new

[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index abc8b8d56c3a2f781d29d8effbbf039cc1ed88f9..788db9ac129fd34557039e46f743d4b9f398221c 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -151,7 +151,7 @@ method:
  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}
  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
+where $z_i^k$ is the $i^{th}$ root of the polynomial $p$ at the

 iteration $k$.
 
 
 iteration $k$.
 
 


@@ -169,7 +169,7 @@ Aberth~\cite{Aberth73} uses a different iteration formula given as:
  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}
  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
+where $p'(z)$ is the polynomial derivative of $p$ evaluated in the

 point $z$.
 
 Aberth, Ehrlich and Farmer-Loizou~\cite{Loizou83} have proved that
 point $z$.
 
 Aberth, Ehrlich and Farmer-Loizou~\cite{Loizou83} have proved that


@@ -191,13 +191,13 @@ 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
 algorithms, in which each processor continues to update its
 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 $z_i((k))$
+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 parallelization for
 a shared memory architecture and for distributed memory one. They were able to
 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 parallelization for
 a shared memory architecture and for distributed memory one. They were able to

-compute the roots of sparse polynomials of degree 10000 in 430 seconds with only 8
+compute the roots of sparse polynomials of degree 10,000 in 430 seconds with only 8

 personal computers and 2 communications per iteration. Comparing to the sequential implementation
 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.
+where it takes up to 3,300 seconds to obtain the same results, the authors show an interesting speedup.

 
 Very few works had been performed since this last work until the appearing of
 the Compute Unified Device Architecture (CUDA)~\cite{CUDA10}, a
 
 Very few works had been performed since this last work until the appearing of
 the Compute Unified Device Architecture (CUDA)~\cite{CUDA10}, a


@@ -211,7 +211,7 @@ 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 about 10 times faster than
 the sequential implementation on a single CPU for  sparse
 Durand-Kerner method on GPU. Their main
 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. 
+polynomials of degree 48,000. 

 
 
 In this paper, we focus on the implementation of the Ehrlich-Aberth
 
 
 In this paper, we focus on the implementation of the Ehrlich-Aberth


@@ -225,15 +225,14 @@ 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
 \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
+and experimental study results. Finally, Section~\ref{sec7} concludes

 this paper and gives some hints for future research directions in this
 topic.
 
 \section{Ehrlich-Aberth method}
 \label{sec1}
 A cubically convergent iteration method for finding zeros of
 this paper and gives some hints for future research directions in this
 topic.
 
 \section{Ehrlich-Aberth method}
 \label{sec1}
 A cubically convergent iteration method for finding zeros of

-polynomials was proposed by O. Aberth~\cite{Aberth73}. In the
-following we present the main stages of our implementation the Ehrlich-Aberth method.
+polynomials was proposed by O. Aberth~\cite{Aberth73}. The Ehrlich-Aberth method contain 4 main steps, presented in the following. 

 %The Aberth method is a purely algebraic derivation. 
 %To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors 
 
 %The Aberth method is a purely algebraic derivation. 
 %To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors 
 


@@ -259,7 +258,7 @@ following we present the main stages of our implementation the Ehrlich-Aberth me
 
 
 \subsection{Polynomials Initialization}
 
 
 \subsection{Polynomials Initialization}

-The initialization of a polynomial p(z) is done by setting each of the $n$ complex coefficients $a_{i}$:
+The initialization of a polynomial $p(z)$ is done by setting each of the $n$ complex coefficients $a_{i}$:

 
 \begin{equation}
 \label{eq:SimplePolynome}
 
 \begin{equation}
 \label{eq:SimplePolynome}


@@ -267,9 +266,9 @@ The initialization of a polynomial p(z) is done by setting each of the $n$ compl
 \end{equation}
 
 
 \end{equation}
 
 

-\subsection{Vector $z^{(0)}$ Initialization}
+\subsection{Vector $Z^{(0)}$ Initialization}

 \label{sec:vec_initialization}
 \label{sec:vec_initialization}

-As 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 Ehrlich-Aberth iteration is started by selecting $n$
 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 Ehrlich-Aberth iteration is started by selecting $n$


