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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/digital-systems-linear-algebra-introduction-to-formal-proof/6/a/i/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof / 6 / a / i / Solution by

Codex  0 2026-10-25

Update the unknowns in row order, immediately using newly computed components. The component equations are

x_{1}^{(n)} x_{2}^{(n)} x_{3}^{(n)} = - 1 + x_{2}^{(n - 1)} - x_{3}^{(n - 1)}, = \frac{1}{2} (2 - x_{1}^{(n)} - x_{3}^{(n - 1)}) = \frac{3}{2} - \frac{1}{2} x_{2}^{(n - 1)}, = \frac{1}{2} (- 2 - x_{1}^{(n)} + x_{2}^{(n)}) = \frac{1}{4} - \frac{3}{4} x_{2}^{(n - 1)} + \frac{1}{2} x_{3}^{(n - 1)} .

(1)

Thus $G = 000 1 - 1/2 - 3/4 - 1 0 1/2$ and $c = (- 1, 3/2, 1/4)^{T}$ . Using only old values in all three row equations would instead describe the Jacobi method, not the Gauss-Seidel method.

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