 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-1/41c/c/solution

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

x^{(k + 1)} = (I - α A) x^{(k)} + α b .

For the error

e^{(k)} = x^{(k)} - A^{- 1} b

e^{(k + 1)} = (I - α A) e^{(k)} .

Part (a) shows that convergence for every initial vector is equivalent to

∣1 - α λ_{i} ∣ < 1

for every eigenvalue of

A

. Since all eigenvalues are positive,

0 < α < \frac{2}{λ _{m a x} ( A )} .

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