


Then A = <=> (D + C_{1 } + C_{2 }) =
(7)
and from this we derive the iteration formula
(7)'
, i.e.,
(8)
, i.e.,
(9)
Note.
The approximations are computed in the order
x_{1 }^{ (k+1)}, x_{2 }^{ (k+1)},
..., x_{n }^{ (k+1)}.
In (9), we use the values x_{1 }^{ (k+1)}, ..., x_{i1 }^{ (k+1)}
already known
to compute x_{i }^{ (k+1)}.
Note. The same assumptions as with the Jacobi method are sufficient to ensure the convergence of the GaussSeidel iteration.
The iteration matrix of the GS is obtained from (7)
Example 1: Solving a system of equations by the GS method
Ratkaisun iteratiivinen tarkentaminen
For the residual _{0 } =  A_{0 }, where _{0 } is an approximation obtained for the solution, we have
i.e., the residual is a solution for the equation. A = _{0 }. Since = _{0 } + , käytetään seuraavaa tarkennusta:
1) Ratk. A = => _{0 } ja _{0 }
2) Ratk. A = _{0 } => _{0 }
3) Aset. _{1 } = _{0 } + _{0 } , _{1 } =  A_{1 }