CAMPARY: Cuda Multiple Precision Arithmetic Library and Applications

Abstract

Abstract. Many scientific computing applications demand massive numerical computations on parallel architectures such as Graphics Processing Units (GPUs). Usually, either floating-point single or double precision arithmetic is used. Higher precision is generally not available in hardware, and software extended precision libraries are much slower and rarely supported on GPUs. We develop CAMPARY: a multipleprecision arithmetic library, using the CUDA programming language for the NVidia GPU platform. In our approach, the precision is extended by representing real numbers as the unevaluated sum of several standard machine precision floating-point numbers. We make use of error-free transforms algorithms, which are based only on native precision operations, but keep track of all rounding errors that occur when performing a sequence of additions and multiplications. This offers the simplicity of using hardware highly optimized floating-point operations, while also allowing for rigorously proven rounding error bounds. This also allows for easy implementation of an interval arithmetic. Currently, all basic multiple-precision arithmetic operations are supported. Our target applications are in chaotic dynamical systems or automatic control.

Is the Hénon attractor chaotic?

“…For the sinks #7 and 8, the results were obtained using multiple precision GPU software. 18 For the other sinks, we were not able to get any significant statistics concerning the convergence times.…”

Tight Interval Inclusions with Compensated Algorithms

“…The QD package [4] provides the double-double and the quad-double data types, that consist of respectively two and four binary64 floating-point numbers. One can also use arbitrary length expansions [5], [6], [7]. If a simple enough computation is performed, its accuracy can be improved thanks to compensated algorithms [8], [9], [10].…”

Algorithms for Triple-Word Arithmetic

“…A generalization of these arithmetics is the notion of floating-point expansion [26], [29], [10], where a highprecision number is represented as the unevaluated sum of floating-point numbers.…”