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 $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$, this function computes the solution $X \in \mathbb{K}^{n \times k}$ of the linear system associated to $A \in \mathbb{K}^{n \times n}, B \in \mathbb{K}^{n \times k}$, which is defined as
$AX = B$If
left
= False, this function returns the matrix $X \in \mathbb{K}^{n \times k}$ that solves the system$XA = B\mathrlap{\qquad A \in \mathbb{K}^{k \times k}, B \in \mathbb{K}^{n \times k}.}$This system of linear equations has one solution if and only if $A$ is invertible. This function assumes that $A$ is invertible.
Supports inputs of float, double, cfloat and cdouble dtypes. Also supports batches of matrices, and if the inputs are batches of matrices then the output has the same batch dimensions.
Letting * be zero or more batch dimensions,
If
A
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 returns X of shape (*, n) or (*, n, k) respectively.Otherwise, if
A
has shape (*, n, n) andB
has shape (n,) or (n, k),B
is 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 computes X =
A
.inverse() @B
in a faster and more numerically stable way than performing the computations separately.Note
It is possible to compute the solution of the system $XA = B$ by passing the inputs
A
andB
transposed and transposing the output returned by this function.Note
When inputs are on a CUDA device, this function synchronizes that device with the CPU.
See also
torch.linalg.solve_triangular()
computes the solution of a triangular system of linear equations with a unique solution.- Parameters:
- Keyword Arguments:
- Raises:
RuntimeError – if the
A
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.shape torch.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.shape torch.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.shape torch.Size([2, 3]) >>> Ax = A @ x.unsqueeze(-1) >>> torch.allclose(Ax, b.unsqueeze(-1).expand_as(Ax)) True