\documentclass[review]{elsarticle}
\usepackage{lineno,hyperref}
-%%\usepackage[utf8]{inputenc}
+\usepackage[utf8]{inputenc}
%%\usepackage[T1]{fontenc}
%%\usepackage[french]{babel}
+\usepackage{float}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage[ruled,vlined]{algorithm2e}
+%\usepackage[french,boxed,linesnumbered]{algorithm2e}
\usepackage{array,multirow,makecell}
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\begin{frontmatter}
-\title{A parallel root finding polynomial on GPU}
+\title{Efficient high degree polynomial root finding using GPU}
%% Group authors per affiliation:
-\author{Elsevier\fnref{myfootnote}}
-\address{Radarweg 29, Amsterdam}
-\fntext[myfootnote]{Since 1880.}
+%\author{Elsevier\fnref{myfootnote}}
+%\address{Radarweg 29, Amsterdam}
+%\fntext[myfootnote]{Since 1880.}
%% or include affiliations in footnotes:
-\author[mymainaddress]{Ghidouche Kahina\corref{mycorrespondingauthor}}
-%%\ead[url]{kahina.ghidouche@gmail.com}
+\author[mymainaddress]{Kahina Ghidouche}
+%%\ead[url]{kahina.ghidouche@univ-bejaia.dz}
\cortext[mycorrespondingauthor]{Corresponding author}
-\ead{kahina.ghidouche@gmail.com}
+\ead{kahina.ghidouche@univ-bejaia.dz}
-\author[mysecondaryaddress]{Couturier Raphael\corref{mycorrespondingauthor}}
+\author[mysecondaryaddress]{Raphaël Couturier\corref{mycorrespondingauthor}}
%%\cortext[mycorrespondingauthor]{Corresponding author}
\ead{raphael.couturier@univ-fcomte.fr}
-\author[mymainaddress]{Abderrahmane Sider\corref{mycorrespondingauthor}}
+\author[mymainaddress]{Abderrahmane Sider}
%%\cortext[mycorrespondingauthor]{Corresponding author}
\ead{ar.sider@univ-bejaia.dz}
-\address[mymainaddress]{Department of informatics,University of Bejaia,Algeria}
-\address[mysecondaryaddress]{FEMTO-ST Institute, University of Franche-Compté }
+\address[mymainaddress]{Laboratoire LIMED, Faculté des sciences
+ exactes, Université de Bejaia, 06000, Algeria}
+\address[mysecondaryaddress]{FEMTO-ST Institute, University of
+ Bourgogne Franche-Comte, France }
\begin{abstract}
-in this article we present a parallel implementation
-of the Aberth algorithm for the problem root finding for
-high degree polynomials on GPU architecture (Graphics
-Processing Unit).
+Polynomials are mathematical algebraic structures that play a great
+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
+GPU architectures. The main result of this
+work is to be able to solve high degree polynomials (up
+to 1,000,000) efficiently. We also compare the results with a
+sequential implementation and the Durand-Kerner method on full and
+sparse polynomials.
\end{abstract}
\begin{keyword}
-root finding of polynomials, high degree, iterative methods, Durant-Kerner, GPU, CUDA, CPU , Parallelization
+Polynomial root finding, Iterative methods, Ehrlich-Aberth, Durand-Kerner, GPU
\end{keyword}
\end{frontmatter}
\linenumbers
-\section{Root finding problem}
-We consider a polynomial of degree \textit{n} having coefficients
-in the complex \textit{C} and zeros $\alpha_{i},\textit{i=1,...,n}$.
+\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{a_{i}x^{i}}=a_{n}\prod(x-\alpha_{i}),a_{0} a_{n}\neq 0}
+ {\Large p(x)=\sum_{i=0}^{n}{a_{i}x^{i}}}.
\end{equation}
%%\end{center}
- The root finding problem consist to find
-all n root of \textit{p(x)}. the problem of finding a root is
-equivalent to the problem of finding a fixed-point. To see this
-consider the fixed-point problem of finding the n-dimensional
-vector x such that
+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}
+
+The problem of finding a root is equivalent to that of solving a fixed-point problem. To see this, consider the fixed-point problem of finding the $n$-dimensional
+vector $x$ such that :
\begin{center}
-$x=g(x). $
+$x=g(x)$
\end{center}
-Where $g : C^{n}\longrightarrow C^{n}$. Note that we can easily
+where $g : C^{n}\longrightarrow C^{n}$. Usually, we can easily
rewrite this fixed-point problem as a root-finding problem by
-setting $f (x) = x-g(x)$ and likewise we can recast the
-root-finding problem into a fixed-point problem by setting
+setting $f(x) = x-g(x)$ and likewise we can recast the
+root-finding problem into a fixed-point problem by setting :
\begin{center}
-$g(x)= f(x)-x$
+$g(x)= f(x)-x$.
\end{center}
-Often it will not be possible 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 numerically can be divided into
+
+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.
-\\
- 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
-be solved by directs methods. Since then researchers have
-concentrated on numerical (iterative) methods such as the famous
-Newton's method, Bernoulli's method of the 18th, and Graeffe's.
-With the advent of electronic computers, different methods has
-been developed such as the Jenkins-Traub method, Larkin s method,
-Muller's method, and several methods for simultaneous
-approximation of all the roots, starting with the Durand-Kerner
+
+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
+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,
+the Muller method, and several other methods for the simultaneous
+approximation of all the roots, starting with the Durand-Kerner (DK)
method:
%%\begin{center}
\begin{equation}
- Z_{i}=Z_{i}-\frac{P(Z_{i})}{\prod_{i\neq j}(z_{i}-z_{j})}
+\label{DK}
+ DK: z_i^{k+1}=z_{i}^{k}-\frac{P(z_i^{k})}{\prod_{i\neq j}(z_i^{k}-z_j^{k})}, i = 1, . . . , n,
\end{equation}
%%\end{center}
+where $z_i^k$ is the $i^{th}$ root of the polynomial $p$ at the
+iteration $k$.
+
-This formula is mentioned for the first time from
+This formula is mentioned for the first time by
Weiestrass~\cite{Weierstrass03} as part of the fundamental theorem
-of Algebra and is rediscovered from Ilieff~\cite{Ilie50},
+of Algebra and is rediscovered by Ilieff~\cite{Ilie50},
Docev~\cite{Docev62}, Durand~\cite{Durand60},
-Kerner~\cite{Kerner66}. Another method discovered from
+Kerner~\cite{Kerner66}. Another method discovered by
Borsch-Supan~\cite{ Borch-Supan63} and also described and brought
-in the following form from Ehrlich~\cite{Ehrlich67} and
-Aberth~\cite{Aberth73}.
+in the following form by Ehrlich~\cite{Ehrlich67} and
+Aberth~\cite{Aberth73} uses a different iteration formula given as:
%%\begin{center}
\begin{equation}
- Z_{i}=Z_{i}-\frac{1}{{\frac {P'(Z_{i})} {P(Z_{i})}}-{\sum_{i\neq j}(z_{i}-z_{j})}}
+\label{Eq:EA}
+ EA: z_i^{k+1}=z_i^{k}-\frac{1}{{\frac {P'(z_i^{k})} {P(z_i^{k})}}-{\sum_{i\neq j}\frac{1}{(z_i^{k}-z_j^{k})}}}, i = 1, . . . , n,
\end{equation}
%%\end{center}
+where $p'(z)$ is the polynomial derivative of $p$ evaluated in the
+point $z$.
