Inverse iteration

Inverse iteration repeatedly solves $(A-sI)y_{k+1}=q_k$ and normalizes. It is the power method for $(A-sI{)}^{-1}$ and converges toward an eigenvector whose eigenvalue is closest to the shift $s$.

If $A$ is real symmetric, $s$ is not an eigenvalue, one eigenvalue ${\lambda }_{\ast }$ is uniquely closest to $s$, and the initial vector has a nonzero component in its eigenspace, fixed-shift inverse iteration converges to that eigenspace. Its asymptotic direction-error ratio iswhere ${\lambda }_{\mathrm{next}}$ is second closest to $s$.

If the closest eigenvalue ${\lambda }_{\ast }$ lies above the shift, the dominant eigenvalue $({\lambda }_{\ast }-s{)}^{-1}$ of the inverse is positive and normalized iterates approach one fixed sign of its eigenvector. If ${\lambda }_{\ast }<s$, that inverse eigenvalue is negative, so the direction converges but consecutive normalized vectors alternate signs. At a shift exactly midway between two simple eigenvalues, the two inverse eigenvalues have equal modulus and opposite signs, so generic iterates alternate between two mixtures instead of selecting one eigenvector.