 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-1/40c/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Here

D = I

, so the Jacobi iteration matrix is

H = I - A = 0 - α - α - α 0 - α - α - α 0 .

- 2 α

. On its two-dimensional orthogonal complement, the coordinates sum to zero and

H v = αv

, so the other eigenvalue is

α

with multiplicity two. Therefore

ρ (H) = max {2∣ α ∣, ∣ α ∣} = 2∣ α ∣.

- \frac{1}{2} < α < \frac{1}{2} .

At either endpoint the spectral radius is one, so convergence for arbitrary initial data fails.

 New to topics? Read the docs here!