pyIA

PyIA: Interval Arithmetic Library

PyIA is a Python library for interval arithmetic. Unlike standard floating-point arithmetic, which is prone to rounding errors, PyIA uses Interval Arithmetic with IEEE 754 directed rounding to produce mathematically guaranteed bounds for every computation.

If PyIA returns an interval [a, b], the true mathematical result is guaranteed to be inside it.

🚀 Key Features

• Native Directed Rounding: Custom C-extension (rounding.c) to control the FPU for rigorous lower/upper bound calculation.
• Interval Arithmetic: Full operator overloading (+, -, *, /, **, @) for scalars, vectors, and matrices.
• Sparse Support: Memory-efficient SparseIntervalMatrix class supporting operations on systems with 50,000+ variables.
• Automatic Differentiation (AD): Vectorised forward-mode AD (Gradient class) capable of handling mixed Interval/Float operations.
• Verified Solvers:
  • Linear Systems ($Ax=b$): Uses Rump's algorithm (Hansen-Bliek-Rump) to bound solutions to ill-conditioned systems (e.g., Hilbert matrices).
  • Nonlinear Systems ($F(x)=0$): Implements the Newton-Krawczyk operator to prove the existence and uniqueness of roots.

🛠️ Installation

PyIA requires a C compiler (GCC/Clang) to build the rounding control module.

1. Clone the repository:

   git clone [https://github.com/YOUR_USERNAME/PyIA.git](https://github.com/YOUR_USERNAME/PyIA.git)
   cd PyIA

2. Install Python dependencies:

   pip install numpy scipy pytest

3. Compile the Rounding Kernel: The Python logic relies on this C-extension to switch rounding modes (Up/Down/Nearest).

   python build_rounding.py

   Success Check: You should see a .so file generated in the directory (e.g., rounding.cpython-312-darwin.so).

⚡ Technical Report and Quick Start

Please view the attached PDF 'pyIA Technical Report.pdf' for a technical report, detailing the algorithms used, performance benchmarks, solvers of systems such as Ax=b and a quickstart guide.