Numerical continuation methods apply predictor-corrector algorithms to track a solution path defined by a family of systems, the so-called homotopy. The systems we consider are defined by polynomials in several variables with complex coefficients. For larger dimensions and degrees, the numerical conditioning worsens and hardware double precision becomes often insufficient to reach the end of […]

January 28, 2015 by hgpu

Many numerical problems require higher precision than the conventional floating-point (single, double) formats. One solution is to use multiple precision libraries such as GNU MPFR, which allow the manipulation of very high precision numbers. But their generality (they are able to handle numbers with millions of digits), is a quite heavy alternative when medium precision […]

July 26, 2014 by hgpu

FPGAs and GPUs are increasingly used in a range of high performance computing applications. When implementing numerical algorithms on either platform, we can choose to represent operands with different levels of accuracy. A trade-off exists between the numerical accuracy of arithmetic operators and the resources needed to implement them. Where algorithmic requirements for numerical stability […]

May 25, 2014 by hgpu

Today, GPUs represent an important hardware development platform for many problems in dynamical systems, where massive parallel computations are needed. Beside that, many numerical studies of chaotic dynamical systems require a computing precision higher than common floating point (FP) formats. One such application is locating invariant sets for chaotic dynamical systems. In particular, we focus […]

March 15, 2014 by hgpu

On modern multi-core, many-core, and heterogeneous architectures, floating-point computations, especially reductions, may become non-deterministic and thus non-reproducible mainly due to non-associativity of floating-point operations. We introduce a solution to compute deterministic sums of floating-point numbers efficiently and with the best possible accuracy. Our multi-level algorithm consists of two main stages: a filtering stage that uses […]

February 26, 2014 by hgpu

A general method to produce uniformly distributed pseudorandom numbers with extended precision by combining two pseudorandom numbers with lower precision is proposed. In particular, this method can be used for pseudorandom number generation with extended precision on graphics processing units (GPU), where the performance of single and double precision operations can vary significantly.

February 12, 2014 by hgpu

In order to compensate for the higher cost of double double and quad double arithmetic when solving large polynomial systems, we investigate the application of NVIDIA Tesla C2050, K20C, and K40 general purpose graphics processing units. As the dimension equals several thousands, the cost to compute one QR decomposition is sufficiently large so that the […]

January 26, 2014 by hgpu

Our problem is to accurately solve linear systems on a general purpose graphics processing unit with double double and quad double arithmetic. The linear systems originate from the application of Newton’s method on polynomial systems. Newton’s method is applied as a corrector in a path following method, so the linear systems are solved in sequence […]

January 25, 2013 by hgpu

A high-performance shooting algorithm is developed to compute time-periodic solutions of the free-surface Euler equations with spectral accuracy in double and quadruple precision. The method is used to study resonance and its effect on standing water waves. We identify new nucleation mechanisms in which isolated large-amplitude solutions, and closed loops of such solutions, suddenly exist […]

October 23, 2012 by hgpu

Our problem is to accurately solve linear systems of modest dimensions (typically, the number of variables equals 32) on a general purpose graphics processing unit. The linear systems originate from the application of Newton’s method on polynomial systems of (moderately) large degrees. Newton’s method is applied as a corrector in a path following method, so […]

October 3, 2012 by hgpu

Multiple results in the literature exist that indicate that all computed solutions to chaotic dynamical systems are time-step dependent. That is, solutions with small but different time steps will decouple from each other after a certain (small) finite amount of simulation time. When using double precision floating point arithmetic time step independent solutions have been […]

December 15, 2011 by hgpu

In this paper we evaluate the potential for using an NVIDIA graphics processing unit (GPU) to accelerate high precision integer multiplication. The reported peak vector performance for a typical GPU appears to offer considerable potential for accelerating such a regular computation. Because of limitations in the on-chip memory, the high cost of kernel launches, and […]

July 7, 2011 by hgpu