Add logabsdet method for UmfpackLU

[ElOceanografo]

Replacement for #38476 after I borked a rebase. Will fix JuliaLang/LinearAlgebra.jl#788. This new method is a direct translation of the dense logabsdet(::LU) method in LinearAlgebra:

julia/stdlib/LinearAlgebra/src/lu.jl

Line 457 in 3365b4e

function logabsdet(F::LU{T}) where T # return log(abs(det)) and sign(det)

Performance problems mentioned in the previous PR should have been fixed by #40663.

[dkarrasch]

I have another way to compute the sign of the permutation, inspired by code from Permutations.jl. Should I push it here and you profile your benchmark cases with it?

[ElOceanografo]

Yes, that would be great!

[ElOceanografo]

That's an improvement! I also did some more testing and you were right, the call to diag was taking longer than indexing directly into U as before, so I switched back.

This method is still a bit slower than the default version for a dense LinearAlgebra.LU object...see simple benchmarks below. Profiling shows most of that time is spent getting U, Rs, p, and q out of the UMFPACK object. But when you factor in the time spent performing the factorization, calling logabsdet on a sparse matrix is now a) possible, b) correct, and c) much faster than the dense version. So I think this is probably ok to merge at this point, unless you've got any other ideas for improvement?

using LinearAlgebra, SparseArrays, BenchmarkTools
n = 2000
pnz = 0.0001
A = sprandn(n, n, pnz); A = A'A + I
Adense = Matrix(A)
F = lu(A)
Fdense = lu(Adense)


@btime logabsdet($F) # 58.337 μs (13 allocations: 127.23 KiB)
@btime logabsdet($Fdense) # 30.058 μs (0 allocations: 0 bytes)
@btime logabsdet($A) # 252.128 μs (77 allocations: 1.05 MiB)
@btime logabsdet($Adense) # 43.563 ms (3 allocations: 30.53 MiB)