YouTip

## PyTorch torch.igamma The `torch.igamma` function in PyTorch computes the element-wise regularized lower incomplete gamma function. It is widely used in statistics, probability theory, and various scientific computing applications. --- ## Introduction The regularized lower incomplete gamma function is mathematically defined as: $$P(a, x) = \frac{1}{\Gamma(a)} \int_{0}^{x} t^{a-1} e^{-t} dt$$ Where: * $a$ is the shape parameter (must be positive). * $x$ is the upper limit of integration (must be non-negative). * $\Gamma(a)$ is the complete gamma function. In PyTorch, `torch.igamma(input, other)` computes this function element-wise, where `input` corresponds to the parameter $a$, and `other` corresponds to the parameter $x$. --- ## Syntax and Parameters ### Syntax ```python torch.igamma(input, other, *, out=None) -> Tensor ``` ### Parameters | Parameter | Type | Description | | :--- | :--- | :--- | | **`input`** | Tensor | The first input tensor ($a$), representing the shape parameter. Must contain positive real numbers. | | **`other`** | Tensor | The second input tensor ($x$), representing the upper integration limit. Must contain non-negative real numbers. | | **`out`** | Tensor (optional) | The output tensor where the result will be written. | ### Return Value * Returns a new tensor containing the element-wise evaluation of the regularized lower incomplete gamma function. * The shape of the returned tensor is determined by broadcasting `input` and `other`. --- ## Code Examples ### Example 1: Basic Usage with 1D Tensors This example demonstrates how to compute the regularized lower incomplete gamma function using two 1D tensors of the same shape. ```python import torch # Define the shape parameter 'a' (input) and evaluation point 'x' (other) a = torch.tensor([1.0, 1.0, 1.0]) x = torch.tensor([1.0, 2.0, 3.0]) # Compute the regularized lower incomplete gamma function result = torch.igamma(a, x) print("Input (a): ", a) print("Other (x): ", x) print("Result: ", result) ``` **Output:** ```text Input (a): tensor([1., 1., 1.]) Other (x): tensor([1., 2., 3.]) Result: tensor([0.6321, 0.8647, 0.9502]) ``` *(Note: When $a = 1$, the function simplifies to $1 - e^{-x}$. For $x=1$, $1 - e^{-1} \approx 0.6321$.)* ### Example 2: Broadcasting Support PyTorch supports broadcasting for `torch.igamma`. You can pass tensors of different shapes as long as they are broadcastable. ```python import torch # 'a' is a 1D tensor of shape (3,) a = torch.tensor([1.0, 2.0, 3.0]) # 'x' is a 2D tensor of shape (2, 1) x = torch.tensor([[1.0], [2.0]]) # The resulting tensor will have a broadcasted shape of (2, 3) result = torch.igamma(a, x) print("Resulting Tensor:\n", result) ``` **Output:** ```text Resulting Tensor: tensor([[0.6321, 0.2642, 0.0803], [0.8647, 0.5940, 0.3233]]) ``` --- ## Considerations and Rules 1. **Domain Restrictions**: * The parameter $a$ (`input`) must be strictly positive ($a > 0$). * The parameter $x$ (`other`) must be non-negative ($x \ge 0$). * If these conditions are violated, the behavior is undefined or may return `NaN`. 2. **Data Types**: Both input tensors must have floating-point data types (e.g., `torch.float32` or `torch.float64`). 3. **Complementary Function**: If you need to compute the regularized *upper* incomplete gamma function ($Q(a, x) = 1 - P(a, x)$), use [`torch.igammac`](https://pytorch.org/docs/stable/generated/torch.igammac.html).