biblio

[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 00d805d1522d68313e14c38ea1de9909d677c012..2dcee5775eefb549e1dbff4c00c96a5267b763e2 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -300,7 +300,7 @@ Here we give a second form of the iterative function used by Ehrlich-Aberth meth
 \begin{equation}
 \label{Eq:Hi}
 EA2: z^{k+1}_{i}=z_{i}^{k}-\frac{\frac{p(z_{i}^{k})}{p'(z_{i}^{k})}}
 \begin{equation}
 \label{Eq:Hi}
 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
+{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=1,. . . .,n

 \end{equation}
 It can be noticed that this equation is equivalent to Eq.~\ref{Eq:EA},
 but we prefer the latter one because we can use it to improve the
 \end{equation}
 It can be noticed that this equation is equivalent to Eq.~\ref{Eq:EA},
 but we prefer the latter one because we can use it to improve the


@@ -345,14 +345,12 @@ propose to use the logarithm and the exponential of a complex in order to comput
 Using the logarithm (eq.~\ref{deflncomplex}) and the exponential (eq.~\ref{defexpcomplex}) operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations
 manipulate lower absolute values and the roots for large polynomial's degrees can be looked for successfully~\cite{Karimall98}.
 
 Using the logarithm (eq.~\ref{deflncomplex}) and the exponential (eq.~\ref{defexpcomplex}) operators, we can replace any multiplications and divisions with additions and subtractions. Consequently, computations
 manipulate lower absolute values and the roots for large polynomial's degrees can be looked for successfully~\cite{Karimall98}.
 

-Applying this solution for the Ehrlich-Aberth method we obtain the
-iteration function with exponential and logarithm:
+Applying this solution for the iteration function Eq.~\ref{Eq:Hi} of Ehrlich-Aberth method, we obtain the 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}
 EA.EL: z^{k+1}_{i}=z_{i}^{k}-\exp \left(\ln \left(
 %%$$ \exp \bigl(  \ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'}))- \ln(1- \exp(\ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'})+\ln\sum_{i\neq j}^{n}\frac{1}{z_{k}-z_{j}})$$
 \begin{equation}
 \label{Log_H2}
 EA.EL: z^{k+1}_{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),
+p(z_{i}^{k})\right)-\ln\left(p'(z^{k}_{i})\right)- \ln\left(1-Q(z^{k}_{i})\right)\right),

 \end{equation}
 
 where:
 \end{equation}
 
 where:


@@ -363,11 +361,12 @@ Q(z^{k}_{i})=\exp\left( \ln (p(z^{k}_{i}))-\ln(p'(z^{k}_{i}))+\ln \left(
 \sum_{i\neq j}^{n}\frac{1}{z^{k}_{i}-z^{k}_{j}}\right)\right)i=1,...,n,
 \end{equation}
 
 \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:
+This solution is applied when the root except the circle unit, represented by the radius $R$ evaluated in C language as :
+\begin{equation}
+\label{R.EL}
+R = exp(log(DBL\_MAX)/(2*n) );
+\end{equation}

 
 

-\begin{verbatim}
-R = exp(log(DBL_MAX)/(2*n) );
-\end{verbatim} 

 
 %\begin{equation}
 
 
 %\begin{equation}
 


@@ -385,7 +384,7 @@ Authors usually adopt one of the two following approaches to parallelize root
 finding algorithms. The first approach aims at reducing the total number of
 iterations as by Miranker
 ~\cite{Mirankar68,Mirankar71}, Schedler~\cite{Schedler72} and
 finding algorithms. The first approach aims at reducing the total number of
 iterations as by Miranker
 ~\cite{Mirankar68,Mirankar71}, Schedler~\cite{Schedler72} and

-Winogard~\cite{Winogard72}. The second approach aims at reducing the
+Winograd~\cite{Winogard72}. The second approach aims at reducing the

 computation time per iteration, as reported
 in~\cite{Benall68,Jana06,Janall99,Riceall06}. 
 
 computation time per iteration, as reported
 in~\cite{Benall68,Jana06,Janall99,Riceall06}. 
 


