nitraiddQ
2020-10-18
Answered

Find a least squares solution of Ax=b by constructing and solving the normal equations.

$A=\left[\begin{array}{cc}3& 1\\ 1& 1\\ 1& 4\end{array}\right],b\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$

$\stackrel{\u2015}{x}=$ ?

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nitruraviX

Answered 2020-10-19
Author has **101** answers

Step 1

We have to find the least square solution of Ax = B, by constructing the normal equations, where

$A=\left[\begin{array}{cc}3& 1\\ 1& 1\\ 1& 4\end{array}\right],b\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$

The set of solutions of the non-empty solutions is given by

$A}^{T}Ax={A}^{T$ b.To solve this normal equations, we first compute the relevant matrices.

${A}^{T}A=\left[\begin{array}{ccc}3& 1& 1\\ 1& 1& 4\end{array}\right]\left[\begin{array}{cc}3& 1\\ 1& 1\\ 1& 4\end{array}\right]=\left[\begin{array}{cc}11& 8\\ 8& 18\end{array}\right]$

${A}^{T}b=\left[\begin{array}{ccc}3& 1& 1\\ 1& 1& 4\end{array}\right]\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}5\\ 6\end{array}\right]$

Step 2

Now, we need to solve$\left[\begin{array}{cc}11& 8\\ 8& 18\end{array}\right]x=\left[\begin{array}{c}5\\ 6\end{array}\right]$ .

The augmented matrix is given by

$\left[\begin{array}{ccc}11& 8& 5\\ 8& 18& 6\end{array}\right]\to \left[\begin{array}{ccc}-3& 10& 1\\ 8& 18& 6\end{array}\right]$

$\to \left[\begin{array}{ccc}1& 10& -3\\ 6& 18& 8\end{array}\right]$

$\to \left[\begin{array}{ccc}1& 10& -3\\ 0& -42& -10\end{array}\right]$

$\to \left[\begin{array}{ccc}1& 10& -3\\ 0& 21& 5\end{array}\right]$

Step 3

From the final matrix, we get the following equations${x}_{1}+10{x}_{2}=-3$

$21{x}_{2}=5$

$\Rightarrow {x}_{2}=\frac{5}{21},{x}_{1}=-\frac{113}{21}$

$\Rightarrow x=\left[\begin{array}{c}\frac{-113}{21}\\ \frac{5}{21}\end{array}\right]$

We have to find the least square solution of Ax = B, by constructing the normal equations, where

The set of solutions of the non-empty solutions is given by

Step 2

Now, we need to solve

The augmented matrix is given by

Step 3

From the final matrix, we get the following equations

Jeffrey Jordon

Answered 2021-11-03
Author has **2313** answers

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