular values and see if its greater than the precision of matrix entries. Is there any other way I can do this ? My matrix is huge so it will take lots of time and RAM to computer Singular values.

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Umm, I don't know if it will be more or less expensive but you can probably solve the system using TDMA or whatever and solve again with a perturbation and figure out the change in solution to estimate the degree of ill-conditioning. If you matrix is sparse, which it is, AFAIK, there shouldn't be much memory usage to compute stuff (eigenvalues, singular values whatever). Time, maybe but space, I'm not sure. –  Inquest Dec 3 '12 at 16:28

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