-Aberth, Ehrlich and Farmer-Loizou~\cite{Loizon83} have proved that
-the above method has cubic order of convergence for simple roots.
-
-
-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
-drastically increase like the degrees of high polynomials. The
-parallelization of these algorithms will improve the convergence
-time.
-
-Many authors have treated the problem of parallelization of
-simultaneous methods. Freeman~\cite{Freeman89} has tested the DK
-method, EA method and another method of the fourth order proposed
-from Farmer and Loizou~\cite{Loizon83},on a 8- processor linear
-chain, for polynomial of degree up to 8. The third method often
-diverges, but the first two methods have speed-up 5.5
-(speed-up=(Time on one processor)/(Time on p processors)). Later
-Freeman and Bane~\cite{Freemanall90} consider asynchronous
+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.
+
+
+Moreover, the convergence times of iterative methods
+drastically increases like the degrees of high polynomials. It is expected that the
+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
+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
-approximations even although the latest values of other $z_i((k))$
-have not received from the other processors, in difference with
-the synchronous version where it would wait.
-in~\cite{Raphaelall01}proposed two methods of parallelization for
-architecture with shared memory and distributed memory,it able to
-compute the root of polynomial degree 10000 on 430 s with only 8
-pc and 2 communications per iteration. Compare to the sequential
-it take 3300 s to obtain the same results.
-
-After this few works discuses this problem until the apparition of
-the Compute Unified Device Architecture (CUDA)~\cite{CUDA10},a
+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
+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.
+
+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 ability of GPU has exceeded the counterpart
-of CPU. It is a waste of resource to be just a graphics card for
-GPU. CUDA adopts a totally new computing architecture to use the
+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
hardware resources provided by GPU in order to offer a stronger
computing ability to the massive data computing.
-Indeed,~\cite{Kahinall14}proposed the implementation of the
-Durand-Kerner method on GPU (Graphics Processing Unit). The main
-result prove that a parallel implementation is 10 times as fast as
-the sequential implementation on a single CPU for high degree
-polynomials that is greater than about 48000.
-\paragraph{}
-The mean part of our work is to implement the Aberth method for the problem root finding for
-high degree polynomials on GPU architecture (Graphics Processing Unit). Initially we present the Aberth method in section 1. Amelioration of Aberth method was proposed in section 2. A related works for the implementation of simultaneous methods in a parallel computer was discuss in section 3. Section 4 we propose a parallel implementation of Aberth method on GPU. Section 5, we present our result and discuss it. Finally, in Section 6, we present our conclusions and future research directions.
+Ghidouche and al~\cite{Kahinall14} proposed an implementation of the
+Durand-Kerner method on GPU. Their main
+result showed that a parallel CUDA implementation is about 10 times faster than
+the sequential implementation on a single CPU for sparse
+polynomials of degree 48,000.
-\section{Aberth method}
-A cubically convergent iteration method for finding zeros of
-polynomials was proposed by O.Aberth~\cite{Aberth73}. The Aberth
-method is a purely algebraic derivation.To illustrate the
-derivation, we let $w_{i}(z)$ be the product of linear factor
-\begin{equation}
-w_{i}(z)=\prod_{j=1,j \neq i}^{n} (z-x_{j})
-\end{equation}
+In this paper, we focus on the implementation of the Ehrlich-Aberth
+method for high degree polynomials on GPU. We propose an adaptation of
+the exponential logarithm in order to be able to solve sparse and full
+polynomial of degree up to $1,000,000$. The paper is organized as
+follows. Initially, we recall the Ehrlich-Aberth method in
+Section~\ref{sec1}. Improvements for the Ehrlich-Aberth method are
+proposed in Section \ref{sec2}. Related work to the implementation of
+simultaneous methods using a parallel approach is presented in Section
+\ref{secStateofArt}. In Section~\ref{sec5} we propose a parallel
+implementation of the Ehrlich-Aberth method on GPU and discuss
+it. Section~\ref{sec6} presents and investigates our implementation
+and experimental study results. Finally, Section~\ref{sec7} concludes
+this paper and gives some hints for future research directions in this
+topic.
-And rational function $R_{i}(z)$ be the correction term of
-Weistrass method~\cite{Weierstrass03}:
+\section{Ehrlich-Aberth method}
+\label{sec1}
+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.
-\begin{equation}
-R_{i}(z)=\frac{p(z)}{w_{i}(z)} , i=1,2,...,n
-\end{equation}
+%The Aberth method is a purely algebraic derivation.
+%To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors
-Differentiating the rational function $R_{i}(z)$ and applying the
-Newton method, we have:
+%\begin{equation}
+%w_{i}(z)=\prod_{j=1,j \neq i}^{n} (z-x_{j})
+%\end{equation}
-\begin{equation}
-\frac{R_{i}(z)}{R_{i}^{'}(z)}= \frac{p(z)}{p^{'}(z)-p(z)\frac{w_{i}(z)}{w_{i}^{'}(z)}}= \frac{p(z)}{p^{'}(z)-p(z) \sum _{j=1,j \neq i}^{n}\frac{1}{z-x_{i}}}, i=1,2,...,n
-\end{equation}
+%And let a rational function $R_{i}(z)$ be the correction term of the
+%Weistrass method~\cite{Weierstrass03}
+
+%\begin{equation}
+%R_{i}(z)=\frac{p(z)}{w_{i}(z)} , i=1,2,...,n.
+%\end{equation}
-Substituting $x_{j}$ for z we obtain the Aberth iteration method
+%Differentiating the rational function $R_{i}(z)$ and applying the
+%Newton method, we have:
+
+%\begin{equation}
+%\frac{R_{i}(z)}{R_{i}^{'}(z)}= \frac{p(z)}{p^{'}(z)-p(z)\frac{w_{i}(z)}{w_{i}^{'}(z)}}= \frac{p(z)}{p^{'}(z)-p(z) \sum _{j=1,j \neq i}^{n}\frac{1}{z-x_{j}}}, i=1,2,...,n
+%\end{equation}
+%where R_{i}^{'}(z)is the rational function derivative of F evaluated in the point z
+%Substituting $x_{j}$ for $z_{j}$ we obtain the Aberth iteration method.%
-Let present the means stages of Aberth method.
\subsection{Polynomials Initialization}
- The initialization of polynomial P(z) with complex coefficients
- are given by:
+The initialization of a polynomial $p(z)$ is done by setting each of the $n$ complex coefficients $a_{i}$:
\begin{equation}
+\label{eq:SimplePolynome}
p(z)=\sum{a_{i}z^{n-i}} , a_{n} \neq 0,a_{0}=1, a_{i}\subset C
\end{equation}
\subsection{Vector $Z^{(0)}$ Initialization}
-
-The choice of the initial points $z^{(0)}_{i}, i = 1, . . . , n.$
-from which starting the iteration (2) or (3), is rather delicate
-since the number of steps needed by the iterative method to reach
+\label{sec:vec_initialization}
+As for any iterative method, we need to choose $n$ initial guess points $z^{0}_{i}, i = 1, . . . , n.$
+The initial guess is very important since the number of steps needed by the iterative method to reach
a given approximation strongly depends on it.
-In~\cite{Aberth73}the Aberth iteration is started by selecting n
-equispaced points on a circle of center 0 and radius r, where r is
-an upper bound to the moduli of the zeros. After,~\cite{Bini96}
-performs this choice by selecting complex numbers along different
-circles and relies on the result of~\cite{Ostrowski41}.
