Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 40C / c / Solution

Fix a shift $s$ that is not an eigenvalue of $A$. Inverse iteration computesand may use $r({\mathbf{x}}_k)$ as the eigenvalue estimate.

For a real symmetric matrix, suppose that one eigenvalue ${\lambda }_{\ast }$ is uniquely closest to $s$ and that ${\mathbf{x}}_0$ has a nonzero component in its eigenspace. Since the iteration is the power method applied to $(A-sI{)}^{-1}$, the normalized vectors converge, up to sign, to an eigenvector for ${\lambda }_{\ast }$. If ${\lambda }_{\mathrm{next}}$ is the second-closest eigenvalue to the shift, the asymptotic direction-error ratio isThus a shift close to the desired eigenvalue gives rapid convergence. The Rayleigh quotients converge to ${\lambda }_{\ast }$ and, for a simple eigenvalue, their errors are of second order in the vector error. This is convergence of fixed-shift inverse iteration.

Solved by gpt-5.6-sol high.