|---|
| [ View source on GitHub](https://github.com/tensorflow/tensorflow/blob/v2.15.1/tensorflow/python/ops/linalg/linalg_impl.py#L350-L443) |
Solve triangular systems of equations with a banded solver.
```python
tf.linalg.banded_triangular_solve(
bands, rhs, lower=True, adjoint=False, name=None
)
```
`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.
```python
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
soln = tf.linalg.banded_triangular_solve(x, tf.ones([3, 1]))
soln
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.
```python
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
soln = tf.linalg.banded_triangular_solve(x, tf.ones([4, 1]), lower=False)
soln
are_equal = (soln == tf.linalg.triangular_solve(
z, tf.ones([4, 1]), lower=False))
tf.reduce_all(are_equal).numpy()
True
```
| ## Args ||
|---|---|
| `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`](https://www.tensorflow.org/versions/r2.15/api_docs/python/tf/linalg/set_diag.md.txt) 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). |
| ## Returns ||
|---|---|
| A `Tensor` of shape \[..., M\] or \[..., M, N\] containing the solutions. ||