+In~\cite{Aberth73} the Ehrlich-Aberth iteration is started by selecting $n$
+equi-spaced points on a circle of center 0 and radius r, where r is
+an upper bound to the moduli of the zeros. Later, Bini and al.~\cite{Bini96}
+performed this choice by selecting complex numbers along different
+circles which relies on the result of~\cite{Ostrowski41}.
\begin{equation}
+\label{eq:radiusR}
%%\begin{align}
\sigma_{0}=\frac{u+v}{2};u=\frac{\sum_{i=1}^{n}u_{i}}{n.max_{i=1}^{n}u_{i}};
v=\frac{\sum_{i=0}^{n-1}v_{i}}{n.min_{i=0}^{n-1}v_{i}};\\
v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
\end{equation}
-\subsection{Iterative Function $H_{i}$}
-The operator used with Aberth method is corresponding to the
-following equation which will enable the convergence towards
-polynomial solutions, provided all the roots are distinct.
+\subsection{Iterative Function}
+%The operator used by the Aberth method is corresponding to the
+%following equation~\ref{Eq:EA} which will enable the convergence towards
+%polynomial solutions, provided all the roots are distinct.
+
+Here we give a second form of the iterative function used by the Ehrlich-Aberth method:
\begin{equation}
-H_{i}(z)=z_{i}-\frac{1}{\frac{P^{'}(z_{i})}{P(z_{i})}-\sum_{j\neq
-i}{\frac{1}{z_{i}-z_{j}}}}
+\label{Eq:Hi}
+EA2: z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
+{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, i=1,. . . .,n
\end{equation}
-
-\subsection{Convergence condition}
-Determines the success of the termination. It consists in stopping
-the iterative function $H_{i}(z)$ when the root are stable, the method
-converge sufficiently:
+It can be noticed that this equation is equivalent to Eq.~\ref{Eq:EA},
+but we prefer the latter one because we can use it to improve the
+Ehrlich-Aberth method and find the roots of high degree polynomials. More
+details are given in Section~\ref{sec2}.
+\subsection{Convergence Condition}
+The convergence condition determines the termination of the algorithm. It consists in stopping the iterative function when the roots are sufficiently stable. We consider that the method converges sufficiently when:
\begin{equation}
-\forall i \in
-[1,n];\frac{z_{i}^{(k)}-z_{i}^{(k-1)}}{z_{i}^{(k)}}<\xi
+\label{eq:Aberth-Conv-Cond}
+\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi
\end{equation}
-\section{Amelioration of Aberth method }
-The Aberth method implementation suffer of overflow problems. This
-situation occurs, for instance, in the case where a polynomial
-having positive coefficients and large degree is computed at a
-point $\xi$ where $|\xi| > 1$. Indeed the limited number in the
-mantissa of floating takings the computation of P(z) wrong when z
-is large. for example $(10^{50}) +1+ (- 10^{50})$ will give result
-0 instead of 1 in reality. Consequently we can not compute the roots
-for large polynomial's degree. This problem was discuss in
-~\cite{Karimall98} for the Durand-Kerner method, the authors
-propose to use the logarithm and the exponential of a complex:
+\section{Improving the Ehrlich-Aberth Method for high degree polynomials with exp-log formulation}
+\label{sec2}
+With high degree polynomial, the Ehrlich-Aberth method implementation,
+as well as the Durand-Kerner implementation, suffers from overflow problems. This
+situation occurs, for instance, in the case where a polynomial,
+having positive coefficients and a large degree, is computed at a
+point $\xi$ where $|\xi| > 1$, where $|z|$ stands for the modolus of a complex $z$. Indeed, the limited number in the
+mantissa of floating points representations makes the computation of $p(z)$ wrong when z
+is large. For example $(10^{50}) +1+ (- 10^{50})$ will give the wrong result
+of $0$ instead of $1$. Consequently, we can not compute the roots
+for large degrees. This problem was discussed earlier in
+~\cite{Karimall98} for the Durand-Kerner method. The authors
+propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent.
\begin{equation}
+\label{deflncomplex}
\forall(x,y)\in R^{*2}; \ln (x+i.y)=\ln(x^{2}+y^{2})
2+i.\arcsin(y\sqrt{x^{2}+y^{2}})_{\left] -\pi, \pi\right[ }
\end{equation}
%%\begin{equation}
\begin{align}
+\label{defexpcomplex}
\forall(x,y)\in R^{*2}; \exp(x+i.y) & = \exp(x).\exp(i.y)\\
- & =\exp(x).\cos(y)+i.\exp(x).\sin(y)
+ & =\exp(x).\cos(y)+i.\exp(x).\sin(y)\label{defexpcomplex1}
\end{align}
%%\end{equation}
-The application of logarithm can replace any multiplications and
-divisions with additions and subtractions. Consequently, it
-manipulates lower absolute values and can be compute the roots for
-large polynomial's degree exceed~\cite{Karimall98}.
+Using the logarithm (eq.~\ref{deflncomplex}) and the exponential (eq.~\ref{defexpcomplex}) operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations
+manipulate lower absolute values and the roots for large polynomial degrees can be looked for successfully~\cite{Karimall98}.
-Applying this solution for the Aberth method we obtain the
-iteration function with logarithm:
+Applying this solution for the iteration function Eq.~\ref{Eq:Hi} of
+Ehrlich-Aberth method, we obtain the following iteration function with exponential and logarithm:
%%$$ \exp \bigl( \ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'}))- \ln(1- \exp(\ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'})+\ln\sum_{i\neq j}^{n}\frac{1}{z_{k}-z_{j}})$$
\begin{equation}
-H_{i}(z)=z_{i}^{k}-\exp \left(\ln \left(
-p(z_{k})\right)-\ln\left(p(z_{k}^{'})\right)- \ln
-\left(1-Q(z_{k})\right)\right)
+\label{Log_H2}
+EA.EL: z^{k+1}_{i}=z_{i}^{k}-\exp \left(\ln \left(
+p(z_{i}^{k})\right)-\ln\left(p'(z^{k}_{i})\right)- \ln\left(1-Q(z^{k}_{i})\right)\right),
\end{equation}
-Where:
+
+where:
\begin{equation}
-Q(z_{k})=\exp\left( \ln (p(z_{k}))-\ln(p(z_{k}^{'}))+\ln \left(
-\sum_{k\neq j}^{n}\frac{1}{z_{k}-z_{j}}\right)\right)
+\label{Log_H1}
+Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}_{i}))+\ln \left(
+\sum_{i\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right)i=1,...,n,
\end{equation}
-This solution is applying when it is necessary
+This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as :
+\begin{equation}
+\label{R.EL}
+R = exp(log(DBL\_MAX)/(2*n) );
+\end{equation}
+
+
+%\begin{equation}
-\section{The implementation of simultaneous methods in a parallel computer}
- The main problem of the simultaneous methods is that the necessary
-time needed for the convergence is increased with the increasing
-of the degree of the polynomial. The parallelization of these
-algorithms will improve the convergence time. Researchers usually
-adopt one of the two following approaches to parallelize root
-finding algorithms. One approach is to reduce the total number of
-iterations as implemented by Miranker
+%R = \exp( \log(DBL\_MAX) / (2*n) )
+%\end{equation}
+ where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.
+
+\section{Implementation of simultaneous methods in a parallel computer}
+\label{secStateofArt}
+The main problem of simultaneous methods is that the
+time needed for convergence is increased when we increase
+the degree of the polynomial. The parallelization of these
+algorithms is expected to improve the convergence time.
