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CHOLMOD doesn't have in-place solves. So if you use sparse Cholesky or LDL factorizations you should use \ for the backward and forward sweeps.
On Slack, TIm Davis explained that the issue is to apply the permutation in-place before the triangular solves. They don't have routines for that kind of operations.
Yes I know, but LDLFactorizations is not used in LinearOperators.jl.
The problem of Tim is that they didn't implemented permute! and invpermute! routines in CHOLMOD. He also said that it could be very slow for some permutation to apply it in-place.
I would be nice to add in-place solves to LinearOperators in the linalg.jl file.
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