torch.linalg.cond — PyTorch 2.7 documentation (original) (raw)
torch.linalg.cond(A, p=None, *, out=None) → Tensor¶
Computes the condition number of a matrix with respect to a matrix norm.
Letting K\mathbb{K} be R\mathbb{R} or C\mathbb{C}, the condition number κ\kappa of a matrixA∈Kn×nA \in \mathbb{K}^{n \times n} is defined as
κ(A)=∥A∥p∥A−1∥p\kappa(A) = \|A\|_p\|A^{-1}\|_p
The condition number of A measures the numerical stability of the linear system AX = Bwith respect to a matrix norm.
Supports input of float, double, cfloat and cdouble dtypes. Also supports batches of matrices, and if A is a batch of matrices then the output has the same batch dimensions.
p defines the matrix norm that is computed. The following norms are supported:
| p | matrix norm |
|---|---|
| None | 2-norm (largest singular value) |
| ‘fro’ | Frobenius norm |
| ‘nuc’ | nuclear norm |
| inf | max(sum(abs(x), dim=1)) |
| -inf | min(sum(abs(x), dim=1)) |
| 1 | max(sum(abs(x), dim=0)) |
| -1 | min(sum(abs(x), dim=0)) |
| 2 | largest singular value |
| -2 | smallest singular value |
where inf refers to float(‘inf’), NumPy’s inf object, or any equivalent object.
For p is one of (‘fro’, ‘nuc’, inf, -inf, 1, -1), this function usestorch.linalg.norm() and torch.linalg.inv(). As such, in this case, the matrix (or every matrix in the batch) A has to be square and invertible.
For p in (2, -2), this function can be computed in terms of the singular valuesσ1≥…≥σn\sigma_1 \geq \ldots \geq \sigma_n
κ2(A)=σ1σnκ−2(A)=σnσ1\kappa_2(A) = \frac{\sigma_1}{\sigma_n}\qquad \kappa_{-2}(A) = \frac{\sigma_n}{\sigma_1}
In these cases, it is computed using torch.linalg.svdvals(). For these norms, the matrix (or every matrix in the batch) A may have any shape.
Note
When inputs are on a CUDA device, this function synchronizes that device with the CPU if p is one of (‘fro’, ‘nuc’, inf, -inf, 1, -1).
Parameters
- A (Tensor) – tensor of shape (*, m, n) where * is zero or more batch dimensions for
pin (2, -2), and of shape (*, n, n) where every matrix is invertible forpin (‘fro’, ‘nuc’, inf, -inf, 1, -1). - p (int, inf , -inf , 'fro' , 'nuc' , optional) – the type of the matrix norm to use in the computations (see above). Default: None
Keyword Arguments
out (Tensor, optional) – output tensor. Ignored if None. Default: None.
Returns
A real-valued tensor, even when A is complex.
Raises
RuntimeError – if p is one of (‘fro’, ‘nuc’, inf, -inf, 1, -1) and the A matrix or any matrix in the batch A is not square or invertible.
Examples:
A = torch.randn(3, 4, 4, dtype=torch.complex64) torch.linalg.cond(A) A = torch.tensor([[1., 0, -1], [0, 1, 0], [1, 0, 1]]) torch.linalg.cond(A) tensor([1.4142]) torch.linalg.cond(A, 'fro') tensor(3.1623) torch.linalg.cond(A, 'nuc') tensor(9.2426) torch.linalg.cond(A, float('inf')) tensor(2.) torch.linalg.cond(A, float('-inf')) tensor(1.) torch.linalg.cond(A, 1) tensor(2.) torch.linalg.cond(A, -1) tensor(1.) torch.linalg.cond(A, 2) tensor([1.4142]) torch.linalg.cond(A, -2) tensor([0.7071])
A = torch.randn(2, 3, 3) torch.linalg.cond(A) tensor([[9.5917], [3.2538]]) A = torch.randn(2, 3, 3, dtype=torch.complex64) torch.linalg.cond(A) tensor([[4.6245], [4.5671]])