An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks

We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of points. First, the amount of purely symbolic operations is significantly reduced, that is, only resultant computation and square-free factorization is still needed. Second, our algorithm neither assumes generic position of the input system nor demands for any change of the coordinate system. The latter is due to a novel inclusion predicate to certify that a certain region is isolating for a solution. Our implementation exploits graphics hardware to expedite the resultant computation. Furthermore, we integrate a number of filtering techniques to improve the overall performance. Efficiency of the proposed method is proven by a comparison of our implementation with two state-of-the-art implementations, that is, LPG and Maple's isolate. For a series of challenging benchmark instances, experiments show that our implementation outperforms both contestants.Comment: 16 pages with appendix, 1 figure, submitted to ALENEX 201

Two parallel implementations of Ehrlich-Aberth algorithm for root-finding of polynomials on multiple GPUs with OpenMP and MPI

International audienceFinding the roots of polynomials is a very important part of solving real-life problems but the higher the degree of the polynomials is, the less easy it becomes. In this paper, we present two different parallel algorithms of the Ehrlich-Aberth method to find roots of sparse and fully defined polynomials of high degrees. Both algorithms are based on CUDA technology to be implemented on multi-GPU computing platforms but each use different parallel paradigms: OpenMP or MPI. The experiments show a quasi-linear speedup by using up-to 4 GPU devices compared to 1 GPU to find the roots of polynomials of degree up-to 1.4 million. Moreover, other experiments show it is possible to find the roots of polynomials of degree up-to 5 million

A parallel implementation of the Durand-Kerner algorithm for polynomial root-finding on GPU

International audienceIn this article we present a parallel implementation of the Durand-Kerner algorithm to find roots of polynomials of high degree on a GPU architecture (Graphics Processing Unit). We have implemented both a CPU version in and a GPU compatible version with CUDA. The main result of our work is a parallel implementation that is 10 times as fast as its sequential counterpart on a single CPU for high degree polynomials that is greater than about 48,000

Interactive Isocontouring of High-Order Surfaces

Scientists and engineers are making increasingly use of hp-adaptive discretization methods to compute simulations. While techniques for isocontouring the high-order data generated by these methods have started to appear, they typically do not facilitate interactive data exploration. This work presents a novel interactive approach for approximate isocontouring of high-order data. The method is based on a two-phase hybrid rendering algorithm. In the first phase, coarsely seeded particles are guided by the gradient of the field for obtaining an initial sampling of the isosurface in object space. The second phase performs ray casting in the image space neighborhood of the initial samples. Since the neighborhood is small, the initial guesses tend to be close to the isosurface, leading to accelerated root finding and thus efficient rendering. The object space phase affects the density of the coarse samples on the isosurface, which can lead to holes in the final rendering and overdraw. Thus, we also propose a heuristic, based on dynamical systems theory, that adapts the neighborhood of the seeds in order to obtain a better coverage of the surface. Results for datasets from computational fluid dynamics are shown and performance measurements for our GPU implementation are given

Interactive ray shading of FRep objects

In this paper we present a method for interactive rendering general procedurally defined functionally represented (FRep) objects using the acceleration with graphics hardware, namely Graphics Processing Units (GPU). We obtain interactive rates by using GPU acceleration for all computations in rendering algorithm, such as ray-surface intersection, function evaluation and normal computations. We compute primary rays as well as secondary rays for shadows, reflection and refraction for obtaining high quality of the output visualization and further extension to ray-tracing of FRep objects. The algorithm is well-suited for modern GPUs and provides acceptable interactive rates with good quality of the results. A wide range of objects can be rendered including traditional skeletal implicit surfaces, constructive solids, and purely procedural objects such as 3D fractals