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 pointsI 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) linearly, so  could 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, such that:$$\begin{cases}y(x) = (bx^2 + cx) \; \text{explains the blue data}\\y(x) = a'(bx^2 + cx) \; \text{for red data}\\y(x) = a''(bx^2 + cx) \; \text{for green data}\end{cases}$$I am looking for a (if it exists) linear way to solve for $a', a'', b$ and $c$ minimizing the overall sum of squares for the three functions.Any idea on how this problem can be approached?Also approximate solutions are welcome...
linear solution of curve fitting on multiple linear functions differing by a multiplier
linear algebra;regression;least squares