Convergence of fixed-shift inverse iteration

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 $s$.