# tensorplay.functional.cholesky_solve

Source: https://www.tensorplay.cn/docs/generated/tensorplay.functional.cholesky_solve.html

```python
tensorplay.functional.cholesky_solve(B, L, upper=False, *, out=None) → Tensor
```

Computes the solution of a system of linear equations with complex Hermitian
or real symmetric positive-definite lhs given its Cholesky decomposition.

Let $A$ be a complex Hermitian or real symmetric positive-definite matrix,
and $L$ its Cholesky decomposition such that:

$$A = LL^{\text{H}}$$

where $L^{\text{H}}$ is the conjugate transpose when $L$ is complex,
and the transpose when $L$ is real-valued.

Returns the solution $X$ of the following linear system:

$$AX = B$$

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

Parameters:

- B ([Tensor](/docs/generated/tensorplay.Tensor.html#tensorplay.Tensor)) – right-hand side tensor of shape (*, n, k) where $*$ is zero or more batch dimensions

- L ([Tensor](/docs/generated/tensorplay.Tensor.html#tensorplay.Tensor)) – tensor of shape (*, n, n) where * is zero or more batch dimensions consisting of lower or upper triangular Cholesky decompositions of symmetric or Hermitian positive-definite matrices.

- upper ([bool](https://docs.python.org/3/builtins/functions.html#bool), optional) – flag that indicates whether $L$ is lower triangular or upper triangular. Default: False.

Keyword Arguments:

out ([Tensor](/docs/generated/tensorplay.Tensor.html#tensorplay.Tensor), optional) – output tensor. Ignored if None. Default: None.

Example:

```
>>> A = tensorplay.randn(3, 3)
>>> A = A @ A.T + tensorplay.eye(3) * 1e-3 # Creates a symmetric positive-definite matrix
>>> L = tensorplay.linalg.cholesky(A) # Extract Cholesky decomposition
>>> B = tensorplay.randn(3, 2)
>>> tensorplay.cholesky_solve(B, L)
tensor([[ -8.1625,  19.6097],
        [ -5.8398,  14.2387],
        [ -4.3771,  10.4173]])
>>> A.inverse() @  B
tensor([[ -8.1626,  19.6097],
        [ -5.8398,  14.2387],
        [ -4.3771,  10.4173]])

>>> A = tensorplay.randn(3, 2, 2, dtype=tensorplay.complex64)
>>> A = A @ A.mH + tensorplay.eye(2) * 1e-3 # Batch of Hermitian positive-definite matrices
>>> L = tensorplay.linalg.cholesky(A)
>>> B = tensorplay.randn(2, 1, dtype=tensorplay.complex64)
>>> X = tensorplay.cholesky_solve(B, L)
>>> tensorplay.dist(X, A.inverse() @ B)
tensor(1.6881e-5)
```