# Which pivoting strategy is used in `LinearAlgebra.bunchkaufman()` function for Bunch-Kaufman factorization when we still not using rook pivoting?

When trying to do Bunch-Kaufman factorization of the symmetric matrix `A` like below:

``````julia> using LinearAlgebra

julia> M = [4 12 -16; 9.5 37 -43; 11 -6.5 9];

julia> A = Symmetric(M, :L)
3×3 Symmetric{Float64, Matrix{Float64}}:
4.0   9.5  11.0
9.5  37.0  -6.5
11.0  -6.5   9.0

julia> S = bunchkaufman(A)    # By default rook::Bool=false
BunchKaufman{Float64, Matrix{Float64}, Vector{Int64}}
D factor:
3×3 Tridiagonal{Float64, Vector{Float64}}:
9.0   0.0        ⋅
0.0  32.3056    0.0
⋅    0.0     -18.8641
L factor:
3×3 UnitLowerTriangular{Float64, Matrix{Float64}}:
1.0        ⋅         ⋅
-0.722222  1.0        ⋅
1.22222   0.539983  1.0
permutation:
3-element Vector{Int64}:
3
2
1

julia> S.P * S.L * S.D * S.L' * S.P' ≈ A
true
``````

We can see to get the matrix `A` back by multiplying the components of the BunchKaufman factorization object `S`. Here we can notice `S.P`, the permutation matrix is not an identity matrix!

``````julia> S.P
3×3 Matrix{Float64}:
0.0  0.0  1.0
0.0  1.0  0.0
1.0  0.0  0.0
``````

Although in this case, if by default rook pivoting is not used in the function, which particular pivoting strategy is used here then?