From: asider Date: Tue, 20 Oct 2015 15:03:08 +0000 (+0100) Subject: section 1 2, et 3 X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/commitdiff_plain/e8348fd796778f1fa87ca325cdb696aabc999afb?hp=--cc section 1 2, et 3 --- e8348fd796778f1fa87ca325cdb696aabc999afb diff --git a/paper.tex b/paper.tex index c213794..a6e1cf8 100644 --- a/paper.tex +++ b/paper.tex @@ -161,7 +161,7 @@ drastically increases like the degrees of high polynomials. It is expected that parallelization of these algorithms will improve the convergence time. -Many authors have dealt with parallelisation of +Many authors have dealt with the parallelisation of simultaneous methods, i.e. that find all the zeros simultaneously. Freeman~\cite{Freeman89} implemeted and compared DK, EA and another method of the fourth order proposed by Farmer and Loizou~\cite{Loizon83}, on a 8- processor linear @@ -341,48 +341,39 @@ Winogard~\cite{Winogard72}. The second approach aims at reducing the computation time per iteration, as reported in~\cite{Benall68,Jana06,Janall99,Riceall06}. -There are many -schemes for simultaneous approximations of all roots of a given +There are many schemes for the simultaneous approximation of all roots of a given polynomial. Several works on different methods and issues of root -finding have been reported in~\cite{Azad07, Gemignani07, Kalantari08, Skachek08, Zhancall08, Zhuall08}. However, Durand-Kerner and Ehrlich methods are the most practical choices among +finding have been reported in~\cite{Azad07, Gemignani07, Kalantari08, Skachek08, Zhancall08, Zhuall08}. However, Durand-Kerner and Ehrlisch-Aberth methods are the most practical choices among them~\cite{Bini04}. These two methods have been extensively -studied for parallelization due to their following advantages. The -computation involved in these methods has some inherent +studied for parallelization due to their intrinsics, i.e. the +computations involved in both methods has some inherent parallelism that can be suitably exploited by SIMD machines. Moreover, they have fast rate of convergence (quadratic for the -Durand-Kerner method and cubic for the Ehrlich). Various parallel +Durand-Kerner and cubic for the Ehrlisch-Aberth). Various parallel algorithms reported for these methods can be found in~\cite{Cosnard90, Freeman89,Freemanall90,,Jana99,Janall99}. Freeman and Bane~\cite{Freemanall90} presented two parallel algorithms on a local memory MIMD computer with the compute-to communication time ratio O(n). However, their algorithms require each processor to communicate its current approximation to all -other processors at the end of each iteration. Therefore they +other processors at the end of each iteration (synchronous). Therefore they cause a high degree of memory conflict. Recently the author in~\cite{Mirankar71} proposed two versions of parallel algorithm -for the Durand-Kerner method, and Aberth method on model of +for the Durand-Kerner method, and Ehrlisch-Aberth method on a model of Optoelectronic Transpose Interconnection System (OTIS).The algorithms are mapped on an OTIS-2D torus using N processors. This -solution need N processors to compute N roots, that it is not -practical (is not suitable to compute large polynomial's degrees). -Until then, the related works are not able to compute the root of -the large polynomial's degrees (higher then 1000) and with small -time. - - Finding polynomial roots rapidly and accurately it is our -objective, with the apparition of the CUDA(Compute Unified Device -Architecture), finding the roots of polynomials becomes rewarding -and very interesting, CUDA adopts a totally new computing -architecture to use the hardware resources provided by GPU in -order to offer a stronger computing ability to the massive data -computing. In~\cite{Kahinall14} we proposed the first implantation -of the root finding polynomials method on GPU (Graphics Processing -Unit),which is the Durand-Kerner method. The main result prove -that a parallel implementation is 10 times as fast as the +solution needs N processors to compute N roots, which is not +practical for solving polynomials with large degrees. +Until very recently, the literature doen not mention implementations able to compute the roots of +large degree polynomials (higher then 1000) and within small or at least tractable times. Finding polynomial roots rapidly and accurately is the main objective of our work. +With the advent of CUDA (Compute Unified Device +Architecture), finding the roots of polynomials receives a new attention because of the new possibilities to solve higher degree polynomials in less time. +In~\cite{Kahinall14} we already proposed the first implementation +of a root finding method on GPUs, that of the Durand-Kerner method. The main result showed +that a parallel CUDA implementation is 10 times as fast as the sequential implementation on a single CPU for high degree -polynomials that is greater than about 48000. Indeed, in this -paper we present a parallel implementation of Aberth method on -GPU, more details are discussed in the following of this paper. +polynomials of 48000. In this paper we present a parallel implementation of Ehlisch-Aberth method on +GPUs, which details are discussed in the sequel. \section {A parallel implementation of Aberth method}