+Authors usually adopt one of the two following approaches to parallelize root
+finding algorithms. The first approach aims at reducing the total number of
+iterations as in Miranker
~\cite{Mirankar68,Mirankar71}, Schedler~\cite{Schedler72} and
-Winogard~\cite{Winogard72}. Another approach is to reduce the
+Winograd~\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
+in~\cite{Benall68,Jana06,Janall99,Riceall06}.
+
+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,
+ Zhancall08, Zhuall08}. However, the Durand-Kerner and the Ehrlich-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 intrinsic parallelism, i.e. the
+computations involved in both methods have 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
+Moreover, they have fast a rate of convergence (quadratic for the
+Durand-Kerner and cubic for the Ehrlich-Aberth). Various parallel
algorithms reported for these methods can be found
-in~\cite{Cosnard90, Freeman89,Freemanall90,,Jana99,Janall99}.
+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
-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
+for the Durand-Kerner method, and the Ehrlich-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 needs $N$ processors to compute $N$ roots, which is not
+practical for solving large degree polynomials.
+
+%Until very recently, the literature did 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.
-
-
-\section {A parallel implementation of Aberth method}
-
-\subsection{Background on the GPU architecture}
-A GPU is viewed as an accelerator for the data-parallel and
-intensive arithmetic computations. It draws its computing power
-from the parallel nature of its hardware and software
-architectures. A GPU is composed of hundreds of Streaming
-Processors (SPs) organized in several blocks called Streaming
-Multiprocessors (SMs). It also has a memory hierarchy. It has a
-private read-write local memory per SP, fast shared memory and
-read-only constant and texture caches per SM and a read-write
-global memory shared by all its SPs~\cite{NVIDIA10}
-
- On a CPU equipped with a GPU, all the data-parallel and intensive
-functions of an application running on the CPU are off-loaded onto
-the GPU in order to accelerate their computations. A similar
-data-parallel function is executed on a GPU as a kernel by
-thousands or even millions of parallel threads, grouped together
-as a grid of thread blocks. Therefore, each SM of the GPU executes
-one or more thread blocks in SIMD fashion (Single Instruction,
-Multiple Data) and in turn each SP of a GPU SM runs one or more
-threads within a block in SIMT fashion (Single Instruction,
-Multiple threads). Indeed at any given clock cycle, the threads
-execute the same instruction of a kernel, but each of them
-operates on different data.
- GPUs only work on data filled in their
-global memories and the final results of their kernel executions
-must be communicated to their CPUs. Hence, the data must be
-transferred in and out of the GPU. However, the speed of memory
-copy between the GPU and the CPU is slower than the memory
-bandwidths of the GPU memories and, thus, it dramatically affects
-the performances of GPU computations. Accordingly, it is necessary
-to limit data transfers between the GPU and its CPU during the
-computations.
-\subsection{Background on the CUDA Programming Model}
-
-The CUDA programming model is similar in style to a single program
-multiple-data (SPMD) softwaremodel. The GPU is treated as a
-coprocessor that executes data-parallel kernel functions. CUDA
-provides three key abstractions, a hierarchy of thread groups,
-shared memories, and barrier synchronization. Threads have a three
-level hierarchy. A grid is a set of thread blocks that execute a
-kernel function. Each grid consists of blocks of threads. Each
-block is composed of hundreds of threads. Threads within one block
-can share data using shared memory and can be synchronized at a
-barrier. All threads within a block are executed concurrently on a
-multithreaded architecture.The programmer specifies the number of
-threads per block, and the number of blocks per grid. A thread in
-the CUDA programming language is much lighter weight than a thread
-in traditional operating systems. A thread in CUDA typically
-processes one data element at a time. The CUDA programming model
-has two shared read-write memory spaces, the shared memory space
-and the global memory space. The shared memory is local to a block
-and the global memory space is accessible by all blocks. CUDA also
-provides two read-only memory spaces, the constant space and the
-texture space, which reside in external DRAM, and are accessed via
-read-only caches.
-
-\subsection{ The implementation of Aberth method on GPU}
+polynomials of 48,000.
+%In this paper we present a parallel implementation of Ehrlich-Aberth
+%method on GPUs for sparse and full polynomials with high degree (up
+%to $1,000,000$).
+
+
+%% \section {A CUDA parallel Ehrlich-Aberth method}
+%% In the following, we describe the parallel implementation of Ehrlich-Aberth method on GPU
+%% for solving high degree polynomials (up to $1,000,000$). First, the hardware and software of the GPUs are presented. Then, the CUDA parallel Ehrlich-Aberth method is presented.
+
+%% \subsection{Background on the GPU architecture}
+%% A GPU is viewed as an accelerator for the data-parallel and
+%% intensive arithmetic computations. It draws its computing power
+%% from the parallel nature of its hardware and software
+%% architectures. A GPU is composed of hundreds of Streaming
+%% Processors (SPs) organized in several blocks called Streaming
+%% Multiprocessors (SMs). It also has a memory hierarchy. It has a
+%% private read-write local memory per SP, fast shared memory and
+%% read-only constant and texture caches per SM and a read-write
+%% global memory shared by all its SPs~\cite{NVIDIA10}.
+
+%% On a CPU equipped with a GPU, all the data-parallel and intensive
+%% functions of an application running on the CPU are off-loaded onto
+%% the GPU in order to accelerate their computations. A similar
+%% data-parallel function is executed on a GPU as a kernel by
+%% thousands or even millions of parallel threads, grouped together
+%% as a grid of thread blocks. Therefore, each SM of the GPU executes
+%% one or more thread blocks in SIMD fashion (Single Instruction,
+%% Multiple Data) and in turn each SP of a GPU SM runs one or more
+%% threads within a block in SIMT fashion (Single Instruction,
+%% Multiple threads). Indeed at any given clock cycle, the threads
+%% execute the same instruction of a kernel, but each of them
+%% operates on different data.
+%% GPUs only work on data filled in their
+%% global memories and the final results of their kernel executions
+%% must be communicated to their CPUs. Hence, the data must be
+%% transferred in and out of the GPU. However, the speed of memory
+%% copy between the GPU and the CPU is slower than the memory
+%% bandwidths of the GPU memories and, thus, it dramatically affects
+%% the performances of GPU computations. Accordingly, it is necessary
+%% to limit as much as possible, data transfers between the GPU and its CPU during the
+%% computations.
+%% \subsection{Background on the CUDA Programming Model}
+
+%% The CUDA programming model is similar in style to a single program
+%% multiple-data (SPMD) software model. The GPU is viewed as a
+%% coprocessor that executes data-parallel kernel functions. CUDA
+%% provides three key abstractions, a hierarchy of thread groups,
+%% shared memories, and barrier synchronization. Threads have a three
+%% level hierarchy. A grid is a set of thread blocks that execute a
+%% kernel function. Each grid consists of blocks of threads. Each
+%% block is composed of hundreds of threads. Threads within one block
+%% can share data using shared memory and can be synchronized at a
+%% barrier. All threads within a block are executed concurrently on a
+%% multithreaded architecture.The programmer specifies the number of
+%% threads per block, and the number of blocks per grid. A thread in
+%% the CUDA programming language is much lighter weight than a thread
+%% in traditional operating systems. A thread in CUDA typically
+%% processes one data element at a time. The CUDA programming model
+%% has two shared read-write memory spaces, the shared memory space
+%% and the global memory space. The shared memory is local to a block
+%% and the global memory space is accessible by all blocks. CUDA also
+%% provides two read-only memory spaces, the constant space and the
+%% texture space, which reside in external DRAM, and are accessed via
+%% read-only caches.
