% file is s32.tex
Apply Newton's method:
\[
\left[
\begin{array}{ccc}
   0 & 0  & \diag(\cdot) \\
    0  &  \mu Z^{-1} \cdot Z^{-1} &   I \\
 \Diag(\cdot) &  I  & 0
\end{array}
\right]
\left(
\begin{array}{c}
   \delta y \\
   \delta Z \\
   \delta X
\end{array}
\right)
= \]
\[\left(
\begin{array}{c}
   ( e- \diag(X))\\
   ( -X + \mu Z^{-1})\\
   (C-\Diag(y) -Z )
\end{array}
\right).
\]
simplify with $Z=C-\Diag(y)$:
\[
\begin{array}{ccc}
 \delta y& = &  (\mu Z^{-1} \circ Z^{-1} )^{-1}
           (e- \mu \diag(Z^{-1}) );\\
\delta Z& =& - \Diag(\delta y); \\
 X+\delta X &= &\mu Z^{-1} ( Z + \Diag(\delta y))Z^{-1}\nonumber.
\end{array}
\]