MAJ figure

diff --git a/figures/Compar_EA_algorithm_CPU_GPU.pdf b/figures/Compar_EA_algorithm_CPU_GPU.pdf
index 5bdd80f26353cad29f827229cf3dad061d83f1d6..81bf3faba5ddf3dc8864dcaa59ed25fabc1058f0 100644 (file)
Binary files a/figures/Compar_EA_algorithm_CPU_GPU.pdf and b/figures/Compar_EA_algorithm_CPU_GPU.pdf differ

diff --git a/figures/Compar_EA_algorithm_CPU_GPU.plot b/figures/Compar_EA_algorithm_CPU_GPU.plot
index c9d1baa5aa08c6e1dd3b9e2dd8f3c63d6ee95317..a92ec7072eb4fcecf62db546c541fd887f373a95 100644 (file)
--- a/figures/Compar_EA_algorithm_CPU_GPU.plot
+++ b/figures/Compar_EA_algorithm_CPU_GPU.plot
@@ -4,8 +4,8 @@ set terminal x11
 set size 1,0.5
 set term postscript enhanced portrait "Helvetica" 12
 
 set size 1,0.5
 set term postscript enhanced portrait "Helvetica" 12
 

-set xlabel "execution times (in s)" 
-set ylabel "polynomial's degrees" 
+set ylabel "execution times (in s)" 
+set xlabel "polynomial's degrees" 

 #set logscale x
 #set logscale y
 
 #set logscale x
 #set logscale y
 

diff --git a/figures/influence_nb_threads.pdf b/figures/influence_nb_threads.pdf
index 8449d4ac44b5191e7c1b0fc9a285573380b027bb..09d0681bb9c91c803ed4bdf470cfe5f3e03a73b4 100644 (file)
Binary files a/figures/influence_nb_threads.pdf and b/figures/influence_nb_threads.pdf differ

diff --git a/figures/influence_nb_threads.txt b/figures/influence_nb_threads.txt
index 787ea7a6ec94388a6148c929ca8f60e9baa4bbe7..ac5fcbd9773c5a7ce2c1fce471701aafa102ba38 100644 (file)
--- a/figures/influence_nb_threads.txt
+++ b/figures/influence_nb_threads.txt
@@ -1,10 +1,10 @@
 #500000                sparse                          full                    #50000          sparse                          full
 #nb threads    times           nb iter         times           nb iter                 times           nb iter         times       nb iter             
 #500000                sparse                          full                    #50000          sparse                          full
 #nb threads    times           nb iter         times           nb iter                 times           nb iter         times       nb iter             

-1024           523             27              545                                                                     8.61        16
-512            449.426         24              520                                                                     9.27        19
-256             440.805        24              480                                                                     7.73        15
-128            456.175         22              560                                                                     8.64        21
-64             472.862         23              603                                                                     7.84        16
-32             830.152         24              920                                                                     11.33       18
-16             1234            24              1870                                                                    20.47       21
-8              2632.78         23              3589                                                                    35.07       26
+1024           523             27              2100.7                                                                  8.61        16
+512            449.426         24              1459.35                                                                 9.27        19
+256             440.805        24              754.24                                                                  7.73        15
+128            456.175         22              718.623         27                                                      8.64        21
+64             472.862         23              715.554         27                                                      7.84        16
+32             830.152         24              1089.61         27                                                      11.33       18
+16             1234            24              1746.53         22                                                      20.47       21
+8              2632.78         23              3112            20                                                      35.07       26

diff --git a/paper.tex b/paper.tex
index 62ded307955714b3c4f5254f8af27286f4e35126..c8a669875ef7f21354062b4e62411a0c98328977 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -6,6 +6,7 @@
 %%\usepackage[french]{babel}
 \usepackage{amsmath,amsfonts,amssymb}
 \usepackage[ruled,vlined]{algorithm2e}
 %%\usepackage[french]{babel}
 \usepackage{amsmath,amsfonts,amssymb}
 \usepackage[ruled,vlined]{algorithm2e}

+%\usepackage[french,boxed,linesnumbered]{algorithm2e}

 \usepackage{array,multirow,makecell}
 \setcellgapes{1pt}
 \makegapedcells
 \usepackage{array,multirow,makecell}
 \setcellgapes{1pt}
 \makegapedcells


@@ -478,7 +479,7 @@ $\Delta z_{max}$=c\;}
 
 ~\\ 
 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.
 
 ~\\ 
 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 :
+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}
 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.
 
 \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.


