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# torch.cholesky_solve¶

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

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

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

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

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

Returns the solution $X$ of the following linear system:

$AX = B$

Supports inputs of float, double, cfloat and cdouble dtypes. Also supports batches of matrices, and if $A$ or $B$ 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 $L$ 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)


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