Gauss-Seidel Method

Numerical Analysis · Axiom Academy

Using new information immediately In Jacobi, we compute all new values using only old values. But once we've computed x₁⁽ᵏ⁺¹⁾, why not use it immediately when computing x₂⁽ᵏ⁺¹⁾? x₁⁽ᵏ⁺¹⁾ uses x₂⁽ᵏ⁾, x₃⁽ᵏ⁾ x₂⁽ᵏ⁺¹⁾ uses x₁⁽ᵏ⁾, x₃⁽ᵏ⁾ x₃⁽ᵏ⁺¹⁾ uses x₁⁽ᵏ⁾, x₂⁽ᵏ⁾ x₁⁽ᵏ⁺¹⁾ uses x₂⁽ᵏ⁾, x₃⁽ᵏ⁾ x₂⁽ᵏ⁺¹⁾ uses x₁⁽ᵏ⁺¹⁾ , x₃⁽ᵏ⁾ x₃⁽ᵏ⁺¹⁾ uses x₁⁽ᵏ⁺¹⁾, x₂⁽ᵏ⁺¹⁾ The key difference from Jacobi: we use x⁽ᵏ⁺¹⁾ values (already computed in this iteration) for j i. Using A = D + L + U, Gauss-Seidel can be written as: Same system, same starting point—see the difference! Gauss-Seidel converges under the same sufficient conditions as Jacobi: Faster convergence than Jacobi Half the memory (in-place updates) You've learned Gauss-Seidel—faster than Jacobi by using new information immediately!

This is the written version of the interactive lesson above. See the full Numerical Analysis course.