Let a linear sytem of equations Ax=b where A=(42−222−3−2−314),b=(10,5,4)T in case we solve this equation system by using Dolittle LU factorization method , find Z and X matrices

Expert answers to "Let a linear sytem of equations Ax=b where..."

Lewis Harvey

Answered question

2021-03-12

Let a linear sytem of equations Ax=b where

A = (\begin{matrix} 4 & 2 & - 2 \\ 2 & 2 & - 3 \\ - 2 & - 3 & 14 \end{matrix}), b = {(\begin{matrix} 10, 5, 4 \end{matrix})}^{T}

in case we solve this equation system by using Dolittle LU factorization method , find Z and X matrices

Answer & Explanation

ottcomn

Skilled2021-03-13Added 97 answers

Step 1
By using Dolitte LU decomposition method ,
A=LU
$[\begin{matrix} 4 & 2 & - 2 \\ 2 & 2 & - 3 \\ - 2 & - 3 & 14 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ L_{21} & 1 & 0 \\ L_{31} & L_{32} & 1 \end{matrix}] [\begin{matrix} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{matrix}]$
$[\begin{matrix} 4 & 2 & - 2 \\ 2 & 2 & - 3 \\ - 2 & - 3 & 14 \end{matrix}] = [\begin{matrix} u_{11} & u_{11} & u_{11} \\ u_{11} L_{21} & L_{21} u_{12} + u_{22} & L_{21} u_{13} + u_{23} \\ L_{31} u_{11} & u_{12} L_{31} + L_{32} u_{22} & L_{31} u_{13} + L_{32} u_{23} + u_{33} \end{matrix}]$
Step 2
By comparing the two matrices we will get,
$u_{11} = 4$
$u_{12} = 2$
$u_{13} = - 2$
$u_{11} L_{21} = 2$
$4 L_{21} = 2$
$L_{21} = 0.5$
$L_{21} u_{12} + u_{22} = 2$
$u_{22} = 2$
$L_{21} u_{13} + u_{23} = - 3$
$u_{23} = - 2$
$L_{31} u_{11} = - 2$
$L_{31} = - 0.5$
$u_{12} L_{31} + L_{32} u_{22} = - 3$
$L_{32} = - 0.5$
$L_{31} u_{13} + L_{32} u_{23} + u_{33} = 14$
$u_{33} = 9$
Step 3
Therefore, the matrices L and U will be ,
$L = [\begin{matrix} 1 & 0 & 0 \\ 0.5 & 1 & 0 \\ - 0.5 & - 2 & 1 \end{matrix}]$
$U = [\begin{matrix} 4 & 2 & - 2 \\ 0 & 1 & - 2 \\ 0 & 0 & 9 \end{matrix}]$ Let Lz=b where $z = [\begin{matrix} z_{1} \\ z_{2} \\ z_{3} \end{matrix}]$
Therefore, we will get ,
$[\begin{matrix} 1 & 0 & 0 \\ 0.5 & 1 & 0 \\ - 0.5 & - 2 & 1 \end{matrix}] [\begin{matrix} z_{1} \\ z_{2} \\ z_{3} \end{matrix}] = [\begin{matrix} 10 \\ 5 \\ 4 \end{matrix}]$
Step 4
From these we get,
$z_{1} = 10$
$0.5 z_{1} + z_{2} = 5$
$z_{2} = 0$
$- \frac{1}{2} z_{1} - 2 z_{2} + z_{3} = 4$
$z_{3} = 9$
Thus t

Jeffrey Jordon

Expert2022-01-29Added 2605 answers

Answer is given below (on video)

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