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+\usepackage{float}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage[ruled,vlined]{algorithm2e}
+%\usepackage[french,boxed,linesnumbered]{algorithm2e}
\usepackage{array,multirow,makecell}
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\begin{frontmatter}
-\title{Parallel polynomial root finding using GPU}
+\title{Rapid solution of very high degree polynomials root finding using GPU}
%% Group authors per affiliation:
\author{Elsevier\fnref{myfootnote}}
\address[mysecondaryaddress]{FEMTO-ST Institute, University of Franche-Compté }
\begin{abstract}
-in this article we present a parallel implementation
-of the Aberth algorithm for the problem root finding for
-high degree polynomials on GPU architecture (Graphics
-Processing Unit).
+Polynomials are mathematical algebraic structures that play a great role in science and engineering. But the process of solving them for high and large degrees is computationally demanding and still not solved. In this paper, we present the results of a parallel implementation of the Ehrlish-Aberth algorithm for the problem root finding for
+high degree polynomials on GPU architectures (Graphics Processing Unit). The main result of this work is to be able to solve high and very large degree polynomials (up to 100000) very efficiently. We also compare the results with a sequential implementation and the Durand-Kerner method on full and sparse polynomials.
\end{abstract}
\begin{keyword}
-root finding of polynomials, high degree, iterative methods, Durant-Kerner, GPU, CUDA, CPU , Parallelization
+root finding of polynomials, high degree, iterative methods, Ehrlish-Aberth, Durant-Kerner, GPU, CUDA, CPU , Parallelization
\end{keyword}
\end{frontmatter}
\linenumbers
\section{The problem of finding roots of a polynomial}
-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 $\alpha_{i},\textit{i=1,...,n}$
+Polynomials are mathematical algebraic structures used in science and engineering to capture physical phenomenons and to express any outcome in the form of a function of some unknown variables. Formally speaking, a polynomial $p(x)$ of degree \textit{n} having $n$ coefficients in the complex plane \textit{C} is :
%%\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}}}.
\end{equation}
%%\end{center}
-The root finding problem consists in finding the values of all the $n$ values of the variable $x$ for which \textit{p(x)} is nullified. Such values are called zeroes of $p$. The problem of finding a root is equivalent to that of solving a fixed-point problem. To see this, consider the fixed-point problem of finding the $n$-dimensional
+The root finding problem consists in finding the values of all the $n$ values of the variable $x$ for which \textit{p(x)} is nullified. Such values are called zeroes of $p$. If zeros are $\alpha_{i},\textit{i=1,...,n}$ the $p(x)$ can be written as :
+\begin{equation}
+ {\Large p(x)=a_{n}\prod_{i=1}^{n}(x-\alpha_{i}), a_{0} a_{n}\neq 0}.
+\end{equation}
+
+The problem of finding a root is equivalent to that of solving a fixed-point problem. To see this, consider the fixed-point problem of finding the $n$-dimensional
vector $x$ such that
\begin{center}
$x=g(x)$
:
\begin{equation}
+\label{eq:SimplePolynome}
p(z)=\sum{a_{i}z^{n-i}} , a_{n} \neq 0,a_{0}=1, a_{i}\subset C
\end{equation}
circles and relies on the result of~\cite{Ostrowski41}.
\begin{equation}
+\label{eq:radiusR}
%%\begin{align}
\sigma_{0}=\frac{u+v}{2};u=\frac{\sum_{i=1}^{n}u_{i}}{n.max_{i=1}^{n}u_{i}};
v=\frac{\sum_{i=0}^{n-1}v_{i}}{n.min_{i=0}^{n-1}v_{i}};\\
\sum_{k\neq j}^{n}\frac{1}{z_{k}-z_{j}}\right)\right).
\end{equation}
-This solution is applied when it is necessary ??? When ??? (SIDER)
+This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated as:
+$$R = \exp( \log(DBL\_MAX) / (2*n) )$$ where $DBL\_MAX$ stands for the maximum representable double value.
\section{The implementation of simultaneous methods in a parallel computer}
\label{secStateofArt}
Moreover, they have fast rate of convergence (quadratic for the
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}.
+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
\section {A CUDA parallel Ehrlisch-Aberth method}
+In the following, we describe the parallel implementation of Ehrlisch-Aberth method on GPU
+for solving high degree polynomials. First, the hardware and software of the GPUs are presented. Then, a CUDA parallel Ehrlisch-Aberth method are presented.
\subsection{Background on the GPU architecture}
A GPU is viewed as an accelerator for the data-parallel and
~\\
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.
\end{equation}
-With the the Gauss-seidel iteration, we have:
+With the Gauss-seidel iteration, we have:
\begin{equation}
\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{algorithm}
-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$), else the kernel executes formulas (13,14). The radius R is evaluated as :
+The first form executes formula \ref{eq:SimplePolynome} if the modulus of the current complex is less than the a certain value called the radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes formulas (Eq.~\ref{deflncomplex},Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as :
$$R = \exp( \log(DBL\_MAX) / (2*n) )$$ where $DBL\_MAX$ stands for the maximum representable double value.
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 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))=.
+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}
- \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.
-\\
- 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}
- {\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}
-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
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)
-
-\subsubsection{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....
