Merge branch 'master' of ssh://info.iut-bm.univ-fcomte.fr/kahina_paper1

[kahina_paper1.git] / paper.tex

\documentclass[review]{elsarticle}

\usepackage{lineno,hyperref}
%%\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}
\setcellgapes{1pt}
\makegapedcells
\newcolumntype{R}[1]{>{\raggedleft\arraybackslash }b{#1}}
\newcolumntype{L}[1]{>{\raggedright\arraybackslash }b{#1}}
\newcolumntype{C}[1]{>{\centering\arraybackslash }b{#1}}
\modulolinenumbers[5]

\journal{Journal of \LaTeX\ Templates}

%%%%%%%%%%%%%%%%%%%%%%%
%% Elsevier bibliography styles
%%%%%%%%%%%%%%%%%%%%%%%
%% To change the style, put a % in front of the second line of the current style and
%% remove the % from the second line of the style you would like to use.
%%%%%%%%%%%%%%%%%%%%%%%

%% Numbered
%\bibliographystyle{model1-num-names}

%% Numbered without titles
%\bibliographystyle{model1a-num-names}

%% Harvard
%\bibliographystyle{model2-names.bst}\biboptions{authoryear}

%% Vancouver numbered
%\usepackage{numcompress}\bibliographystyle{model3-num-names}

%% Vancouver name/year
%\usepackage{numcompress}\bibliographystyle{model4-names}\biboptions{authoryear}

%% APA style
%\bibliographystyle{model5-names}\biboptions{authoryear}

%% AMA style
%\usepackage{numcompress}\bibliographystyle{model6-num-names}

%% `Elsevier LaTeX' style
\bibliographystyle{elsarticle-num}
%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\begin{frontmatter}

\title{Rapid solution of very high degree polynomials root finding using GPU}

%% Group authors per affiliation:
\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}
\cortext[mycorrespondingauthor]{Corresponding author}
\ead{kahina.ghidouche@gmail.com}

\author[mysecondaryaddress]{Couturier Raphael\corref{mycorrespondingauthor}}
%%\cortext[mycorrespondingauthor]{Corresponding author}
\ead{raphael.couturier@univ-fcomte.fr}

\author[mymainaddress]{Abderrahmane Sider\corref{mycorrespondingauthor}}
%%\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 }

\begin{abstract}
Polynomials are mathematical algebraic structures that play a great role in science and engineering. But the process of solving them  for high and large degrees is computationally demanding and still not solved. In this paper, we present the results of a parallel implementation of the Ehrlish-Aberth algorithm for the problem root finding for
high degree polynomials on GPU architectures (Graphics Processing Unit). The main result of this work is to be able to solve high and very large degree polynomials (up to 100000) very 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, Ehrlish-Aberth, Durant-Kerner, GPU, CUDA, CPU , Parallelization
\end{keyword}

\end{frontmatter}

\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 :
%%\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 :
\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}
$x=g(x)$
\end{center}
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
\begin{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
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, mathmathicians have
focussed on numerical (iterative) methods such as the famous
Newton's method, Bernoulli's method of the 18th, and Graeffe's.

Later on, with the advent of electronic computers, other 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 (DK)
method :
%%\begin{center}
\begin{equation}
 Z_{i}=Z_{i}-\frac{P(Z_{i})}{\prod_{i\neq j}(z_{i}-z_{j})}
\end{equation}
%%\end{center}

This formula is mentioned for the first time by
Weiestrass~\cite{Weierstrass03} as part of the fundamental theorem
of Algebra and is rediscovered by Ilieff~\cite{Ilie50},
Docev~\cite{Docev62}, Durand~\cite{Durand60},
Kerner~\cite{Kerner66}. Another method discovered by
Borsch-Supan~\cite{ Borch-Supan63} and also described and brought
in the following form by Ehrlich~\cite{Ehrlich67} and
Aberth~\cite{Aberth73} uses a different iteration formula given as fellows :
%%\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})}}.
\end{equation}
%%\end{center}

Aberth, Ehrlich and Farmer-Loizou~\cite{Loizon83} have proved that
the Ehrlisch-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
drastically increases like the degrees of high polynomials. It is expected that the
parallelization of these algorithms will improve the convergence
time.

