correct

[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 2dcee5775eefb549e1dbff4c00c96a5267b763e2..1a307883c303d75f16d1073e57c98899c90aefd6 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -82,7 +82,7 @@
 
 \begin{abstract}
 Polynomials are mathematical algebraic structures that play a great
 
 \begin{abstract}
 Polynomials are mathematical algebraic structures that play a great

-role in science and engineering. Finding roots of high degree
+role in science and engineering. Finding the roots of high degree

 polynomials is computationally demanding. In this paper, we present
 the results of a parallel implementation of the Ehrlich-Aberth
 algorithm for the root finding problem for high degree polynomials on
 polynomials is computationally demanding. In this paper, we present
 the results of a parallel implementation of the Ehrlich-Aberth
 algorithm for the root finding problem for high degree polynomials on


@@ -101,15 +101,15 @@ Polynomial root finding, Iterative methods, Ehrlich-Aberth, Durand-Kerner, GPU
 
 \linenumbers
 
 
 \linenumbers
 

-\section{The problem of finding roots of a polynomial}
-Polynomials are mathematical algebraic structures used in science and engineering to capture physical phenomenons and to express any outcome in the form of a function of some unknown variables. Formally speaking,  a polynomial $p(x)$ of degree \textit{n} having $n$ coefficients in the complex plane \textit{C} is :
+\section{The problem of finding the roots of a polynomial}
+Polynomials are mathematical algebraic structures used in science and engineering to capture physical phenomena and to express any outcome in the form of a function of some unknown variables. Formally speaking,  a polynomial $p(x)$ of degree \textit{n} having $n$ coefficients in the complex plane \textit{C} is :

 %%\begin{center}
 \begin{equation}
      {\Large p(x)=\sum_{i=0}^{n}{a_{i}x^{i}}}.
 \end{equation}
 %%\end{center}
 
 %%\begin{center}
 \begin{equation}
      {\Large p(x)=\sum_{i=0}^{n}{a_{i}x^{i}}}.
 \end{equation}
 %%\end{center}
 

-The root finding problem consists in finding the values of all the $n$ values of the variable $x$ for which \textit{p(x)} is nullified. Such values are called zeroes of $p$. If zeros are $\alpha_{i},\textit{i=1,...,n}$ the $p(x)$ can be written as :
+The root finding problem consists in finding the values of all the $n$ values of the variable $x$ for which \textit{p(x)} is nullified. Such values are called zeros of $p$. If zeros are $\alpha_{i},\textit{i=1,...,n}$ the $p(x)$ can be written as :

 \begin{equation}
      {\Large p(x)=a_{n}\prod_{i=1}^{n}(x-\alpha_{i}), a_{0} a_{n}\neq 0}.
 \end{equation}
 \begin{equation}
      {\Large p(x)=a_{n}\prod_{i=1}^{n}(x-\alpha_{i}), a_{0} a_{n}\neq 0}.
 \end{equation}


@@ -127,22 +127,22 @@ root-finding problem into a fixed-point problem by setting :
 $g(x)= f(x)-x$.
 \end{center}
 
 $g(x)= f(x)-x$.
 \end{center}
 

-Often it is not be possible to solve such nonlinear equation
-root-finding problems analytically. When this occurs we turn to
+It is often impossible to solve such nonlinear equation
+root-finding problems analytically. When this occurs, we turn to

 numerical methods to approximate the solution. 
 Generally speaking, algorithms for solving problems can be divided into
 two main groups: direct methods and iterative methods.
 numerical methods to approximate the solution. 
 Generally speaking, algorithms for solving problems can be divided into
 two main groups: direct methods and iterative methods.

-\\
-Direct methods exist only for $n \leq 4$, solved in closed form by G. Cardano
-in the mid-16th century. However, N. H. Abel in the early 19th
-century showed that polynomials of degree five or more could not
+
+Direct methods only exist for $n \leq 4$, solved in closed form
+by G. Cardano in the mid-16th century. However, N. H. Abel in the early 19th
+century proved that polynomials of degree five or more could not

 be solved by  direct methods. Since then, mathematicians have
 focussed on numerical (iterative) methods such as the famous
 be solved by  direct methods. Since then, mathematicians have
 focussed on numerical (iterative) methods such as the famous

-Newton method, the Bernoulli method of the 18th, and the Graeffe method.
+Newton method, the Bernoulli method of the 18th century, and the Graeffe method.

 
 Later on, with the advent of electronic computers, other methods have
 been developed such as the Jenkins-Traub method, the Larkin method,
 
 Later on, with the advent of electronic computers, other methods have
 been developed such as the Jenkins-Traub method, the Larkin method,

-the Muller method, and several methods for simultaneous
+the Muller method, and several other methods for the simultaneous

 approximation of all the roots, starting with the Durand-Kerner (DK)
 method:
 %%\begin{center}
 approximation of all the roots, starting with the Durand-Kerner (DK)
 method:
 %%\begin{center}


@@ -176,30 +176,30 @@ Aberth, Ehrlich and Farmer-Loizou~\cite{Loizou83} have proved that
 the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of convergence.
 
 
 the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of convergence.
 
 

-Iterative methods raise several problem when implemented e.g.
-specific sizes of numbers must be used to deal with this
-difficulty. Moreover, the convergence time of iterative methods
+Moreover, the convergence times of iterative methods

 drastically increases like the degrees of high polynomials. It is expected that the
 drastically increases like the degrees of high polynomials. It is expected that the

-parallelization of these algorithms will improve the convergence
-time.
+parallelization of these algorithms will reduce the execution times.

 
 Many authors have dealt with the parallelization of
 simultaneous methods, i.e. that find all the zeros simultaneously. 
 Freeman~\cite{Freeman89} implemented and compared DK, EA and another method of the fourth order proposed
 
 Many authors have dealt with the parallelization of
 simultaneous methods, i.e. that find all the zeros simultaneously. 
 Freeman~\cite{Freeman89} implemented and compared DK, EA and another method of the fourth order proposed

-by Farmer and Loizou~\cite{Loizou83}, on a 8-processor linear
-chain, for polynomials of degree up to 8. The third method often
-diverges, but the first two methods have speed-up equal to 5.5. Later,
+by Farmer and Loizou~\cite{Loizou83}, on an 8-processor linear
+chain, for polynomials of degree 8. The third method often
+diverges, but the first two methods have speed-ups equal to 5.5. Later,

 Freeman and Bane~\cite{Freemanall90}  considered asynchronous
 algorithms, in which each processor continues to update its
 Freeman and Bane~\cite{Freemanall90}  considered asynchronous
 algorithms, in which each processor continues to update its

-approximations even though the latest values of other $z_i^{k}$
-have not been received from the other processors, in contrast with synchronous algorithms where it would wait those values before making a new iteration.
+approximations even though the latest values of other roots
+have not yet been received from the other processors.  In contrast,
+synchronous algorithms   wait the computation of all roots at a given
+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
 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
 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


@@ -231,8 +231,11 @@ topic.
 
 \section{Ehrlich-Aberth method}
 \label{sec1}
 
 \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 
 
 %The Aberth method is a purely algebraic derivation. 
 %To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors