= Solution
Fix a shift $s$ that is not an <eigenvalue> of $A$. <Inverse iteration> computes
$$
(A-sI)\mathbf y_{k+1}=\mathbf x_k,
\qquad
\mathbf x_{k+1}
=\frac{\mathbf y_{k+1}}{\|\mathbf y_{k+1}\|},
$$
and may use $r(\mathbf x_k)$ as the eigenvalue estimate.
For a real symmetric matrix, suppose that one eigenvalue $\lambda_*$ 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 $\lambda_*$. If $\lambda_{\rm next}$ is the second-closest eigenvalue to the shift, the asymptotic direction-error ratio is
$$
\boxed{
\frac{|\lambda_*-s|}
{|\lambda_{\rm next}-s|}
}.
$$
Thus a shift close to the desired eigenvalue gives rapid convergence. The Rayleigh quotients converge to $\lambda_*$ and, for a simple eigenvalue, their errors are of second order in the vector error. This is <convergence of fixed-shift inverse iteration>.
Back to article page
