 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-3/40a/a/i/solution

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

e_{k} = x_{k} - x^{*}

gives

e_{k + 1} = H e_{k}

provided the fixed-point consistency condition

x^{*} = H x^{*} + v

holds. Convergence for every starting vector is therefore equivalent to

H^{k} \to 0

, which in finite dimensions is equivalent to

ρ (H) < 1.

These two conditions are necessary and sufficient.

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