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I have found a simple python implementation, which does not require iteration (although it would be less stable than iterative solutions), but it still requires a scipy.linalg.schur decomposition.
I am the author who touched the DARE solver in SciPy introduced here and modified over the years. The iterative solvers are not more stable in fact it is the other way around but when arrays are too big for dense computations, decompositions become intractable and we resort back to iterative solvers.
Benner's group has lots of experience with those solvers and model reduction problems such as https://github.com/pymor/pymor/
What is the requirement of something to be compatible with PyTorch? You can in fact ask them about this specifically, they are always a helpful bunch https://github.com/pytorch/pytorch