A more scholarly (and older) treatment is in section 3 of this article version of Ch. That allows them to reduce the problem of chol([A, B*; B, C]) to just chol(A) and chol(Q).
The point of the algorithm is that you do not choose A and C to have the same size.
I'll update you all if I have any major breakthroughs.
Sadly, although I have done a fair amount of research on the subject, it does not appear that $C^$ helps wrt computation in the exact calculation of $Q^$.
The best approach appears to be to calculate $Q$ using $C^$ as above and then use cholesky to decompose $Q$ to $Q^$ and then forward substitution to calculate $Q^$.
$Q = Q^Q^ = C^ C^ - (B^ A^)(A^ B)$ = $(C^ B^A^)(C^ - B^A^)^$ My question is this: Given this set up is it possible to algebraicly calculate $Q^$ without having to apply cholesky decomposition to $Q$. If A, C are fixed, and B is variable but nice (low-rank), then you want what is called "Cholesky update".
Or in other words can I use $C^$ to help me in the calculation of $Q^$. If A, B, C are fixed, then probably you should not be picky about how the blocking is done, and you want to use a standard "block Cholesky".