 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-67/5/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Assume

n \geq 3

and use periodic indices for the Jacobi method. At iteration

r

, it is

x_{j}^{(r + 1)} = \frac{1}{2} (x_{j - 1}^{(r)} + x_{j + 1}^{(r)} - b_{j}), x_{0} = x_{n}, x_{n + 1} = x_{1} .

All values on its right side belong to the old iterate. For lexicographic Gauss-Seidel iteration, compute in the order

1, \dots, n

and use a newly available coordinate immediately:

x_{1}^{(r + 1)} x_{j}^{(r + 1)} x_{n}^{(r + 1)} = \frac{1}{2} (x_{2}^{(r)} + x_{n}^{(r)} - b_{1}), = \frac{1}{2} (x_{j - 1}^{(r + 1)} + x_{j + 1}^{(r)} - b_{j}), 2 \leq j \leq n - 1, = \frac{1}{2} (x_{1}^{(r + 1)} + x_{n - 1}^{(r + 1)} - b_{n}) .

The wrap-around term in the last row uses the new

x_{1}

, while the wrap-around term in the first row uses the old

x_{n}

. This is important for the actual iteration matrix and its convergence behavior.

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