## Section: New Results

### Multiprecision arithmetic

Participant : Fredrik Johansson.

In [17], F. Johansson and I. Blagouchine devise an efficient algorithm to compute the generalized Stieltjes constants ${\gamma}_{n}\left(a\right)$ to arbitrary precision with rigorous error bounds, for the first time achieving this with low complexity with respect to the order $n$. The algorithm consists of locating an approximate steepest descent contour and then evaluating the integral numerically in ball arithmetic using the Petras algorithm with a Taylor expansion for bounds near the saddle point. An implementation is provided in the Arb library.

In [26], F. Johansson describes algorithms to compute elliptic functions and their relatives (Jacobi theta functions, modular forms, elliptic integrals, and the arithmetic-geometric mean) numerically to arbitrary precision with rigorous error bounds for arbitrary complex variables. Implementations in ball arithmetic are available in the Arb library. This overview article discusses the standard algorithms from a concrete implementation point of view, and also presents some improvements.

In [21], Fredrik Johansson develops algorithms for real and complex dot product and matrix multiplication in arbitrary-precision floating-point and ball arithmetic. The new methods are implemented in Arb and significantly speed up polynomial operations and linear algebra in high precision.