Jacobijeva OR-metoda (JOR metoda)

Imamo

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

Neka je $D$ regularna matrica. Stavimo

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

Tada je

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

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

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

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

Tako imamo sljedeći algoritam

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

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

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

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

odnosno

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

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

$$\vdots$$

$$x_{n}^{(k+1)} = (1-\omega)\,x_{n}^{(k)} + \alpha_{n\,1}\,x_{1}^{(k)} + \alpha_{n\,2}\,x_{2}^{(k)} + \cdots + \alpha_{n\,n-1}\,x_{{n-1}}^{(k)} + \beta_n,$$

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.