lucid.sqrt¶

lucid.sqrt(a: Tensor, /) → Tensor¶

The sqrt function computes the element-wise square root of each element in the input tensor.

Function Signature¶

def sqrt(a: Tensor) -> Tensor

Parameters¶

a (Tensor): The input tensor for which the square root is computed.

Returns¶

Tensor:
A new tensor containing the element-wise square root of the input tensor. If a requires gradients, the resulting tensor will also require gradients.

Forward Calculation¶

The forward calculation for sqrt is:

\[\mathbf{out}_i = \sqrt{\mathbf{a}_i}\]

where \(\mathbf{a}_i\) is the element of the input tensor a, and \(\mathbf{out}_i\) is the corresponding element of the output tensor.

Backward Gradient Calculation¶

For a tensor a involved in the sqrt operation, the gradient with respect to the output (out) is computed as:

\[\frac{\partial \mathbf{out}_i}{\partial \mathbf{a}_i} = \frac{1}{2 \sqrt{\mathbf{a}_i}}\]

This means that for each element in the input tensor, the gradient is half the reciprocal of the square root of the corresponding value in the tensor.

Example¶

>>> import lucid
>>> a = Tensor([1, 4, 9], requires_grad=True)
>>> out = lucid.sqrt(a)
>>> print(out)
Tensor([1.         2.         3.        ], grad=None)

The sqrt function supports tensors of arbitrary shape:

>>> import lucid
>>> a = Tensor([[1, 4], [9, 16]], requires_grad=True)
>>> out = lucid.sqrt(a)
>>> print(out)
Tensor([[1. 2.] [3. 4.]], grad=None)