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?