torch.linalg.solve#

torch.linalg.solve(A,B,*,left=True,out=None)→Tensor#

Computes the solution of a square system of linear equations with a unique solution.

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

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.

This system of linear equations has one solution if and only if$A$A isinvertible.This function assumes that$A$A is invertible.

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.

Letting* be zero or more batch dimensions,

• IfA has shape(*, n, n) andB has shape(*, n) (a batch of vectors) or shape(*, n, k) (a batch of matrices or “multiple right-hand sides”), this function returnsX of shape(*, n) or(*, n, k) respectively.

• Otherwise, ifA has shape(*, n, n) andB has shape(n,) or(n, k),Bis broadcasted to have shape(*, n) or(*, n, k) respectively.This function then returns the solution of the resulting batch of systems of linear equations.

Note

This function computesX =A.inverse() @B in a faster andmore numerically stable way than performing the computations separately.

Note

It is possible to compute the solution of the system$XA=B$XA=B by passing the inputsA andB transposed and transposing the output returned by this function.

Note

A is allowed to be a non-batchedtorch.sparse_csr_tensor, but only withleft=True.

Note

When inputs are on a CUDA device, this function synchronizes that device with the CPU. For a version of this function that does not synchronize, seetorch.linalg.solve_ex().

See also

torch.linalg.solve_triangular() computes the solution of a triangular system of linearequations with a unique solution.

Parameters

• A (Tensor) – tensor of shape(*, n, n) where* is zero or more batch dimensions.

• B (Tensor) – right-hand side tensor of shape(*, n) or(*, n, k) or(n,) or(n, k)according to the rules described above

Keyword Arguments

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

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

Raises

RuntimeError – if theA matrix is not invertible or any matrix in a batchedA is not invertible.

Examples:

>>>A=torch.randn(3,3)>>>b=torch.randn(3)>>>x=torch.linalg.solve(A,b)>>>torch.allclose(A@x,b)True>>>A=torch.randn(2,3,3)>>>B=torch.randn(2,3,4)>>>X=torch.linalg.solve(A,B)>>>X.shapetorch.Size([2, 3, 4])>>>torch.allclose(A@X,B)True>>>A=torch.randn(2,3,3)>>>b=torch.randn(3,1)>>>x=torch.linalg.solve(A,b)# b is broadcasted to size (2, 3, 1)>>>x.shapetorch.Size([2, 3, 1])>>>torch.allclose(A@x,b)True>>>b=torch.randn(3)>>>x=torch.linalg.solve(A,b)# b is broadcasted to size (2, 3)>>>x.shapetorch.Size([2, 3])>>>Ax=A@x.unsqueeze(-1)>>>torch.allclose(Ax,b.unsqueeze(-1).expand_as(Ax))True