GuideLinear systemsintermediate

Gauss-Seidel method

The choice M=D+L, which reuses each freshly computed component within the same iteration, and why it often converges faster than Jacobi.

Use the new values immediately

Gauss-Seidel takes $M=D+L$ and $N=-U$ . As soon as it computes $x_1^{(k+1)}$ , it uses it to compute $x_2^{(k+1)}$ , and so on within the same iteration.

x^{(k+1)}=-(D+L)^{-1}U\,x^{(k)}+(D+L)^{-1}b\ \Longleftrightarrow\ (D+L)x^{(k+1)}=b-Ux^{(k)}

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)