I'm considering Gauss-Newton algorithm to minimize $\left\Vert \mathbf{f}\left( \mathbf{z}\right) \right\Vert ^{2}=\left\Vert \mathbf{f}\left( \mathbf{x},\mathbf{y}\right) \right\Vert ^{2}=\left\Vert \mathbf{a}-\mathbf{H}\left( \mathbf{x}\right) \mathbf{y}\right\Vert ^{2}$ with respect to $\mathbf{z}=\left( \mathbf{x}',\mathbf{y}' \right) ^{\prime }$ where $\mathbf{a}$ is a constant vector and $\mathbf{H}\left( \mathbf{x}\right) $ is a nonlinear matrix function of $\mathbf{x}.$ The updated $\mathbf{z}$ can be obtained from Taylor expansion, \begin{equation*} \mathbf{f}\left( \mathbf{x},\mathbf{y}\right) \approx \mathbf{f}\left( \mathbf{\bar{x}},\mathbf{\bar{y}}\right) +\frac{\partial \mathbf{f}\left( \mathbf{z}\right) }{\partial \mathbf{z}^{\prime }}\left( \mathbf{z}-\mathbf{\bar{z}}\right) =\mathbf{f}\left( \mathbf{\bar{x}},\mathbf{\bar{y}}\right) +\frac{\partial \mathbf{f}\left( \mathbf{x},\mathbf{y}\right) }{\partial \mathbf{x}^{\prime }}\left( \mathbf{x}-\mathbf{\bar{x}}\right) +\frac{\partial \mathbf{f}\left( \mathbf{x},\mathbf{y}\right) }{\partial \mathbf{y}^{\prime }}\left( \mathbf{y}-\mathbf{\bar{y}}\right) \end{equation*} Instead of updating entire $\mathbf{z,}$ I'm considering updating $\mathbf{x}$ and $\mathbf{y}$ separately to utilize the linearity of $\mathbf{y}$. In this case, the updated $\mathbf{x}$ and $\mathbf{y}$ can be obtained from Taylor expansion
\begin{eqnarray*} \mathbf{f}\left( \mathbf{x},\mathbf{y}\right) &\approx &\mathbf{f}\left( \mathbf{\bar{x}},\mathbf{y}\right) +\frac{\partial \mathbf{f}\left( \mathbf{x},\mathbf{y}\right) }{\partial \mathbf{x}^{\prime }}\left( \mathbf{x}-\mathbf{\bar{x}}\right) \\ \mathbf{f}\left( \mathbf{x},\mathbf{y}\right) &\approx &\mathbf{f}\left( \mathbf{x},\mathbf{\bar{y}}\right) +\frac{\partial \mathbf{f}\left( \mathbf{x},\mathbf{y}\right) }{\partial \mathbf{y}^{\prime }}\left( \mathbf{y}-\mathbf{\bar{y}}\right) \end{eqnarray*} From these, we have \begin{equation*} \mathbf{f}\left( \mathbf{x},\mathbf{y}\right) \approx \left[ \mathbf{f}\left( \mathbf{x},\mathbf{\bar{y}}\right) +\mathbf{f}\left( \mathbf{\bar{x}},\mathbf{y}\right) -\mathbf{f}\left( \mathbf{x},\mathbf{y}\right) \right] +\frac{\partial \mathbf{f}\left( \mathbf{x},\mathbf{y}\right) }{\partial \mathbf{x}^{\prime }}\left( \mathbf{x}-\mathbf{\bar{x}}\right) +\frac{\partial \mathbf{f}\left( \mathbf{x},\mathbf{y}\right) }{\partial \mathbf{y}^{\prime }}\left( \mathbf{y}-\mathbf{\bar{y}}\right) \end{equation*} However, comparing this expression with the above, the term in the bracket does not coincide with $\mathbf{f}\left( \mathbf{\bar{x}},\mathbf{\bar{y}}\right)$. I'm wondering why such a difference apears. Also, I'm wondering if considering updating $\mathbf{x}$ and $\mathbf{y}$ separately is theoretically valid.