5.4. The Gauss-Seidel method

A = a11 ··· 0 : a22 : · 0 ··· an

+ 0 ··· 0 a21 0 ··· 0 : : an1 an,n-1 0

+ 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)

x_i ^(k+1) = - (1 / a_ii ) a_ij x_j ^{(k + 1)} - (1 / a_ii ) a_ij x_j ^(k) + (1 / a_ii ) b_i .

Note. The approximations are computed in the order x₁ ^(k+1), x₂ ^(k+1), ..., x_n ^(k+1). In (9), we use the values x₁ ^(k+1), ..., x_i-1 ^(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 Gauss-Seidel iteration.

The iteration matrix of the G-S is obtained from (7)

^{(k + 1)} = - (D + C₁ ) ^{- 1}C₂ ^(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