 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-4/40c/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

For a nonzero real vector

x

A

R_{A} (x) = \frac{x ^{T} A x}{x ^{T} x} .

v_{0}

. Given

v_{k}

, compute the shift

μ_{k} = R_{A} (v_{k}),

(A - μ_{k} I) w_{k} = v_{k},

and normalize in the Euclidean norm:

v_{k + 1} = \frac{w _{k}}{∥ w _{k} ∥ _{2}} .

The next eigenvalue estimate is

μ_{k + 1} = R_{A} (v_{k + 1})

. The process stops when the eigenpair residual is sufficiently small.

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