Problem: minimize f(x, y) = $|x + y -3|^2 + |2x + y + 1|^2 + |3x + y - 2|^2$ using least squares, x and y complex
I am getting lost on how to approach this problem. In my book, the least squares approximation problem is set up as finding the line of best fit through a fixed set of points $(x_i, y_i$)
The only idea I have is the absolute value squared looks like the inner product <-> length equation... but the terms inside don't factor to anything..
Can someone point me in the right direction?
Update: It seems $f(x, y)$ is the length of the vector $(x + y - 3, 2x + y + 1, 3x + y - 2)$ squared? For least squares approximation, I think I need to write something of the form $Ax = b$... been stuck what to do next though
Let $$A=\begin{bmatrix} 1 & 1 \\ 2 & 1 \\ 3 &1 \end{bmatrix}, \vec{x}=\begin{bmatrix}x \\ y\end{bmatrix}$$ and $$b=\begin{bmatrix} 3 \\ -1 \\ 2 \end{bmatrix}$$
You are trying to minimize $$\| A \vec{x} -b \|^2$$ meaning youneed to find the best approximation (or least square solution) to $$A \vec{x} =b$$