If I have a large (e.g. 6000x6000), sparse, positive definite matrix $M$ (which may have individual entries everywhere, but most non-zero entries are on / around the diagional). Divide $M$ into blocks (with $A$, $D$, $F$ square):
$M = \begin{pmatrix}A&B&C\\B^T&D&E\\C^T&E^T&F\end{pmatrix}$
typically, $A$ is medium-sized (e.g. 200x200), $D$ is very small (e.g. 6x6), and $F$ is very large (e.g. 5794x5794).
Now if I already have the Schur complement of the large block $F$
$S_F = \begin{pmatrix}A'&B'\\B'^T&D'\end{pmatrix} = \begin{pmatrix}A&B\\B^T&D\end{pmatrix} - \begin{pmatrix}C\\E\end{pmatrix} F^{-1} \begin{pmatrix}C^T&E^T\end{pmatrix}$
I can, with little required computations, include more rows / columns in the complement, e.g.
$S_{DF} = A' - B' D'^{-1}B'^T$ ( = complement of $\begin{pmatrix}D&E\\E^T&F\end{pmatrix}$ in $M$)
Is it possible to (efficiently) do the same thing the other way round, that is, compute $S_F$ from $M$ and $S_{DF}$, without having to invert $F$ again?
($M$ comes from pose-graph optimization)