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
chain, for polynomials 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}  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.
Couturier et al. ~\cite{Raphaelall01} proposed two methods of parallelisation for
a shared memory architecture and for distributed memory one. They were able to
compute the roots of polynomials of degree 10000 in 430 seconds with only 8
personal computers and 2 communications per iteration. Comparing to the sequential implementation
where it takes up to 3300 seconds to obtain the same results, the authors show an interesting speedup, indeed.

Very few works had been since this last work until the appearing 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
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.


Ghidouche et al. ~\cite{Kahinall14} proposed an implementation of the
Durand-Kerner method on GPU. Their 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 of about 48000. To our knowledge, it is the first time such high degree polynomials are numerically solved.


In this paper, we focus on the implementation of the Aberth method for
high degree polynomials on GPU. The paper is organised as fellows. Initially, we recall the Aberth method in Section.\ref{sec1}. Improvements for the 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.4 we propose a parallel implementation of the Aberth method on GPU and discuss it. Section 5 presents and investigates our implementation and experimental study results. Finally, Section 6 concludes this paper and gives some hints for future research directions in this topic.  

\section{The Sequential Aberth method}
\label{sec1}
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 factors 

\begin{equation}
w_{i}(z)=\prod_{j=1,j \neq i}^{n} (z-x_{j})
\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}

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_{i}}}, i=1,2,...,n
\end{equation}

Substituting $x_{j}$ for z we obtain the Aberth iteration method.

In the fellowing we present the main stages of the running of the Aberth method.

\subsection{Polynomials Initialization}
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}

Like 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 $n$
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 et al.~\cite{Bini96}
performed this choice by selecting complex numbers along different
circles and 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}};\\
%%\end{align}
\end{equation}
Where:
\begin{equation}
u_{i}=2.|a_{i}|^{\frac{1}{i}};
v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
\end{equation}

\subsection{Iterative Function $H_{i}$}
The operator used by the Aberth method is corresponding to the
following equation which will enable the convergence towards
polynomial solutions, provided all the roots are distinct.

\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}}}}
\end{equation}

\subsection{Convergence Condition}
The convergence condition determines the termination of the algorithm. It consists in stopping from running
the iterative function $H_{i}(z)$ when the roots are sufficiently stable. We consider that the method
converges sufficiently when :

\begin{equation}
\label{eq:Aberth-Conv-Cond}
\forall i \in
[1,n];\frac{z_{i}^{(k)}-z_{i}^{(k-1)}}{z_{i}^{(k)}}<\xi
\end{equation}


\section{Improving the Ehrlisch-Aberth Method}
\label{sec2}
The Ehrlisch-Aberth method implementation suffers of 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 $|x|$ stands for the modolus of a complex $x$. 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 early discussed 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)\label{defexpcomplex}
\end{align}
%%\end{equation}

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's degrees can be looked for successfully~\cite{Karimall98}.

Applying this solution for the Aberth method we obtain the
iteration function with 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),
\end{equation}

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).
\end{equation}

This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated as:

$$R = \exp( \log(DBL\_MAX) / (2*n) )$$ where $DBL\_MAX$ stands for the maximum representable double value.

\section{The implementation of simultaneous methods in a parallel computer}
\label{secStateofArt}   
The main problem of simultaneous methods is that the necessary
time needed for convergence is increased when we increase
the degree of the polynomial. The parallelisation 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 by Miranker
~\cite{Mirankar68,Mirankar71}, Schedler~\cite{Schedler72} and
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 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 Ehrlisch-Aberth methods are the most practical choices among
them~\cite{Bini04}. These two methods have been extensively
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 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 (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 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 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 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 CUDA parallel Ehrlisch-Aberth method}
In the following, we describe the parallel implementation of Ehrlisch-Aberth method on GPU 
for solving high degree polynomials. First, the hardware and software of the GPUs are presented. Then, a CUDA parallel Ehrlisch-Aberth method are 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.

