Computes the solution of a system of linear equations with complex Hermitian or real symmetric positive-definite lhs given its Cholesky decomposition.
Let A A A be a complex Hermitian or real symmetric positive-definite matrix, and L L L its Cholesky decomposition such that:
A = L LH A = LL^{\text{H}} A=LLH
where LH L^{\text{H}} LH is the conjugate transpose when L L L is complex, and the transpose when L L L is real-valued.
Returns the solution X X X of the following linear system:
A X = B AX = B AX=B
Supports inputs of float, double, cfloat and cdouble dtypes. Also supports batches of matrices, and if A A A or B B B is a batch of matrices then the output has the same batch dimensions.
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 L L L is lower triangular or upper triangular. Default: False.
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)
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