From 0b48d3784303d0beafc660614dfc6ec5d1232f01 Mon Sep 17 00:00:00 2001 From: couturie Date: Fri, 6 Nov 2015 11:58:24 +0100 Subject: [PATCH 1/1] correct --- paper.tex | 14 +++++++++----- 1 file changed, 9 insertions(+), 5 deletions(-) diff --git a/paper.tex b/paper.tex index bfbbc31..1a30788 100644 --- a/paper.tex +++ b/paper.tex @@ -195,10 +195,11 @@ iterations before making a new one. Couturier and al.~\cite{Raphaelall01} proposed two methods of parallelization for a shared memory architecture and for distributed memory one. They were able to compute the roots of sparse polynomials of degree 10,000 in 430 seconds with only 8 -personal computers and 2 communications per iteration. Comparing to the sequential implementation -where it takes up to 3,300 seconds to obtain the same results, the authors show an interesting speedup. +personal computers and 2 communications per iteration. Compared to sequential implementations +where it takes up to 3,300 seconds to obtain the same results, the +authors' work experiment show an interesting speedup. -Very few works had been performed since this last work until the appearing of +Few works have been conducted after those works until the appearance of the Compute Unified Device Architecture (CUDA)~\cite{CUDA10}, a parallel computing platform and a programming model invented by NVIDIA. The computing power of GPUs (Graphics Processing Unit) has exceeded that of CPUs. However, CUDA adopts a totally new computing architecture to use the @@ -230,8 +231,11 @@ topic. \section{Ehrlich-Aberth method} \label{sec1} -A cubically convergent iteration method for finding zeros of -polynomials was proposed by O. Aberth~\cite{Aberth73}. The Ehrlich-Aberth method contain 4 main steps, presented in the following. +A cubically convergent iteration method to find zeros of +polynomials was proposed by O. Aberth~\cite{Aberth73}. The +Ehrlich-Aberth method contains 4 main steps, presented in what +follows. + %The Aberth method is a purely algebraic derivation. %To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors -- 2.39.5