Past exam of the numerical analysis course of the University of Oxford / 2014 / 1 / d / iii / Solution

With $A_0=A$, repeat the shifted QR algorithm:The QR shift ${\mu }_k$ is chosen from $A_k$, for example its bottom-right entry or an eigenvalue of a trailing two-by-two block. Explicitly using the orthogonality of $Q_k$ givesThus each update is a matrix similarity preserving eigenvalues. Accumulate $Q_0{Q}_1\cdots Q_k$ if eigenvectors or the full reduction are wanted. Once a subdiagonal entry is negligible, block deflation in the QR algorithm separates the corresponding blocks and iteration continues on the active block.