LU Decomposition, linear equations

With the LU decomposition it is rather simple to solve a system of linear equations $$\begin{align} a_{11}x_1 +a_{12}x_2 +a_{13}x_3 + a_{14}x_4=&w_1 \nonumber \\ a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4=&w_2 \nonumber \\ a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + a_{34}x_4=&w_3 \nonumber \\ a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + a_{44}x_4=&w_4. \nonumber \end{align}$$

This can be written in matrix form as $$\mathbf{Ax}=\mathbf{w}.$$

where $\mathbf{A}$ and $\mathbf{w}$ are known and we have to solve for $\mathbf{x}$. Using the LU dcomposition we write $$\mathbf{A} \mathbf{x} \equiv \mathbf{L} \mathbf{U} \mathbf{x} =\mathbf{w}.$$

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