linear solution of curve fitting on multiple linear functions differing by a multiplier I recently posted this question here but I thought this could be of interest also in mathematics, given I found a partially related question here
I am facing the following problem. I know nonlinear least squares can provide a solution but I am wondering if a linear way to solve this data fitting problem may exists.
This is my input dataset:
I've got three different dataset composed of scattered points

I know a linear equation in the form 
$$
y(x) = bx^2 + cx
$$
can be used to explain any of my dataset. I know how to fit this function to a given dataset (e.g the blue one) using linear least square, so I could fit the above function separately for each dataset, but I am looking for something different.
In the specific case I have an additional constrain: I also know that the three functions describing the three dataset share the $b$ and $c$ parameters, while they differ by a multiplier, something like this:
$$
\begin{cases}
y(x) = (bx^2 + cx) \; \text{explains the blue data}\\
y(x) = a'(bx^2 + cx) \; \text{for red data}\\
y(x) = a''(bx^2 + cx) \; \text{for green data}
\end{cases}
$$
I am looking for a (if it exists) linear way to solve this problem globally. Obtaining $a', a'', b$ and $c$
Maybe I am missing something so also having a different point of view on how to correctly formulate the problem could help.
Also approximate solutions are welcome...

I think it is not possible (but I may be wrong) to solve this directly. But I would do following:


*

*Calculate $b,c$ via least squares with the blue dataset only.


Then iterate over those two steps:


*

*Take $b,c$ from previous step and calculate $a',a''$ via least squares.

*Take $a',b'$ from previous step and calculate $b,c$ via least squares.


I think this will converge pretty quickly when the datapoints are not scattered too much.
