Treat real transposes like adjoint in internal dispatch

[jishnub]

Since real transposes are equivalent to adjoints, we may compile methods only for one of the two types when we are unwrapping a Transpose through the wrapperop mechanism. This will improve the time to the second execution in certain cases (as the same type will be re-used). This is primarily useful in internal method dispatches where the result of wrapperop will not be returned.

For example, on master

julia> using LinearAlgebra

julia> U = UpperTriangular([1 2; 3 4]);

julia> @time transpose(U) * parent(U);
  0.140280 seconds (552.81 k allocations: 27.080 MiB, 99.90% compilation time)

julia> @time adjoint(U) * parent(U);
  0.124338 seconds (404.18 k allocations: 19.600 MiB, 99.92% compilation time)

whereas, on this PR,

julia> @time transpose(U) * parent(U);
  0.161032 seconds (553.25 k allocations: 27.090 MiB, 8.75% gc time, 99.91% compilation time)

julia> @time adjoint(U) * parent(U);
  0.053536 seconds (356.69 k allocations: 17.240 MiB, 99.81% compilation time)

The second execution is noticeably faster.

[dkarrasch]

Nice. Just a quick check: why can't we make this the "official" behaviour for wrapperop? Is it user-facing and would potentially "confuse" users?

[jishnub]

The current behavior of preserving the type in wrapperop is used in certain functions, e.g. similar and getindex. It seems like a useful behavior to preserve, and we may add another function (or method) that is primarily for optimizations. In the indexing context, it is indeed easier to remember that indexing into an Adjoint block matrix always produces an Adjoint matrix, and not a Transpose in certain cases. Moreover, this change would be strictly non-breaking if the returned types are not altered.

However, this may be a method of wrapperop instead of its own function, as they essentially do the same thing.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 92.12%. Comparing base (07725da) to head (ac4fca8).
Report is 3 commits behind head on master.

Additional details and impacted files

@@           Coverage Diff           @@
##           master    #1296   +/-   ##
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  Coverage   92.12%   92.12%           
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  Files          34       34           
  Lines       15509    15511    +2     
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+ Hits        14287    14289    +2     
  Misses       1222     1222           

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