5.4.1 Sequential algorithm

The reason for the better convergence behavior of the Gauß-Seidel iteration compared to the $\omega$-Jacobi iteration (alg. 5.5) consists in the continuous update of the residuum whenever a new component $\underline{u}^{k}_i$ of the $k^{\text{th}}$ iterate has been calculated.

$$u_i^k \;\;:=\;\; u_i^{k-1} + K_{i,i}^{-1} \left( f_i - \sum\limits_{j=1}^{i-1} K_{i,j} u_j^k - \sum\limits_{j=i}^{n} K_{i,j} u_j^{k-1} \right)$$ (5.2)

Now, the iterate $u_i^k$ in (5.2) depends on the numbering of the unknowns in vector  $\underline{u}$, i.e., the whole Gauß-Seidel iteration depends on the numbering of the unknowns.