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 # HyperGraph
 
-Reversed mode second order automatic differentiation for python
+**Reverse-mode second-order automatic differentiation for Python.**
 
-[![Build Status](https://github.com/oberbichler/HyperGraph/workflows/Python%20package/badge.svg?branch=master)](https://github.com/oberbichler/HyperGraph/actions) [![PyPI](https://img.shields.io/pypi/v/hypergraph)](https://pypi.org/project/hypergraph)
+HyperGraph computes exact **gradients** and **Hessians** of scalar expressions using a computational-graph approach based on reverse-mode automatic differentiation. The core is implemented in C++ with [Eigen](https://eigen.tuxfamily.org/) and exposed to Python via [pybind11](https://github.com/pybind/pybind11), making it both fast and easy to use.
+
+[![CI](https://github.com/oberbichler/HyperGraph/actions/workflows/ci.yml/badge.svg)](https://github.com/oberbichler/HyperGraph/actions/workflows/ci.yml)
+[![PyPI](https://img.shields.io/pypi/v/hypergraph)](https://pypi.org/project/hypergraph)
+[![Python](https://img.shields.io/pypi/pyversions/hypergraph)](https://pypi.org/project/hypergraph)
+[![License: MIT](https://img.shields.io/badge/License-MIT-yellow.svg)](LICENSE)
+
+> [!NOTE]
+> This is an experimental project. It implements the graph-based edge-pushing algorithm from Gower & Mello (2010) as a standalone library. The approach works well but is not as widely adopted or battle-tested as other techniques.
+
+---
+
+## Features
+
+- **Second-order derivatives** — computes both the gradient vector and the full Hessian matrix in a single pass
+- **High performance** — C++17 core with Eigen, optional BLAS acceleration
+- **NumPy integration** — HyperGraph variables work seamlessly with `np.sin`, `np.cos`, `np.sqrt`, `np.dot`, `np.cross`, `np.linalg.norm`, and more
+- **Comprehensive math library** — arithmetic, trigonometric, hyperbolic, inverse-hyperbolic, exponential, logarithmic, and power functions
+- **Cross-platform** — tested on Linux, macOS, and Windows with Python 3.12 and 3.13
 
 ## Installation
 
-```
+Install from [PyPI](https://pypi.org/project/hypergraph):
+
+```bash
 pip install hypergraph
 ```
 
+## Quick Start
+
+```python
+import hypergraph as hg
+import numpy as np
+
+# Create a computation graph
+graph = hg.HyperGraph()
+
+# Define variables
+x, y = graph.new_variables([2.0, 3.0])
+
+# Build an expression: f(x, y) = x² · sin(y)
+f = x ** 2 * np.sin(y)
+
+# Compute gradient and Hessian
+graph.compute(f)
+
+print("f =", f.value)           # 4.0 * sin(3.0) ≈ 0.5645
+print("∇f =", graph.g())        # [2x·sin(y), x²·cos(y)]
+print("Hf =\n", graph.h())      # Hessian matrix
+```
+
+## Usage
+
+### Creating Variables
+
+```python
+graph = hg.HyperGraph()
+
+# Single variable
+a = graph.new_variable(5.0)
+
+# Multiple variables at once
+x, y, z = graph.new_variables([1.0, 2.0, 3.0])
+```
+
+### Building Expressions
+
+Variables support all standard arithmetic operators and can be used directly with NumPy functions:
+
+```python
+# Arithmetic
+f = (x + y) * z - 1.0
+
+# NumPy functions
+f = np.sin(x) * np.cos(y) + np.sqrt(z)
+
+# Vector operations
+a = np.array([x, y, z])
+b = np.array([1.0 * x, 2.0 * y, 3.0 * z])
+f = np.linalg.norm(np.cross(a, b))
+```
+
+### Computing Derivatives
+
+```python
+# Evaluate gradient and Hessian for a scalar expression
+graph.compute(f)
+
+# Gradient (as NumPy vector)
+g = graph.g()                      # shape: (n,)
+
+# Hessian (upper-triangular by default)
+h = graph.h()                      # shape: (n, n), upper-triangular
+h_full = graph.h(full=True)        # shape: (n, n), symmetric
+
+# Write into pre-allocated arrays to avoid allocation
+g_out = np.empty(n)
+h_out = np.empty((n, n))
+graph.g(out=g_out)
+graph.h(out=h_out)
+```
+
+## Supported Operations
+
+| Category | Operations |
+|---|---|
+| **Arithmetic** | `+`, `-`, `*`, `/`, `**`, `+=`, `-=`, `*=`, `/=`, unary `-` |
+| **Comparison** | `==`, `!=`, `<`, `>`, `<=`, `>=` |
+| **Trigonometric** | `sin`, `cos`, `tan`, `arcsin`, `arccos`, `arctan`, `atan2` |
+| **Hyperbolic** | `sinh`, `cosh`, `tanh`, `arcsinh`, `arccosh`, `arctanh` |
+| **Exponential** | `exp`, `log` |
+| **Power / Root** | `pow` (`**`), `square`, `sqrt`, `abs` |
+
+Trigonometric and root functions can be called via NumPy (`np.sin(x)`, `np.sqrt(x)`, …) or as methods on a variable (`x.sin()`, `x.sqrt()`, …). The `atan2` function is available as `hg.atan2(y, x)`.
+
+## Building from Source
+
+HyperGraph uses [scikit-build-core](https://github.com/scikit-build/scikit-build-core) and requires a C++17 compiler, CMake ≥ 3.24, and pybind11. [Eigen 3.4](https://eigen.tuxfamily.org/) is fetched automatically during the build.
+
+```bash
+# Clone the repository
+git clone https://github.com/oberbichler/HyperGraph.git
+cd HyperGraph
+
+# Install with uv (recommended)
+uv sync
+
+# Or install with pip
+pip install .
+
+# Run tests
+pytest
+```
+
 ## Credits
+
 - "Hessian Matrices via Automatic Differentiation", Gower and Mello 2010
 - [HAD by Tzu-Mao Li](https://github.com/BachiLi/had)
 
-## Reference
+## Citation
 
-If you use HyperGraph, please refer to the official GitHub repository:
+If you use HyperGraph in your research, please cite:
 
-```
+```bibtex
 @misc{HyperGraph,
-  author = "Thomas Oberbichler",
-  title = "HyperGraph",
-  howpublished = "\url{http://github.com/oberbichler/HyperGraph}",
+  author = {Thomas Oberbichler},
+  title  = {HyperGraph},
+  url    = {https://github.com/oberbichler/HyperGraph},
+  year   = {2019}
 }
 ```
+
+## License
+
+This project is licensed under the [MIT License](LICENSE).