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5.4. The Gauss-Seidel method

Decompose A = (aij ) as follows: A = D + C1 + C2 ,

A = a11 ···0
:a22
:·
0···an
+ 0···0
a21 0···0
::
an1 an,n-10
+ 0 a12 ···a1n
::
an-1,n
0 ···0

Then A = <=> (D + C1 + C2 ) =

(7)

D + C1 = - C2 + <=> + D -1C1 = - D -1C2 + D -1

and from this we derive the iteration formula

(7)'

(k+1) + D -1C1 (k+1) = - D -1C2 (k) + D -1

, i.e.,

(8)

(k+1) = - D -1C1 (k+1) - D -1C2 (k) + D -1

, i.e.,

(9)

xi (k+1) = - (1 / aii ) aij xj (k + 1) - (1 / aii ) aij xj (k) + (1 / aii ) bi .

Note. 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) already known
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)

(k + 1) = - (D + C1 ) - 1C2 (k) + (D + C1 ) - 1

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

0 = - A0 = A - A0 = A ( - 0 ) = A
,

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


Exercises: E56, E57, E58
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