correct

[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index c78df7b0f8e1304ef642b6471ee068440e90627c..8506d8553dc0f2327718d401a98660e5b806b216 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -172,7 +172,7 @@ Aberth~\cite{Aberth73} uses a different iteration formula given as:
 where $P'(z)$ is the polynomial derivative of $P$ evaluated in the
 point $z$.
 
 where $P'(z)$ is the polynomial derivative of $P$ evaluated in the
 point $z$.
 

-Aberth, Ehrlich and Farmer-Loizou~\cite{Loizon83} have proved that
+Aberth, Ehrlich and Farmer-Loizou~\cite{Loizou83} have proved that

 the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of convergence.
 
 
 the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of convergence.
 
 


@@ -186,7 +186,7 @@ time.
 Many authors have dealt with the parallelization of
 simultaneous methods, i.e. that find all the zeros simultaneously. 
 Freeman~\cite{Freeman89} implemented and compared DK, EA and another method of the fourth order proposed
 Many authors have dealt with the parallelization of
 simultaneous methods, i.e. that find all the zeros simultaneously. 
 Freeman~\cite{Freeman89} implemented and compared DK, EA and another method of the fourth order proposed

-by Farmer and Loizou~\cite{Loizon83}, on a 8-processor linear
+by Farmer and Loizou~\cite{Loizou83}, 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 equal to 5.5. Later,
 Freeman and Bane~\cite{Freemanall90}  considered asynchronous
 chain, for polynomials of degree up to 8. The third method often
 diverges, but the first two methods have speed-up equal to 5.5. Later,
 Freeman and Bane~\cite{Freemanall90}  considered asynchronous


@@ -339,7 +339,7 @@ propose to use the logarithm and the exponential of a complex in order to comput
 \begin{align}
 \label{defexpcomplex}
  \forall(x,y)\in R^{*2}; \exp(x+i.y) & = \exp(x).\exp(i.y)\\
 \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}
+                                     & =\exp(x).\cos(y)+i.\exp(x).\sin(y)\label{defexpcomplex1}

 \end{align}
 %%\end{equation}
 
 \end{align}
 %%\end{equation}
 


@@ -391,9 +391,9 @@ 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
 
 There are many schemes for the simultaneous approximation of all roots of a given
 polynomial. Several works on different methods and issues of root

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


@@ -412,21 +412,26 @@ 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.
 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
 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
 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.
+polynomials of 48000.
+%In this paper we present a parallel implementation of Ehrlich-Aberth
+%method on GPUs for sparse and full polynomials with high degree (up
+%to $1,000,000$).

 
 
 \section {A CUDA parallel Ehrlich-Aberth method}
 In the following, we describe the parallel implementation of Ehrlich-Aberth method on GPU 
 
 
 \section {A CUDA parallel Ehrlich-Aberth method}
 In the following, we describe the parallel implementation of Ehrlich-Aberth method on GPU 

-for solving high degree polynomials. First, the hardware and software of the GPUs are presented. Then, a CUDA parallel Ehrlich-Aberth method are presented.
+for solving high degree polynomials (up to $1,000,000$). First, the hardware and software of the GPUs are presented. Then, the CUDA parallel Ehrlich-Aberth method is presented.

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


@@ -484,41 +489,48 @@ 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.
 
 texture space, which reside in external DRAM, and are accessed via
 read-only caches.
 

