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}
-\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}
-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
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}
\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}
-%\LinesNumbered
+
\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 $Z^{0}$\;
+$\Delta z_{max}=0$\;
+ k=0\;
-\While {$\Delta z_{max}\succ \epsilon$}{
+\While {$\Delta z_{max} > \varepsilon$}{
Let $\Delta z_{max}=0$\;
\For{$j \gets 0 $ \KwTo $n$}{
-$ZPrec\left[j\right]=Z\left[j\right]$\;
-$Z\left[j\right]=H\left(j,Z\right)$\;
+$ZPrec\left[j\right]=Z\left[j\right]$;// 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 $n-1$}{
-$c=\frac{\left|Z\left[i\right]-ZPrec\left[i\right]\right|}{Z\left[i\right]}$\;
+$c= testConverge(\Delta z_{max},ZPrec\left[j\right],Z\left[j\right])$\;
\If{$c > \Delta z_{max}$ }{
$\Delta z_{max}$=c\;}
}
+
}
\end{algorithm}
%\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)}
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 $\Delta z_{max}=0$\;
-$ 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}
~\\
