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Backports for 1.10.9 #57182

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Backported PRs:

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Contains multiple commits, manual intervention needed:

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LilithHafner and others added 4 commits January 28, 2025 14:23
The main motivation of this PR is to fix #55807.
dc689fe tries to remove the slow
`may_contain_union_decision` check by re-organizing the code path. Now
the fast path has been removed and most of its optimization has been
integrated into the preserved slow path.
Since the slow path stores all inner ∃ decisions on the outer most R
stack, there might be overflow risk.
aee69a4 should fix that concern.

The reported MWE now becomes
```julia
  0.000002 seconds
  0.000040 seconds (105 allocations: 4.828 KiB, 52.00% compilation time)
  0.000023 seconds (105 allocations: 4.828 KiB, 49.36% compilation time)
  0.000026 seconds (105 allocations: 4.828 KiB, 50.38% compilation time)
  0.000027 seconds (105 allocations: 4.828 KiB, 54.95% compilation time)
  0.000019 seconds (106 allocations: 4.922 KiB, 49.73% compilation time)
  0.000024 seconds (105 allocations: 4.828 KiB, 52.24% compilation time)
```

Local bench also shows that 72855cd slightly accelerates
`OmniPackage.jl`'s loading
```julia
julia> @time using OmniPackage
# v1.11rc4
 20.525278 seconds (25.36 M allocations: 1.606 GiB, 8.48% gc time, 12.89% compilation time: 77% of which was recompilation)
# v1.11rc4+aee69a4+72855cd
 19.527871 seconds (24.92 M allocations: 1.593 GiB, 8.88% gc time, 15.13% compilation time: 82% of which was recompilation)
```

(cherry picked from commit f3a36d7)
This PR is a potential fix for #54833.

## Description
The function
https://github.com/JuliaLang/julia/blob/2a06376c18afd7ec875335070743dcebcd85dee7/stdlib/LinearAlgebra/src/triangular.jl#L2220
computes $\boldsymbol{A}^{\dfrac{1}{2^s}} - \boldsymbol{I}$ for a
real-valued $2\times 2$ matrix $\boldsymbol{A}$ using Algorithm 5.1 in
[R1]. However, the algorithm in [R1] as well as the above function do
not handle the case $s=0.$ This fix extends the function to compute
$\boldsymbol{A}^{\dfrac{1}{2^s}} - \boldsymbol{I} \Bigg|_{s=0} =
\boldsymbol{A} - \boldsymbol{I}.$

## Checklist
- [X] Fix code: `stdlib\LinearAlgebra\src\triangular.jl` in function
`_sqrt_pow_diag_block_2x2!(A, A0, s)`.
- [X] Add test case: `stdlib\LinearAlgebra\test\triangular.jl`.
- [X] Update `NEWS.md`.
- [X] Testing and self review.

|  Tag  | Reference |
| --- | --- |
| <nobr>[R1]</nobr> | Al-Mohy, Awad H. and Higham, Nicholas J. "Improved
Inverse Scaling and Squaring Algorithms for the Matrix Logarithm", 2011,
url: https://eprints.maths.manchester.ac.uk/1687/1/paper11.pdf |

---------

Co-authored-by: Daniel Karrasch <[email protected]>
Co-authored-by: Oscar Smith <[email protected]>
(cherry picked from commit 2cdfe06)
@KristofferC KristofferC added the release Release management and versioning. label Jan 28, 2025
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