torch.linalg.lu_solve#

torch.linalg.lu_solve(LU,pivots,B,*,left=True,adjoint=False,out=None)→Tensor#

Computes the solution of a square system of linear equations with a unique solution given an LU decomposition.

Letting$\mathbb{K}$K be$\mathbb{R}$R or$\mathbb{C}$C,this function computes the solution$X\in {\mathbb{K}}^{n\times k}$X∈Kn×k of thelinear system associated to$A\in {\mathbb{K}}^{n\times n},B\in {\mathbb{K}}^{n\times k}$A∈Kn×n,B∈Kn×k, which is defined as

$$AX=B$$AX=B

where$A$A is given factorized as returned bylu_factor().

Ifleft= False, this function returns the matrix$X\in {\mathbb{K}}^{n\times k}$X∈Kn×k that solves the system

$$XA=BA\in {\mathbb{K}}^{k\times k},B\in {\mathbb{K}}^{n\times k}\mathrm{.}$$XA=BA∈Kk×k,B∈Kn×k.

Ifadjoint= True (andleft= True), given an LU factorization of$A$Athis function function returns the$X\in {\mathbb{K}}^{n\times k}$X∈Kn×k that solves the system

$$A^{\text{H}}X=BA\in {\mathbb{K}}^{k\times k},B\in {\mathbb{K}}^{n\times k}\mathrm{.}$$AHX=BA∈Kk×k,B∈Kn×k.

where$A^{\text{H}}$AH is the conjugate transpose when$A$A is complex, and thetranspose when$A$A is real-valued. Theleft= False case is analogous.

Supports inputs of float, double, cfloat and cdouble dtypes.Also supports batches of matrices, and if the inputs are batches of matrices thenthe output has the same batch dimensions.

Parameters

• LU (Tensor) – tensor of shape(*, n, n) (or(*, k, k) ifleft= True)where* is zero or more batch dimensions as returned bylu_factor().

• pivots (Tensor) – tensor of shape(*, n) (or(*, k) ifleft= True)where* is zero or more batch dimensions as returned bylu_factor().

• B (Tensor) – right-hand side tensor of shape(*, n, k).

Keyword Arguments

• left (bool,optional) – whether to solve the system$AX=B$AX=B or$XA=B$XA=B. Default:True.

• adjoint (bool,optional) – whether to solve the system$AX=B$AX=B or$A^{\text{H}}X=B$AHX=B. Default:False.

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

Examples:

>>>A=torch.randn(3,3)>>>LU,pivots=torch.linalg.lu_factor(A)>>>B=torch.randn(3,2)>>>X=torch.linalg.lu_solve(LU,pivots,B)>>>torch.allclose(A@X,B)True>>>B=torch.randn(3,3,2)# Broadcasting rules apply: A is broadcasted>>>X=torch.linalg.lu_solve(LU,pivots,B)>>>torch.allclose(A@X,B)True>>>B=torch.randn(3,5,3)>>>X=torch.linalg.lu_solve(LU,pivots,B,left=False)>>>torch.allclose(X@A,B)True>>>B=torch.randn(3,3,4)# Now solve for A^T>>>X=torch.linalg.lu_solve(LU,pivots,B,adjoint=True)>>>torch.allclose(A.mT@X,B)True