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TensorFlow 1 version
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View source on GitHub
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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: 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.
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
Args | |
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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. 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.
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diagonals_format
|
one of matrix, sequence, or compact. Default is
compact.
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transpose_rhs
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If True, rhs is transposed before solving (has no effect
if the shape of rhs is [..., M]).
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conjugate_rhs
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If True, rhs is conjugated before solving.
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name
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A name to give this Op (optional).
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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]).
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Returns | |
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A Tensor of shape [..., M] or [..., M, K] containing the solutions.
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Raises | |
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ValueError
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An unsupported type is provided as input, or when the input tensors have incorrect shapes. |
UnimplementedError
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Whenever partial_pivoting is true and the backend is
XLA.
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[1] Nicholas J. Higham (2002). Accuracy and Stability of Numerical Algorithms: Second Edition. SIAM. p. 175. ISBN 978-0-89871-802-7.