I need to compute the Cholesky decomposition $A=LL^T$ in order to obtain the matrix $L$.

My matrix $A$ is often badly conditioned (non-positive definite). I am now trying an approach where I normalize $A$ with a matrix $N$ before performing the decomposition. I want to reapply $N$ to the obtained result in order to find the correct matrix $L$.

Is such a method possible, and if so, how and why would it work? Are there any alternative ways to deal with non-positive definite matrices in a Cholesky decomposition?

  • $\begingroup$ If the non psd-ness stems only from numerical imprecision, you could simply stop the Cholesky process when the square root of a very slightly negative number occurs. At that point you have an approximate partial, low rank decomposition, which might be good enough for your purposes. $\endgroup$ – kimchi lover Sep 3 '18 at 13:03

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