[kahina_paper1.git] / paper.tex

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\begin{document}

\begin{frontmatter}

\title{A parallel  root finding polynomial on 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 Raphaël\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
  Béjaia, Algeria}
\address[mysecondaryaddress]{FEMTO-ST Institute, University of
  Bourgogne Franche-Comte }

\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).
\end{abstract}

\begin{keyword}
root finding of polynomials, high degree, iterative methods, Durant-Kerner, GPU, CUDA, CPU , Parallelization
\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 $\alpha_{i},\textit{i=1,...,n}$. 
%%\begin{center}
\begin{equation}
     {\Large p(x)=\sum{a_{i}x^{i}}=a_{n}\prod(x-\alpha_{i}),a_{0} a_{n}\neq 0}
\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
\begin{center}
$x=g(x).  $
\end{center}
Where $g : C^{n}\longrightarrow C^{n}$. Note that 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 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
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
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 from
Weiestrass~\cite{Weierstrass03} as part of the fundamental theorem
of Algebra and is rediscovered from Ilieff~\cite{Ilie50},
Docev~\cite{Docev62}, Durand~\cite{Durand60},
Kerner~\cite{Kerner66}. Another method discovered from
Borsch-Supan~\cite{ Borch-Supan63} and also described and brought
in the following form from Ehrlich~\cite{Ehrlich67} and
Aberth~\cite{Aberth73}.
%%\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 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
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
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
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.  

\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}

And rational function $R_{i}(z)$ be the correction term of
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

Let present the means stages of Aberth method.

\subsection{Polynomials Initialization}
 The initialization of polynomial P(z) with complex coefficients
 are given by:

\begin{equation}
  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
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}.

\begin{equation}
%%\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 with 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}
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:

\begin{equation}
\forall i \in
[1,n];\frac{z_{i}^{(k)}-z_{i}^{(k-1)}}{z_{i}^{(k)}}<\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:

\begin{equation}
 \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}
 \forall(x,y)\in R^{*2}; \exp(x+i.y) & = \exp(x).\exp(i.y)\\
                                     & =\exp(x).\cos(y)+i.\exp(x).\sin(y)
\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}.

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 applying when it is necessary

\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
~\cite{Mirankar68,Mirankar71}, Schedler~\cite{Schedler72} and
Winogard~\cite{Winogard72}. Another approach is to reduce 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
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
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
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
algorithms reported for these methods can be found
in~\cite{Cosnard90, Freeman89,Freemanall90,,Jana99,Janall99}.
Freeman and Bane~\cite{Freemanall90} presented two parallel
algorithms on a local memory MIMD computer with the compute-to
communication time ratio O(n). However, their algorithms require
each processor to communicate its current approximation to all
other processors at the end of each iteration. Therefore they
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
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}
%%\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)}

\BlankLine

Initialization of the parameter 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 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:

\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}

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}

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.

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.

\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.

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.

The execution time for both steps 3 and 4 can see like:
\begin{equation}
T_{exe}=N(T_{i}(N)+T_{j})+O(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:

\begin{equation}
Total\_time_{exe}=\left[N\left(T_{i}(N)+T_{j}\right)+O(n)\right].Nbr\_iter
\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.

Let nbr\_thread be the number of threads executed in parallel, so you can easily transform the (18)formula like this: 

\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:

\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)}

\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\;

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

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: 

\begin{algorithm}[H]
\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 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:
\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:
%%\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}} 
\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}
 
\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}



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