@@ -408,9 +407,9 @@ other processors at the end of each iteration (synchronous). Therefore they
 cause a high degree of memory conflict. Recently the author
 in~\cite{Mirankar71} proposed two versions of parallel algorithm
 for the Durand-Kerner method, and Ehrlich-Aberth method on a model of
 cause a high degree of memory conflict. Recently the author
 in~\cite{Mirankar71} proposed two versions of parallel algorithm
 for the Durand-Kerner method, and Ehrlich-Aberth method on a model of

-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
+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.
 %Until very recently, the literature did not mention implementations
 %able to compute the roots of large degree polynomials (higher then
 practical for solving polynomials with large degrees.
 %Until very recently, the literature did not mention implementations
 %able to compute the roots of large degree polynomials (higher then


@@ -423,7 +422,7 @@ 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
 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.
+polynomials of 48,000.

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


@@ -541,20 +540,27 @@ polynomials of 48000.
 \subsection{Parallel implementation with CUDA }
 
 In order to implement the Ehrlich-Aberth method in CUDA, it is
 \subsection{Parallel implementation with CUDA }
 
 In order to implement the Ehrlich-Aberth method in CUDA, it is

-possible to use the Jacobi scheme or the Gauss Seidel one.  With the
+possible to use the Jacobi scheme or the Gauss-Seidel one.  With the

 Jacobi iteration, at iteration $k+1$ we need all the previous values
 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 :
+$z^{k}_{i}$ to compute the new values $z^{k+1}_{i}$, that is :

 
 \begin{equation}
 
 \begin{equation}

-EAJ: z^{k+1}_{i}=\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.
+EAJ: 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=1,. . . .,n.

 \end{equation}
 
 With the Gauss-Seidel iteration, we have:
 \end{equation}
 
 With the Gauss-Seidel iteration, we have:

+%\begin{equation}
+%\label{eq:Aberth-H-GS}
+%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}
+

 \begin{equation}
 \label{eq:Aberth-H-GS}
 \begin{equation}
 \label{eq:Aberth-H-GS}

-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.
+EAGS: 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^{i-1}_{j=1}\frac{1}{z^{k}_{i}-z^{k+1}_{j}}+\sum_{j=i+1}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}})}, i=1,. . . .,n

 \end{equation}
 \end{equation}

-%%Here a finiched my revision %%
+

 Using Eq.~\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 $z^{k+1}_{i}$.
 Using Eq.~\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 $z^{k+1}_{i}$.


@@ -574,42 +580,52 @@ quickly because, just as any Jacobi algorithm (for solving linear systems of equ
 
 %In CUDA programming, all the instructions of the  \verb=for= loop are executed by the GPU as a kernel. A kernel is a function written in CUDA and defined by the  \verb=__global__= qualifier added before a usual \verb=C= function, which instructs the compiler to generate appropriate code to pass it to the CUDA runtime in order to be executed on the GPU. 
 
 
 %In CUDA programming, all the instructions of the  \verb=for= loop are executed by the GPU as a kernel. A kernel is a function written in CUDA and defined by the  \verb=__global__= qualifier added before a usual \verb=C= function, which instructs the compiler to generate appropriate code to pass it to the CUDA runtime in order to be executed on the GPU. 
 

-Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth algorithm using CUDA.
+Algorithm~\ref{alg2-cuda} shows sketch of the Ehrlich-Aberth method using CUDA.

