8,148 research outputs found
Computing the Lambert W function in arbitrary-precision complex interval arithmetic
We describe an algorithm to evaluate all the complex branches of the Lambert
W function with rigorous error bounds in interval arithmetic, which has been
implemented in the Arb library. The classic 1996 paper on the Lambert W
function by Corless et al. provides a thorough but partly heuristic numerical
analysis which needs to be complemented with some explicit inequalities and
practical observations about managing precision and branch cuts.Comment: 16 pages, 4 figure
Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq
International audienceThe verification of floating-point mathematical libraries requires computing numerical bounds on approximation errors. Due to the tightness of these bounds and the peculiar structure of approximation errors, such a verification is out of the reach of generic tools such as computer algebra systems. In fact, the inherent difficulty of computing such bounds often mandates a formal proof of them. In this paper, we present a tactic for the Coq proof assistant that is designed to automatically and formally prove bounds on univariate expressions. It is based on a formalization of floating-point and interval arithmetic, associated with an on-the-fly computation of Taylor expansions. All the computations are performed inside Coq's logic, in a reflexive setting. This paper also compares our tactic with various existing tools on a large set of examples
Secure Numerical and Logical Multi Party Operations
We derive algorithms for efficient secure numerical and logical operations
using a recently introduced scheme for secure multi-party
computation~\cite{sch15} in the semi-honest model ensuring statistical or
perfect security. To derive our algorithms for trigonometric functions, we use
basic mathematical laws in combination with properties of the additive
encryption scheme in a novel way. For division and logarithm we use a new
approach to compute a Taylor series at a fixed point for all numbers. All our
logical operations such as comparisons and large fan-in AND gates are perfectly
secure. Our empirical evaluation yields speed-ups of more than a factor of 100
for the evaluated operations compared to the state-of-the-art
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