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 direct methods. Since then, mathmathicians have
+be solved by direct methods. Since then, mathematicians have
focussed on numerical (iterative) methods such as the famous
Newton method, the Bernoulli method of the 18th, and the Graeffe method.
method:
%%\begin{center}
\begin{equation}
- 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,
+\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
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 :
+Aberth~\cite{Aberth73} uses a different iteration formula given as:
%%\begin{center}
\begin{equation}
\label{Eq: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,
+ 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
parallelization of these algorithms will improve the convergence
time.
-Many authors have dealt with the parallelisation of
+Many authors have dealt with the parallelization 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
+Freeman~\cite{Freeman89} implemented 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,
+diverges, but the first two methods have speed-up equal to 5.5. 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 and al~\cite{Raphaelall01} proposed two methods of parallelisation for
+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 polynomials of degree 10000 in 430 seconds with only 8
+compute the roots of sparse 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.
+where it takes up to 3300 seconds to obtain the same results, the authors show an interesting speedup.
-Very few works had been since this last work until the appearing of
+Very few works had been performed 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 CPUs. However, CUDA adopts a totally new computing architecture to use the
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 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 Ehrlich-Aberth method for
-high degree polynomials on GPU. The paper is organized as fellows. 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} 6 concludes this paper and gives some hints for future research directions in this topic.
-
-\section{The Sequential Aberth method}
+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 48000.
+
+
+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} 6 concludes
+this paper and gives some hints for future research directions in this
+topic.
+
+\section{The Sequential Ehrlich-Aberth method}
\label{sec1}
A cubically convergent iteration method for finding zeros of
-polynomials was proposed by O. Aberth~\cite{Aberth73}. In the fellowing we present the main stages of the running of the Aberth method.
+polynomials was proposed by O. Aberth~\cite{Aberth73}. In the
+following we present the main stages of our implementation the Ehrlich-Aberth method.
%The Aberth method is a purely algebraic derivation.
%To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors
\subsection{Vector $z^{(0)}$ Initialization}
-Like for any iterative method, we need to choose $n$ initial guess points $z^{(0)}_{i}, i = 1, . . . , n.$
+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 $n$
+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
v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
\end{equation}
-\subsection{Iterative Function $H_{i}(z^{k})$}
-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.
+\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 Ehrlich-Aberth method:
\begin{equation}
\label{Eq:Hi}
-H_{i}(z^{k+1})=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
+EA2: z^{k+1}=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=0,. . . .,n
\end{equation}
-we notice that the function iterative $H_{i}$ in Eq.~\ref{Eq:Hi} it the same those presented in Eq.~\ref{Eq:EA}, but we prefer used the last one seen the advantage of its use to improve the Aberth method. More detail in the section ~\ref{sec2}.
+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 very high degrees 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 from running
-the iterative function $H_{i}(z)$ when the roots are sufficiently stable. We consider that the method
-converges sufficiently when :
+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}
\label{eq:Aberth-Conv-Cond}
\end{equation}
-\section{Improving the Ehrlich-Aberth Method}
+\section{Improving the Ehrlich-Aberth Method for high degree polynomials with exp.log formulation}
\label{sec2}
-The Ehrlich-Aberth method implementation suffers of overflow problems. This
+With high degree polynomial, the Ehrlich-Aberth method implementation,
+as well as the Durand-Kerner implement, 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 $|x|$ stands for the modolus of a complex $x$. Indeed, the limited number in the
manipulate lower absolute values and the roots for large polynomial's degrees can be looked for successfully~\cite{Karimall98}.
Applying this solution for the Ehrlich-Aberth method we obtain the
-iteration function with logarithm:
+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}
\label{Log_H2}
-H_{i}(z^{k+1})=z_{i}^{k}-\exp \left(\ln \left(
+EA.EL: z^{k+1}=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}
\begin{equation}
\label{Log_H1}
-Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}^{i}))+\ln \left(
+Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}_{i}))+\ln \left(
\sum_{k\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right).
\end{equation}
%\end{equation}
where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.
-\section{The implementation of simultaneous methods in a parallel computer}
+\section{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
+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
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-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
+studied for parallelization due to their intrinsics parallelism, 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
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.
+%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
texture space, which reside in external DRAM, and are accessed via
read-only caches.
-\section{ The implementation of Aberth method on GPU}
+\section{ The implementation of 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 main steps of Aberth method are shown in Algorithm.~\ref{alg1-seq} :
+\subsection{A sequential Ehrlich-Aberth algorithm}
+The main steps of Ehrlich-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)}
+\caption{A sequential algorithm to find roots with the Ehrlich-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
\For{$i \gets 0 $ \KwTo $n-1$}{
$c=\frac{\left|Z\left[i\right]-ZPrec\left[i\right]\right|}{Z\left[i\right]}$\;
-\If{$c\succ\Delta z_{max}$ }{
+\If{$c > \Delta z_{max}$ }{
$\Delta z_{max}$=c\;}
}
}
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}
-z^{k+1}_{i}=\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.
+EAJ: z^{k+1}_{i}=\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 Gauss-Seidel iteration, we have:
\begin{equation}
\label{eq:Aberth-H-GS}
-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.
+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}
%%Here a finiched my revision %%
-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 $z^{k+1}_{i}$.
+Using Equation.~\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 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 parallelize these steps in order to reduce the global execution time. In the following, we explain how we did parallelize these steps on a GPU architecture using the CUDA platform.
