The Guass-Seidal Method is an [[Iterative Method]] to find the solution to a [[System of Linear Equations]]. It shares many similarities with the [[Jacobi Method]]. The procedure follows the [[Jacobi Method]], but when going down the linear equations we use the newly updated $x_{n}^{(k+1)}$ value. $ \huge \begin{cases} a_{11}x_{1}^{(k+1)} + a_{12}x_{2}^{(k)} + \cdots + a_{1n} x_{n}^{(k)} &= b_{1} \\ a_{21}x_{1}^{(k+1)} + a_{22}x_{2}^{(k+1)} + \cdots + a_{1n} x_{n}^{(k)} &= b_{2} \\ \vdots &\\ a_{n_{1}} x_{1}^{(k+1)} + a_{n {2}}x_{2}^{(k+1)} + \cdots + a_{n n}x_{n}^{(k+1)} &= b_{n} \end{cases} $ If $A$ is [[Diagonally Dominant]], then $||T|| < 1$. $\huge \norm{ T_{GS} } \leq \max_{i} \frac{ \sum_{j>i} |a_{ij}| }{ |a_{i i}| - \sum_{j < i} |a_{i j} } $ This can also be shown the following relation to the [[Jacobi Method]], $ \huge ||T_{GS}|| \leq ||T_{J}|| < 1 $