1. Iterative Methods Good for sparse matrices Jacobi Iteration
Gauss-Seidel Iteration

2. Iterative Method Concept
Given a system of linear equation
Direct solution for x;
𝑎 11 𝑥 1  𝑎 12 𝑥 2 ⋯ 𝑎 1n 𝑥 𝑛 = 𝑏 1 ⋮ 𝑎 𝑛1 𝑥 1  𝑎 𝑛2 𝑥 2 ⋯ 𝑎 𝑛𝑛 𝑥 𝑛 = 𝑏 𝑛
𝑥 1 = 1 𝑎 𝑏 1 − 𝑎 12 𝑥 2 −⋯ 𝑎 1n 𝑥 𝑛 ⋮ 𝑥 𝑛 = 1 𝑎 𝑛𝑛 𝑏 𝑛 − 𝑎 𝑛1 𝑥 1 −⋯ 𝑎 𝑛,𝑛−1 𝑥 𝑛−1

3. Compact form Iteration
𝑥 𝑖 = 1 𝑎 𝑖𝑖 𝑏 𝑖 − 𝑗=1,𝑗≠𝑖 𝑛 𝑎 𝑖𝑗 𝑥 𝑗 ;𝑖=1,2,⋯𝑛
Iteration
𝑥 𝑖 2 = 1 𝑎 𝑖𝑖 𝑏 𝑖 − 𝑗=1,𝑗≠𝑖 𝑛 𝑎 𝑖𝑗 𝑥 𝑗 1 ;𝑖=1,2,⋯𝑛

4. Solve Iteratively Start with initial guess for x's, simplest is xi=0
Iterate using new values comparing error with previous values
Convergence when
∣  𝑎,𝑖 ∣= ∣ 𝑥 𝑖 𝑗 − 𝑥 𝑖 𝑗−1 𝑥 𝑖 𝑗 ∣ 100%  𝑠

5. Jacobi Iteration Gauss-Seidel Iteration
All x's for each iteration unmodified during iteration
Gauss-Seidel Iteration
x's modified during iteration as new values of x's are calculated

6. Convergence Gauss-Seidel may not converge
Diagonally dominant (guaranteed convergence)
Relaxation for improved convergence
New value of x is modified by a weighted average
Weighting factor = relaxation factor = λ