Iterative linear solvers
Objective. Apply Jacobi and Gauss-Seidel methods.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
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Learn: the key idea
Iterative linear solvers approximate the solution of Ax = b by generating a sequence of vectors that converges to the true answer, rather than solving directly. The Jacobi method updates each component using only values from the previous iteration, while Gauss-Seidel immediately uses newly updated components within the same sweep, usually converging faster. Both methods are guaranteed to converge if A is strictly diagonally dominant. For example, solving 4x − y = 3 and −x + 4y = 3 with x₀ = y₀ = 0 by Jacobi gives x₁ = 0.75, y₁ = 0.75, then x₂ = 0.9375, y₂ = 0.9375, approaching x = y = 1. These methods suit large sparse systems where direct elimination is too costly.
Worked example
Perform one Jacobi iteration for 5x − y = 9, −x + 5y = 9 starting from x₀ = 0, y₀ = 0, giving x₁ and y₁.
- 1Rearrange: x = (9 + y)/5, y = (9 + x)/5.
- 2Substitute x₀ = 0, y₀ = 0 into both formulas simultaneously.
- 3x₁ = (9 + 0)/5 = 1.8, y₁ = (9 + 0)/5 = 1.8.
- 4Check: both use only iteration-0 values, consistent with Jacobi's definition.
Answer: x₁ = 1.8, y₁ = 1.8
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Start practisingCommon mistake. Students apply Jacobi or Gauss-Seidel to a system that is not diagonally dominant and assume convergence is guaranteed; it may diverge. Always check |a_ii| ≥ Σ|a_ij| for j ≠ i before trusting the iteration.
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