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Sep 19, 2020 at 20:27 comment added Damien I have asked a related question on condition number relationships of $H$ and its blocks over at [math](math.stackexchange.com/questions/3832653/… )
Sep 19, 2020 at 20:22 comment added Damien Extending for pivoting is useful - thanks. Also, for future readers, the condition number estimate from an LU decomposition can be found in Algorithm 3.5.1 of Golub and Van Loan.
Sep 17, 2020 at 15:23 comment added vibe It seems OP is happy with this answer. But I'll mention a few avenues for improving things further. It may be possible to scale the original matrix to reduce the condition number before doing the LU factorization. See LAPACK's DGEEQU routine for an example. If the matrix is still close to singular, you could use a partial or full pivoting approach in the $L_{22} U_{22}$ step to help, and also reveal the rank of the matrix.
Sep 17, 2020 at 14:53 comment added Damien I will have a look at some of the failure cases I have and see if there are any problems with the $L_{22} U_{22}$.
Sep 17, 2020 at 14:50 comment added Damien I actually did not know about the LU condition number trick! I have gotten the details from Golub and Van Loan.
Sep 17, 2020 at 14:46 vote accept Damien
Sep 17, 2020 at 13:01 comment added rchilton1980 Agreed with Federico, if the input $\mathbf H$ is ill conditioned, you might encounter cancellation problems when forming $\mathbf A - \mathbf B^T \mathbf D^{-1} \mathbf B$
Sep 17, 2020 at 11:52 comment added Federico Poloni Isn't this basically the same thing as the Schur complement approach that OP wanted to get away from? $L_{22}U_{22}$ is the Schur complement.
Sep 17, 2020 at 1:25 history edited vibe CC BY-SA 4.0
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Sep 17, 2020 at 1:16 history answered vibe CC BY-SA 4.0