I have a symmetric indefinite matrix $A$ which is banded, and I want to compute the $LDL^T$ factorization, however there do not appear to exist codes to do this in standard libraries (such as LAPACK, matlab, etc). I'd prefer to use an efficient algorithm rather than use a standard $LDL^T$ code on a full/dense matrix.
I am wondering if perhaps the reason codes don't exist is because the $LDL^T$ factorization of a symmetric banded matrix does not have the same banded structure? For a symmetric positive definite banded matrix, it is true that the Cholesky factor shares the same band structure as the original matrix, and indeed there exists a LAPACK routine to compute this factorization (DPBTRF). It seems the same should be true for $LDL^T$ but am I missing something?
Also, I found this a reference here which describes the Bunch-Kaufman algorithm which computes $LDL^T$ for a symmetric indefinite band matrix. The abstract says that the algorithm does not preserve band structure. But what about the usual diagonal pivoting algorithm? Wouldn't that preserve band structure?