torch.cholesky_solve — PyTorch 2.7 documentation (original) (raw)

torch.cholesky_solve(B, L, upper=False, *, out=None) → Tensor¶

Computes the solution of a system of linear equations with complex Hermitian or real symmetric positive-definite lhs given its Cholesky decomposition.

Let AA be a complex Hermitian or real symmetric positive-definite matrix, and LL its Cholesky decomposition such that:

A=LLHA = LL^{\text{H}}

where LHL^{\text{H}} is the conjugate transpose when LL is complex, and the transpose when LL is real-valued.

Returns the solution XX of the following linear system:

AX=BAX = B

Supports inputs of float, double, cfloat and cdouble dtypes. Also supports batches of matrices, and if AA or BB is a batch of matrices then the output has the same batch dimensions.

Parameters

• B (Tensor) – right-hand side tensor of shape (*, n, k)where ∗* is zero or more batch dimensions
• L (Tensor) – tensor of shape (*, n, n) where * is zero or more batch dimensions consisting of lower or upper triangular Cholesky decompositions of symmetric or Hermitian positive-definite matrices.
• upper (bool, optional) – flag that indicates whether LL is lower triangular or upper triangular. Default: False.

Keyword Arguments

out (Tensor, optional) – output tensor. Ignored if None. Default: None.

Example:

A = torch.randn(3, 3) A = A @ A.T + torch.eye(3) * 1e-3 # Creates a symmetric positive-definite matrix L = torch.linalg.cholesky(A) # Extract Cholesky decomposition B = torch.randn(3, 2) torch.cholesky_solve(B, L) tensor([[ -8.1625, 19.6097], [ -5.8398, 14.2387], [ -4.3771, 10.4173]]) A.inverse() @ B tensor([[ -8.1626, 19.6097], [ -5.8398, 14.2387], [ -4.3771, 10.4173]])

A = torch.randn(3, 2, 2, dtype=torch.complex64) A = A @ A.mH + torch.eye(2) * 1e-3 # Batch of Hermitian positive-definite matrices L = torch.linalg.cholesky(A) B = torch.randn(2, 1, dtype=torch.complex64) X = torch.cholesky_solve(B, L) torch.dist(X, A.inverse() @ B) tensor(1.6881e-5)