Gauss–Seidel method
Description

Let A x = b {\textstyle \mathbf {A} \mathbf {x} =\mathbf {b} } be a square system of n linear equations, where:

A =, x =, b =. {\displaystyle \mathbf {A} ={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\cdots &a_{nn}\end{bmatrix}},\qquad \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\qquad \mathbf {b} ={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\end{bmatrix}}.}

When A {\displaystyle \mathbf {A} } and b {\displaystyle \mathbf {b} } are known, and x {\displaystyle \mathbf {x} } is unknown, we can use the Gauss–Seidel method to approximate x {\displaystyle \mathbf {x} }. The vector x (0) {\displaystyle \mathbf {x} ^{(0)}} denotes our initial guess for x {\displaystyle \mathbf {x} } (often x i (0) = 0 {\displaystyle \mathbf {x} _{i}^{(0)}=0} for i = 1, 2,..., n {\displaystyle i=1,2,...,n}). We denote by x (k) {\displaystyle \mathbf {x} ^{(k)}} the k {\displaystyle k} -th approximation or iteration of x {\displaystyle \mathbf {x} }, and by x (k + 1) {\displaystyle \mathbf {x} ^{(k+1)}} the approximation of x {\displaystyle \mathbf {x} } at the next (or k + 1 {\displaystyle k+1}) iteration.
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