@@ -579,11 +580,12 @@ The last kernel verifies the convergence of the roots after each update of $Z^{(
 The kernels terminate it computations when all the roots converge. Finally, the solution of the root finding problem is copied back from GPU global memory to CPU memory. We use the communication functions of CUDA for the memory allocation in the GPU \verb=(cudaMalloc())= and for data transfers from the CPU memory to the GPU memory \verb=(cudaMemcpyHostToDevice)=
 or from GPU memory to CPU memory \verb=(cudaMemcpyDeviceToHost))=. 
 %%HIER END MY REVISIONS (SIDER)
 The kernels terminate it computations when all the roots converge. Finally, the solution of the root finding problem is copied back from GPU global memory to CPU memory. We use the communication functions of CUDA for the memory allocation in the GPU \verb=(cudaMalloc())= and for data transfers from the CPU memory to the GPU memory \verb=(cudaMemcpyHostToDevice)=
 or from GPU memory to CPU memory \verb=(cudaMemcpyDeviceToHost))=. 
 %%HIER END MY REVISIONS (SIDER)

-\subsection{Experimental study}
+\section{Experimental study}

 
 

-\subsubsection{Definition of the polynomial used}
-We use a polynomial of the following form for which the
-roots are distributed on 2 distinct circles:
+\subsection{Definition of the polynomial used}
+We use two forms of  polynomials:
+\paragraph{sparse polynomial}:
+in this following form, 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}
 \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}


@@ -591,18 +593,19 @@ roots are distributed on 2 distinct circles:
 This form makes it possible to associate roots having two
 different modules and thus to work on a polynomial constitute
 of four non zero terms.
 This form makes it possible to associate roots having two
 different modules and thus to work on a polynomial constitute
 of four non zero terms.

-\\
- An other form of the polynomial to obtain  a full polynomial is:
+
+\paragraph{Full polynomial}:
+ 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}
 
 \begin{equation}
 %%\begin{equation}
        %%\forall \alpha_{i} \in C,\forall n_{i}\in N^{*}; P(z)= \sum^{n}_{i=1}(z^{n^{i}}.a_{i})
 %%\end{equation}
 
 \begin{equation}

-     {\Large \forall a_{i} \in C; p(x)=\sum^{n-1}_{i=1} a_{i}.x^{i}} 
+     {\Large \forall a_{i} \in C, i\in N; p(x)=\sum^{n-1}_{i=1} a_{i}.x^{i}} 

 \end{equation}
 \end{equation}

-with this formula, we can have until \textit{n} non zero terms.
+with this form, we can have until \textit{n} non zero terms.

 
 

-\subsubsection{The study condition} 
+\subsection{The study condition} 

 In order to have representative average values, for each
 point of our curves we measured the roots finding of 10
 different polynomials.
 In order to have representative average values, for each
 point of our curves we measured the roots finding of 10
 different polynomials.


@@ -614,12 +617,13 @@ to validate that our algorithm is powerful with high degree
 polynomials. The execution time remains the
 element-key which justifies our work of parallelization.
        For our tests we used a CPU Intel(R) Xeon(R) CPU
 polynomials. The execution time remains the
 element-key which justifies our work of parallelization.
        For our tests we used a CPU Intel(R) Xeon(R) CPU

-E5620@2.40GHz and a GPU Tesla C2070 (with 6 Go of ram)
+E5620@2.40GHz and a GPU K40 (with 6 Go of ram)
+

 
 

-\subsubsection{Comparative study}
+\subsection{Comparative study}

 We initially carried out the convergence of Aberth algorithm with various sizes of polynomial, in second we evaluate the influence of the size of the threads per block....
 
 We initially carried out the convergence of Aberth algorithm with various sizes of polynomial, in second we evaluate the influence of the size of the threads per block....
 

-\paragraph{Aberth algorithm on CPU and GPU}
+\subsubsection{Aberth algorithm on CPU and GPU}

 
 %\begin{table}[!ht]
 %      \centering
 
 %\begin{table}[!ht]
 %      \centering


@@ -646,7 +650,7 @@ We initially carried out the convergence of Aberth algorithm with various sizes
 \end{figure}
 
 
 \end{figure}
 
 

-\paragraph{The impact of the thread's number into the convergence of Aberth  algorithm}
+\subsubsection{The impact of the thread's number into the convergence of Aberth  algorithm}

 
 %\begin{table}[!h]
 %      \centering
 
 %\begin{table}[!h]
 %      \centering


@@ -676,7 +680,7 @@ We initially carried out the convergence of Aberth algorithm with various sizes
 
 
 
 
 
 

-\paragraph{A comparative study between Aberth and Durand-kerner algorithm}
+\subsubsection{A comparative study between Aberth and Durand-kerner algorithm}

 \begin{table}[htbp]
        \centering
                \begin{tabular} {|R{2cm}|L{2.5cm}|L{2.5cm}|L{1.5cm}|L{1.5cm}|}
 \begin{table}[htbp]
        \centering
                \begin{tabular} {|R{2cm}|L{2.5cm}|L{2.5cm}|L{1.5cm}|L{1.5cm}|}