Computing hypergeometric functions rigorously (1606.06977v2)

Abstract: We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions ${}_0F_1$, ${}_1F_1$, ${}_2F_1$ and ${}_2F_0$ (or the Kummer $U$-function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software, and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function ${}_pF_q$ and computation of high-order parameter derivatives.

• The paper presents a rigorous framework for evaluating hypergeometric functions using arbitrary-precision interval arithmetic.
• It employs series acceleration techniques like binary splitting to manage computational complexity and improve numerical stability.
• The study offers significant performance gains and reliable error bounds, benefiting applications in physics and computational mathematics.

Evaluation and Implementation of Hypergeometric Functions

The paper under review, authored by Fredrik Johansson, provides a comprehensive exploration and implementation of hypergeometric functions using arbitrary-precision interval arithmetic, specifically within the Arb library framework. The paper presents both performance enhancements and methodological rigor for the computation of these functions, anchoring its contributions significantly within mathematical and computational research domains.

Overview

Hypergeometric functions, denoted as ${}_pF_q$, are fundamental in various fields of science and engineering due to their wide-ranging applicability in expressing solutions of linear differential equations. Johansson addresses the evaluation of these functions, notably ${}_0F_1$, ${}_1F_1$, ${}_2F_1$, and ${}_2F_0$ (Kummer $U$-function), alongside support for general ${}_pF_q$ functions for complex parameters and arguments. This work extends to cover error functions, Bessel functions, Legendre functions, and more. The implementation utilizes Arb, an open-source C library, which integrates efficient methods for accurate hypergeometric function computation, emphasizing competitive performance on par with or exceeding existing arbitrary-precision libraries.

Technical Contribution

The intricacies of evaluating hypergeometric functions often involve handling convergence and growth challenges, especially within complex domains. The paper elaborates on a numerically stable framework to achieve accurate evaluations using direct series expansions and asymptotic series, complemented by rigorous error bounds. Critical to this implementation is the use of interval arithmetic for error management, particularly beneficial in preserving precision and correctness across computations.

Implementation methodologies include various series acceleration techniques like binary splitting and rectangular splitting, which are pivotal for managing computational complexity, particularly at high precision. Additionally, Johansson introduces innovative methods for evaluating parameter derivatives using truncated power series arithmetic, facilitating the computation of complex-parameter limits with efficiency and precision.

Implications and Future Directions

Johansson's work has immediate implications for experts requiring high-precision computations of special functions across simulations and analytical computations in physics, number theory, and computational mathematics. The implementation's performance gains offer practical benefits, engaging directly with applications demanding precision, such as electromagnetic simulations and integer relation searches.

The paper also speculates on future trajectories needing attention, such as extending rigorous bounds for asymptotic expansions of generalized hypergeometric functions beyond the presently covered cases, enhancing handling for large parameter values, and improving machine-level arithmetic efficiency for lower precision use cases.

Conclusion

The exploration of hypergeometric functions in this paper stands as a significant contribution to mathematical software, delivering robust methods for exact arithmetic operations. Johansson sets a foundation for ongoing enhancements in numerical software tools, addressing theoretical gaps and practical needs of researchers who engage with complex mathematical functions routinely. This work not only provides immediate benefits through improved computational methods but also establishes a base for future explorations aimed at broadening the scope of special function evaluations. As computational demands grow, such rigorous frameworks will become indispensable, coloring the landscape of both mathematical computation and its applications in scientific research.