MatrixFactorizations.jl
A Julia package to contain non-standard matrix factorizations. At the moment it implements the QL, RQ, and UL factorizations, a combined Cholesky factorization with inverse, and polar decompositions. In the future it may include other factorizations such as the LQ factorization.
QL Factorization
The QL Factorization is analogous to the QR factorization but with a lower-triangular matrix:
julia> using MatrixFactorizations
julia> A = randn(5,5);
julia> ql(A)
MatrixFactorizations.QL{Float64,Array{Float64,2}}
Q factor:
5×5 MatrixFactorizations.QLPackedQ{Float64,Array{Float64,2}}:
0.155574 -0.112555 -0.686362 -0.701064 0.0233276
0.363037 -0.0583277 -0.0329428 0.152682 0.916736
-0.670742 0.294355 -0.547867 0.347157 0.206843
-0.61094 -0.566041 0.324432 -0.353128 0.276396
0.144425 -0.759527 -0.349868 0.489877 -0.199681
L factor:
5×5 Array{Float64,2}:
-1.75734 0.0 0.0 0.0 0.0
-0.58336 -2.08515 0.0 0.0 0.0
0.0213518 0.0722477 -2.01325 0.0 0.0
0.930364 -0.46796 -0.322493 -2.9091 0.0
1.08725 0.746217 0.549688 -1.10194 -2.0581
julia> b = randn(5); ql(A) \ b ≈ A \ b
true