Section: New Results

Symbolic computation

The article [27], of which F. Johansson is a coauthor, was published. The article describes SymPy, an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy.

Hypergeometric functions are among the most important mathematical functions, with a wide range of applications in everything from physics to number theory. The practical computation of such functions is a challenging problem. The preprint [26]. presents an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions ${}_0{F}_1$, ${}_1{F}_1$, ${}_2{F}_1$ and ${}_2{F}_0$ (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 ${}_p{F}_q$ and computation of high-order parameter derivatives.

The preprint [25] is the corresponding paper for the software Arb developed by F. Johansson. Arb is a C library for arbitrary-precision interval arithmetic using the midpoint-radius representation, also known as ball arithmetic. It supports real and complex numbers, polynomials, power series, matrices, and evaluation of many special functions. The core number types are designed for versatility and speed in a range of scenarios, allowing performance that is competitive with non-interval arbitrary-precision types such as MPFR and MPC floating-point numbers. This paper discusses the low-level number representation, strategies for precision and error bounds, and the implementation of efficient polynomial arithmetic with interval coefficients.