Then A = <=> (D + C1 + C2 ) =
and from this we derive the iteration formula
The approximations are computed in the order
x1 (k+1), x2 (k+1),
..., xn (k+1).
In (9), we use the values x1 (k+1), ..., xi-1 (k+1)
to compute xi (k+1).
Note. The same assumptions as with the Jacobi method are sufficient to ensure the convergence of the Gauss-Seidel iteration.
The iteration matrix of the G-S is obtained from (7)
Example 1: Solving a system of equations by the G-S method
Ratkaisun iteratiivinen tarkentaminen
For the residual 0 = - A0 , 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 = - A1