@@ -300,7 +299,7 @@ Here we give a second form of the iterative function used by Ehrlich-Aberth meth
 
 \begin{equation}
 \label{Eq:Hi}
 
 \begin{equation}
 \label{Eq:Hi}

-EA2: z^{k+1}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
+EA2: z^{k+1}_{i}=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}
 It can be noticed that this equation is equivalent to Eq.~\ref{Eq:EA},
 {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}
 It can be noticed that this equation is equivalent to Eq.~\ref{Eq:EA},


@@ -322,8 +321,8 @@ 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
 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
-mantissa of floating points representations makes the computation of p(z) wrong when z
+point $\xi$ where $|\xi| > 1$, where $|z|$ stands for the modolus of a complex $z$. Indeed, the limited number in the
+mantissa of floating points representations makes the computation of $p(z)$ wrong when z

 is large. For example $(10^{50}) +1+ (- 10^{50})$ will give the wrong result
 of $0$ instead of $1$. Consequently, we can not compute the roots
 for large degrees. This problem was early discussed in
 is large. For example $(10^{50}) +1+ (- 10^{50})$ will give the wrong result
 of $0$ instead of $1$. Consequently, we can not compute the roots
 for large degrees. This problem was early discussed in


@@ -351,7 +350,7 @@ 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}
 %%$$ \exp \bigl(  \ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'}))- \ln(1- \exp(\ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'})+\ln\sum_{i\neq j}^{n}\frac{1}{z_{k}-z_{j}})$$
 \begin{equation}
 \label{Log_H2}

-EA.EL: z^{k+1}=z_{i}^{k}-\exp \left(\ln \left(
+EA.EL: z^{k+1}_{i}=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}
 p(z_{i}^{k})\right)-\ln\left(p'(z^{k}_{i})\right)- \ln
 \left(1-Q(z^{k}_{i})\right)\right),
 \end{equation}


@@ -361,7 +360,7 @@ where:
 \begin{equation}
 \label{Log_H1}
 Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}_{i}))+\ln \left(
 \begin{equation}
 \label{Log_H1}
 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).
+\sum_{i\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right)i=1,...,n,

 \end{equation}
 
 This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as:
 \end{equation}
 
 This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as:


@@ -583,9 +582,9 @@ Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth algorithm using C
 \caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
 
 \KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (error tolerance
 \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), Pu (the derivative of P), $n$ (Polynomial's degrees), $\Delta z_{max}$ (maximum value of stop condition)}
+  threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial's degrees), $\Delta z_{max}$ (Maximum value of stop condition)}

 
 

-\KwOut {$Z$ (The solution root's vector), $ZPrec$ (the previous solution root's vector)}
+\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}

 
 \BlankLine
 
 
 \BlankLine
 


@@ -606,11 +605,19 @@ Copy results from GPU memory to CPU memory\;
 \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 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}).
+After the initialization step, all data of the root finding problem
+must be copied from the CPU memory to the GPU global memory. Next, all
+the data-parallel arithmetic operations inside the main loop
+\verb=(while(...))= are executed as kernels by the GPU. The
+first kernel named \textit{save} in line 6 of
+Algorithm~\ref{alg2-cuda} consists in saving the vector of
+polynomial's root found at the previous time-step in GPU memory, in
+order to check the convergence of the roots after each iteration (line
+8, Algorithm~\ref{alg2-cuda}).

 
 The second kernel executes the iterative function $H$ and updates
 
 The second kernel executes the iterative function $H$ and updates

-$d\_Z$, according to Algorithm~\ref{alg3-update}. We notice that the
-update kernel is called in two forms, separated with the value of
+Z, according to Algorithm~\ref{alg3-update}. We notice that the
+update kernel is called in two forms, according to the value

 \emph{R} which determines the radius beyond which we apply the
 exponential logarithm algorithm. 
 
 \emph{R} which determines the radius beyond which we apply the
 exponential logarithm algorithm. 
 


