Adding specialized ≈ and norm for structured matrices.

[mcognetta]

the == half of this PR was moved to #30108.

The logic for isapprox here is not correct yet. The way I plan to proceed is to implement specialized norm for structured matrices and then the fallback to the generic isapprox can be made much faster even if it does not fall back to element wise comparison.

---

Currently, == and ≈ fall back to generic methods. This PR adds special methods for structured matrix types. It also fixes an error in bidiag.jl where == fails:

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

---

(Forgot to time ≈):

v1.0

julia> @time D ≈ B
  1.253851 seconds (10 allocations: 24.047 KiB)
true

julia> @time D ≈ T
  0.029518 seconds (12 allocations: 39.953 KiB)
true

julia> @time B ≈ T
  0.669546 seconds (11 allocations: 32.016 KiB)
true

This PR

julia> @time D ≈ B
  0.000044 seconds (6 allocations: 8.109 KiB)
true

julia> @time D ≈ T
  0.000045 seconds (6 allocations: 8.109 KiB)
true

julia> @time B ≈ T
  0.000036 seconds (7 allocations: 16.047 KiB)
true

[mschauer]

Should this be a test for approximately zero when atol!=0?

[mschauer]

The norm argument is unused

[simonbyrne]

For vecnorm it would be hypot(vecnorm(B.ev), vecnorm(A.diag .- B.dv)) <= ..., but to deal with the general case I guess you need a fallback.

[mcognetta]

vecnorm was deprecated with v0.7. The problem with using a recursive isapprox fallback is that isapprox doesn't have a kwarg for norm when the eltype is Number.

[andreasnoack]

This needs a rebase.

[andreasnoack]

@mcognetta would you be able to rebase this one?

[mcognetta]

Yes, I will try. I haven't looked at this in quite a while though. Does it otherwise seem ok?

[andreasnoack]

I haven't looked at this in quite a while though.

Same here. I looked at the list of PRs that I was requested for review I found this old one.

Does it otherwise seem ok?

Yeah. I think so.