Add lu decomposition for Tensors

[jofrevalles]

Summary

This PR adds the new lu function for tensors, extending the LinearAlgebra.lu function (resolve #26). The new lu function returns the LU decomposition of a Tensor, where the tensor can be recovered by contracting the permutation tensor P, the tensor L, and tensor U. The tensors L and U are reshaped versions of the original lower and upper triangular matrices obtained during the decomposition process, respectively.

This implementation is inspired by the LU decomposition in the scipy library, as it returns the permutation tensor P allowing the original tensor A to be recovered with the contraction A = P * L * U. This contrasts with LinearAlgebra, where the permutation vector p is returned, and the original matrix can be recovered with P' * A = L * U (where P' is the permutation matrix built from p).

Please let me know if there are any concerns or issues with extending the LinearAlgebra library in this manner.

We have also added tests for this new function.

Example

A usage example of the lu function:

julia> using Tenet; using LinearAlgebra; using Test

julia> tensor = Tensor(rand(4, 4, 4), (:i, :j, :k))
4×4×4 Tensor{Float64, 3, Array{Float64, 3}}:
...

julia> P, L, U = lu(tensor, left_inds = labels(tensor)[1:2])
...

julia> @test contract(contract(P, L), U) ≈ tensor
Test Passed

[codecov]

Codecov Report

Attention: 1 lines in your changes are missing coverage. Please review.

Comparison is base (5b3e11b) 87.71% compared to head (525f14e) 88.18%.
Report is 1 commits behind head on develop.

Additional details and impacted files

@@             Coverage Diff             @@
##           develop      #94      +/-   ##
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+ Coverage    87.71%   88.18%   +0.46%     
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  Files           10       10              
  Lines          570      584      +14     
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+ Hits           500      515      +15     
+ Misses          70       69       -1     

Files	Coverage Δ
src/Tensor.jl	88.99% <100.00%> (ø)
src/Transformations.jl	98.41% <100.00%> (ø)
src/Numerics.jl	92.20% <97.72%> (+3.31%)	⬆️

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[mofeing]

@jofrevalles I've updated the code to the recent changes. Also, I've refactored the SVD and QR factorizations. Would you mind giving a quick review?

[jofrevalles]

@jofrevalles I've updated the code to the recent changes. Also, I've refactored the SVD and QR factorizations. Would you mind giving a quick review?

I have checked it and everything looks good to me!