Si \( \omega < 1 \), factor de subrelajación
Si \( \omega > 1 \), factor de sobrerelajación
Calcular \( \omega_{0} \) de \( \mathbf{A} \) $$ \begin{equation*} \mathbf{A} = \begin{bmatrix} 10 & -1 & 2 & 0 \\ -1 & 11 & -1 & 3 \\ 2 & -1 & 10 & -1 \\ 0 & 3 & -1 & 8 \end{bmatrix} \end{equation*} $$
La matriz diagonal es $$ \begin{equation*} \mathbf{D} = \begin{bmatrix} 10 & 0 & 0 & 0 \\ 0 & 11 & 0 & 0 \\ 0 & 0 & 10 & 0 \\ 0 & 0 & 0 & 8 \end{bmatrix} \end{equation*} $$
La matriz triangular inferior es $$ \begin{equation*} \mathbf{L} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 2 & -1 & 0 & 0 \\ 0 & 3 & -1 & 0 \end{bmatrix} \end{equation*} $$
La matriz triangular superior es $$ \begin{equation*} \mathbf{U} = \begin{bmatrix} 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 3 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 \end{bmatrix} \end{equation*} $$
Reemplazando $$ \begin{equation*} \mathbf{T} = \begin{bmatrix} \frac{1}{10} & 0 & 0 & 0 \\ 0 & \frac{1}{11} & 0 & 0 \\ 0 & 0 & \frac{1}{10} & 0 \\ 0 & 0 & 0 & \frac{1}{8} \end{bmatrix} \begin{bmatrix} 0 & -1 & 2 & 0 \\ -1 & 0 & -1 & 3 \\ 2 & -1 & 0 & -1 \\ 0 & 3 & -1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -\frac{1}{10} & \frac{1}{5} & 0 \\ -\frac{1}{11} & 0 & -\frac{1}{11} & \frac{3}{11} \\ \frac{1}{5} & -\frac{1}{10} & 0 & -\frac{1}{10} \\ 0 & \frac{3}{8} & -\frac{1}{8} & 0 \end{bmatrix} \end{equation*} $$
El radio espectral de \( \mathbf{T} \) es el mayor en valor absoluto de los valores propios $$ \begin{equation*} \rho(\mathbf{T}) = 0.426437 \end{equation*} $$
Reemplazando $$ \begin{equation*} \omega_{0} = \frac{2}{1 + \sqrt{1 - 0.426437^{2}}} = 1.05 \end{equation*} $$
Si \( w_{1} = \omega \) $$ \begin{equation*} x_{i}^{(k)} = \omega x_{i}^{(k)} + w_{2} x_{i}^{(k-1)} \end{equation*} $$
Si \( x_{i}^{(k)} \approx x_{i}^{(k-1)} \) $$ \begin{equation*} x_{i}^{(k)} = \omega x_{i}^{(k)} + w_{2} x_{i}^{(k)} \end{equation*} $$
Despejando \( w_{2} \) $$ \begin{equation*} w_{2} = 1 - \omega \end{equation*} $$
Reemplazando en (16.3) $$ \begin{equation} x_{i}^{(k)} = \omega x_{i}^{(k)} + (1 - \omega) x_{i}^{(k-1)} \tag{16.4} \end{equation} $$