+
+\section{ Implementation of the Ehrlich-Aberth method on GPU}
+\label{sec5}
%%\subsection{A CUDA implementation of the Aberth's method }
%%\subsection{A GPU implementation of the Aberth's method }
-\subsubsection{A sequential Aberth algorithm}
-The means steps of Aberth method can expressed as an algorithm
-like:
-
-\begin{algorithm}[H]
-\LinesNumbered
-\caption{Algorithm to find root polynomial with Aberth method}
-
-\KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error
-tolerance threshold),P(Polynomial to solve)}
-
-\KwOut {Z(The solution root's vector)}
+%% \subsection{Sequential Ehrlich-Aberth algorithm}
+%% The main steps of Ehrlich-Aberth method are shown in Algorithm.~\ref{alg1-seq} :
+%% %\LinesNumbered
+%% \begin{algorithm}[H]
+%% \label{alg1-seq}
-\BlankLine
+%% \caption{A sequential algorithm to find roots with the Ehrlich-Aberth method}
-Initialization of the parameter of the polynomial to solve\;
-Initialization of the solution vector $Z^{0}$\;
+%% \KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (error tolerance
+%% threshold), $P$ (Polynomial to solve),$Pu$ (the derivative of P) $\Delta z_{max}$ (maximum value
+%% of stop condition), k (number of iteration), n (Polynomial's degrees)}
+%% \KwOut {$Z$ (The solution root's vector), $ZPrec$ (the previous solution root's vector)}
-\While {$\Delta z_{max}\succ \epsilon$}{
- Let $\Delta z_{max}=0$\;
-\For{$j \gets 0 $ \KwTo $n$}{
-$ZPrec\left[j\right]=Z\left[j\right]$\;
-$Z\left[j\right]=H\left(j,Z\right)$\;
-}
+%% \BlankLine
-\For{$i \gets 0 $ \KwTo $n-1$}{
-$c=\frac{\left|Z\left[i\right]-ZPrec\left[i\right]\right|}{Z\left[i\right]}$\;
-\If{$c\succ\Delta z_{max}$ }{
-$\Delta z_{max}$=c\;}
-}
-}
-\end{algorithm}
+%% Initialization of $P$\;
+%% Initialization of $Pu$\;
+%% Initialization of the solution vector $Z^{0}$\;
+%% $\Delta z_{max}=0$\;
+%% k=0\;
-~\\
-In this sequential algorithm one thread CPU execute all steps. Let see the step 3 the execution of the iterative function, 2 instructions are needed, the first instruction \textit{save} the solution vector for the previous iteration, the second instruction \textit{update} or compute a new values of the roots.
-We have two manner to execute the iterative function, taking a Jacobi iteration who need all the previous value $z^{(k)}_{i}$ to compute the new value $z^{(k+1)}_{i}$we have:
+%% \While {$\Delta z_{max} > \varepsilon$}{
+%% Let $\Delta z_{max}=0$\;
+%% \For{$j \gets 0 $ \KwTo $n$}{
+%% $ZPrec\left[j\right]=Z\left[j\right]$;// save Z at the iteration k.\
-\begin{equation}
-H(i,z^{k+1})=\frac{p(z^{(k)}_{i})}{p'(z^{(k)}_{i})-p(z^{(k)}_{i})\sum^{n}_{j=1 j\neq i}\frac{1}{z^{(k)}_{i}-z^{(k)}_{j}}}, i=1,...,n.
-\end{equation}
+%% $Z\left[j\right]=H\left(j, Z, P, Pu\right)$;//update Z with the iterative function.\
+%% }
+%% k=k+1\;
-Or with the Gauss-seidel iteration, we have:
-\begin{equation}
-H(i,z^{k+1})=\frac{p(z^{(k)}_{i})}{p'(z^{(k)}_{i})-p(z^{(k)}_{i})\sum^{i-1}_{j=1}\frac{1}{z^{(k)}_{i}-z^{(k+1)}_{j}}+\sum^{n}_{j=i+1}\frac{1}{z^{(k)}_{i}-z^{(k)}_{j}}}, i=1,...,n.
-\end{equation}
+%% \For{$i \gets 0 $ \KwTo $n-1$}{
+%% $c= testConverge(\Delta z_{max},ZPrec\left[j\right],Z\left[j\right])$\;
+%% \If{$c > \Delta z_{max}$ }{
+%% $\Delta z_{max}$=c\;}
+%% }
-In formula(16), the Gauss-seidel iteration converge more quickly because they used the most fresh computed root $z^{k+1}_{i}$ , at this reason we used Gauss-seidel iteration.
+%% }
+%% \end{algorithm}
-The steps 4 of the Aberth method compute the convergence of the roots, using(9) formula.
-Both steps 3 and 4 use 1 thread to compute N roots on CPU, which is harmful for the large polynomial's roots finding.
+%% ~\\
+%% In this sequential algorithm, one CPU thread executes all the steps. Let us look to the $3^{rd}$ step i.e. the execution of the iterative function, 2 sub-steps are needed. The first sub-step \textit{save}s the solution vector of the previous iteration, the second sub-step \textit{update}s or computes the new values of the roots vector.
-\paragraph{The execution time}
-Let $T_{i}(N)$: the time to compute one new root's value of the step 3,$T_{i}$ depend on the polynomial's degrees N, when N increase $T_{i}$ increase to. We need $N.T_{i}(N)$ to compute all the new root's value in one iteration on the step 3.
+\subsection{Parallel implementation with CUDA }
-Let $T_{j}$: the time to compute one root's convergence value of the step 4, we need $N.T_{j}$ to compute all the root's convergence value in one iteration on the step 4.
+In order to implement the Ehrlich-Aberth method in CUDA, it is
+possible to use the Jacobi scheme or the Gauss-Seidel one. With the
+Jacobi iteration, at iteration $k+1$ we need all the previous values
+$z^{k}_{i}$ to compute the new values $z^{k+1}_{i}$, that is :
-The execution time for both steps 3 and 4 can see like:
\begin{equation}
-T_{exe}=N(T_{i}(N)+T_{j})+O(n).
+EAJ: z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
+{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}\sum_{j=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}}}, i=1,. . . .,n.
\end{equation}
-Let Nbr\_iter the number of iteration necessary to compute all the roots, so the total execution time $Total\_time_{exe}$ can give like:
+
+With the Gauss-Seidel iteration, we have:
+%\begin{equation}
+%\label{eq:Aberth-H-GS}
+%EAGS: z^{k+1}_{i}=\frac{p(z^{k}_{i})}{p'(z^{k}_{i})-p(z^{k}_{i})(\sum^{i-1}_{j=1}\frac{1}{z^{k}_{i}-z^{k+1}_{j}}+\sum^{n}_{j=i+1}\frac{1}{z^{k}_{i}-z^{k}_{j}})}, i=1,...,n.
+%\end{equation}
\begin{equation}
-Total\_time_{exe}=\left[N\left(T_{i}(N)+T_{j}\right)+O(n)\right].Nbr\_iter
+\label{eq:Aberth-H-GS}
+EAGS: z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
+{1-\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}(\sum^{i-1}_{j=1}\frac{1}{z^{k}_{i}-z^{k+1}_{j}}+\sum_{j=i+1}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}})}, i=1,. . . .,n
\end{equation}
-The execution time increase with the increasing of the polynomial's root, which take necessary to parallelize this step to reduce the execution time. In the following paper you explain how we parrallelize this step using GPU architecture with CUDA platform.
