Question

Find a least squares solution of Ax=b by constructing and solving the normal equations. A=[(3,1),(1,1),(1,4)], b[(1),(1),(1)] bar(x)=?

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asked 2020-10-18
Find a least squares solution of Ax=b by constructing and solving the normal equations.
\(\displaystyle{A}={\left[\begin{array}{cc} {3}&{1}\\{1}&{1}\\{1}&{4}\end{array}\right]},{b}{\left[\begin{array}{c} {1}\\{1}\\{1}\end{array}\right]}\)
\(\displaystyle\overline{{{x}}}=\)?

Answers (1)

2020-10-19
Step 1
We have to find the least square solution of Ax = B, by constructing the normal equations, where
\(\displaystyle{A}={\left[\begin{array}{cc} {3}&{1}\\{1}&{1}\\{1}&{4}\end{array}\right]},{b}{\left[\begin{array}{c} {1}\\{1}\\{1}\end{array}\right]}\)
The set of solutions of the non-empty solutions is given by
\(\displaystyle{A}^{{T}}{A}{x}={A}^{{T}}\) b.To solve this normal equations, we first compute the relevant matrices.
\(\displaystyle{A}^{{T}}{A}={\left[\begin{array}{ccc} {3}&{1}&{1}\\{1}&{1}&{4}\end{array}\right]}{\left[\begin{array}{cc} {3}&{1}\\{1}&{1}\\{1}&{4}\end{array}\right]}={\left[\begin{array}{cc} {11}&{8}\\{8}&{18}\end{array}\right]}\)
\(\displaystyle{A}^{{T}}{b}={\left[\begin{array}{ccc} {3}&{1}&{1}\\{1}&{1}&{4}\end{array}\right]}{\left[\begin{array}{c} {1}\\{1}\\{1}\end{array}\right]}={\left[\begin{array}{c} {5}\\{6}\end{array}\right]}\)
Step 2
Now, we need to solve \(\displaystyle{\left[\begin{array}{cc} {11}&{8}\\{8}&{18}\end{array}\right]}{x}={\left[\begin{array}{c} {5}\\{6}\end{array}\right]}\).
The augmented matrix is given by
\(\displaystyle{\left[\begin{array}{ccc} {11}&{8}&{5}\\{8}&{18}&{6}\end{array}\right]}\rightarrow{\left[\begin{array}{ccc} -{3}&{10}&{1}\\{8}&{18}&{6}\end{array}\right]}\)
\(\displaystyle\rightarrow{\left[\begin{array}{ccc} {1}&{10}&-{3}\\{6}&{18}&{8}\end{array}\right]}\)
\(\displaystyle\rightarrow{\left[\begin{array}{ccc} {1}&{10}&-{3}\\{0}&-{42}&-{10}\end{array}\right]}\)
\(\displaystyle\rightarrow{\left[\begin{array}{ccc} {1}&{10}&-{3}\\{0}&{21}&{5}\end{array}\right]}\)
Step 3
From the final matrix, we get the following equations \(\displaystyle{x}_{{1}}+{10}{x}_{{2}}=-{3}\)
\(\displaystyle{21}{x}_{{2}}={5}\)
\(\displaystyle\Rightarrow{x}_{{2}}=\frac{{5}}{{21}},{x}_{{1}}=-\frac{{113}}{{21}}\)
\(\displaystyle\Rightarrow{x}={\left[\begin{array}{c} \frac{{-{113}}}{{21}}\\\frac{{5}}{{21}}\end{array}\right]}\)
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