torch.cholesky_solve#

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

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

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

$$A=L{L}^{\text{H}}$$A=LLH

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

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

$$AX=B$$AX=B

Supports inputs of float, double, cfloat and cdouble dtypes.Also supports batches of matrices, and if$A$A or$B$B is a batch of matricesthen the output has the same batch dimensions.

Parameters

• B (Tensor) – right-hand side tensor of shape(*, n, k)where$\ast$∗ is zero or more batch dimensions

• L (Tensor) – tensor of shape(*, n, n) where* is zero or more batch dimensionsconsisting of lower or upper triangular Cholesky decompositions ofsymmetric or Hermitian positive-definite matrices.

• upper (bool,optional) – flag that indicates whether$L$L is lower triangularor upper triangular. Default:False.

Keyword Arguments

out (Tensor,optional) – output tensor. Ignored ifNone. 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()@Btensor([[ -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)