specialize copyto! and multiplication by numbers for Q from qr

[ranocha]

This fixes two performance bugs reported in JuliaLang/LinearAlgebra.jl#800 and JuliaLang/LinearAlgebra.jl#800 (multiplication of Q from qr by a Diagonal or UniformScaling). In particular, it improves the performance of generating random orthogonal matrices as described in JuliaLang/LinearAlgebra.jl#800.

Latest Julia nightly (v1.7.0-DEV.448)

julia> using LinearAlgebra

julia> n = 200; Q, R = qr(randn(n, n)); A = randn(n, n); D = Diagonal(randn(n));

julia> @time Q * A;
  0.073661 seconds (263.33 k allocations: 16.445 MiB, 98.74% compilation time)

julia> @time A * Q;
  0.007283 seconds (4.03 k allocations: 885.695 KiB, 86.00% compilation time)

julia> @time Q * D;
  1.402949 seconds (718.24 k allocations: 241.546 MiB, 1.77% gc time, 12.63% compilation time)

julia> @time D * Q;
  1.262098 seconds (386.20 k allocations: 222.074 MiB, 0.53% gc time, 3.97% compilation time)

julia> @time Q * I;
  1.255886 seconds (363.48 k allocations: 221.299 MiB, 0.56% gc time, 3.27% compilation time)

julia> @time I * Q;
  1.250831 seconds (330.64 k allocations: 219.089 MiB, 0.54% gc time, 3.02% compilation time)

This PR

julia> using LinearAlgebra

julia> n = 200; Q, R = qr(randn(n, n)); A = randn(n, n); D = Diagonal(randn(n));

julia> @time Q * A;
  0.079362 seconds (263.33 k allocations: 16.442 MiB, 98.72% compilation time)

julia> @time A * Q;
  0.008170 seconds (4.03 k allocations: 885.695 KiB, 85.55% compilation time)

julia> @time Q * D;
  0.117407 seconds (366.77 k allocations: 21.280 MiB, 9.75% gc time, 98.93% compilation time)

julia> @time D * Q;
  0.083525 seconds (265.85 k allocations: 15.456 MiB, 98.40% compilation time)

julia> @time Q * I;
  0.013162 seconds (15.95 k allocations: 1.567 MiB, 90.46% compilation time)

julia> @time I * Q;
  0.010540 seconds (8.82 k allocations: 1.140 MiB, 88.74% compilation time)

Of course, further performance optimizations are likely possibly by introducing better specializations of various multiplication methods of Q by something from the left/right as mentioned in JuliaLang/LinearAlgebra.jl#800, but this should be a good first step.

I cannot add labels to this, but "linear algebra" and "performance" would be appropriate, I think.

[sostock]

The same could be done for LQPackedQ (which is not a subtype of AbstractQ, I don’t know whether it should be). I just did some benchmarks and it leads to massive performance improvements.

[ranocha]

I would be happy to include these modifications here (if you create a PR into my branch or post them as suggested changes as review). Otherwise, it would also be great if you could open a new PR to Julia. Anyway, it would be nice if someone could review this and we could get this merged.

[ranocha]

Bump. Is there anything that needs to be done before this can be merged?