integration

Project Title: Adaptive Quadrature Library for Scientific Computing

Project Description:

Develop a comprehensive, high-performance adaptive quadrature library in Modern C++ for numerical integration of complex scientific functions. This library should be capable of handling a wide range of integration challenges, including singularities, oscillatory functions, and multi-dimensional integrals.

Objectives:

1. Implement multiple adaptive quadrature algorithms
2. Create a flexible, extensible architecture for easy addition of new methods
3. Optimize performance for both single-threaded and parallel execution
4. Provide comprehensive error estimation and control
5. Develop a user-friendly API for easy integration into scientific applications

Expected Features:

• Support for various adaptive quadrature methods (e.g., Gauss-Kronrod, Clenshaw-Curtis, Tanh-Sinh)
• Capability to handle 1D, 2D, and 3D integrals
• Automatic algorithm selection based on integrand properties
• Support for vector-valued integrands
• Ability to handle infinite and semi-infinite intervals
• Comprehensive error estimation and convergence criteria
• Parallelization support using modern C++ concurrency features

Suggested Tools/Libraries:

• Eigen for linear algebra operations
• Boost for special functions and utilities
• Google Test for unit testing
• Doxygen for documentation
• CMake for build system

Potential Challenges:

• Efficiently handling highly oscillatory or singular integrands
• Balancing accuracy and performance across diverse problem types
• Implementing robust error estimation for adaptive refinement
• Designing an intuitive yet flexible API for scientific users

Deliverables:

1. Source code repository on GitHub
2. Comprehensive documentation (API reference, usage guide, algorithm descriptions)
3. Unit tests and integration tests
4. Benchmarking suite comparing performance against existing libraries
5. Sample applications demonstrating library usage in scientific contexts
6. Technical report detailing design decisions, performance analysis, and future work

Additional Considerations:

• Ensure thread-safety for parallel scientific computing environments
• Implement efficient memory management for large-scale problems
• Consider support for GPU acceleration of suitable algorithms
• Explore potential for automatic differentiation integration
• Investigate adaptability for use in solving differential equations

This project challenges students to delve deep into numerical analysis, algorithm design, and high-performance computing while creating a practical tool for scientific computing. It encourages exploration of modern C++ features, software engineering best practices, and the intricacies of numerical integration methods.