adding == for structured matrices (split off of #29724)

[mcognetta]

This is the completed half of #29724

The logic I implemented in #29724 is flawed for isapprox and several new methods (specifically norms) need to be implemented. This PR contains just the exact equality code for structured matrices. It also includes a bug fix for comparing two Bidiagonal matrices when they have different uplo, which shouldn't be held up by the other PR.

The Bidiagonal bug:

julia> x = rand(3)
3-element Array{Float64,1}:
 0.7279081769922535
 0.8949580887222155
 0.7781661021218007

julia> Bu = Bidiagonal(x, zeros(2), 'U')
3×3 Bidiagonal{Float64,Array{Float64,1}}:
 0.727908  0.0        ⋅      
  ⋅        0.894958  0.0     
  ⋅         ⋅        0.778166

julia> Bl = Bidiagonal(x, zeros(2), 'L')
3×3 Bidiagonal{Float64,Array{Float64,1}}:
 0.727908   ⋅         ⋅      
 0.0       0.894958   ⋅      
  ⋅        0.0       0.778166

julia> Bu == Bl
false

julia> Matrix(Bu) == Matrix(Bl)
true

This PR

julia> x = rand(3)
3-element Array{Float64,1}:
 0.3172029052612273 
 0.8635761133663176 
 0.17013855660697486

julia> Bu = Bidiagonal(x, zeros(2), 'U')
3×3 Bidiagonal{Float64,Array{Float64,1}}:
 0.317203  0.0        ⋅      
  ⋅        0.863576  0.0     
  ⋅         ⋅        0.170139

julia> Bl = Bidiagonal(x, zeros(2), 'L')
3×3 Bidiagonal{Float64,Array{Float64,1}}:
 0.317203   ⋅         ⋅      
 0.0       0.863576   ⋅      
  ⋅        0.0       0.170139

julia> Bu == Bl
true

julia> Matrix(Bu) == Matrix(Bl)
true

---

Some timings (these are pessimal cases):
v1.0

julia> D=Diagonal(zeros(1000));B=Bidiagonal(zeros(1000), zeros(999), 'U');T=Tridiagonal(zeros(999), zeros(1000), zeros(999))

julia> @time D == B
  0.636254 seconds (4 allocations: 160 bytes)
true

julia> @time D == T
  0.008160 seconds (4 allocations: 160 bytes)
true

julia> @time B == T
  0.629008 seconds (4 allocations: 160 bytes)
true

This PR

julia> D=Diagonal(zeros(1000));B=Bidiagonal(zeros(1000), zeros(999), 'U');T=Tridiagonal(zeros(999), zeros(1000), zeros(999))

julia> @time D == B
  0.000007 seconds (4 allocations: 160 bytes)
true

julia> @time D == T
  0.000005 seconds (4 allocations: 160 bytes)
true

julia> @time B == T
  0.000012 seconds (4 allocations: 160 bytes)
true

[mcognetta]

The CI failure seems erroneous. Also, should this be back ported since the Bidiagonal bug appears in 1.0 and previous versions?

[StefanKarpinski]

I've restarted the macOS CI run.

[fredrikekre]

I've restarted the macOS CI run.

#30111

[StefanKarpinski]

Yeah, I hadn't seen that it was a repeat issue at this point. At least it fails fast.