Algebraic formulations for the solution of the nullspace-free eigenvalue problem using the inexact shift-and-invert Lanczos method

Given the generalized symmetric eigenvalue problem $Ax=\lambda Mx$, with A semidefinite and M definite, we analyze some algebraic formulations for the approximation of the smallest nonzero eigenpairs, assuming that a sparse basis for the null space is available. In particular, we consider the inexact version of the Shift--and--Invert Lanczos method, and we show that apparently different algebraic formulations provide the same approximation iterates, under some natural hypotheses. Our results suggest that alternative strategies need to be explored to really take advantage of the special problem setting, other than reformulating the algebraic problem. Experiments on a real application problem corroborate our theoretical findings.