/TensorFlow 1.15


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Solves tridiagonal systems of equations.

The input can be supplied in various formats: matrix, sequence and compact, specified by the diagonals_format arg.

In matrix format, diagonals must be a tensor of shape [..., M, M], with two inner-most dimensions representing the square tridiagonal matrices. Elements outside of the three diagonals will be ignored.

In sequence format, diagonals are supplied as a tuple or list of three tensors of shapes [..., N], [..., M], [..., N] representing superdiagonals, diagonals, and subdiagonals, respectively. N can be either M-1 or M; in the latter case, the last element of superdiagonal and the first element of subdiagonal will be ignored.

In compact format the three diagonals are brought together into one tensor of shape [..., 3, M], with last two dimensions containing superdiagonals, diagonals, and subdiagonals, in order. Similarly to sequence format, elements diagonals[..., 0, M-1] and diagonals[..., 2, 0] are ignored.

The compact format is recommended as the one with best performance. In case you need to cast a tensor into a compact format manually, use tf.gather_nd. An example for a tensor of shape [m, m]:

rhs = tf.constant([...])
matrix = tf.constant([[...]])
m = matrix.shape[0]
dummy_idx = [0, 0]  # An arbitrary element to use as a dummy
indices = [[[i, i + 1] for i in range(m - 1)] + [dummy_idx],  # Superdiagonal
         [[i, i] for i in range(m)],                          # Diagonal
         [dummy_idx] + [[i + 1, i] for i in range(m - 1)]]    # Subdiagonal
diagonals=tf.gather_nd(matrix, indices)
x = tf.linalg.tridiagonal_solve(diagonals, rhs)

Regardless of the diagonals_format, rhs is a tensor of shape [..., M] or [..., M, K]. The latter allows to simultaneously solve K systems with the same left-hand sides and K different right-hand sides. If transpose_rhs is set to True the expected shape is [..., M] or [..., K, M].

The batch dimensions, denoted as ..., must be the same in diagonals and rhs.

The output is a tensor of the same shape as rhs: either [..., M] or [..., M, K].

The op isn't guaranteed to raise an error if the input matrix is not invertible. tf.debugging.check_numerics can be applied to the output to detect invertibility problems.

Note: with large batch sizes, the computation on the GPU may be slow, if either partial_pivoting=True or there are multiple right-hand sides (K > 1). If this issue arises, consider if it's possible to disable pivoting and have K = 1, or, alternatively, consider using CPU.

On CPU, solution is computed via Gaussian elimination with or without partial pivoting, depending on partial_pivoting parameter. On GPU, Nvidia's cuSPARSE library is used: https://docs.nvidia.com/cuda/cusparse/index.html#gtsv

diagonals A Tensor or tuple of Tensors describing left-hand sides. The shape depends of diagonals_format, see description above. Must be float32, float64, complex64, or complex128.
rhs A Tensor of shape [..., M] or [..., M, K] and with the same dtype as diagonals.
diagonals_format one of matrix, sequence, or compact. Default is compact.
transpose_rhs If True, rhs is transposed before solving (has no effect if the shape of rhs is [..., M]).
conjugate_rhs If True, rhs is conjugated before solving.
name A name to give this Op (optional).
partial_pivoting whether to perform partial pivoting. True by default. Partial pivoting makes the procedure more stable, but slower. Partial pivoting is unnecessary in some cases, including diagonally dominant and symmetric positive definite matrices (see e.g. theorem 9.12 in [1]).
A Tensor of shape [..., M] or [..., M, K] containing the solutions.
ValueError An unsupported type is provided as input, or when the input tensors have incorrect shapes.

[1] Nicholas J. Higham (2002). Accuracy and Stability of Numerical Algorithms: Second Edition. SIAM. p. 175. ISBN 978-0-89871-802-7.

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Code samples licensed under the Apache 2.0 License.