Updated derivative rules for complex QR decomposition

[IvanYashchuk]

Updated qr_backward to work correctly for complex-valued inputs.
Added torch.qr to list of complex tests.

The previous implementation for real-valued differentiation used equation 42 from https://arxiv.org/abs/1001.1654
The current implementation is a bit simpler but the result for the real-valued input case is the same and all tests still pass.
Derivation of complex-valued QR differentiation https://giggleliu.github.io/2019/04/02/einsumbp.html

Ref. #33152

[dr-ci]

💊 CI failures summary and remediations

As of commit 630991a (more details on the Dr. CI page):

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• 2/2 failures possibly* introduced in this PR
  • 1/2 non-CircleCI failure(s)

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1 failure not recognized by patterns:

Job	Step	Action
pytorch_linux_xenial_py3_clang5_mobile_build	Set Up CI Environment After attach_workspace	🔁 rerun

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This comment has been revised 17 times.

[albanD]

Thanks for looking into this and updating the formula!

There were discussions as well about the numerical stability of the current implementation as well as limitations for degenerate matrices. Does this new algorithm have the same limitations?

[IvanYashchuk]

There were discussions as well about the numerical stability of the current implementation as well as limitations for degenerate matrices. Does this new algorithm have the same limitations?

Yes, I think it carries the same set of limitations as the previous implementation because the new one still doesn't use pivoting or any other techniques for improving numerical stability.

[codecov]

Codecov Report

Merging #48489 (02ef819) into master (c7b8f3e) will increase coverage by 55.81%.
The diff coverage is 100.00%.

@@             Coverage Diff             @@
##           master   #48489       +/-   ##
===========================================
+ Coverage   24.94%   80.76%   +55.81%     
===========================================
  Files         467     1867     +1400     
  Lines       58746   201579   +142833     
===========================================
+ Hits        14656   162809   +148153     
+ Misses      44090    38770     -5320     

[albanD]

LGTM thanks!

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[albanD]

@IvanYashchuk this has merge conflict with master because the other PR landed first and modified the exact same lines.
Could you rebase on top of master please?

[IvanYashchuk]

@albanD, done.

[albanD]

Thanks for the quick rebase! The lint is not happy with the extra space in the test file though

[IvanYashchuk]

Oh, should be fixed now.

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[albanD]

Thanks!

[IvanYashchuk]

Resolved another merge conflict.

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[albanD]

@IvanYashchuk sorry about this one, it keeps failing to land and then have merge conflicts...