LU decomposition without pivoting in JULIA - matlab

Question

Trying to rewrite the lu_nopivot from this answer https://stackoverflow.com/a/41151228 into JULIA and use only one loop.
The julia code I wrote
using LinearAlgebra
function lu_nopivot(A)
n = size(A, 1)
L = Matrix{eltype(A)}(I, n, n)
U = copy(A)
for k = 1:n
L[k+1:n,k] = U[k+1:n,k] / U[k,k]
U[k+1:n,:] = U[k+1:n,:] - L[k+1:n,k]*U[k,:]
end
return L, U
end
But calling the function
L, U = lu_nopivot(A) gives an error MethodError: no method matching *(::Vector{Float64}, ::Vector{Float64}) on L[k+1:n,k]*U[k,:]. Any reason why this is?

Answer 1

The problem is that when you do U[k, :], even though you're extracting out a row, what you get back is a column vector. So L[k+1:n,k]*U[k,:] becomes an attempt to multiply a column vector by a column vector.
One way to get a row vector aka a 1 x N Matrix in Julia (though I don't know if this is the idiomatic way) is to do U[k:k, :] instead:
U[k+1:n,:] = U[k+1:n,:] - L[k+1:n,k] * U[k:k,:]
Note however that Julia's implementation of lu already has a no-pivot option:
Pivoting can be turned off by passing
pivot = NoPivot().
julia> M = rand(1.0:100.0, 3, 3)
3×3 Matrix{Float64}:
71.0 50.0 23.0
82.0 63.0 37.0
96.0 28.0 5.0
julia> L, U = lu_nopivot(M); # after the above k:k change has been made
julia> L
3×3 Matrix{Float64}:
1.0 0.0 0.0
1.15493 1.0 0.0
1.35211 -7.53887 1.0
julia> U
3×3 Matrix{Float64}:
71.0 50.0 23.0
0.0 5.25352 10.4366
0.0 0.0 52.5818
julia> lu(M, NoPivot())
LU{Float64, Matrix{Float64}}
L factor:
3×3 Matrix{Float64}:
1.0 0.0 0.0
1.15493 1.0 0.0
1.35211 -7.53887 1.0
U factor:
3×3 Matrix{Float64}:
71.0 50.0 23.0
0.0 5.25352 10.4366
0.0 0.0 52.5818
Using this is likely more performant, and more robust too (eg. your current implementation cannot handle matrices of eltype Int, since L and U are given the same type as the input but will need to contain Floats).