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Jacobijeva OR-metoda (JOR metoda)

Imamo

$\displaystyle A = L + D + R.$

Neka je $ D$ regularna matrica. Stavimo

$\displaystyle B(\omega{}) = \frac{1}{\omega{}}\,D.$

Tada je

$\displaystyle B(\omega{})^{-1} = \omega{}\,D^{-1},$

$\displaystyle C(\omega{}) = \left(\frac{1-\omega{}}{\omega{}}\right)\,D - (L + R),$

$\displaystyle G(\omega{}) = B(\omega{})^{-1}\,C(\omega{}) = (1 - \omega{})\,I -
\omega{}\,D^{-1}\,(L + R),$

$\displaystyle \boldsymbol{g}(\omega{}) = B(\omega{})^{-1}\,\boldsymbol{b} =
\omega{}\,D^{-1}\,\boldsymbol{b}.$

Tako imamo sljedeći algoritam

Algoritam 7   (Jacobijeva OR-metoda) Proizvoljno izaberemo početnu aproksimaciju

% latex2html id marker 38940
$\displaystyle \boldsymbol{x}^{(0)}=\left[
\begin{array}{c}
x_1^{(0)} \\  x_2^{(0)} \\  \vdots \\  x_n^{(0)}
\end{array}
\right],$

i zatim računamo sljedeće aproksimacije $ \boldsymbol{x}^{(n)}$ po formuli

$\displaystyle \boldsymbol{x}^{(k+1)} = \left[(1 - \omega{})\,I -
\omega{}\,D^{-1}\,(L + R)\right]\,\boldsymbol{x}^{(k)} +
\omega{}\,D^{-1}\,\boldsymbol{b},$

odnosno
$\displaystyle x_{1}^{(k+1)}$ $\displaystyle =$ $\displaystyle (1-\omega)\,x_{1}^{(k)} +
\alpha_{1\,2}\,x_{2}^{(k)} +
\alpha_{1\,3}\,x_{3}^{(k)} + \cdots +
\alpha_{1\,n}\,x_{n}^{(k)} +
\beta_1$  
$\displaystyle x_{2}^{(k+1)}$ $\displaystyle =$ $\displaystyle (1-\omega)\,x_{2}^{(k)} +
\alpha_{2\,1}\,x_{1}^{(k)} +
\alpha_{2\,3}\,x_{3}^{(k)} + \cdots +
\alpha_{2\,n}\,x_{n}^{(k)} +
\beta_2$  
    $\displaystyle \vdots$  
$\displaystyle x_{n}^{(k+1)}$ $\displaystyle =$ $\displaystyle (1-\omega)\,x_{n}^{(k)} +
\alpha_{n\,1}\,x_{1}^{(k)} +
\alpha_{n\,2}\,x_{2}^{(k)} + \cdots +
\alpha_{n\,n-1}\,x_{{n-1}}^{(k)} +
\beta_n,$  

gdje je $ \alpha_{i\,j}=-\frac{\omega}{\alpha_{i\,i}}\,a_{i\,j},$ i $ \beta_i=\frac{\omega}{\alpha_{i\,i}}\,b_i.$

Primijetimo da za $ \omega{}=1$ Jacobijeva OR metoda postaje Jacobijeva metoda.


next up previous contents index
Next: Gauss-Seidelova OR-metoda (SOR metoda) Up: OR (overrelaxation) metode Previous: OR (overrelaxation) metode   Sadržaj   Indeks
2001-10-26