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Hi,

my question is similar to this one. I have to compute $B^TC^{−1}B$, where $C$ is a strictly positive definite $n\times n$ matrix and $B$ is $n\times m$. The matrix $C$ is huge ($n$ up to a hundred thousand) and sparse. If $m$ is small I compute $b^TC^{−1}b$ for all columns $b$ of $B$ using CG which is really fast (much faster than Cholesky decomposition). However this becomes problematic if $m$ gets larger.

Are there methods from numerical optimization to compute $B^TC^{−1}B$ more efficient?

Thank you very much!

Manuel

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  • $\begingroup$ It's no clear what you mean by "solve... directly". Do you mean "compute"? $\endgroup$
    – Dima Pasechnik
    Commented Dec 14, 2012 at 13:50
  • $\begingroup$ @Dima Pasechnik: Yes, I changed it to make it clear $\endgroup$
    – Manuel Schmidt
    Commented Dec 14, 2012 at 14:48
  • $\begingroup$ I would suggest first check what is wrong with CG. If for the bigger systems, it takes the similar number of iterations to convergence compared to the smaller system, then I don't know anything you can do about it. But if the bigger system takes a lot more iterations, you should look for additoinal structures of the system and find a good preconditioner. $\endgroup$
    – Lizao Li
    Commented Feb 4, 2013 at 5:03

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