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# numpy.tril_indices

numpy.tril_indices(n, k=0, m=None)

  Return the indices for the lower-triangle of an (n, m) array.

  Parameters:
    n int

      The row dimension of the arrays for which the returned indices will be valid.

    k int, optional

      Diagonal offset (see `tril` for details).

    m int, optional

      The column dimension of the arrays for which the returned arrays will be valid. By default m is taken equal to n.

  Returns:
    inds tuple of arrays

      The row and column indices, respectively. The row indices are sorted in non-decreasing order, and the correspdonding column indices are strictly increasing for each row.

  See also

  `triu_indices`

    similar function, for upper-triangular.

  `mask_indices`

    generic function accepting an arbitrary mask function.

  `tril`, `triu`

  Examples

      >>> import numpy as np

  Compute two different sets of indices to access 4x4 arrays, one for the lower triangular part starting at the main diagonal, and one starting two diagonals further right:

      >>> il1 = np.tril_indices(4)
      >>> il1
      (array([0, 1, 1, 2, 2, 2, 3, 3, 3, 3]), array([0, 0, 1, 0, 1, 2, 0, 1, 2, 3]))

  Note that row indices (first array) are non-decreasing, and the corresponding column indices (second array) are strictly increasing for each row. Here is how they can be used with a sample array:

      >>> a = np.arange(16).reshape(4, 4)
      >>> a
      array([[ 0,  1,  2,  3],
             [ 4,  5,  6,  7],
             [ 8,  9, 10, 11],
             [12, 13, 14, 15]])

  Both for indexing:

      >>> a[il1]
      array([ 0,  4,  5, ..., 13, 14, 15])

  And for assigning values:

      >>> a[il1] = -1
      >>> a
      array([[-1,  1,  2,  3],
             [-1, -1,  6,  7],
             [-1, -1, -1, 11],
             [-1, -1, -1, -1]])

  These cover almost the whole array (two diagonals right of the main one):

      >>> il2 = np.tril_indices(4, 2)
      >>> a[il2] = -10
      >>> a
      array([[-10, -10, -10,   3],
             [-10, -10, -10, -10],
             [-10, -10, -10, -10],
             [-10, -10, -10, -10]])