In my research I need to calculate $\operatorname{Trace}(A^{-1} C)$ where $A$ is given by two large, but sparse, matrices $B$ and $C$ by $A=B+\lambda C$.
I need to do this inversion many times, so I hope that an analytic approximation exists such that I need only invert a few matrices once. $B$ and $C$ are constant but $\lambda$ varies.
If it's helpful, the norm of $B$ is $\approx 2000$ times the norm of $C$, and $\lambda$ will typically be $\approx 400$.
Unfortunately, neither Tr$(B^{-1} C)$ nor Tr$(C^{-1} B)$ are small.
What you can consider is using the Taylor expansion of $f\colon \lambda \mapsto (B + \lambda C)^{-1}$. We have by the chain rule \begin{align*} f'(\lambda) &= D(\cdot)^{-1}(B + \lambda C)[C]\\ &= -(B+\lambda C)^{-1}C(B+\lambda C)^{-1}\\ f''(\lambda) &= -2(B + \lambda C)^{-1}C(B + \lambda C)^{-1}C(B+\lambda C)^{-1} \end{align*} So $$ (B + \lambda C)^{-1} = B^{-1} - B^{-1}CB^{-1}\lambda - B^{-1}CB^{-1}CB^{-1}\lambda^2 + o(\lambda^2) $$ But I don't if this is helpful in your case.
You can of course also write $f(\lambda) = \lambda^{-1} (C + \lambda^{-1}B)^{-1}$, giving $$ (B + \lambda C)^{-1} = \lambda^{-1}C^{-1} - C^{-1}BC^{-1} - C^{-1}BC^{-1}BC^{-1}\lambda + o(\lambda) $$