MAJ des figures

[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 62ded307955714b3c4f5254f8af27286f4e35126..9897244a20b19a6506dc8eeb096f446bec2fbf4d 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


@@ -95,7 +96,7 @@ root finding of polynomials, high degree, iterative methods, Durant-Kerner, GPU,
 Polynomials are algebraic structures used in mathematics that capture physical phenomenons and that express the outcome in the form of a function of some unknown variable. Formally speaking,  a polynomial $p(x)$ of degree \textit{n} having $n$ coefficients in the complex plane \textit{C} and zeros $\alpha_{i},\textit{i=1,...,n}$ 
 %%\begin{center}
 \begin{equation}
 Polynomials are algebraic structures used in mathematics that capture physical phenomenons and that express the outcome in the form of a function of some unknown variable. Formally speaking,  a polynomial $p(x)$ of degree \textit{n} having $n$ coefficients in the complex plane \textit{C} and zeros $\alpha_{i},\textit{i=1,...,n}$ 
 %%\begin{center}
 \begin{equation}

-     {\Large p(x)=\sum{a_{i}x^{i}}=a_{n}\prod(x-\alpha_{i}),a_{0} a_{n}\neq 0}.
+     {\Large p(x)=\sum_{i=0}^{n}{a_{i}x^{i}}=a_{n}\prod_{i=1}^{n}(x-\alpha_{i}), a_{0} a_{n}\neq 0}.

 \end{equation}
 %%\end{center}
 
 \end{equation}
 %%\end{center}
 


@@ -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.


@@ -579,34 +580,30 @@ 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 study two forms of  polynomials the sparse polynomials and the full polynomials:
+\paragraph{Sparse polynomial}: in this following form, the roots are distributed on 2 distinct circles:

 \begin{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})
+       \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}
 
 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.
 \end{equation}
 
 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}_{i=0} a_{i}.x^{i}} 

 \end{equation}
 \end{equation}

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

 
 

+\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
 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


@@ -614,12 +611,13 @@ 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 +644,7 @@ 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


@@ -674,9 +672,14 @@ We initially carried out the convergence of Aberth algorithm with various sizes
 \label{fig:01}
 \end{figure}
 
 \label{fig:01}
 \end{figure}
 

+\begin{figure}[htbp]
+\centering
+  \includegraphics[width=0.8\textwidth]{figures/log_exp}
+\caption{The impact of exp-log solution to compute very high degrees of  polynomial.}
+\label{fig:01}
+\end{figure}

 
 

-
-\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}|}