#### 25.3.5.7 Substitution

We introduced the LU-decomposition as a method for solving a linear system of equations. However, until now we have only decomposed the matrix A into L and U. Obviously, we now want to take the second step and see that this matrix decomposition helps us in solving a linear system of equations.

For this we return to the system of Eq. 25.2 from which we also took the matrix. The right-hand side is

$\stackrel{\to }{b}=\left(\begin{array}{c}2\\ 4\\ 0\end{array}\right)$

We will now use the function DoLUDecomposition in combination with Eq. 25.20 and Eq. 25.21 for solving the linear system of equations

$\begin{array}{c}\mathbf{L}\stackrel{\to }{y}=\stackrel{\to }{b}\\ \mathbf{U}\stackrel{\to }{x}=\stackrel{\to }{y}\end{array}$

described by Eq. 25.18 and Eq. 25.19.

Calculating $\stackrel{\to }{y}$. We first solve Eq. ...

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