torch.cholesky_solve

torch.cholesky_solve(input, input2, upper=False, *, out=None) → Tensor

Solves a linear system of equations with a positive semidefinite matrix to be inverted given its Cholesky factor matrix $u$.

If upper is False, $u$ is and lower triangular and c is returned such that:

$$c = (u u^T)^{{-1}} b$$

If upper is True or not provided, $u$ is upper triangular and c is returned such that:

$$c = (u^T u)^{{-1}} b$$

torch.cholesky_solve(b, u) can take in 2D inputs b, u or inputs that are batches of 2D matrices. If the inputs are batches, then returns batched outputs c

Supports real-valued and complex-valued inputs. For the complex-valued inputs the transpose operator above is the conjugate transpose.

Parameters

• input (Tensor) – input matrix $b$ of size $(*, m, k)$, where $*$ is zero or more batch dimensions

• input2 (Tensor) – input matrix $u$ of size $(*, m, m)$, where $*$ is zero of more batch dimensions composed of upper or lower triangular Cholesky factor

• upper (bool, optional) – whether to consider the Cholesky factor as a lower or upper triangular matrix. Default: False.

Keyword Arguments

out (Tensor, optional) – the output tensor for c

Example:

>>> a = torch.randn(3, 3)
>>> a = torch.mm(a, a.t()) # make symmetric positive definite
>>> u = torch.linalg.cholesky(a)
>>> a
tensor([[ 0.7747, -1.9549,  1.3086],
        [-1.9549,  6.7546, -5.4114],
        [ 1.3086, -5.4114,  4.8733]])
>>> b = torch.randn(3, 2)
>>> b
tensor([[-0.6355,  0.9891],
        [ 0.1974,  1.4706],
        [-0.4115, -0.6225]])
>>> torch.cholesky_solve(b, u)
tensor([[ -8.1625,  19.6097],
        [ -5.8398,  14.2387],
        [ -4.3771,  10.4173]])
>>> torch.mm(a.inverse(), b)
tensor([[ -8.1626,  19.6097],
        [ -5.8398,  14.2387],
        [ -4.3771,  10.4173]])