在Jacobi method中，\(x^{(n+1)}\)是由\(x^{(n)}\)全部一口氣帶入原式求得，而Gauss–Seidel method則是依序將已知的\(x_{j}^{(n+1)}\)帶入得到新的\(x_{j+1}^{(n+1)}\)。
如果讀者對machine learning還有些印象，就好比batch (Jacobi method)和online (Gauss–Seidel method) learning的差別，方便做理解記憶
假設有一方程組
\[Ax=b\]
這邊我們舉例同Jacobi method
\[\begin{bmatrix}{A}_{11}&{A}_{12}&{A}_{13}\\{A}_{21}&{A}_{22}&{A}_{23}\\{A}_{31}&{A}_{32}&{A}_{33}\end{bmatrix}\begin{bmatrix}{x}_{1}\\{x}_{2}\\{x}_{3}\end{bmatrix}=\begin{bmatrix}{b}_{1}\\{b}_{2}\\{b}_{3}\end{bmatrix}\]
Jacobi method的最後式子如下，帶入舊的\(x^{(n)}\)的到新的\(x^{(n+1)}\)
\({x_{1}^{(n+1)}}=\frac{1}{{A_{11}}}\left(b_{1}-{A_{12}}{x_{2}^{(n)}}-{A_{13}}{x_{3}^{(n)}}\right)\)
\({x_{2}^{(n+1)}}=\frac{1}{{A_{22}}}\left(b_{2}-{A_{21}}{x_{1}^{(n)}}-{A_{23}}{x_{3}^{(n)}}\right)\)
\({x_{3}^{(n+1)}}=\frac{1}{{A_{33}}}\left(b_{3}-{A_{31}}{x_{1}^{(n)}}-{A_{32}}{x_{2}^{(n)}}\right)\)
Gauss–Seidel method將最後的式子修改
\({x_{1}^{(n+1)}}=\frac{1}{{A_{11}}}\left(b_{1}-{A_{12}}{x_{2}^{(n)}}-{A_{13}}{x_{3}^{(n)}}\right)\)
\({x_{2}^{(n+1)}}=\frac{1}{{A_{22}}}\left(b_{2}-{A_{21}}{x_{1}^{(n+1)}}-{A_{23}}{x_{3}^{(n)}}\right)\)
\({x_{3}^{(n+1)}}=\frac{1}{{A_{33}}}\left(b_{3}-{A_{31}}{x_{1}^{(n+1)}}-{A_{32}}{x_{2}^{(n+1)}}\right)\)
注意等號右邊的\(x_{j}^{(n)}\)逐式被換成已知的\(x_{j}^{(n+1)}\)
將其整理成matrix form，將\(x^{(n+1)}\)項移到等號左邊，\(x^{(n)}\)項在等號右邊
\[L_*x^{(n+1)}=b-Ux^{(n)}\]
\[x^{(n+1)}=L_*^{-1}\left(b-Ux^{(n)}\right)\]
\[A={L}_{*}+U\]
其中\({L}_{*}\)表包含主對角線之下三角矩陣，\(U\)則表不含主對角線之上三角矩陣