I have seen many resources about Bunch-Kaufman factorization but am still not sure about how it works exactly. For example, I often see a similar description:
The Bunch-Kaufman Decomposition of a square symmetric matrix $A$ is $A = P L D L' P'$ where $P$ is a permutation matrix, $L$ is unit-lower triangular and $D$ is block-diagonal with blocks of dimension $1 \times 1$ or $2 \times 2$.
However, when I find any pseudocode algorithm description (eg. https://books.google.cz/books?id=X5YfsuCWpxMC&pg=PA158&redir_esc=y&hl=cs#v=onepage&q&f=false), I see a different version which returns a normal diagonal matrix $D$, and not block-diagonal matrix. This version also has no permutation matrix $P$.
Is LDLT with and without permutation a different algorithm (I guess so)? Both are called Bunch-Kaufman? Or is the version with permutation called in a different way? Why permutation helps and how do I find out whether I need a version with permutation, or if this pseudocode description that I have found is sufficient for my application ($6 \times 6$ matrices decomposition)?