MAJ Mybibfile

[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 76b626e2b46a919c49ea888d44f23733381c28dd..4213238dc15bba747cab568e22b4b4baecab2c2e 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -308,18 +308,18 @@ but we prefer the latter one because we can use it to improve the
 Ehrlich-Aberth method and find the roots of very high degrees polynomials. More
 details are given in Section ~\ref{sec2}.
 \subsection{Convergence Condition}
 Ehrlich-Aberth method and find the roots of very high degrees polynomials. More
 details are given in Section ~\ref{sec2}.
 \subsection{Convergence Condition}

-The convergence condition determines the termination of the algorithm. It consists in stopping from running the iterative function  when the roots are sufficiently stable. We consider that the method converges sufficiently when:
+The convergence condition determines the termination of the algorithm. It consists in stopping the iterative function  when the roots are sufficiently stable. We consider that the method converges sufficiently when:

 
 \begin{equation}
 \label{eq:Aberth-Conv-Cond}
 
 \begin{equation}
 \label{eq:Aberth-Conv-Cond}

-\forall i \in
-[1,n];\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}<\xi
+\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi

 \end{equation}
 
 
 \section{Improving the Ehrlich-Aberth Method for high degree polynomials with exp.log formulation}
 \label{sec2}
 \end{equation}
 
 
 \section{Improving the Ehrlich-Aberth Method for high degree polynomials with exp.log formulation}
 \label{sec2}

-The Ehrlich-Aberth method implementation suffers of overflow problems. This
+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
 point $\xi$ where $|\xi| > 1$, where $|x|$ stands for the modolus of a complex $x$. Indeed, the limited number in the
 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


@@ -347,7 +347,7 @@ Using the logarithm (eq.~\ref{deflncomplex}) and the exponential (eq.~\ref{defex
 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
 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 logarithm:
+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}


@@ -493,32 +493,37 @@ read-only caches.
 
 \subsection{A sequential Ehrlich-Aberth algorithm}
 The main steps of Ehrlich-Aberth method are shown in Algorithm.~\ref{alg1-seq} :
 
 \subsection{A sequential Ehrlich-Aberth algorithm}
 The main steps of Ehrlich-Aberth method are shown in Algorithm.~\ref{alg1-seq} :

-  
+%\LinesNumbered  

 \begin{algorithm}[H]
 \label{alg1-seq}
 \begin{algorithm}[H]
 \label{alg1-seq}

-%\LinesNumbered
+

 \caption{A sequential algorithm to find roots with the Ehrlich-Aberth method}
 
 \caption{A sequential algorithm to find roots with the Ehrlich-Aberth method}
 

-\KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error tolerance threshold),P(Polynomial to solve)}
-\KwOut {Z(The solution root's vector)}
+\KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error tolerance threshold), P(Polynomial to solve),$\Delta z_{max}$ (maximum value of stop condition),k (number of iteration),n(Polynomial's degrees)}
+\KwOut {Z (The solution root's vector),ZPrec (the previous solution root's vector)}

 
 \BlankLine
 
 Initialization of the coefficients of the polynomial to solve\;
 Initialization of the solution vector $Z^{0}$\;
 
 \BlankLine
 
 Initialization of the coefficients of the polynomial to solve\;
 Initialization of the solution vector $Z^{0}$\;

+$\Delta z_{max}=0$\;
+ k=0\;

 
 

-\While {$\Delta z_{max}\succ \epsilon$}{
+\While {$\Delta z_{max} > \varepsilon$}{

  Let $\Delta z_{max}=0$\;
 \For{$j \gets 0 $ \KwTo $n$}{
  Let $\Delta z_{max}=0$\;
 \For{$j \gets 0 $ \KwTo $n$}{

-$ZPrec\left[j\right]=Z\left[j\right]$\;
-$Z\left[j\right]=H\left(j,Z\right)$\;
+$ZPrec\left[j\right]=Z\left[j\right]$;// save Z at the iteration k.\
+
+$Z\left[j\right]=H\left(j,Z\right)$;//update Z with the iterative function.\

 }
 }

+k=k+1\;

 
 \For{$i \gets 0 $ \KwTo $n-1$}{
 
 \For{$i \gets 0 $ \KwTo $n-1$}{

-$c=\frac{\left|Z\left[i\right]-ZPrec\left[i\right]\right|}{Z\left[i\right]}$\;
+$c= testConverge(\Delta z_{max},ZPrec\left[j\right],Z\left[j\right])$\;

 \If{$c > \Delta z_{max}$ }{
 $\Delta z_{max}$=c\;}
 }
 \If{$c > \Delta z_{max}$ }{
 $\Delta z_{max}$=c\;}
 }

+

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


@@ -555,8 +560,7 @@ Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth algorithm using C
 %\LinesNumbered
 \caption{CUDA Algorithm to find roots with the Ehrlich-Aberth method}
 
 %\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)}
+\KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error tolerance threshold), P(Polynomial to solve), $\Delta z_{max}$ (maximum value of stop condition)}

 
 \KwOut {Z(The solution root's vector)}
 
 
 \KwOut {Z(The solution root's vector)}
 


@@ -565,12 +569,14 @@ tolerance threshold),P(Polynomial to solve)}
 Initialization of the coeffcients of the polynomial to solve\;
 Initialization of the solution vector $Z^{0}$\;
 Allocate and copy initial data to the GPU global memory\;
 Initialization of the coeffcients of the polynomial to solve\;
 Initialization of the solution vector $Z^{0}$\;
 Allocate and copy initial data to the GPU global memory\;

-
+k=0\;

 \While {$\Delta z_{max}\succ \epsilon$}{
  Let $\Delta z_{max}=0$\;
 \While {$\Delta z_{max}\succ \epsilon$}{
  Let $\Delta z_{max}=0$\;

-$ kernel\_save(d\_z^{k-1})$\;
-$ kernel\_update(d\_z^{k})$\;
-$kernel\_testConverge(\Delta z_{max},d_z^{k},d_z^{k-1})$\;
+$ kernel\_save(d\_Z^{k-1})$\;
+k=k+1\;
+$ kernel\_update(d\_Z^{k})$\;
+$kernel\_testConverge(\Delta z_{max},d\_Z^{k},d\_Z^{k-1})$\;
+

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