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forward and reverse mode automatic differentiation primitives for Julia Base + StdLibs

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- https://people.maths.ox.ac.uk/gilesm/files/NA-08-01.pdf - QR: https://arxiv.org/pdf/1001.1654.pdf

enhancement
help wanted
good first issue
docs
missing rule

There are currently instances when we want to use arrays of zeros in which size information is retained - this is not something supported by the `Zero` differential, but which...

performance
design

Since checkpointing can be implemented as a sensitivity, it may very well belong here. See: - Nabla: https://github.com/invenia/Nabla.jl/pull/171 - Zygote: https://fluxml.ai/Zygote.jl/dev/adjoints/#Checkpointing-1 - Tracker: https://github.com/FluxML/Tracker.jl/blob/bd98dab59026eb753c296a5f19367a408c2cb8b6/src/Tracker.jl#L82-L89

performance
design

Quoting Will in #29: > FWIW, the other thing to think about is what is actually happening computationally under the hood. Ultimately the `Diagonal` matrix type doesn't use any off-diagonal...

design

Some of the code used `zero(X)`, or `@thunk(zero(X))` I feel like if `Zero` is working right we should be able to replace that with `Zero()` E.g. https://github.com/JuliaDiff/ChainRules.jl/blob/master/test/rules/linalg/factorization.jl

performance
design

Suppose we have a function `f: ℝᵐ → ℝⁿ` that for some reason we want to differentiate in forward mode, which will require calling all `frule`s `m` times. This seems...

performance
design

This follows up a discussion on Slack. @sethaxen: > The pushforward of a function that produces a `Symmetric` matrix should also produce a `Symmetric` matrix. > Is it also true...

design

We already have support for this via `svd(X).S`, but support for `svdvals` directly would be nice. It'd also be great to support GenericLinearAlgebra, which I don't think we do at...

missing rule

not done in https://github.com/JuliaDiff/ChainRules.jl/pull/173 because https://github.com/JuliaDiff/ChainRulesTestUtils.jl/issues/22#issuecomment-615961039 (which blocks this issue)

testing

We still use a explict seed for testing `mod` and `rem`. During #185 I removed it and those tests failed. I have not investigated closely, it could be they are...

testing