I am trying to calculate all of the eigenvectors/eigenvalues of large (40000x40000), sparse, asymmetric matrix.  I am using MATLAB and have 3 GB of working memory.  The way I am calculating them is by sweeping through (as suggested here -http://www.mathworks.com/support/solutions/en/data/1-A0QRGR/index.html?product=SL&solution=1-A0QRGR). All of my eigenvalues are negative. I start from near the most negative eigenvalue, use eigs, then select the maximum. I repeat eigs on that maximum plus 1E-5 to avoid using eigs with a shift equal to an eigenvalue and then collect the eigenvalues (and the corresponding eigenvectors) that are greater than that maximum plus 1E-14 to avoid duplicate eigenvalues due to numerical error. I repeat this until the maximum of a new set of eigenvalues reaches close to 0. However, I always seem to fall a few (~5) eigenvalues short of the total. I have checked for the possibility of a maximum being a complex number and making sure that the complex conjugate has been tabulated, but that does not do anything. Moreover, some (~2) of the zero eigenvalues have corresponding eigenvectors that are entirely NaN.  I am currently at a complete loss of how to proceed, and any suggestions regarding useful texts, etc., would be greatly appreciated.
How to calculate ALL of the eigenvalues/eigenvectors of a large, sparse, asymmetric matrix?
matrices;sparse