-\section{ The implementation of Ehrlich-Aberth method on GPU}
+\section{ 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 }
 
 
 
 \label{sec5}
 %%\subsection{A CUDA implementation of the Aberth's method }
 %%\subsection{A GPU implementation of the Aberth's method }
 
 
 

-\subsection{A sequential Ehrlich-Aberth algorithm}
+\subsection{Sequential Ehrlich-Aberth algorithm}

 The main steps of Ehrlich-Aberth method are shown in Algorithm.~\ref{alg1-seq} :
 The main steps of Ehrlich-Aberth method are shown in Algorithm.~\ref{alg1-seq} :

-  
+%\LinesNumbered  

 \begin{algorithm}[H]
 \label{alg1-seq}
 \begin{algorithm}[H]
 \label{alg1-seq}

-%\LinesNumbered
+

 \caption{A sequential algorithm to find roots with the Ehrlich-Aberth method}
 
 \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)}
+\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (error tolerance
+  threshold), P (Polynomial to solve), $\Delta z_{max}$ (maximum value
+  of stop condition), k (number of iteration), n (Polynomial's degrees)}
+\KwOut {Z (The solution root's vector), ZPrec (the previous solution root's vector)}

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

+$\Delta z_{max}=0$\;
+ k=0\;

 
 

-\While {$\Delta z_{max}\succ \epsilon$}{
+\While {$\Delta z_{max} > \varepsilon$}{

  Let $\Delta z_{max}=0$\;
 \For{$j \gets 0 $ \KwTo $n$}{
  Let $\Delta z_{max}=0$\;
 \For{$j \gets 0 $ \KwTo $n$}{

-$ZPrec\left[j\right]=Z\left[j\right]$\;
-$Z\left[j\right]=H\left(j,Z\right)$\;
+$ZPrec\left[j\right]=Z\left[j\right]$;// save Z at the iteration k.\
+
+$Z\left[j\right]=H\left(j,Z\right)$;//update Z with the iterative function.\

 }
 }

+k=k+1\;

 
 \For{$i \gets 0 $ \KwTo $n-1$}{
 
 \For{$i \gets 0 $ \KwTo $n-1$}{

-$c=\frac{\left|Z\left[i\right]-ZPrec\left[i\right]\right|}{Z\left[i\right]}$\;
+$c= testConverge(\Delta z_{max},ZPrec\left[j\right],Z\left[j\right])$\;

 \If{$c > \Delta z_{max}$ }{
 $\Delta z_{max}$=c\;}
 }
 \If{$c > \Delta z_{max}$ }{
 $\Delta z_{max}$=c\;}
 }

+

 }
 \end{algorithm}
 
 }
 \end{algorithm}
 


@@ -536,13 +548,15 @@ With the Gauss-Seidel iteration, we have:
 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 %%
 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} 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 $z^{k+1}_{i}$.
+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 any Jacobi algorithm (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.
 
 
 
 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.
 
 

-\subsection{A Parallel implementation with CUDA }
+\subsection{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.
 
 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.
 


@@ -555,8 +569,7 @@ Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth algorithm using C
 %\LinesNumbered
 \caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
 
 %\LinesNumbered
 \caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
 

-\KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error
-tolerance threshold),P(Polynomial to solve)}
+\KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error tolerance threshold), P(Polynomial to solve), $\Delta z_{max}$ (maximum value of stop condition)}

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


@@ -565,12 +578,14 @@ tolerance threshold),P(Polynomial to solve)}
 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\;
 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\;

-
+k=0\;

 \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\_update(d\_z^{k})$\;
-$kernel\_testConverge(\Delta z_{max},d_z^{k},d_z^{k-1})$\;
+$ kernel\_save(d\_Z^{k-1})$\;
+k=k+1\;
+$ kernel\_update(d\_Z^{k})$\;
+$kernel\_testConverge(\Delta z_{max},d\_Z^{k},d\_Z^{k-1})$\;
+

 }
 \end{algorithm}
 ~\\ 
 }
 \end{algorithm}
 ~\\ 


@@ -640,12 +655,6 @@ All experimental results obtained from the simulations are made in double precis
 \subsection{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]
-%\centering
- % \includegraphics[width=0.8\textwidth]{figures/Compar_EA_algorithm_CPU_GPU}
-%\caption{The execution time in seconds of Ehrlich-Aberth algorithm on CPU core vs. on a Tesla GPU}
-%\label{fig:01}
-%\end{figure}

 
 \begin{figure}[H]
 \centering
 
 \begin{figure}[H]
 \centering


@@ -666,7 +675,7 @@ For that, we notice that the maximum number of threads per block for the Nvidia
 \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)}
 \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}
+\label{fig:02}

 \end{figure}
 
 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.
 \end{figure}
 
 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.