 
 

+\begin{enumerate}

 \begin{algorithm}[H]
 \label{alg2-cuda}
 %\LinesNumbered
 \caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
 
 \begin{algorithm}[H]
 \label{alg2-cuda}
 %\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), Pu (the derivative of P), $n$ (Polynomial's degrees), $\Delta z_{max}$ (maximum value of stop condition)}
+\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (Error tolerance
+  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
 

-Initialization of the of P\;
-Initialization of the of Pu\;
-Initialization of the solution vector $Z^{0}$\;
-Allocate and copy initial data to the GPU global memory\;
-k=0\;
+\item Initialization of the of P\;
+\item Initialization of the of Pu\;
+\item Initialization of the solution vector $Z^{0}$\;
+\item Allocate and copy initial data to the GPU global memory\;
+\item k=0\;

 \While {$\Delta z_{max} > \epsilon$}{
 \While {$\Delta z_{max} > \epsilon$}{

- Let $\Delta z_{max}=0$\;
-$ kernel\_save(ZPrec,Z)$\;
-k=k+1\;
-$ kernel\_update(Z,P,Pu)$\;
-$kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;
+\item Let $\Delta z_{max}=0$\;
+\item $ kernel\_save(ZPrec,Z)$\;
+\item  k=k+1\;
+\item $ kernel\_update(Z,P,Pu)$\;
+\item $kernel\_testConverge(\Delta z_{max},Z,ZPrec)$\;

 
 }
 
 }

-Copy results from GPU memory to CPU memory\;
+\item Copy results from GPU memory to CPU memory\;

 \end{algorithm}
 \end{algorithm}

+\end{enumerate}

 ~\\ 
 
 ~\\ 
 

-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}).
-
-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
+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 7 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
+10, Algorithm~\ref{alg2-cuda}).
+
+The second kernel executes the iterative function and updates
+$Z$, 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. 
 


@@ -619,9 +635,9 @@ 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}
 


@@ -629,12 +645,10 @@ The first form executes formula the EA function Eq.~\ref{Eq:Hi} if the modulus
 of the current complex is less than the a certain value called the
 radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes the EA.EL
 function Eq.~\ref{Log_H2}
 of the current complex is less than the a certain value called the
 radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes the EA.EL
 function Eq.~\ref{Log_H2}

-(with 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.
+(with Eq.~\ref{deflncomplex}, Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as in Eq.~\ref{R.EL}.

 
 The last kernel checks the convergence of the roots after each update
 
 The last kernel checks the convergence of the roots after each update

-of $Z^{(k)}$, according to formula Eq.~\ref{eq:Aberth-Conv-Cond}. We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. 
+of $Z^{k}$, according to formula Eq.~\ref{eq:Aberth-Conv-Cond}. We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. 

 
 The kernel terminates its computations when all the roots have
 converged. It should be noticed that, as blocks of threads are
 
 The kernel terminates its computations when all the roots have
 converged. It should be noticed that, as blocks of threads are


@@ -683,7 +697,7 @@ all the coefficients are not null. A full polynomial is defined by:
 %polynomials. The execution time remains the
 %element-key which justifies our work of parallelization.
 For our tests, a CPU Intel(R) Xeon(R) CPU
 %polynomials. The execution time remains the
 %element-key which justifies our work of parallelization.
 For our tests, a CPU Intel(R) Xeon(R) CPU

-E5620@2.40GHz and a GPU K40 (with 6 Go of ram) is used. 
+E5620@2.40GHz and a GPU K40 (with 6 Go of ram) are used. 

 
 
 %\subsection{Comparative study}
 
 
 %\subsection{Comparative study}


@@ -718,7 +732,7 @@ of the methods are given in Section~\ref{sec:vec_initialization}.
 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
 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
+Tesla K40 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
 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


@@ -835,7 +849,8 @@ numerical applications on GPU.
 
 In future works, we plan to investigate the possibility of using
 several multiple GPUs simultaneously, either with multi-GPU machine or
 
 In future works, we plan to investigate the possibility of using
 several multiple GPUs simultaneously, either with multi-GPU machine or

-with cluster of GPUs.
+with cluster of GPUs. It may also be interesting to study the
+implementation of other root finding polynomial methods on GPU.