The speed $v$ of a ship is related to the power $p$ of its engine by the empirical formula $$p=a+bv^3$$ Determine the values of $a$ and $b$ in the least squares sense from the following experimental data:
$$\begin{array}{c|cccc} \hline v & 6 & 8 & 10 & 12 \\ \hline p & 420 & 805 & 1370 & 2370 \\ \hline \end{array}$$
[ Source: $\small\textit{ A Theoretical Introduction to Numerical Analysis, Ryaben'kii & Tsyknov} ]$
I know how to solve a least squares system where the variables are linear. However, $v^3$ is not linear and I'm not sure what to do. Do I do the same thing, using:
$$\phi(a,b) = \sum_{k=1}^{4} [a+bv_k^3-p]^2 = [a+6^3b-420]^2...$$
and solve? Or is there a way to account for the cube?
Let $w=v^3$, for each $v$, you should be able to compute the corresponding $w$.
Now solve for $a$ and $b$ in $p=a+bw$.
While the problem is not linear in $v$, it is linear in $v^3=w$.
Your approach is correct.