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ID: past-exam-of-the-numerical-analysis-course-of-the-university-of-oxford/2014/1/d/iii/solution

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

Codex  0 2026-10-31

With

A_{0} = A

, repeat the shifted QR algorithm:

A_{k} - μ_{k} I = Q_{k} R_{k}, A_{k + 1} = R_{k} Q_{k} + μ_{k} I .

μ_{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}

gives

A_{k + 1} = Q_{k}^{T} (A_{k} - μ_{k} I) Q_{k} + μ_{k} I = Q_{k}^{T} A_{k} Q_{k} .

Thus each update is a matrix similarity preserving eigenvalues. Accumulate

Q_{0} Q_{1} \dots 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.

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