YouTipPytorch Torch Lu_Unpack
## PyTorch torch.lu_unpack
`torch.lu_unpack` is a PyTorch function used to unpack the packed LU factorization (and its associated pivot indices) into the explicit permutation matrix $P$, lower triangular matrix $L$, and upper triangular matrix $U$.
This is particularly useful when you need to perform explicit matrix operations with individual components of the LU decomposition, or when verifying mathematical identities.
---
### Function Definition
```python
torch.lu_unpack(LU_data, LU_pivots, unpack_data=True, unpack_pivots=True, out=None)
```
#### Parameters:
* **`LU_data`** (*Tensor*): A tensor containing the packed LU factorization (where $L$ and $U$ are stored in a single matrix).
* **`LU_pivots`** (*Tensor*): The pivot indices obtained from the LU decomposition.
* **`unpack_data`** (*bool, optional*): Whether to unpack the $L$ and $U$ matrices. If set to `False`, the returned $L$ and $U$ will be `None`. Default is `True`.
* **`unpack_pivots`** (*bool, optional*): Whether to unpack the permutation matrix $P$ from the pivot indices. If set to `False`, the returned $P$ will be `None`. Default is `True`.
* **`out`** (*tuple, optional*): An optional output tuple of three tensors to store the results.
#### Returns:
* **`tuple`**: A tuple of three tensors: `(P, L, U)`.
* **`P`**: The permutation matrix (or `None` if `unpack_pivots=False`).
* **`L`**: The lower triangular matrix with unit diagonal elements (or `None` if `unpack_data=False`).
* **`U`**: The upper triangular matrix (or `None` if `unpack_data=False`).
---
## Code Example
The following example demonstrates how to perform LU decomposition on a 2D tensor using `torch.linalg.lu_factor` (or the legacy `torch.lu`), unpack the results using `torch.lu_unpack`, and verify that $A = P \times L \times U$.
```python
import torch
# 1. Create a 3x3 matrix A
A = torch.tensor([[1.0, 2.0, 3.0],
[4.0, 5.0, 6.0],
[7.0, 8.0, 9.0]])
# 2. Perform LU decomposition
# Note: torch.linalg.lu_factor is the preferred modern alternative to torch.lu
LU, pivots = torch.linalg.lu_factor(A)
# 3. Unpack the packed LU and pivots into P, L, and U matrices
P, L, U = torch.lu_unpack(LU, pivots)
# 4. Print the results
print("Original Matrix A:")
print(A)
print("\nPermutation Matrix P:")
print(P)
print("\nLower Triangular Matrix L:")
print(L)
print("\nUpper Triangular Matrix U:")
print(U)
# 5. Verify the decomposition: P @ L @ U should equal A
print("\nVerification: P @ L @ U =")
print(P @ L @ U)
```
### Output
```text
Original Matrix A:
tensor([[1., 2., 3.],
[4., 5., 6.],
[7., 8., 9.]])
Permutation Matrix P:
tensor([[0., 0., 1.],
[1., 0., 0.],
[0., 1., 0.]])
Lower Triangular Matrix L:
tensor([[1.0000, 0.0000, 0.0000],
[0.1429, 1.0000, 0.0000],
[0.5714, 0.5000, 1.0000]])
Upper Triangular Matrix U:
tensor([[7.0000, 8.0000, 9.0000],
[0.0000, 0.8571, 1.7143],
[0.0000, 0.0000, 0.0000]])
Verification: P @ L @ U =
tensor([[1., 2., 3.],
[4., 5., 6.],
[7., 8., 9.]])
```
---
## Important Considerations
1. **API Deprecation Warning**:
While `torch.lu` is still available in legacy codebases, it is deprecated in modern PyTorch versions. You should use `torch.linalg.lu_factor` to compute the packed LU decomposition, and then pass its outputs directly to `torch.lu_unpack`.
2. **Batch Support**:
`torch.lu_unpack` fully supports batched inputs. If your input matrix `A` has a shape of `(*, M, N)`, the unpacked matrices `P`, `L`, and `U` will preserve the batch dimensions `*` accordingly.
3. **Memory Efficiency**:
If you only need the permutation matrix $P$ or only the triangular matrices $L$ and $U$, you can set `unpack_data=False` or `unpack_pivots=False` respectively to save memory and computation time.