Fix precision of 2x2 eigenvectors

[KnutAM]

Fixes #956 by subtracting mean diagonal (draft because for now only for A.uplo == 'U', see question below)

a = SMatrix{2,2}((1.0, 1e-100, 1e-100, 1.0))
Master (fast, but incorrect)

@btime eigen($a)
9.510 ns (0 allocations: 0 bytes)
eigen(a)
Eigen{Float64, Float64, SMatrix{2, 2, Float64, 4}, SVector{2, Float64}}
values:
2-element SVector{2, Float64} with indices SOneTo(2):
 1.0
 1.0
vectors:
2×2 SMatrix{2, 2, Float64, 4} with indices SOneTo(2)×SOneTo(2):
 0.0  0.0
 1.0  1.0

PR (slower, but correct)

julia> @btime eigen($a)
  11.011 ns (0 allocations: 0 bytes)

julia> eigen(a)
Eigen{Float64, Float64, SMatrix{2, 2, Float64, 4}, SVector{2, Float64}}
values:
2-element SVector{2, Float64} with indices SOneTo(2):
 1.0
 1.0
vectors:
2×2 SMatrix{2, 2, Float64, 4} with indices SOneTo(2)×SOneTo(2):
 -0.707107  0.707107
  0.707107  0.707107

Question

Why is the algorithm split between upper and lower diagonal, should be easier to do

a21 = A.uplo == 'U' ? a[3]' : a[2]

And then the algorithms are equal.

For now, I have only changed the algorithm for upper diagonal entry (hence draft), + no tests are added yet. Once I understand this choice, I can finish it up!

[mateuszbaran]

I think the added accuracy is worth sacrificing a bit of performance. The implementation is probably split because no one noticed before that defining a21 like you did would be enough -- I think this would be a good improvement.

[mateuszbaran]

BTW, could you fix the issue on Julia 1.10? You just need to replace

@test_throws ArgumentError SA[1;2;;3]

with

if VERSION < v"1.10-DEV"
    @test_throws ArgumentError SA[1;2;;3]
else
    @test_throws DimensionMismatch SA[1;2;;3]
end

in initializers.jl.

[KnutAM]

Thanks!

[mateuszbaran]

LGTM 👍 . Could you just bump patch version?