Is there any Riemann sum approximation for $\int_{A}^{B} tr(X^{-1} dX)$? Here $A, B$ are PSD matrices and $B \succ A$.
I'm thinking of $\sum_{t=1}^T tr\left(X(t)^{-1} (X(t) - X(t-1))\right)$, but how do I choose $X(t)$ in order to guarantee the error decays as $O(1/T)$? If I choose $X(t) \succ X(t-1)$ and $\|X(t) - X(t-1)\|_2 \sim \frac{1}{T}$, would the approximation error decays with the rate $O(1/T)$?
By Jacobi's formula
$$\eqalign{ {\rm Tr}(X^{-1}dX) &= d\,{\rm Tr}(\log(X)) \\ &= d\log(\det(X)) \\ }$$ Therefore the integral can be evaluated exactly without needing approximations $$\eqalign{ {\cal J} &= \int d\log(\det(X)) \\ &= \log(\det(B)) - \log(\det(A)) \\ &= \log(\det(BA^{-1})) \\ }$$