-
-\paragraph{Aberth algorithm on CPU and GPU}
-
-\begin{table}[!ht]
- \centering
- \begin{tabular} {|R{2cm}|L{2.5cm}|L{2.5cm}|L{1.5cm}|L{1.5cm}|}
- \hline Polynomial's degrees & $T_{exe}$ on CPU & $T_{exe}$ on GPU & CPU iteration & GPU iteration\\
- \hline 5000 & 1.90 & 0.40 & 18 & 17\\
- \hline 10000 & 172.723 & 0.59 & 21 & 24\\
- \hline 20000 & 172.723 & 1.52 & 21 & 25\\
- \hline 30000 & 172.723 & 2.77 & 21 & 33\\
- \hline 50000 & 172.723 & 3.92 & 21 & 18\\
- \hline 500000 & $>$1h & 497.109 & & 24\\
- \hline 1000000 & $>$1h & 1,524.51& & 24\\
- \hline
- \end{tabular}
- \caption{the convergence of Aberth algorithm}
- \label{tab:theConvergenceOfAberthAlgorithm}
-\end{table}
+E5620@2.40GHz and a GPU K40 (with 6 Go of ram).
+
+
+\subsection{Comparative study}
+In this section, we discuss the performance Ehrlish-Aberth method of root finding polynomials implemented on CPUs and on GPUs.
+
+We performed a set of experiments on the sequential and the parallel algorithms, for both sparse and full polynomials and different sizes. We took into account the execution time,the polynomial size and the number of threads per block performed by sum or each experiment on CPUs and on GPUs.
+
+All experimental results obtained from the simulations are made in double precision data, for a convergence tolerance of the methods set to $10^{-7}$. Since we were more interested in the comparison of the performance behaviors of Ehrlish-Aberth and Durand-Kerner methods on CPUs versus on GPUs.
+
+\subsubsection{Aberth algorithm on CPU and GPU}
+
+%\begin{table}[!ht]
+% \centering
+% \begin{tabular} {|R{2cm}|L{2.5cm}|L{2.5cm}|L{1.5cm}|L{1.5cm}|}
+% \hline Polynomial's degrees & $T_{exe}$ on CPU & $T_{exe}$ on GPU & CPU iteration & GPU iteration\\
+% \hline 5000 & 1.90 & 0.40 & 18 & 17\\
+% \hline 10000 & 172.723 & 0.59 & 21 & 24\\
+% \hline 20000 & 172.723 & 1.52 & 21 & 25\\
+% \hline 30000 & 172.723 & 2.77 & 21 & 33\\
+% \hline 50000 & 172.723 & 3.92 & 21 & 18\\
+% \hline 500000 & $>$1h & 497.109 & & 24\\
+% \hline 1000000 & $>$1h & 1,524.51& & 24\\
+% \hline
+% \end{tabular}
+% \caption{the convergence of Aberth algorithm}
+% \label{tab:theConvergenceOfAberthAlgorithm}
+%\end{table}
-\paragraph{The impact of the thread's number into the convergence of Aberth algorithm}
-
-\begin{table}[!h]
- \centering
- \begin{tabular} {|R{2.5cm}|L{2.5cm}|L{2.5cm}|}
- \hline Thread's numbers & Execution time &Number of iteration\\
- \hline 1024 & 523 & 27\\
- \hline 512 & 449.426 & 24\\
- \hline 256 & 440.805 & 24\\
- \hline 128 & 456.175 & 22\\
- \hline 64 & 472.862 & 23\\
- \hline 32 & 830.152 & 24\\
- \hline 8 & 2632.78 & 23 \\
- \hline
- \end{tabular}
- \caption{The impact of the thread's number into the convergence of Aberth algorithm}
- \label{tab:Theimpactofthethread'snumberintotheconvergenceofAberthalgorithm}
-
-\end{table}
-
-
-\begin{figure}[htbp]
+\begin{figure}[H]
+\centering
+ \includegraphics[width=0.8\textwidth]{figures/Compar_EA_algorithm_CPU_GPU}
+\caption{Aberth algorithm on CPU and GPU}
+\label{fig:01}
+\end{figure}
+
+
+\subsubsection{The impact of the thread's number into the convergence of Aberth algorithm}
+
+%\begin{table}[!h]
+% \centering
+% \begin{tabular} {|R{2.5cm}|L{2.5cm}|L{2.5cm}|}
+% \hline Thread's numbers & Execution time &Number of iteration\\
+% \hline 1024 & 523 & 27\\
+% \hline 512 & 449.426 & 24\\
+% \hline 256 & 440.805 & 24\\
+% \hline 128 & 456.175 & 22\\
+% \hline 64 & 472.862 & 23\\
+% \hline 32 & 830.152 & 24\\
+% \hline 8 & 2632.78 & 23 \\
+% \hline
+% \end{tabular}
+% \caption{The impact of the thread's number into the convergence of Aberth algorithm}
+% \label{tab:Theimpactofthethread'snumberintotheconvergenceofAberthalgorithm}
+%
+%\end{table}
+
+
+\begin{figure}[H]
\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}
+\subsubsection{The impact of exp-log solution to compute very high degrees of polynomial}
+\begin{figure}[H]
+\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}
+
+\subsubsection{A comparative study between Aberth and Durand-kerner algorithm}
-\paragraph{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}|}
- \hline Polynomial's degrees & Aberth $T_{exe}$ & D-Kerner $T_{exe}$ & Aberth iteration & D-Kerner iteration\\
- \hline 5000 & 0.40 & 3.42 & 17 & 138 \\
- \hline 50000 & 3.92 & 385.266 & 17 & 823\\
- \hline 500000 & 497.109 & 4677.36 & 24 & 214\\
- \hline
- \end{tabular}
- \caption{Aberth algorithm compare to Durand-Kerner algorithm}
- \label{tab:AberthAlgorithCompareToDurandKernerAlgorithm}
-\end{table}
+\begin{figure}[H]
+\centering
+ \includegraphics[width=0.8\textwidth]{figures/EA_DK}
+\caption{Ehrlisch-Aberth and Durand-Kerner algorithm on GPU}
+\label{fig:01}
+\end{figure}
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