Enhance qr function with mode argument

[jofrevalles]

Summary

Currently, the qr decomposition on a Tensor t defaults to the "full" mode, which given a matrix (M, N) with M>N we get a Q matrix with size (M, M) and R with size (N, N). This is counter-intuitive since these two tensors share a virtual index but they do not have the same dimensions.

In this PR we fix this by adding a mode argument to the qr function (which by default is set to :reduced). With mode=:reduced, it will ensure that the dimensions of the virtual index match and with mode=:full the qr functions returns the full matrices.

Example of usage

This enhancement facilitates more intuitive operations as illustrated below:

 julia> using Tenet

 julia> A = Tensor(rand(32, 8), (:i, :j))
32×8 Tensor{Float64, 2, Matrix{Float64}}: ...

julia> Q, R = qr(A, :reduced; left_inds=(:i,))
...

julia> size(Q), size(R)
((32, 8), (8, 8))

julia> Q, R = qr(A, :full; left_inds=(:i,))
...

julia> size(Q), size(R)
((32, 32), (8, 8))

[mofeing]

I see the problem but I don't think this is the solution. In the documentation of LinearAlgebra.qr function you can find the following paragraph:

...
Multiplication with respect to either full/square or non-full/square Q is allowed, i.e. both F.Q*F.R and F.Q*A are supported. A Q matrix can be converted into a regular matrix with Matrix. This operation returns the "thin" Q factor, i.e., if A is m×n with m>=n, then Matrix(F.Q) yields an m×n matrix with orthonormal columns. To retrieve the "full" Q factor, an m×m orthogonal matrix, use F.Q*I. If m<=n, then Matrix(F.Q) yields an m×m orthogonal matrix.
...

I would default to retrieving the thin Q matrix without a mode argument.

[jofrevalles]

I would default to retrieving the thin Q matrix without a mode argument.

Okay, this also makes sense. In this way we always get size(Q, virtualind)=size(R, virtualind) which follows our implementation.

[codecov]

Codecov Report

Attention: 3 lines in your changes are missing coverage. Please review.

Comparison is base (e7e5fae) 0.23% compared to head (baf5d24) 0.23%.

Additional details and impacted files

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  Lines          862     862           
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  Misses         860     860           

Files	Coverage Δ
src/Numerics.jl	0.00% <0.00%> (ø)

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[mofeing]

Is the PR ready?

[jofrevalles]

Is the PR ready?

Yes!