- #1

- 746

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## Homework Statement

Five measurements of fuel consumption of a random Fiat car gives us data:

v (km/h) l/100 km

70 --> 3.9+-0.1

90 --> 4.8+-0.1

120 --> 6.0+-0.1

150 --> 8.1+-0.1

160 --> 9.0+-0.1

We would like to check if for velocities greater than 70 km/h the fuel consumption obeys ##\Phi =\Phi _0 +\alpha v^2##. Find ##\Phi _0## and ##\alpha ## using least-squares estimation.

## Homework Equations

## The Attempt at a Solution

Ok, I tried to do something but I assume there exist an easier way than mine...

Firstly, the law that fuel consumption should obey is ##\Phi =\Phi _0 +\beta v +\alpha v^2##. So $$

\begin{bmatrix}

\Phi _0\\

\beta\\

\alpha

\end{bmatrix}=\begin{bmatrix}

\Phi _0\\

0\\

\alpha

\end{bmatrix}=(H^TH)^{-1}H^Tz$$ Where IF I am not mistaken $$H=

\begin{bmatrix}

1 & 0& 70^2\\

1& 0 & 90^2\\

1& 0 & 120^2\\

1& 0 & 150^2\\

1& 0 & 160^2

\end{bmatrix}$$ meaning $$H^TH=

\begin{bmatrix}

5 &0 & 75500\\

0 & 0&0 \\

75500& 0&14.6\cdot 10^8

\end{bmatrix}$$ Now here the problems start. Am I really supposed to find the inverse of this? Is there really no better/faster solution to this problem?