|TensorFlow 2.0 version||View source on GitHub|
Solves tridiagonal systems of equations.
tf.linalg.tridiagonal_solve( diagonals, rhs, diagonals_format='compact', transpose_rhs=False, conjugate_rhs=False, name=None, partial_pivoting=True )
The input can be supplied in various formats:
compact, specified by the
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.
diagonals are supplied as a tuple or list of three
tensors of shapes
[..., N] representing
superdiagonals, diagonals, and subdiagonals, respectively.
N can be either
M; in the latter case, the last element of superdiagonal and the
first element of subdiagonal will be ignored.
compact format the three diagonals are brought together into one tensor
[..., 3, M], with last two dimensions containing superdiagonals,
diagonals, and subdiagonals, in order. Similarly to
diagonals[..., 0, M-1] and
diagonals[..., 2, 0] are ignored.
compact format is recommended as the one with best performance. In case
you need to cast a tensor into a compact format manually, use
An example for a tensor of shape [m, m]:
rhs = tf.constant([...]) matrix = tf.constant([[...]]) m = matrix.shape 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
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
is set to
True the expected shape is
[..., M] or
[..., K, M].
The batch dimensions, denoted as
..., must be the same in
The output is a tensor of the same shape as
[..., M] or
[..., M, K].
The op isn't guaranteed to raise an error if the input matrix is not
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
partial_pivoting=True or there are multiple right-hand sides
K > 1). If this issue arises, consider if it's possible to disable pivoting
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
Tensoror tuple of
Tensors describing left-hand sides. The shape depends of
diagonals_format, see description above. Must be
Tensorof shape [..., M] or [..., M, K] and with the same dtype as
diagonals_format: one of
compact. Default is
rhsis transposed before solving (has no effect if the shape of rhs is [..., M]).
rhsis conjugated before solving.
name: A name to give this
partial_pivoting: whether to perform partial pivoting.
Trueby 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 ).
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.
 Nicholas J. Higham (2002). Accuracy and Stability of Numerical Algorithms: Second Edition. SIAM. p. 175. ISBN 978-0-89871-802-7.