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[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index c213794d35699bf2504b82e3ecaaa3681f8c45b0..104bcbe3f09960c129674f962433e64c584e739c 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -161,7 +161,7 @@ drastically increases like the degrees of high polynomials. It is expected that
 parallelization of these algorithms will improve the convergence
 time.
 
 parallelization of these algorithms will improve the convergence
 time.
 

-Many authors have dealt with parallelisation of
+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
 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


@@ -272,9 +272,10 @@ i}{\frac{1}{z_{i}-z_{j}}}}
 \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
 \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:
+converges sufficiently when :

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


@@ -306,7 +307,7 @@ propose to use the logarithm and the exponential of a complex in order to comput
 \end{align}
 %%\end{equation}
 
 \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
+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
 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


@@ -341,51 +342,42 @@ Winogard~\cite{Winogard72}. The second approach aims at reducing the
 computation time per iteration, as reported
 in~\cite{Benall68,Jana06,Janall99,Riceall06}. 
 
 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
+There are many schemes for the simultaneous approximation of all roots of a given

 polynomial. Several works on different methods and issues of root
 polynomial. Several works on different methods and issues of root

-finding have been reported in~\cite{Azad07, Gemignani07, Kalantari08, Skachek08, Zhancall08, Zhuall08}. However, Durand-Kerner and Ehrlich methods are the most practical choices among
+finding have been reported in~\cite{Azad07, Gemignani07, Kalantari08, 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
 them~\cite{Bini04}. These two methods have been extensively

-studied for parallelization due to their following advantages. The
-computation involved in these methods has some inherent
+studied for parallelization due to their 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
 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
+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
 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
+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
 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
+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
 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
+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
 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.
+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 parallel implementation of Aberth method}
+\section {A CUDA parallel Ehrlisch-Aberth method}

 
 \subsection{Background on the GPU architecture}
 A GPU is viewed as an accelerator for the data-parallel and
 
 \subsection{Background on the GPU architecture}
 A GPU is viewed as an accelerator for the data-parallel and


@@ -396,9 +388,9 @@ 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
 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}
+global memory shared by all its SPs~\cite{NVIDIA10}.

 
 

-    On a CPU equipped with a GPU, all the data-parallel and intensive
+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
 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


@@ -417,12 +409,12 @@ 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
 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
+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
 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
+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
 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


@@ -450,12 +442,12 @@ read-only caches.
 
 
 \subsubsection{A sequential Aberth algorithm}
 
 
 \subsubsection{A sequential Aberth algorithm}

-The means steps of Aberth method can expressed as an algorithm
-like:
+The main steps of Aberth method are shown in Algorithm.~\ref{alg1-seq} :

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

+\label{alg1-seq}

 %\LinesNumbered
 %\LinesNumbered

-\caption{Algorithm to find root polynomial with Aberth method}
+\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)}
 
 \KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error
 tolerance threshold),P(Polynomial to solve)}


@@ -464,7 +456,7 @@ tolerance threshold),P(Polynomial to solve)}
 
 \BlankLine
 
 
 \BlankLine
 

-Initialization of the parameter of the polynomial to solve\;
+Initialization of the coefficients of the polynomial to solve\;

 Initialization of the solution vector $Z^{0}$\;
 
 \While {$\Delta z_{max}\succ \epsilon$}{
 Initialization of the solution vector $Z^{0}$\;
 
 \While {$\Delta z_{max}\succ \epsilon$}{


@@ -483,57 +475,62 @@ $\Delta z_{max}$=c\;}
 \end{algorithm}
 
 ~\\ 
 \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:
+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 $z^{(k+1)}_{i}$, taht 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}
 
 
 \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:
+With the the Gauss-seidel iteration, we have:

 \begin{equation}
 \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.
+\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}
 
 \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.
+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 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.
+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}
 
 \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 $T_{i}(n)$ 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 $T_{j}$: 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.
+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.

 
 

-The execution time for both steps 3 and 4 can see like:
+Thus, the execution time for both steps 3 and 4 is:

 \begin{equation}
 \begin{equation}

-T_{exe}=N(T_{i}(N)+T_{j})+O(n).
+T_{iter}=n(T_{i}(n)+T_{j})+O(n).

