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# torch.cholesky_inverse

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

  Computes the inverse of a symmetric positive-definite matrix A A A using its Cholesky factor u u u: returns matrix `inv`. The inverse is computed using LAPACK routines `dpotri` and `spotri` (and the corresponding MAGMA routines).

  If `upper` is `False`, u u u is lower triangular such that the returned tensor is

  i n v=(u u T)−1 inv = (uu^{{T}})^{{-1}} in v=(u u T)−1

  If `upper` is `True` or not provided, u u u is upper triangular such that the returned tensor is

  i n v=(u T u)−1 inv = (u^T u)^{{-1}} in v=(u T u)−1

  Supports input of float, double, cfloat and cdouble dtypes. Also supports batches of matrices, and if A A A is a batch of matrices then the output has the same batch dimensions.

  Parameters:

    - input (Tensor) – the input tensor A A A of size (∗,n,n)(*, n, n)(∗,n,n), consisting of symmetric positive-definite matrices where ∗*∗ is zero or more batch dimensions.

    - upper (bool, optional) – flag that indicates whether to return a upper or lower triangular matrix. Default: False

  Keyword Arguments:

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

  Example:

      >>> a = torch.randn(3, 3)
      >>> a = torch.mm(a, a.t()) + 1e-05 * torch.eye(3) # make symmetric positive definite
      >>> u = torch.linalg.cholesky(a)
      >>> a
      tensor([[  0.9935,  -0.6353,   1.5806],
              [ -0.6353,   0.8769,  -1.7183],
              [  1.5806,  -1.7183,  10.6618]])
      >>> torch.cholesky_inverse(u)
      tensor([[ 1.9314,  1.2251, -0.0889],
              [ 1.2251,  2.4439,  0.2122],
              [-0.0889,  0.2122,  0.1412]])
      >>> a.inverse()
      tensor([[ 1.9314,  1.2251, -0.0889],
              [ 1.2251,  2.4439,  0.2122],
              [-0.0889,  0.2122,  0.1412]])
      >>> a = torch.randn(3, 2, 2) # Example for batched input
      >>> a = a @ a.mT + 1e-03 # make symmetric positive-definite
      >>> l = torch.linalg.cholesky(a)
      >>> z = l @ l.mT
      >>> torch.dist(z, a)
      tensor(3.5894e-07)

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