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

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 11 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

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

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

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

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