 \end{equation}
 \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:
+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}
 
 \begin{equation}

-Total\_time_{exe}=\left[N\left(T_{i}(N)+T_{j}\right)+O(n)\right].Nbr\_iter
+\label{eq:T-global}
+T=\left[n\left(T_{i}(n)+T_{j}\right)+O(n)\right].K

 \end{equation}
 \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.
+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{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.
+\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 nbr\_thread be the number of threads executed in parallel, so you can easily transform the (18)formula like this: 
+Let N be the number of threads executed in parallel,  Equation.~\ref{eq:T-global} becomes then : 

 
 \begin{equation}
 
 \begin{equation}

-Total\_time_{exe}=\left[\frac{N}{nbr\_thread}\left(T_{i}(N)+T_{j}\right)+O(n)\right].Nbr\_iter.
+T=\left[\frac{n}{N}\left(T_{i}(n)+T_{j}\right)+O(n)\right].K.

 \end{equation}
 
 \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 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 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:
+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]
 
 \begin{algorithm}[H]

+\label{alg2-cuda}

 %\LinesNumbered
 %\LinesNumbered

-\caption{Algorithm to find root polynomial with Aberth method}
+\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)}
 
 \KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error
 tolerance threshold),P(Polynomial to solve)}


@@ -542,24 +539,25 @@ tolerance threshold),P(Polynomial to solve)}
 
 \BlankLine
 
 
 \BlankLine
 

-Initialization of the parameter of the polynomial to solve\;
+Initialization of the coeffcients of the polynomial to solve\;

 Initialization of the solution vector $Z^{0}$\;
 Initialization of the solution vector $Z^{0}$\;

-Allocate and fill the data in the global memory GPU\;
+Allocate and copy initial data to the GPU global memory\;

 
 \While {$\Delta z_{max}\succ \epsilon$}{
  Let $\Delta z_{max}=0$\;
 
 \While {$\Delta z_{max}\succ \epsilon$}{
  Let $\Delta z_{max}=0$\;

-$ kernel\_save(d\_Z^{k-1})$\;
+$ kernel\_save(d\_z^{k-1})$\;

 $ kernel\_update(d\_z^{k})$\;
 $ kernel\_update(d\_z^{k})$\;

-$kernel\_testConverge (d_?z_{max},d_Z^{k},d_Z^{k-1})$\;
+$kernel\_testConverge(\Delta z_{max},d_z^{k},d_z^{k-1})$\;

 }
 \end{algorithm}
 ~\\ 
 
 }
 \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).
+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 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: 
+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]
 
 \begin{algorithm}[H]

+\label{alg3-update}

 %\LinesNumbered
 \caption{A global Algorithm for the iterative function}
 
 %\LinesNumbered
 \caption{A global Algorithm for the iterative function}
 


@@ -570,14 +568,15 @@ $kernel\_update\_Log(d\_z^{k})$\;
 }
 \end{algorithm}
 
 }
 \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:
+The first form executes the formula (8) if the modulus is of the current complex is less than the radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes formulas (13,14). The radius R is evaluated as :

 
 

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

 
 

-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 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 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)=
+The kernels terminates it 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))=. 
 or from the GPU memory to the CPU memory \verb=(cudaMemcpyDeviceToHost))=. 

+%%HIER END MY REVISIONS (SIDER)

 \subsection{Experimental study}
 
 \subsubsection{Definition of the polynomial used}
 \subsection{Experimental study}
 
 \subsubsection{Definition of the polynomial used}


@@ -657,6 +656,16 @@ We initially carried out the convergence of Aberth algorithm with various sizes
                
 \end{table}
 
                
 \end{table}
 

+
+\begin{figure}[htbp]
+\centering
+  \includegraphics[width=0.8\textwidth]{figures/influence_nb_threads}
+\caption{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
+\label{fig:01}
+\end{figure}
+
+
+

 \paragraph{A comparative study between Aberth and Durand-kerner algorithm}
 \begin{table}[htbp]
        \centering
 \paragraph{A comparative study between Aberth and Durand-kerner algorithm}
 \begin{table}[htbp]
        \centering


@@ -672,7 +681,6 @@ We initially carried out the convergence of Aberth algorithm with various sizes
 \end{table}
 
 
 \end{table}
 
 

-

 \bibliography{mybibfile}
 
 \end{document}
 \bibliography{mybibfile}
 
 \end{document}