right_svd
=========

.. py:function:: t3toolbox.backend.linalg.right_svd(G0_i_a_j, min_rank = None, max_rank = None, rtol = None, atol = None)

   .. code-block:: python

      def right_svd(
              G0_i_a_j: NDArray, # shape=(ni, na, nj)
              min_rank: int = None, # 1 <= min_rank <= max_rank <= minimum(ni*na, nj)
              max_rank: int = None, # 1 <= min_rank <= max_rank <= minimum(ni*na, nj)
              rtol: float = None, # removes singular values satisfying sigma < maximum(atol, rtol*sigma1)
              atol: float = None, # removes singular values satisfying sigma < maximum(atol, rtol*sigma1)
      ) -> typ.Tuple[
          NDArray, # U_i_x,       shape=(ni, nx)
          NDArray, # ss_x,        shape=(nx,)
          NDArray, # Vt_x_a_j,    shape=(nx, na, nj)
      ]:


   Compute (truncated) singular value decomposition of 3-tensor right unfolding.

   Last two indices of the tensor are grouped for the SVD: ``G[i,a,j] = sum_x U[i,x] ss[x] Vt[x,a,j]``,
   with ``Vt`` orthonormal in its grouped ``(a,j)`` rows. Truncation args behave as in
   :py:func:`truncated_svd`.

   .. rubric:: Examples

   >>> import numpy as np
   >>> import t3toolbox.backend.linalg as linalg
   >>> np.random.seed(0)
   >>> G = np.random.randn(4, 5, 6)                      # (ni, na, nj)
   >>> U, ss, Vt = linalg.right_svd(G)
   >>> print(U.shape, ss.shape, Vt.shape)               # U is the 2d left factor; Vt keeps (a,j)
   (4, 4) (4,) (4, 5, 6)
   >>> print(np.allclose(np.einsum('ix,x,xaj->iaj', U, ss, Vt), G))   # reconstructs G
   True
   >>> Vm = Vt.reshape(Vt.shape[0], -1)
   >>> print(np.allclose(Vm @ Vm.T, np.eye(Vm.shape[0])))   # right unfolding of Vt is orthonormal
   True