\subsection{ The implementation of Aberth method on GPU}
%%\subsection{A CUDA implementation of the Aberth's method }
%%\subsection{A GPU implementation of the Aberth's method }



\subsubsection{A sequential Aberth algorithm}
The main steps of Aberth method are shown in Algorithm.~\ref{alg1-seq} :
  
\begin{algorithm}[H]
\label{alg1-seq}
%\LinesNumbered
\caption{A sequential algorithm to find roots with the Aberth method}

\KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error
tolerance threshold),P(Polynomial to solve)}

\KwOut {Z(The solution root's vector)}

\BlankLine

Initialization of the coefficients of the polynomial to solve\;
Initialization of the solution vector $Z^{0}$\;

\While {$\Delta z_{max}\succ \epsilon$}{
 Let $\Delta z_{max}=0$\;
\For{$j \gets 0 $ \KwTo $n$}{
$ZPrec\left[j\right]=Z\left[j\right]$\;
$Z\left[j\right]=H\left(j,Z\right)$\;
}

\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}

~\\ 
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.
There exists two ways to execute the iterative function that we call a Jacobi one and a 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 :

\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}

With the the Gauss-seidel iteration, we have:
\begin{equation}
\label{eq:Aberth-H-GS}
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}

Using Equation.~\ref{eq:Aberth-H-GS} for the update sub-step of $H(i,z^{k+1})$, we expect the Gauss-Seidel iteration to converge more quickly because, just as its ancestor (for solving linear systems of equations), it uses the most fresh computed roots $z^{k+1}_{i}$.

The $4^{th}$ step of the algorithm checks the convergence condition using Equation.~\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.

\paragraph{The execution time}
Let $T_{i}(n)$ be the time to compute one new root value at step 3, $T_{i}$ depends on the polynomial's degree $n$. When $n$ increase $T_{i}(n)$ increases too. We need $n.T_{i}(n)$ to compute all the new values in one iteration at step 3.

Let $T_{j}$ be the time needed to check the convergence of one root value at the step 4, so we need $n.T_{j}$ to compute global convergence condition in each iteration at step 4.

Thus, the execution time for both steps 3 and 4 is:
\begin{equation}
T_{iter}=n(T_{i}(n)+T_{j})+O(n).
\end{equation}
Let $K$ be the number of iterations necessary to compute all the roots, so the total execution time $T$ can be given as:

\begin{equation}
\label{eq:T-global}
T=\left[n\left(T_{i}(n)+T_{j}\right)+O(n)\right].K
\end{equation}
The execution time increases with the increasing of the polynomial degree, which justifies to parallelise these steps  in order to reduce the global execution time. In the following,  we explain how we did parrallelize these steps on a GPU architecture using the CUDA platform.

\subsubsection{A Parallel implementation with CUDA }
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 schduler 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. It's up to the programmer, to describe the execution context, that is the size of the Grid, the number of blocks and the number of threads per block upon the call of a given kernel, according to a special syntax defined by CUDA.

Let N be the number of threads executed in parallel,  Equation.~\ref{eq:T-global} becomes then : 

\begin{equation}
T=\left[\frac{n}{N}\left(T_{i}(n)+T_{j}\right)+O(n)\right].K.
\end{equation}

In theory, total execution time $T$ on GPU is speed up $N$ times as  $T$ on CPU. We will see at what extent this is true in the experimental study hereafter. 
~\\
~\\
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 ``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 Aberth algorithm usind CUDA.

\begin{algorithm}[H]
\label{alg2-cuda}
%\LinesNumbered
\caption{CUDA Algorithm to find roots with the Aberth method}

\KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error
tolerance threshold),P(Polynomial to solve)}

