Fast Hankel Transform Algorithm: Efficient Computation and Implementation

The Fast Hankel Transform Algorithm is a numerical computation method designed for efficient calculation of Hankel transforms, which are specialized integral transformations commonly used in physics, engineering, and signal processing. This algorithm is particularly valuable for solving problems with radial symmetry, such as electromagnetic field analysis, optical imaging, and seismic wave data interpretation. The primary computational challenge in Hankel transforms lies in efficiently evaluating the integral form, as direct numerical integration typically requires substantial computational resources. The algorithm overcomes this by employing mathematical approximations and discretization techniques. Common implementation approaches include utilizing Fast Fourier Transform (FFT) for indirect computation or employing discrete sampling points of special basis functions (like Bessel functions) to enhance both accuracy and speed. In code implementations, this often involves precomputing transformation kernels and optimizing memory access patterns for large datasets. Compared to traditional point-by-point integration methods, the Fast Hankel Transform Algorithm significantly reduces computational complexity from O(N²) to O(N log N), making it suitable for large-scale data processing applications such as high-resolution spectral analysis and image reconstruction. Advanced implementations incorporate adaptive sampling techniques and error control mechanisms, where programmers typically implement threshold-based sampling density adjustment and iterative refinement loops to maintain numerical stability while preserving computational efficiency. Key functions in practical implementations often include kernel generation routines, symmetry optimization handlers, and precision control modules that dynamically adjust calculation parameters based on error tolerance settings.