@@ -678,7 +687,7 @@ In this experiment we report the performance of log.exp solution describe in ~\r
 \centering
   \includegraphics[width=0.8\textwidth]{figures/sparse_full_explog}
 \caption{The impact of exp-log solution to compute very high degrees of  polynomial.}
 \centering
   \includegraphics[width=0.8\textwidth]{figures/sparse_full_explog}
 \caption{The impact of exp-log solution to compute very high degrees of  polynomial.}

-\label{fig:01}
+\label{fig:03}

 \end{figure}
 
 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.
 \end{figure}
 
 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.


@@ -686,16 +695,7 @@ The figure 3, show a comparison between the execution time of the Ehrlich-Aberth
 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 . 
 
 
 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 . 
 
 

-%\begin{figure}[H]
-\%centering
-  %\includegraphics[width=0.8\textwidth]{figures/log_exp_Sparse}
-%\caption{The impact of exp-log solution to compute very high degrees of  polynomial.}
-%\label{fig:01}
-%\end{figure}
-
-%we report the performances of the exp.log for the Ehrlich-Aberth algorithm for solving very high degree of polynomial. 

 
 

- 

 \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.  
 
 \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.  
 


@@ -703,7 +703,7 @@ In this part, we are interesting to compare the simultaneous methods, Ehrlich-Ab
 \centering
   \includegraphics[width=0.8\textwidth]{figures/EA_DK}
 \caption{The execution time of Ehrlich-Aberth versus Durand-Kerner algorithm on GPU}
 \centering
   \includegraphics[width=0.8\textwidth]{figures/EA_DK}
 \caption{The execution time of Ehrlich-Aberth versus Durand-Kerner algorithm on GPU}

-\label{fig:01}
+\label{fig:04}

 \end{figure}
 
 This figure show the execution time of the both algorithm EA and DK with sparse polynomial degrees ranging from 1000 to 1000000. We can see that the Ehrlich-Aberth algorithm are faster than Durand-Kerner algorithm, with an average of 25 times as fast. Then, when degrees of polynomial exceed 500000 the execution time with EA is of the order 100 whereas DK passes in the order 1000. %with double precision not exceed $10^{-5}$.
 \end{figure}
 
 This figure show the execution time of the both algorithm EA and DK with sparse polynomial degrees ranging from 1000 to 1000000. We can see that the Ehrlich-Aberth algorithm are faster than Durand-Kerner algorithm, with an average of 25 times as fast. Then, when degrees of polynomial exceed 500000 the execution time with EA is of the order 100 whereas DK passes in the order 1000. %with double precision not exceed $10^{-5}$.


@@ -712,7 +712,7 @@ This figure show the execution time of the both algorithm EA and DK with sparse
 \centering
   \includegraphics[width=0.8\textwidth]{figures/EA_DK_nbr}
 \caption{The iteration number of Ehrlich-Aberth versus Durand-Kerner algorithm}
 \centering
   \includegraphics[width=0.8\textwidth]{figures/EA_DK_nbr}
 \caption{The iteration number of Ehrlich-Aberth versus Durand-Kerner algorithm}

-\label{fig:01}
+\label{fig:05}

 \end{figure}
 
 %\subsubsection{The execution time of Ehrlich-Aberth algorithm on OpenMP(1 core, 4 cores) vs. on a Tesla GPU}
 \end{figure}
 
 %\subsubsection{The execution time of Ehrlich-Aberth algorithm on OpenMP(1 core, 4 cores) vs. on a Tesla GPU}