2 Gauss-Seidel iteration

If the vector ${\rm\bf x}=\{x_1,\ldots ,x_m\}$ we used Jacobi type iteration then when we come to evaluate $x_s^{(k+1)}$ we have already evaluated $x_1^{(k+1)},\ldots ,x_{s-1}^{(k+1)}$, so we could evaluate the $s^{th}$ row of ${\rm\bf L}{\rm\bf x}^{(k+1)}$. In matrix motation

$${\rm\bf x}^{(k+1)}={\rm\bf L}{\rm\bf x}^{(k+1)}+{\rm\bf U}{\rm\bf x}^{(k)}+{\rm\bf D}^{-1}{\rm\bf f},$$ (136)

$${\rm\bf x}^{(k+1)}=({\rm\bf I}-{\rm\bf L})^{-1}({\rm\bf U}{\rm\bf x}^{(k)}+{\rm\bf D}^{-1}{\rm\bf f}),$$ (137)

so that the iteration matrix is

$${\rm\bf H}=({\rm\bf I}-{\rm\bf L})^{-1}{\rm\bf U},\ \ {\rm Gauss-Seidel\ iteration\ matrix}.$$ (138)