\KwOut {Z(The solution root's vector)}

\BlankLine

Initialization of the coeffcients of the polynomial to solve\;
Initialization of the solution vector $Z^{0}$\;
Allocate and copy initial data to the GPU global memory\;

\While {$\Delta z_{max}\succ \epsilon$}{
 Let $\Delta z_{max}=0$\;
$ kernel\_save(d\_z^{k-1})$\;
$ kernel\_update(d\_z^{k})$\;
$kernel\_testConverge(\Delta z_{max},d_z^{k},d_z^{k-1})$\;
}
\end{algorithm}
~\\ 

After the initialisation 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 access data already present 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 named \textit{save} in line 6 of Algorithm~\ref{alg2-cuda} consists in saving the vector of polynomial's root found at the previous time-step in GPU memory, in order to check the convergence of the roots after each iteration (line 8, Algorithm~\ref{alg2-cuda}).

The second kernel executes the iterative function $H$ and updates $z^{k}$, according to Algorithm~\ref{alg3-update}. We notice that the update kernel is called in two forms, separated with the value of \emph{R} which determines the radius beyond which we apply the logarithm computation of the power of a complex. 

\begin{algorithm}[H]
\label{alg3-update}
%\LinesNumbered
\caption{A global Algorithm for the iterative function}

\eIf{$(\left|Z^{(k)}\right|<= R)$}{
$kernel\_update(d\_z^{k})$\;}
{
$kernel\_update\_Log(d\_z^{k})$\;
}
\end{algorithm}

The first form executes formula \ref{eq:SimplePolynome} if the modulus of the current complex is less than the a certain value called the radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes formulas (Eq.~\ref{deflncomplex},Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as :

$$R = \exp( \log(DBL\_MAX) / (2*n) )$$ where $DBL\_MAX$ stands for the maximum representable double value.

The last kernel verifies the convergence of the roots after each update of $Z^{(k)}$, according to formula. We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. 

The kernels terminate it computations when all the roots converge. Finally, the solution of the root finding problem is copied back from GPU global memory to CPU memory. We use the communication functions of CUDA for the memory allocation in the GPU \verb=(cudaMalloc())= and for data transfers from the CPU memory to the GPU memory \verb=(cudaMemcpyHostToDevice)=
or from GPU memory to CPU memory \verb=(cudaMemcpyDeviceToHost))=. 
%%HIER END MY REVISIONS (SIDER)
\section{Experimental study}

\subsection{Definition of the polynomial used}
We study two forms of  polynomials the sparse polynomials and the full polynomials:
\paragraph{Sparse polynomial}: in this following form, 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.

\paragraph{Full polynomial}: the second form used to obtain a full polynomial is:
%%\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, i\in N;  p(x)=\sum^{n}_{i=0} a_{i}.x^{i}} 
\end{equation}
with this form, we can have until \textit{n} non zero terms.

\subsection{The study condition} 
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 K40 (with 6 Go of ram).


\subsection{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....

\subsubsection{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}
 
\begin{figure}[H]
\centering
  \includegraphics[width=0.8\textwidth]{figures/Compar_EA_algorithm_CPU_GPU}
\caption{Aberth algorithm on CPU and GPU}
\label{fig:01}
\end{figure}


\subsubsection{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}


\begin{figure}[H]
\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:01}
\end{figure}

\subsubsection{The impact of exp-log solution to compute very high degrees of  polynomial}
\begin{figure}[H]
\centering
  \includegraphics[width=0.8\textwidth]{figures/log_exp}
\caption{The impact of exp-log solution to compute very high degrees of  polynomial.}
\label{fig:01}
\end{figure}

\subsubsection{A comparative study between Aberth and Durand-kerner algorithm}


\begin{figure}[H]
\centering
  \includegraphics[width=0.8\textwidth]{figures/EA_DK}
\caption{Ehrlisch-Aberth and Durand-Kerner algorithm on GPU}
\label{fig:01}
\end{figure}


\bibliography{mybibfile}

\end{document}