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tf.linalg.banded_triangular_solve

Solve triangular systems of equations with a banded solver.

bands is a tensor of shape [..., K, M], where K represents the number of bands stored. This corresponds to a batch of M by M matrices, whose K subdiagonals (when lower is True) are stored.

This operator broadcasts the batch dimensions of bands and the batch dimensions of rhs.

Examples:

Storing 2 bands of a 3x3 matrix. Note that first element in the second row is ignored due to the 'LEFT_RIGHT' padding.

x = [[2., 3., 4.], [1., 2., 3.]]
x2 = [[2., 3., 4.], [10000., 2., 3.]]
y = tf.zeros([3, 3])
z = tf.linalg.set_diag(y, x, align='LEFT_RIGHT', k=(-1, 0))
z
<tf.Tensor: shape=(3, 3), dtype=float32, numpy=
array([[2., 0., 0.],
       [2., 3., 0.],
       [0., 3., 4.]], dtype=float32)>
soln = tf.linalg.banded_triangular_solve(x, tf.ones([3, 1]))
soln
<tf.Tensor: shape=(3, 1), dtype=float32, numpy=
array([[0.5 ],
       [0.  ],
       [0.25]], dtype=float32)>
are_equal = soln == tf.linalg.banded_triangular_solve(x2, tf.ones([3, 1]))
tf.reduce_all(are_equal).numpy()
True
are_equal = soln == tf.linalg.triangular_solve(z, tf.ones([3, 1]))
tf.reduce_all(are_equal).numpy()
True

Storing 2 superdiagonals of a 4x4 matrix. Because of the 'LEFT_RIGHT' padding the last element of the first row is ignored.

x = [[2., 3., 4., 5.], [-1., -2., -3., -4.]]
y = tf.zeros([4, 4])
z = tf.linalg.set_diag(y, x, align='LEFT_RIGHT', k=(0, 1))
z
<tf.Tensor: shape=(4, 4), dtype=float32, numpy=
array([[-1.,  2.,  0.,  0.],
       [ 0., -2.,  3.,  0.],
       [ 0.,  0., -3.,  4.],
       [ 0.,  0., -0., -4.]], dtype=float32)>
soln = tf.linalg.banded_triangular_solve(x, tf.ones([4, 1]), lower=False)
soln
<tf.Tensor: shape=(4, 1), dtype=float32, numpy=
array([[-4.       ],
       [-1.5      ],
       [-0.6666667],
       [-0.25     ]], dtype=float32)>
are_equal = (soln == tf.linalg.triangular_solve(
  z, tf.ones([4, 1]), lower=False))
tf.reduce_all(are_equal).numpy()
True

bands A Tensor describing the bands of the left hand side, with shape [..., K, M]. The K rows correspond to the diagonal to the K - 1-th diagonal (the diagonal is the top row) when lower is True and otherwise the K - 1-th superdiagonal to the diagonal (the diagonal is the bottom row) when lower is False. The bands are stored with 'LEFT_RIGHT' alignment, where the superdiagonals are padded on the right and subdiagonals are padded on the left. This is the alignment cuSPARSE uses. See tf.linalg.set_diag for more details.
rhs A Tensor of shape [..., M] or [..., M, N] and with the same dtype as diagonals. Note that if the shape of rhs and/or diags isn't known statically, rhs will be treated as a matrix rather than a vector.
lower An optional bool. Defaults to True. Boolean indicating whether bands represents a lower or upper triangular matrix.
adjoint An optional bool. Defaults to False. Boolean indicating whether to solve with the matrix's block-wise adjoint.
name A name to give this Op (optional).

A Tensor of shape [..., M] or [..., M, N] containing the solutions.