torch.linalg.ldl_factor

torch.linalg.ldl_factor(A, *, hermitian=False, out=None)

Computes a compact representation of the LDL factorization of a Hermitian or symmetric (possibly indefinite) matrix.

When A is complex valued it can be Hermitian (hermitian= True) or symmetric (hermitian= False).

The factorization is of the form the form $A = L D L^T$. If hermitian is True then transpose operation is the conjugate transpose.

$L$ (or $U$) and $D$ are stored in compact form in LD. They follow the format specified by LAPACK’s sytrf function. These tensors may be used in torch.linalg.ldl_solve() to solve linear systems.

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.

Note

When inputs are on a CUDA device, this function synchronizes that device with the CPU. For a version of this function that does not synchronize, see torch.linalg.ldl_factor_ex().

Parameters

A (Tensor) – tensor of shape (*, n, n) where * is zero or more batch dimensions consisting of symmetric or Hermitian matrices.

Keyword Arguments

• hermitian (bool, optional) – whether to consider the input to be Hermitian or symmetric. For real-valued matrices, this switch has no effect. Default: False.

• out (tuple, optional) – tuple of two tensors to write the output to. Ignored if None. Default: None.

Returns

A named tuple (LD, pivots).

Examples:

>>> A = torch.randn(3, 3)
>>> A = A @ A.mT # make symmetric
>>> A
tensor([[7.2079, 4.2414, 1.9428],
        [4.2414, 3.4554, 0.3264],
        [1.9428, 0.3264, 1.3823]])
>>> LD, pivots = torch.linalg.ldl_factor(A)
>>> LD
tensor([[ 7.2079,  0.0000,  0.0000],
        [ 0.5884,  0.9595,  0.0000],
        [ 0.2695, -0.8513,  0.1633]])
>>> pivots
tensor([1, 2, 3], dtype=torch.int32)