torch.linalg.cond¶
- torch.linalg.cond(A, p=None, *, out=None) Tensor ¶
Computes the condition number of a matrix with respect to a matrix norm.
Letting be or , the condition number of a matrix is defined as
The condition number of
A
measures the numerical stability of the linear system AX = B with 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()
andtorch.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 valuesIn 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).See also
torch.linalg.solve()
for a function that solves linear systems of square matrices.torch.linalg.lstsq()
for a function that solves linear systems of general matrices.- Parameters
A (Tensor) – tensor of shape (*, m, n) where * is zero or more batch dimensions for
p
in (2, -2), and of shape (*, n, n) where every matrix is invertible forp
in (‘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 theA
matrix or any matrix in the batchA
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]])