Question

Compute the least-squares error associated with the least square solution found in Exercise 4.

Short answer

The least-square error is \(\sqrt 6 \).

Step-by-step solution

Write the results of Exercise 3

The normal equation is:

\(\left[ {\begin{aligned}{{}{}}3&3\\3&{11}\end{aligned}} \right]\left[ {\begin{aligned}{{}{}}{{x_1}}\\{{x_2}}\end{aligned}} \right] = \left[ {\begin{aligned}{{}{}}6\\{14}\end{aligned}} \right]\)

The \({\bf{\hat x}}\) component is \(\left[ {\begin{aligned}{{}{}}1\\1\end{aligned}} \right]\).

Find the matrix \(A{\bf{\hat x}} - {\bf{b}}\)

\[\begin{aligned}{}A{\bf{\hat x}} - {\bf{b}} &= \left[ {\begin{aligned}{{}{}}1&3\\1&{ - 1}\\1&1\end{aligned}} \right]\left[ {\begin{aligned}{{}{}}1\\1\end{aligned}} \right] - \left[ {\begin{aligned}{{}{}}5\\1\\0\end{aligned}} \right]\\ &= \left[ {\begin{aligned}{{}{}}4\\0\\2\end{aligned}} \right] - \left[ {\begin{aligned}{{}{}}5\\1\\0\end{aligned}} \right]\\ &= \left[ {\begin{aligned}{{}{}}{ - 1}\\{ - 1}\\2\end{aligned}} \right]\end{aligned}\]

Find the magnitude of \(\left| {A{\bf{\hat x}} - {\bf{b}}} \right|\)

The value of \(\left| {A{\bf{\hat x}} - {\bf{b}}} \right|\) can be calculated as:

\(\begin{aligned}{}\left| {A{\bf{\hat x}} - {\bf{b}}} \right| &= \sqrt {{{\left( { - 1} \right)}^2} + {{\left( { - 1} \right)}^2} + {{\left( 2 \right)}^2}} \\ &= \sqrt {1 + 1 + 4} \\ &= \sqrt 6 \end{aligned}\)

Thus, the least square error is \(\sqrt 6 \).