-\subsubsection{Parallelize the steps on GPU }
-On the CPU Aberth algorithm both steps 3 and 4 contain the loop \verb=for=, it use one thread to execute all the instruction in the loop N times. Here we explain how the GPU architecture can compute this loop and reduce the execution time.
-The GPU architecture assign the execution of this loop to a groups of parallel threads organized as a grid of blocks each block contain a number of threads. All threads within a block are executed concurrently in parallel. The instruction are executed as a kernel.
+Using Eq.~\ref{eq:Aberth-H-GS} to update the vector solution
+\textit{Z}, we expect the Gauss-Seidel iteration to converge more
+quickly because, just as any Jacobi algorithm (for solving linear
+systems of equations), it uses the freshest computed roots $z^{k+1}_{i}$.
-Let nbr\_thread be the number of threads executed in parallel, so you can easily transform the (18)formula like this:
+%The $4^{th}$ step of the algorithm checks the convergence condition using Eq.~\ref{eq:Aberth-Conv-Cond}.
+%Both steps 3 and 4 use 1 thread to compute all the $n$ roots on CPU, which is very harmful for performance in case of the large degree polynomials.
-\begin{equation}
-Total\_time_{exe}=\left[\frac{N}{nbr\_thread}\left(T_{i}(N)+T_{j}\right)+O(n)\right].Nbr\_iter.
-\end{equation}
-In theory, the $Total\_time_{exe}$ on GPU is speed up nbr\_thread times as a $Total\_time_{exe}$ on CPU. We show more details in the experiment part.
-~\\
-~\\
-In CUDA platform, All the instruction of the loop \verb=for= are executed by the GPU as a kernel form. A kernel is a procedure written in CUDA and defined by a heading \verb=__global__=, which means that it is to be executed by the GPU. The following algorithm see the Aberth algorithm on GPU:
+%On the CPU, both steps 3 and 4 contain the loop \verb=for= and a single thread executes all the instructions in the loop $n$ times. In this subsection, we explain how the GPU architecture can compute this loop and reduce the execution time.
+%In the GPU, the scheduler assigns the execution of this loop to a
+%group of threads organised as a grid of blocks with block containing a
+%number of threads. All threads within a block are executed
+%concurrently in parallel. The instructions run on the GPU are grouped
+%in special function called kernels. With CUDA, a programmer must
+%describe the kernel execution context: the size of the Grid, the number of blocks and the number of threads per block.
+
+%In CUDA programming, all the instructions of the \verb=for= loop are executed by the GPU as a kernel. A kernel is a function written in CUDA and defined by the \verb=__global__= qualifier added before a usual \verb=C= function, which instructs the compiler to generate appropriate code to pass it to the CUDA runtime in order to be executed on the GPU.
+
+Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth method using CUDA.
+
+\begin{enumerate}
\begin{algorithm}[H]
-\LinesNumbered
-\caption{Algorithm to find root polynomial with Aberth method}
+\label{alg2-cuda}
+%\LinesNumbered
+\caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
-\KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error
-tolerance threshold),P(Polynomial to solve)}
+\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
+ threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degrees), $\Delta z_{max}$ (Maximum value of stop condition)}
-\KwOut {Z(The solution root's vector)}
+\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
\BlankLine
-Initialization of the parameter of the polynomial to solve\;
-Initialization of the solution vector $Z^{0}$\;
-Allocate and fill the data in the global memory GPU\;
+\item Initialization of the of P\;
+\item Initialization of the of Pu\;
+\item Initialization of the solution vector $Z^{0}$\;
+\item Allocate and copy initial data to the GPU global memory\;
+\item k=0\;
+\While {$\Delta z_{max} > \epsilon$}{
+\item Let $\Delta z_{max}=0$\;
+\item $ kernel\_save(ZPrec,Z)$\;
+\item k=k+1\;
+\item $ kernel\_update(Z,P,Pu)$\;
+\item $kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;
-\While {$\Delta z_{max}\succ \epsilon$}{
- Let $\Delta z_{max}=0$\;
-$ kernel\_save(d\_Z^{k-1})$\;
-$ kernel\_update(d\_z^{k})$\;
-$kernel\_testConverge (d_?z_{max},d_Z^{k},d_Z^{k-1})$\;
}
+\item Copy results from GPU memory to CPU memory\;
\end{algorithm}
+\end{enumerate}
~\\
-After the initialization step, all data of the root finding problem to be solved must be copied from the CPU memory to the GPU global memory, because the GPUs only work on the data filled in their memories. Next, all the data-parallel arithmetic operations inside the main loop \verb=(do ... while(...))= are executed as kernels by the GPU. The first kernel \textit{save} in line( 6, Algorithm 2) consist to save the vector of polynomial's root found at the previous time step on GPU memory, in order to test the convergence of the root at each iteration in line (8, Algorithm 2).
-
-The second kernel executes the iterative function and update Z(k),as formula (), we notice that the kernel update are called in two forms, separated with the value of \emph{R} which determines the radius beyond which we apply the logarithm formula like this:
+After the initialization step, all data of the root finding problem
+must be copied from the CPU memory to the GPU global memory. Next, all
+the data-parallel arithmetic operations inside the main loop
+\verb=(while(...))= are executed as kernels by the GPU. The
+first kernel named \textit{save} in line 7 of
+Algorithm~\ref{alg2-cuda} consists in saving the vector of
+polynomial roots found at the previous time-step in GPU memory, in
+order to check the convergence of the roots after each iteration (line
+10, Algorithm~\ref{alg2-cuda}).
+
+The second kernel executes the iterative function and updates
+$Z$, according to Algorithm~\ref{alg3-update}. We notice that the
+update kernel is called in two forms, according to the value
+\emph{R} which determines the radius beyond which we apply the
+exponential logarithm algorithm.
\begin{algorithm}[H]
-\LinesNumbered
-\caption{A global Algorithm for the iterative function}
+\label{alg3-update}
+%\LinesNumbered
+\caption{Kernel update}
-\eIf{$(\left|Z^{(k)}\right|<= R)$}{
-$kernel\_update(d\_z^{k})$\;}
+\eIf{$(\left|Z\right|<= R)$}{
+$kernel\_update(Z,P,Pu)$\;}
{
-$kernel\_update\_Log(d\_z^{k})$\;
+$kernel\_update\_ExpoLog(Z,P,Pu)$\;
}
\end{algorithm}
-The first form execute the formula(8) if all the module's $( |Z(k)|<= R)$, else the kernel execute the formulas(13,14).the radius R was computed like:
-
-$$R = \exp( \log(DBL\_MAX) / (2*(double)P.degrePolynome) )$$
-
-The last kernel verify the convergence of the root after each update of $Z^{(k)}$, as formula(), we used the function of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel.
-
-The kernels terminates its computations when all the root are converged. Finally, the solution of the root finding problem is copied back from the GPU global memory to the CPU memory. We use the communication functions of CUDA for the memory allocations in the GPU \verb=(cudaMalloc())= and the data transfers from the CPU memory to the GPU memory \verb=(cudaMemcpyHostToDevice)=
-or from the GPU memory to the CPU memory \verb=(cudaMemcpyDeviceToHost))=.
