\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), $\Delta z_{max}$ (maximum value
+ threshold), $P$ (Polynomial to solve),$Pu$ (the derivative of P) $\Delta z_{max}$ (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)}
+\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 $P$\;
+Initialization of $Pu$\;
Initialization of the solution vector $Z^{0}$\;
$\Delta z_{max}=0$\;
k=0\;
\For{$j \gets 0 $ \KwTo $n$}{
$ZPrec\left[j\right]=Z\left[j\right]$;// save Z at the iteration k.\
-$Z\left[j\right]=H\left(j,Z\right)$;//update Z with the iterative function.\
+$Z\left[j\right]=H\left(j, Z, P, Pu\right)$;//update Z with the iterative function.\
}
k=k+1\;
%\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), $\Delta z_{max}$ (maximum value of stop condition)}
+\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)}
-\KwOut {Z (The solution root's vector)}
+\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 of P\;
+Initialization of the of Pu\;
Initialization of the solution vector $Z^{0}$\;
-Allocate and copy initial data to the GPU global memory\;
+Allocate and copy initial data to the GPU global memory ($d\_Z,d\_ZPrec,d\_P,d\_Pu$)\;
k=0\;
-\While {$\Delta z_{max}\succ \epsilon$}{
+\While {$\Delta z_{max} > \epsilon$}{
Let $\Delta z_{max}=0$\;
-$ kernel\_save(d\_Z^{k-1})$\;
+$ kernel\_save(d\_ZPrec,d\_Z)$\;
k=k+1\;
-$ kernel\_update(d\_Z^{k})$\;
-$kernel\_testConverge(\Delta z_{max},d\_Z^{k},d\_Z^{k-1})$\;
+$ kernel\_update(d\_Z,d\_P,d\_Pu)$\;
+$kernel\_testConverge(\Delta z_{max},d\_Z,d\_ZPrec)$\;
}
+Copy results from GPU memory to CPU memory\;
\end{algorithm}
~\\
-After the initialisation 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}).
+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
-$z^{k}$, according to Algorithm~\ref{alg3-update}. We notice that the
+$d\_Z$, according to Algorithm~\ref{alg3-update}. We notice that the
update kernel is called in two forms, separated with the value of
\emph{R} which determines the radius beyond which we apply the
exponential logarithm algorithm.
%\LinesNumbered
\caption{Kernel update}
-\eIf{$(\left|Z^{(k)}\right|<= R)$}{
-$kernel\_update(d\_z^{k})$\;}
+\eIf{$(\left|d\_Z\right|<= R)$}{
+$kernel\_update((d\_Z,d\_Pcoef,d\_Pdegres,d\_Pucoef,d\_Pudegres)$\;}
{
-$kernel\_update\_ExpoLog(d\_z^{k})$\;
+$kernel\_update\_ExpoLog((d\_Z,d\_Pcoef,d\_Pdegres,d\_Pucoef,d\_Pudegres))$\;
}
\end{algorithm}
