I have this equalities, with $\Delta X$ and $\Delta S$ unknown matrices, $X$, $S$ and $\tau$ known. $$W^{-1}\Delta X W^{-1}+\Delta S=\tau X^{-1}-S$$or equivalently $$\Delta X + W\Delta S W=\tau S^{-1}-X$$
My first question is how can i compute, or how can i find $\Delta X$.
My second question is, that given the following sets: $$N_\mathcal{F}(\theta):=\{(X,y,S)\in\mathcal{F}^0 : d_\mathcal{F}(X,S)\leq\theta\mu(X,S)\}$$ and $$\mathcal{F}^0:=\{(X,y,S):\textbf{tr}(A_iX)=b_i; \sum_{i=1}^my_iA_i+S=C; X,S\succ 0\;, i=1,...,m\}$$ Wher $\theta, C, A_i$ are given, and $$d_\mathcal{F}(X,S)=||S^{1/}2XS^{1/2}-\mu(X,S)I||_F$$ $$\mu(X,S)=\textbf{tr}(XS)/n$$ With $n=size(X)$.
How can i find, or compute in matlab, a fesasible pair $(X^0,S^0)\in N_\mathcal{F}(\theta)$