Implemented Cholesky on CPU #1026

[arn4]

Proposed changes

Implemented Cholesky Decomposition for real matrix on CPU, as requested in #1026.

There is nothing done for the GPU.

Checklist

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• I have read the CONTRIBUTING document
• I have run pre-commit run --all-files to format my code / installed pre-commit prior to committing changes
• I have added tests that prove my fix is effective or that my feature works
• I have updated the necessary documentation (if needed)

[angeloskath]

This looks fantastic! Left some tiny nitpicky comments for small typos and such.

[angeloskath]

It currently crashes. Maybe undefined is indeed better. We could just not throw in that case.

[arn4]

Yes, it does crash if the matrix is not positive semidefinite. The behavior I wanted was

• raise when the matrix is not positive semidefinite, since it is directly detected by Lapack
• undefined if the matrix is not symmetric, since adding a check will come an unnecessary overhead.

Maybe I should specify it in the doc, or do you still prefer not raising at all?

[angeloskath]

I changed it to do nothing and added a comment to add some error checking if we can throw from the implementation in the future. Otherwise, potentially crashing the program in an unrecoverable way based on the values of the input matrix seems an extremely bad idea to me.

[angeloskath]

I missed it in the first review but here you were zeroing column-wise which is really inefficient especially for large matrices. I also use std::fill which may be faster if the compiler couldn't figure out that the previous loop was vectorizable.