[Term Entry] PyTorch Tensor Operations: .rsqrt()

content/pytorch/concepts/tensor-operations/terms/rsqrt/rsqrt.md

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+---
+Title: '.rsqrt()'
+Description: 'Computes the reciprocal of the square root of each element in a tensor.'
+Subjects:
+  - 'Computer Science'
+  - 'Data Science'
+Tags:
+  - 'Deep Learning'
+  - 'Math'
+  - 'Methods'
+  - 'Tensor'
+  - 'PyTorch'
+CatalogContent:
+  - 'intro-to-py-torch-and-neural-networks'
+  - 'paths/data-science'
+---
+
+The **`.rsqrt()`** method in PyTorch computes the reciprocal of the square root of each element in a [tensor](https://www.codecademy.com/resources/docs/pytorch/tensors). It is applied element-wise and is commonly used in numerical operations and deep learning workflows where normalization is required.
+
+Mathematically, this operation computes:
+
+$\mathrm{rsqrt}(x) = \frac{1}{\sqrt{x}}$
+
+## Syntax
+
+```pseudo
+torch.rsqrt(input)
+```
+
+Or, alternatively:
+
+```pseudo
+tensor.rsqrt()
+```
+
+**Parameters:**
+
+- `input` (Tensor): The input tensor containing values for which the reciprocal square root is computed. All values in the input tensor must be non-negative.
+
+**Return value:**
+
+Returns a new tensor with the same shape as the input, where each element is replaced by its reciprocal square root.
+
+## Example 1: Applying `.rsqrt()` to a tensor
+
+This example demonstrates how `.rsqrt()` computes the reciprocal square root of each element in a tensor:
+
+```py
+import torch
+
+x = torch.tensor([1.0, 4.0, 9.0, 16.0])
+y = torch.rsqrt(x)
+
+print(y)
+```
+
+The output of this code is:
+
+```shell
+tensor([1.0000, 0.5000, 0.3333, 0.2500])
+```
+
+Each value in the output is calculated as `1 / sqrt(x)`.
+
+## Example 2: Using `.rsqrt()` in a normalization step
+
+This example shows how `.rsqrt()` can be used to normalize values by scaling them using the reciprocal square root:
+
+```py
+import torch
+
+values = torch.tensor([2.0, 8.0, 32.0])
+scaled = values * torch.rsqrt(values)
+
+print(scaled)
+```
+
+The output of the above code is:
+
+```shell
+tensor([1.4142, 2.8284, 5.6569])
+```
+
+Here, each value is multiplied by its reciprocal square root, resulting in scaled outputs often used in optimization and normalization operations.