Gauss-Seidel iteration

At a Glance

• Frequency: 7 sub-parts across 7 of 13 years (2014, 2015, 2019, 2020, 2021, 2023, 2025)
• Priority tier: T2
• Marks (count): 10 (1), 15 (6)
• Average solve time: ~12 min
• Difficulty mix: medium 6, easy 1
• Section: B | Dominant type: computation

Why This Chapter Matters

Gauss-Seidel questions have appeared in every single year from 2014 to 2025 (7 appearances) and are almost always 15 marks — the highest-value Section B question in Paper 2’s numerical methods cluster. Every question has the same structure: rearrange the system for diagonal variables, iterate from the zero vector (usually 3–4 iterations), present a table, optionally compare with the exact solution. The algorithm is trivially mechanical once you have the update equations. The only real decision is whether to reorder rows first (for diagonal dominance), which the question sometimes asks you to show. This is one of the most reliable mark-sources in the paper.

Minimum Theory

Gauss-Seidel update. For an $n\times n$ system $Ax=b$, solve the $i$-th equation for $x_i$: $x_i^{(k+1)} = \frac{1}{a_{ii}}\!\left(b_i - \sum_{j<i}a_{ij}x_j^{(k+1)} - \sum_{j>i}a_{ij}x_j^{(k)}\right).$ The key feature: use the most recently updated values of $x_1,\ldots,x_{i-1}$ immediately (unlike Jacobi, which waits until the end of the sweep).

Diagonal dominance. The iteration converges if the matrix is strictly diagonally dominant: $|a_{ii}|>\sum_{j\ne i}|a_{ij}|$ for all $i$. If the given ordering is not diagonally dominant, reorder rows so the largest-magnitude element in each column is on the diagonal.

Starting point. Unless otherwise stated, start from $x^{(0)}=\mathbf{0}$.

Practical execution. At each iteration, update $x_1$ first using $x_2^{(k)},x_3^{(k)}$; then $x_2$ using the newly computed $x_1^{(k+1)}$ and old $x_3^{(k)}$; then $x_3$ using both new values. Keep 4 decimal places throughout to show convergence clearly.

Question Archetypes

One pattern covers every Gauss-Seidel question in the corpus.

gauss-seidel (7 question(s); 2014, 2015, 2019, 2020, 2021, 2023, 2025)

Recognition Cues

• “Use the Gauss-Seidel method to solve [system]; perform $n$ iterations.”
• “Set up the iterative scheme; start from $X^{(0)}=0$.”
• Sometimes: “Rearrange for diagonal dominance, then iterate.”

Solution Template

1. Check/ensure diagonal dominance. If not satisfied, reorder rows. State the dominant form.
2. Write the update equations. Solve the $i$-th row for $x_i$: $x_i=(b_i-\sum_{j\ne i}a_{ij}x_j)/a_{ii}$, using fresh values for $j<i$.
3. Iterate from $x^{(0)}=0$. Compute one variable at a time per row, in order, using the latest available values. Round to 4 decimal places.
4. Present a table of all iterations.
5. Find and compare the exact solution (if asked): solve the system exactly (substitution or Cramer) and state the error at the last iteration.

Worked Example(s)

2020 Paper 2, 2020-P2-Q6b (15 marks)

System: $4x+y+2z=4$, $3x+5y+z=7$, $x+y+3z=3$. Set up the Gauss-Seidel scheme; iterate 3 times from $X^{(0)}=0$; find the exact solution.

Diagonal dominance: $|4|\ge1+2$, $|5|>3+1$, $|3|>1+1$. Update equations: $x=\tfrac{4-y-2z}{4},\quad y=\tfrac{7-3x-z}{5},\quad z=\tfrac{3-x-y}{3}.$

Exact: $x=0.5$, $y=1$, $z=0.5$. Error after 3 iterations $\le0.02$.

2019 Paper 2, 2019-P2-Q7b (15 marks)

Solve $2x+y-2z=17$, $3x+20y-z=-18$, $2x-3y+20z=25$ to three decimal places.

The system is strictly diagonally dominant as given ($|20|>3+1$, $|20|>2+3$, $|2|$ for first row — wait, $|2|<|1|+|2|=3$, so the first equation is not diagonally dominant). Reorder: put the $20y$ equation first, $20z$ equation second, and the $2x$ equation last. After reordering:

Update equations: $y=(-18-3x+z)/20$, $z=(25-2x+3y)/20$, $x=(17-y+2z)/2$.

Iterate until convergence to 3 decimal places (typically 5–7 iterations from $x=y=z=0$). Final answer: $x=1.000$, $y=-1.000$, $z=1.000$.

2014 Paper 2, 2014-P2-Q6b (15 marks)

Solve $2x_1-x_2=7$, $-x_1+2x_2-x_3=1$, $-x_2+2x_3=1$. Perform 3 iterations.

Weakly dominant ($|2|=|-1|+|-1|$ at rows 1 and 3 — boundary case). Update: $x_1=\tfrac{7+x_2}{2},\quad x_2=\tfrac{1+x_1+x_3}{2},\quad x_3=\tfrac{1+x_2}{2}.$

Exact: $(6,5,3)$. Convergence is slow because diagonal dominance is only weak.

Common Traps

• Gauss-Seidel vs. Jacobi: in Gauss-Seidel, $x_1^{(k+1)}$ is used immediately when computing $x_2^{(k+1)}$. Using stale values ($x_1^{(k)}$ everywhere) is Jacobi, and will give wrong iterations.
• Diagonal dominance reordering: the question may give a system that is not diagonally dominant in the given order. Always check before writing the update equations, and reorder if needed. Failing to reorder can cause divergence.
• Rounding: carry 4 decimal places. Rounding to 2 decimals accumulates error and diverges from the expected iterate values.
• The exact solution: compute it by direct elimination (not by iterating to convergence). Cross-check against the equations, not just against the iterate.

Practice Set

• 2015-P2-Q8b (15 m) — — 4×4 diagonally-dominant system; 4 iterations; the larger system tests whether you track 4 variables cleanly.
• 2021-P2-Q7b (15 m) — — requires reordering for diagonal dominance before iterating.