Relation within Gauss-Newton method for minimization

[i_a_n]

If we study model fit on a nonlinear regression model $Y_i=f(z_i,\theta)+\epsilon_i$, $i=1,...,n$, and in the Gauss-Newton method, the update on the parameter $\theta$ from step $t$ to $t+1$ is to minimize the sum of squares $\sum_{i=1}^{n}[Y_i-f(z_i,\theta^{(t)})-(\theta-\theta^{(t)})^Tf'(z_i,\theta^{(t)})]^2$. Can we prove that (why) (part 1) the update is given in the following form: $\theta^{(t+1)}=\theta^{(t)}+[(A^{(t)})^TA^{(t)}]^{-1}(A^{(t)})^Tx^{(t)}$,(part 2) where $A^{(t)}$ is a matrix whose $i$-th row is $f'(z_i,\theta^{(t)})^T$, and $x^{(t)}$ is a column vector whose $i$-th entry is $Y_i-f(z_i,\theta^{(t)})$.
Any solution or hints? How to derive those relationships?

Thanks in advance!

 

[Siron]

If we study model fit on a nonlinear regression model $Y_i=f(z_i,\theta)+\epsilon_i$, $i=1,...,n$, and in the Gauss-Newton method, the update on the parameter $\theta$ from step $t$ to $t+1$ is to minimize the sum of squares $\sum_{i=1}^{n}[Y_i-f(z_i,\theta^{(t)})-(\theta-\theta^{(t)})^Tf'(z_i,\theta^{(t)})]^2$. Can we prove that (why) (part 1) the update is given in the following form: $\theta^{(t+1)}=\theta^{(t)}+[(A^{(t)})^TA^{(t)}]^{-1}(A^{(t)})^Tx^{(t)}$,(part 2) where $A^{(t)}$ is a matrix whose $i$-th row is $f'(z_i,\theta^{(t)})^T$, and $x^{(t)}$ is a column vector whose $i$-th entry is $Y_i-f(z_i,\theta^{(t)})$.
Any solution or hints? How to derive those relationships?

Thanks in advance!

How are those iterations (or updates) defined in the Gauss-Newton method?