@@ -620,9 +627,9 @@ exponential logarithm algorithm.
 \caption{Kernel update}
 
 \eIf{$(\left|Z\right|<= R)$}{
 \caption{Kernel update}
 
 \eIf{$(\left|Z\right|<= R)$}{

-$kernel\_update((Z,P,Pu)$\;}
+$kernel\_update(Z,P,Pu)$\;}

 {
 {

-$kernel\_update\_ExpoLog((Z,P,Pu))$\;
+$kernel\_update\_ExpoLog(Z,P,Pu)$\;

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


@@ -712,13 +719,33 @@ of the methods are given in Section~\ref{sec:vec_initialization}.
   on a CPU with OpenMP (1 core, 4 cores) and on a Tesla GPU}
 \label{fig:01}
 \end{figure}
   on a CPU with OpenMP (1 core, 4 cores) and on a Tesla GPU}
 \label{fig:01}
 \end{figure}

-%%Figure 1 %%show a comparison of execution time between the parallel and sequential version of the Ehrlich-Aberth algorithm with sparse polynomial exceed 100000, 
-In Figure~\ref{fig:01}, we report respectively the execution time of the Ehrlich-Aberth method implemented initially on one core of the Quad-Core Xeon E5620 CPU than on four cores of the same machine with \textit{OpenMP} platform and the execution time of the same method implemented on one Nvidia Tesla K40c GPU, with sparse polynomial degrees ranging from 100,000 to 1,000,000. We can see that the method implemented on the GPU are faster than those implemented on the CPU (4 cores). This is due to the GPU ability to compute the data-parallel functions faster than its CPU counterpart. However, the execution time for the CPU(4 cores) implementation exceed 5,000 s for 250,000 degrees polynomials, in counterpart  the GPU implementation for the same polynomials not reach 100 s, more than again, with an execution time under to 2,500 s CPU (4 cores) implementation can resolve polynomials degrees of only 200,000, whereas GPU implementation can resolve polynomials more than 1,000,000 degrees. We can also notice that the GPU implementation are almost 47 faster then those implementation on the CPU(4 cores). However the CPU(4 cores) implementation are almost 4 faster then his implementation on CPU (1 core). Furthermore, we verify that the number of iterations and the convergence precision is the same for the both CPU and GPU implementation. %This reduction of time allows us to compute roots of polynomial of more important degree at the same time than with a CPU.
+%%Figure 1 %%show a comparison of execution time between the parallel
+%%and sequential version of the Ehrlich-Aberth algorithm with sparse
+%%polynomial exceed 100000,
+
+In Figure~\ref{fig:01}, we report the execution times of the
+Ehrlich-Aberth method on one core of a Quad-Core Xeon E5620 CPU, on
+four cores on the same machine with \textit{OpenMP} and on a Nvidia
+Tesla K40c GPU.  We chose different sparse polynomials with degrees
+ranging from 100,000 to 1,000,000. We can see that the implementation
+on the GPU is faster than those implemented on the CPU.
+However, the execution time for the
+CPU (4 cores) implementation exceed 5,000s for 250,000 degrees
+polynomials. In counterpart, the GPU implementation for the same
+polynomials do not take more 100s. With the GPU
+we can solve high degrees polynomials very quickly up to degree
+ of 1,000,000. We can also notice that the GPU implementation are
+ almost 40 faster then those implementation on the CPU (4 cores).
+
+
+ 
+
+%This reduction of time allows us to compute roots of polynomial of more important degree at the same time than with a CPU.

  
  %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.
 
 \subsection{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
  
  %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.
 
 \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. 
+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). 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 1,024, so we varied the number of threads per block from 8 to 1,024. We took into account the execution time for both sparse and full of 10 different polynomials of size 50,000 and 10 different polynomials of size 500,000 degrees.
 
 \begin{figure}[htbp]
 For that, we notice that the maximum number of threads per block for the Nvidia Tesla K40 GPU is 1,024, so we varied the number of threads per block from 8 to 1,024. We took into account the execution time for both sparse and full of 10 different polynomials of size 50,000 and 10 different polynomials of size 500,000 degrees.
 
 \begin{figure}[htbp]