Simplify small Bidiagonal-AbstractVecOrMat multiplication

[jishnub]

Writing the base case out as an explicit loop helps with TTFX

julia> using LinearAlgebra

julia> T = Bidiagonal(ones(4), ones(3), :U); AT = Matrix(T); x = UpperTriangular(AT);

julia> @time T * x;
  0.195728 seconds (522.75 k allocations: 25.519 MiB, 8.07% gc time, 99.96% compilation time) # master
  0.126475 seconds (404.02 k allocations: 19.361 MiB, 99.96% compilation time) # this PR

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 92.00%. Comparing base (243efdb) to head (2f98c8b).
Report is 6 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1278      +/-   ##
==========================================
- Coverage   92.03%   92.00%   -0.04%     
==========================================
  Files          34       34              
  Lines       15488    15488              
==========================================
- Hits        14254    14249       -5     
- Misses       1234     1239       +5     

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[dkarrasch]

So the 2x2 and 3x3 cases are handled just as well by the loops in _mul_bitrisym in terms of runtime performance?

[jishnub]

Yes, in fact, it should be faster, as there are fewer operations to carry out if we use the structure of the matrix.

julia> T = Bidiagonal(ones(2), ones(1), :U); TM = Matrix(T); C = similar(TM);

julia> @btime mul!($C, $T, $TM);
  56.527 ns (0 allocations: 0 bytes) # master
  23.549 ns (0 allocations: 0 bytes) # this PR

julia> T = Bidiagonal(ones(3), ones(2), :U); TM = Matrix(T); C = similar(TM);

julia> @btime mul!($C, $T, $TM);
  116.016 ns (0 allocations: 0 bytes) # master
  37.616 ns (0 allocations: 0 bytes) # this PR