Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 41E (Numerical Analysis)

For a fixed shift $s$ that is not an eigenvalue, inverse iteration is the power method applied to $(A-sI{)}^{-1}$. Before normalization, the $k$th vector is proportional toIf $s<{\lambda }_1$, factor out the first coefficient:Every ratio in parentheses lies strictly between zero and one. The bracket therefore tends to $w_1$, and normalization in the Euclidean norm givesThis proves the stated convergence to $w_1$ or $-w_1$ and exhibits the convergence of fixed-shift inverse iteration rate.

Now suppose ${\lambda }_m<s<{\lambda }_{m+1}$. The closest eigenvalue to $s$ is one of ${\lambda }_m$ and ${\lambda }_{m+1}$.

Finally, if ${\lambda }_n<s$, the closest eigenvalue is ${\lambda }_n$ and its inverse eigenvalue is negative. Thus the direction approaches the line spanned by $w_n$, while normalized vectors alternate between the two signs. These cases are the sign behavior of fixed-shift inverse iteration.

PutThe Rayleigh quotient shift is exactlyExpanding the denominator givesThe question prints this as $s_k={\lambda }_1-K{ϵ}^2+O({ϵ}^4)$, so its sign convention requiresunless every $a_i$ vanishes. Equivalently, $-K$ is the positive quadratic Rayleigh-quotient error.

The shifted inverse solve has the spectral decompositionOnly relative coefficients matter after normalization, so factor out $({\lambda }_1-s_k{)}^{-1}$:From the shift expansion,whereas, for $i\ge 2$,ConsequentlyAfter the final normalization,Thus an $O(ϵ)$ eigenvector-direction error becomes $O({ϵ}^3)$ in one step, proving the Local cubic convergence of Rayleigh quotient iteration for this symmetric matrix.