Gauss Newton method for affine functions I have the function f(x) = $$\frac 1 2 \sum_{j=1}^m r_j(x)^2$$where $r_j(x)$ is an affine function in x for all j. When I minimize this function, using the Gauss Newton approximation, it is said to converge to the minimum in a single itteration. How can this be explained or proved?

$$f(x]=\frac 1 2 \sum_{j=1}^m r_j(x)^2\implies f'(x)=\sum_{j=1}^m r_j(x)r'_j(x)$$
$$r_j(x)=a_j+b_j x\implies f'(x)=\sum_{j=1}^m(a_j+b_j x)b_j=0\implies x=-\frac{\sum_{j=1}^m a_j b_j } {\sum_{j=1}^m b_j^2  }$$ and $$f''(x)=\sum_{j=1}^m b_j^2 >0$$
This explains that (I hope).