-\subsection{Experimental study}
-
-\subsubsection{Definition of the polynomial used}
-We use a polynomial of the following form for which the
-roots are distributed on 2 distinct circles:
+If the modulus
+of the current complex is less than a given value called the
+radius i.e. ($ |z^{k}_{i}|<= R$), then the classical form of the EA
+function Eq.~\ref{Eq:Hi} is executed else the EA.EL
+function Eq.~\ref{Log_H2} is executed.
+(with Eq.~\ref{deflncomplex}, Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as in Eq.~\ref{R.EL}.
+
+The last kernel checks the convergence of the roots after each update
+of $Z^{k}$, according to formula Eq.~\ref{eq:Aberth-Conv-Cond}. We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel.
+
+The kernel terminates its computations when all the roots have
+converged. It should be noticed that, as blocks of threads are
+scheduled automatically by the GPU, we have absolutely no control on
+the order of the blocks. Consequently, our algorithm is executed more
+or less with the asynchronous iteration model, where blocks of roots
+are updated in a non deterministic way. As the Durand-Kerner method
+has been proved to converge with asynchronous iterations, we think it
+is similar with the Ehrlich-Aberth method, but we did not try to prove
+this in that paper. Another consequence of that, is that several
+executions of our algorithm with the same polynomial do not
+necessarily give the same result (but roots have the same accuracy)
+and the same number of iterations (even if the variation is not very
+significant).
+
+
+
+
+
+%%HIER END MY REVISIONS (SIDER)
+\section{Experimental study}
+\label{sec6}
+%\subsection{Definition of the used polynomials }
+We study two categories of polynomials: sparse polynomials and the full polynomials.\\
+{\it A sparse polynomial} is a polynomial for which only some
+coefficients are not null. In this paper, we consider sparse polynomials for which the roots are distributed on 2 distinct circles:
\begin{equation}
- \forall \alpha_{1} \alpha_{2} \in C,\forall n_{1},n_{2} \in N^{*}; P(z)= (z^{n^{1}}-\alpha_{1})(z^{n^{2}}-\alpha_{2})
-\end{equation}
-
-This form makes it possible to associate roots having two
-different modules and thus to work on a polynomial constitute
-of four non zero terms.
-\\
- An other form of the polynomial to obtain a full polynomial is:
+ \forall \alpha_{1} \alpha_{2} \in C,\forall n_{1},n_{2} \in N^{*}; P(z)= (z^{n_{1}}-\alpha_{1})(z^{n_{2}}-\alpha_{2})
+\end{equation}\noindent
+{\it A full polynomial} is, in contrast, a polynomial for which
+all the coefficients are not null. A full polynomial is defined by:
%%\begin{equation}
%%\forall \alpha_{i} \in C,\forall n_{i}\in N^{*}; P(z)= \sum^{n}_{i=1}(z^{n^{i}}.a_{i})
%%\end{equation}
\begin{equation}
- {\Large \forall a_{i} \in C; p(x)=\sum^{n-1}_{i=1} a_{i}.x^{i}}
+ {\Large \forall a_{i} \in C, i\in N; p(x)=\sum^{n}_{i=0} a_{i}.x^{i}}
\end{equation}
-with this formula, we can have until \textit{n} non zero terms.
-
-\subsubsection{The study condition}
-In order to have representative average values, for each
-point of our curves we measured the roots finding of 10
-different polynomials.
-
-The our experiences results concern two parameters which are
-the polynomial degree and the execution time of our program
-to converge on the solution. The polynomial degree allows us
-to validate that our algorithm is powerful with high degree
-polynomials. The execution time remains the
-element-key which justifies our work of parallelization.
- For our tests we used a CPU Intel(R) Xeon(R) CPU
-E5620@2.40GHz and a GPU Tesla C2070 (with 6 Go of ram)
-
-\subsubsection{Comparative study}
-We initially carried out the convergence of Aberth algorithm with various sizes of polynomial, in second we evaluate the influence of the size of the threads per block....
-
-\paragraph{Aberth algorithm on CPU and GPU}
-
-\begin{table}[!ht]
- \centering
- \begin{tabular} {|R{2cm}|L{2.5cm}|L{2.5cm}|L{1.5cm}|L{1.5cm}|}
- \hline Polynomial's degrees & $T_{exe}$ on CPU & $T_{exe}$ on GPU & CPU iteration & GPU iteration\\
- \hline 5000 & 1.90 & 0.40 & 18 & 17\\
- \hline 10000 & 172.723 & 0.59 & 21 & 24\\
- \hline 20000 & 172.723 & 1.52 & 21 & 25\\
- \hline 30000 & 172.723 & 2.77 & 21 & 33\\
- \hline 50000 & 172.723 & 3.92 & 21 & 18\\
- \hline 500000 & $>$1h & 497.109 & & 24\\
- \hline 1000000 & $>$1h & 1,524.51& & 24\\
- \hline
- \end{tabular}
- \caption{the convergence of Aberth algorithm}
- \label{tab:theConvergenceOfAberthAlgorithm}
-\end{table}
+%With this form, we can have until \textit{n} non zero terms whereas the sparse ones have just two non zero terms.
+
+%\subsection{The study condition}
+%Two parameters are studied are
+%the polynomial degree and the execution time of our program
+%to converge on the solution. The polynomial degree allows us
+%to validate that our algorithm is powerful with high degree
+%polynomials. The execution time remains the
+%element-key which justifies our work of parallelization.
+For our tests, a CPU Intel(R) Xeon(R) CPU
+E5620@2.40GHz and a GPU K40 (with 6 Go of ram) are used.
+
+
+%\subsection{Comparative study}
+%First, performances of the Ehrlich-Aberth method of root finding polynomials
+%implemented on CPUs and on GPUs are studied.
+
+We performed a set of experiments on the sequential and the parallel algorithms, for both sparse and full polynomials of different sizes. We took into account the execution times, the polynomial size and the number of threads per block performed by sum or each experiment on CPU and on GPU.
+
+All experimental results obtained from the simulations are made in
+double precision data, the convergence threshold of the methods is set
+to $10^{-7}$.
+%Since we were more interested in the comparison of the
+%performance behaviors of Ehrlich-Aberth and Durand-Kerner methods on
+%CPUs versus on GPUs.
+The initialization values of the vector solution
+of the methods are given in Section~\ref{sec:vec_initialization}.
+
+\subsection{Comparison of execution times of the Ehrlich-Aberth method
+ on a CPU with OpenMP (1 core and 4 cores) vs. on a Tesla GPU}
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[width=0.8\textwidth]{figures/openMP-GPU}
+\caption{Comparison of execution times of the Ehrlich-Aberth method
+ on a CPU with OpenMP (1 core, 4 cores) and on a Tesla GPU}
+\label{fig:01}
+\end{figure}
+%%Figure 1 %%show a comparison of execution time between the parallel
+%%and sequential version of the Ehrlich-Aberth algorithm with sparse
+%%polynomial exceed 100000,
+
+In Figure~\ref{fig:01}, we report the execution times of the
+Ehrlich-Aberth method on one core of a Quad-Core Xeon E5620 CPU, on
+four cores on the same machine with \textit{OpenMP} and on a Nvidia
+Tesla K40 GPU. We chose different sparse polynomials with degrees
+ranging from 100,000 to 1,000,000. We can see that the implementation
+on the GPU is faster than those implemented on the CPU.
+However, the execution time for the
+CPU (4 cores) implementation exceed 5,000s for 250,000 degrees
+polynomials. On the other hand, the GPU implementation for the same
+polynomials do not take more 100s. With the GPU
+we can solve high degree polynomials very quickly up to degree 1,000,000. We can also notice that the GPU implementation are
+ almost 40 times faster then the implementation on the CPU (4 cores).