-\subsubsection{A Parallel implementation with CUDA }
+\subsection{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 :
+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.
-\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.
+Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth algorithm using CUDA.
\begin{algorithm}[H]
\label{alg2-cuda}
%\LinesNumbered
-\caption{CUDA Algorithm to find roots with the Aberth method}
+\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)}
\begin{algorithm}[H]
\label{alg3-update}
%\LinesNumbered
-\caption{A global Algorithm for the iterative function}
+\caption{Kernel update}
\eIf{$(\left|Z^{(k)}\right|<= R)$}{
$kernel\_update(d\_z^{k})$\;}
%%HIER END MY REVISIONS (SIDER)
\section{Experimental study}
\label{sec6}
-\subsection{Definition of the used polynomials }
+%\subsection{Definition of the used polynomials }
We study two categories of polynomials : the sparse polynomials and the full polynomials.
-\paragraph{A sparse polynomial}: is a polynomial for which only some coefficients are not null. We use in the following polonymial for which the roots are distributed on 2 distinct circles :
+\paragraph{A sparse polynomial}:is a polynomial for which only some coefficients are not null. We use in the following polynomial 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}
-\paragraph{A full polynomial} is in contrast, a polynomial for which all the coefficients are not null. the second form used to obtain a full polynomial is:
+\paragraph{A full polynomial}:is in contrast, a polynomial for which all the coefficients are not null. 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}
\end{equation}
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}
+%\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
E5620@2.40GHz and a GPU K40 (with 6 Go of ram).
-\subsection{Comparative study}
+%\subsection{Comparative study}
In this section, we discuss the performance Ehrlich-Aberth method of root finding polynomials implemented on CPUs and on GPUs.
We performed a set of experiments on the sequential and the parallel algorithms, for both sparse and full polynomials and different sizes. We took into account the execution time, the polynomial size and the number of threads per block performed by sum or each experiment on CPUs and on GPUs.
All experimental results obtained from the simulations are made in double precision data, for a convergence tolerance of the methods 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 Ehrlich-Aberth method are given in section 2.2.
-\subsubsection{The execution time in seconds of Ehrlich-Aberth algorithm on CPU OpenMP (1 core, 4 cores) vs. on a Tesla GPU}
+\subsection{The execution time in seconds of Ehrlich-Aberth algorithm on CPU OpenMP (1 core, 4 cores) vs. on a Tesla GPU}
%\begin{figure}[H]
%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.
-\subsubsection{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
+\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) and to optimize the use of the various memoirs GPU. 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 notice that the maximum number of threads per block for the Nvidia Tesla K40 GPU is 1024, so we varied the number of threads per block from 8 to 1024. We took into account the execution time for both sparse and full of 10 different polynomials of size 50000 and 10 different polynomials of size 500000 degrees.
The figure 2 show that, the best execution time for both sparse and full polynomial are given when the threads number varies between 64 and 256 threads per bloc. We notice that with small polynomials the best number of threads per block is 64, Whereas, the large polynomials the best number of threads per block is 256. However,In the following experiments we specify that the number of thread by block is 256.
-\subsubsection{The impact of exp-log solution to compute very high degrees of polynomial}
+\subsection{The impact of exp-log solution to compute very high degrees of polynomial}
In this experiment we report the performance of log.exp solution describe in ~\ref{sec2} to compute very high degrees polynomials.
\begin{figure}[H]
\label{fig:01}
\end{figure}
-The figure 3, show a comparison between the execution time of the Ehrlich-Aberth algorithm applying log-exp solution and the execution time of the Ehrlich-Aberth algorithm without applying log-exp solution, with full and sparse polynomials degrees. We can see that the execution time for the both algorithms are the same while the full polynomials degrees are less than 4000 and full polynomials are less than 150,000. After,we show clearly that the classical version of Ehrlich-Aberth algorithm (without applying log.exp) stop to converge and can not solving any polynomial sparse or full. In counterpart, the new version of Ehrlich-Aberth algorithm (applying log.exp solution) can solve very high and large full polynomial exceed 100,000 degrees.
+The figure 3, show a comparison between the execution time of the Ehrlich-Aberth algorithm applying exp.log solution and the execution time of the Ehrlich-Aberth algorithm without applying exp.log solution, with full and sparse polynomials degrees. We can see that the execution time for the both algorithms are the same while the full polynomials degrees are less than 4000 and full polynomials are less than 150,000. After,we show clearly that the classical version of Ehrlich-Aberth algorithm (without applying log.exp) stop to converge and can not solving any polynomial sparse or full. In counterpart, the new version of Ehrlich-Aberth algorithm (applying log.exp solution) can solve very high and large full polynomial exceed 100,000 degrees.
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 log.exp 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 .
%we report the performances of the exp.log for the Ehrlich-Aberth algorithm for solving very high degree of polynomial.
-\subsubsection{A comparative study between Ehrlich-Aberth algorithm and Durand-kerner algorithm}
+\subsection{A comparative study between Ehrlich-Aberth algorithm and Durand-kerner algorithm}
In this part, we are interesting to compare the simultaneous methods, Ehrlich-Aberth and Durand-Kerner in parallel computer using GPU. We took into account the execution time, the number of iteration and the polynomial's size. for the both sparse and full polynomials.
\begin{figure}[H]
\section{Conclusion and perspective}
+
\label{sec7}
\bibliography{mybibfile}
