Added a Multivariate Log Gamma Function File

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

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+---
+Title: '.mvlgamma()'
+Description: 'Computes the multivariate log-gamma function for tensor inputs.'
+Subjects:
+  - 'Computer Science'
+  - 'Data Science'
+  - 'Machine Learning'
+Tags:
+  - 'PyTorch'
+  - 'Tensor Operations'
+  - 'Math Functions'
+  - 'Probability'
+  - 'Statistics'
+CatalogContent:
+  - 'intro-to-py-torch-and-neural-networks'
+  - 'paths/data-science'
+---
+
+The **`torch.mvlgamma()`** function in PyTorch computes the **multivariate log-gamma function** for tensor inputs. It is widely used in multivariate statistics, particularly to calculate normalization constants in distributions such as the Wishart or multivariate Gamma distributions. Denoted as ${(\ln(\Gamma_p(a))\)}$, the multivariate log-gamma function generalizes the standard log-gamma function to matrix-valued arguments and is evaluated **element-wise** for each tensor value. It is defined for ${\(a > \frac{p-1}{2}\)}$, where ${\(p\)}$ is the specified dimension.
+
+The function is commonly applied in probability distributions over matrices, with the Wishart distribution being a frequent use case.
+
+The formula for the multivariate log-gamma function is:
+
+$$\ln(\Gamma_p(a)) = \frac{p(p-1)}{4} \ln(\pi) + \sum_{i=1}^{p} \ln\left(\Gamma\left(a - \frac{i-1}{2}\right)\right)$$
+
+where $\Gamma(\cdot)$ is the standard gamma function.
+
+## Syntax
+
+```pseudo
+torch.mvlgamma(input, p, out = None)
+```
+
+**Parameters:**
+
+- `input` (Tensor): Tensor of values for which to compute the multivariate log-gamma function.
+- `p` (int): Number of dimensions in the multivariate gamma function. Must be a positive integer.
+- `out` (Tensor, optional): Tensor to store the output. If provided, must be broadcastable to the shape of the result.
+
+**Return value:**
+
+Returns a tensor containing the multivariate log-gamma values for each element in `input`, evaluated element-wise according to the specified dimension `p`.
+
+## Example 1: Element-wise computation
+
+In this example, `torch.mvlgamma()` computes the multivariate log-gamma function for a 1D tensor:
+
+```py
+import torch
+
+# Input tensor (values must be > (p-1)/2)
+# Here p = 3, so input elements must be > (3-1)/2 = 1
+a = torch.tensor([1.5, 2.0, 3.5])
+p_dimension = 3
+
+# Compute the multivariate log gamma function
+result = torch.mvlgamma(a, p=p_dimension)
+
+print(f"Input tensor: {a}")
+print(f"Dimension p: {p_dimension}")
+print(f"Result (mvlgamma): {result}")
+```
+
+The output of this code is:
+
+```shell
+Input tensor: tensor([1.5000, 2.0000, 3.5000])
+Dimension p: 3
+Result (mvlgamma): tensor([2.1687, 1.5963, 3.8959])
+```
+
+## Example 2: Comparison with standard lgamma
+
+In this example, `torch.mvlgamma()` is computed with `p=1` to show it matches the standard `torch.lgamma()`:
+
+```py
+import torch
+
+a_scalar = torch.tensor([2.5, 4.0])
+p1_result = torch.mvlgamma(a_scalar, p=1)
+lgamma_result = torch.lgamma(a_scalar)
+
+print(f"Input (p=1): {a_scalar}")
+print(f"mvlgamma (p=1): {p1_result}")
+print(f"lgamma: {lgamma_result}")
+```
+
+The output of this code is:
+
+```shell
+Input (p=1): tensor([2.5000, 4.0000])
+mvlgamma (p=1): tensor([0.2847, 1.7918])
+lgamma: tensor([0.2847, 1.7918])
+```