+
+
+
+
+%This reduction of time allows us to compute roots of polynomial of more important degree at the same time than with a CPU.
-\paragraph{The impact of the thread's number into the convergence of Aberth algorithm}
-
-\begin{table}[!h]
- \centering
- \begin{tabular} {|R{2.5cm}|L{2.5cm}|L{2.5cm}|}
- \hline Thread's numbers & Execution time &Number of iteration\\
- \hline 1024 & 523 & 27\\
- \hline 512 & 449.426 & 24\\
- \hline 256 & 440.805 & 24\\
- \hline 128 & 456.175 & 22\\
- \hline 64 & 472.862 & 23\\
- \hline 32 & 830.152 & 24\\
- \hline 8 & 2632.78 & 23 \\
- \hline
- \end{tabular}
- \caption{The impact of the thread's number into the convergence of Aberth algorithm}
- \label{tab:Theimpactofthethread'snumberintotheconvergenceofAberthalgorithm}
-
-\end{table}
-
-\paragraph{A comparative study between Aberth and Durand-kerner algorithm}
-\begin{table}[htbp]
- \centering
- \begin{tabular} {|R{2cm}|L{2.5cm}|L{2.5cm}|L{1.5cm}|L{1.5cm}|}
- \hline Polynomial's degrees & Aberth $T_{exe}$ & D-Kerner $T_{exe}$ & Aberth iteration & D-Kerner iteration\\
- \hline 5000 & 0.40 & 3.42 & 17 & 138 \\
- \hline 50000 & 3.92 & 385.266 & 17 & 823\\
- \hline 500000 & 497.109 & 4677.36 & 24 & 214\\
- \hline
- \end{tabular}
- \caption{Aberth algorithm compare to Durand-Kerner algorithm}
- \label{tab:AberthAlgorithCompareToDurandKernerAlgorithm}
-\end{table}
+ %We notice that the convergence precision is a round $10^{-7}$ for the both implementation on CPU and GPU. Consequently, we can conclude that Ehrlich-Aberth on GPU are faster and accurately then CPU implementation.
+
+\subsection{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
+To optimize the performances of an algorithm on a GPU, it is necessary to maximize the use of cores GPU (maximize the number of threads executed in parallel). In fact, it is interesting to see the influence of the number of threads per block on the execution time of Ehrlich-Aberth algorithm.
+For that, we noticed that the maximum number of threads per block for
+the Nvidia Tesla K40 GPU is 1,024, so we varied the number of threads
+per block from 8 to 1,024. We took into account the execution time for
+10 different sparse and full polynomials of degree 50,000 and of degree 500,000.
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[width=0.8\textwidth]{figures/influence_nb_threads}
+\caption{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
+\label{fig:02}
+\end{figure}
+
+Figure~\ref{fig:02} shows that, the best execution time for both
+sparse and full polynomial are given when the threads number varies
+between 64 and 256 threads per block. We notice that with small
+polynomials the best number of threads per block is 64, whereas for large polynomials the best number of threads per block is
+256. However, in the following experiments we specify that the number
+of threads per block is 256.
+
+
+\subsection{Influence of exp-log solution to compute high degree polynomials}
+
+In this experiment we report the performance of the exp-log solution described in Section~\ref{sec2} to compute high degree polynomials.
+\begin{figure}[htbp]
+\centering
+ \includegraphics[width=0.8\textwidth]{figures/sparse_full_explog}
+\caption{The impact of exp-log solution to compute high degree polynomials}
+\label{fig:03}
+\end{figure}
+
+
+Figure~\ref{fig:03} shows a comparison between the execution time of
+the Ehrlich-Aberth method using the exp-log solution and the
+execution time of the Ehrlich-Aberth method without this solution,
+with full and sparse polynomials degrees. We can see that the
+execution times for both algorithms are the same with full polynomials
+degree inferior to 4,000 and sparse polynomials inferior to 150,000. We
+also clearly show that the classical version (without exp-log) of
+Ehrlich-Aberth algorithm does not converge after these degrees with
+sparse and full polynomials. On the contrary, the new version of
+the Ehrlich-Aberth algorithm with the exp-log solution can solve
+high degree polynomials.
+
+%in fact, when the modulus of the roots are up than \textit{R} given in ~\ref{R},this exceed the limited number in the mantissa of floating points representations and can not compute the iterative function given in ~\ref{eq:Aberth-H-GS} to obtain the root solution, who justify the divergence of the classical Ehrlich-Aberth algorithm. However, applying exp-log solution given in ~\ref{sec2} took into account the limit of floating using the iterative function in(Eq.~\ref{Log_H1},Eq.~\ref{Log_H2} and allows to solve a very large polynomials degrees .
+
+
+
+
+\subsection{Comparison of the Durand-Kerner and the Ehrlich-Aberth methods}
+
+In this part, we compare the Durand-Kerner and the Ehrlich-Aberth
+methods on GPU. We took into account the execution times, the number of iterations and the polynomials size for both sparse and full polynomials.
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[width=0.8\textwidth]{figures/EA_DK}
+\caption{Execution times of the Durand-Kerner and the Ehrlich-Aberth methods on GPU}
+\label{fig:04}
+\end{figure}
+
+Figure~\ref{fig:04} shows the execution times of both methods with
+sparse polynomial degrees ranging from 1,000 to 1,000,000. We can see
+that the Ehrlich-Aberth algorithm is faster than Durand-Kerner
+algorithm, being on average 25 times faster. Then, when degrees of
+polynomials exceed 500,000 the execution times with DK are very long.
+
+%with double precision not exceed $10^{-5}$.
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[width=0.8\textwidth]{figures/EA_DK_nbr}
+\caption{The number of iterations to converge for the Ehrlich-Aberth
+ and the Durand-Kerner methods}
+\label{fig:05}
+\end{figure}
+
+Figure~\ref{fig:05} shows the evaluation of the number of iterations according
+to the degree of polynomials for both EA and DK algorithms. We can see
+that the number of iterations of DK is of order 100 while EA is of order
+10. Indeed the computation of the derivative of P in the iterative function (Eq.~\ref{Eq:Hi}) executed by EA
+allows the algorithm to converge faster. On the contrary, the
+DK operator (Eq.~\ref{DK}) needs low operations, consequently low
+execution times per iteration, but it needs more iterations to converge.
+
+
+
+
+ \section{Conclusion and perspectives}
+\label{sec7}
+In this paper we have presented the parallel implementation
+Ehrlich-Aberth method on GPU for the problem of finding roots
+polynomial. Moreover, we have improved the classical Ehrlich-Aberth
+method which suffers from overflow problems, the exp-log solution
+applied to the iterative function allows to solve high degree
+polynomials.
+
+We have performed many experiments with the Ehrlich-Aberth method in
+GPU. These experiments highlight that this method is more efficient in
+GPU than all the other implementations. The improvement with
+the exponential logarithm solution allows us to solve sparse and full
+high degree polynomials up to 1,000,000 degree. Hence, it may be
+possible to consider using polynomial root finding methods in other
+numerical applications on GPU.
+
+
+In future works, we plan to investigate the possibility of using
+several multiple GPUs simultaneously, either with a multi-GPU machine or
+with a cluster of GPUs. It may also be interesting to study the
+implementation of other root finding polynomial methods on GPU.
\bibliography{mybibfile}
-\end{document}
\ No newline at end of file
+\end{document}
