Et lineært ligningssystem er givet ved:
A
x
=
b
{\displaystyle Ax=b}
hvor
A
{\displaystyle A}
n
×
n
{\displaystyle n\times n}
matrix ,
b
{\displaystyle b}
n
×
1
{\displaystyle n\times 1}
vektor , og
x
{\displaystyle x}
n
×
1
{\displaystyle n\times 1}
x
{\displaystyle x}
A
{\displaystyle A}
inverteres , men det kan være svært eller umuligt. I stedet deler man i Gauss–Seidel-metoden
A
{\displaystyle A}
A
=
L
+
U
hvor
L
=
[
a
11
0
⋯
0
a
21
a
22
⋯
0
⋮
⋮
⋱
⋮
a
n
1
a
n
2
⋯
a
n
n
]
,
U
=
[
0
a
12
⋯
a
1
n
0
0
⋯
a
2
n
⋮
⋮
⋱
⋮
0
0
⋯
0
]
.
{\displaystyle A=L+U\qquad {\text{hvor}}\qquad L={\begin{bmatrix}a_{11}&0&\cdots &0\\a_{21}&a_{22}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\cdots &a_{nn}\end{bmatrix}},\quad U={\begin{bmatrix}0&a_{12}&\cdots &a_{1n}\\0&0&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &0\end{bmatrix}}.}
Dermed bliver ligningssystemet:
L
x
+
U
x
=
b
L
x
=
b
−
U
x
{\displaystyle {\begin{aligned}Lx+Ux&=b\\Lx&=b-Ux\end{aligned}}}
Ved at invertere
L
{\displaystyle L}
x
k
+
1
=
L
−
1
(
b
−
U
x
k
)
{\displaystyle x_{k+1}=L^{-1}(b-Ux_{k})}
hvor
x
k
{\displaystyle x_{k}}
x
k
+
1
{\displaystyle x_{k+1}}
I praksis er metoden dog bedre gengivet elementvist som: [ 1]
x
i
(
k
+
1
)
=
1
a
i
i
(
b
i
−
∑
j
=
1
i
−
1
a
i
j
x
j
(
k
+
1
)
−
∑
j
=
i
+
1
n
a
i
j
x
j
(
k
)
)
,
i
=
1
,
2
,
…
,
n
.
{\displaystyle x_{i}^{(k+1)}={\frac {1}{a_{ii}}}\left(b_{i}-\sum _{j=1}^{i-1}a_{ij}x_{j}^{(k+1)}-\sum _{j=i+1}^{n}a_{ij}x_{j}^{(k)}\right),\quad i=1,2,\dots ,n.}
