\begin{abstract}
Polynomials are mathematical algebraic structures that play a great
-role in science and engineering. Finding roots of high degree
+role in science and engineering. Finding the roots of high degree
polynomials is computationally demanding. In this paper, we present
the results of a parallel implementation of the Ehrlich-Aberth
algorithm for the root finding problem for high degree polynomials on
\linenumbers
-\section{The problem of finding roots of a polynomial}
-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 :
+\section{The problem of finding the roots of a polynomial}
+Polynomials are mathematical algebraic structures used in science and engineering to capture physical phenomena 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_{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 $p$. If zeros are $\alpha_{i},\textit{i=1,...,n}$ the $p(x)$ can be written as :
+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 zeros 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}
$g(x)= f(x)-x$.
\end{center}
-Often it is not be possible to solve such nonlinear equation
-root-finding problems analytically. When this occurs we turn to
+It is often impossible to solve such nonlinear equation
+root-finding problems analytically. When this occurs, we turn to
numerical methods to approximate the solution.
Generally speaking, algorithms for solving problems can be divided into
two main groups: direct methods and iterative methods.
-\\
-Direct methods exist only for $n \leq 4$, solved in closed form by G. Cardano
-in the mid-16th century. However, N. H. Abel in the early 19th
-century showed that polynomials of degree five or more could not
+
+Direct methods only exist for $n \leq 4$, solved in closed form
+by G. Cardano in the mid-16th century. However, N. H. Abel in the early 19th
+century proved that polynomials of degree five or more could not
be solved by direct methods. Since then, mathematicians have
focussed on numerical (iterative) methods such as the famous
-Newton method, the Bernoulli method of the 18th, and the Graeffe method.
+Newton method, the Bernoulli method of the 18th century, and the Graeffe method.
Later on, with the advent of electronic computers, other methods have
been developed such as the Jenkins-Traub method, the Larkin method,
-the Muller method, and several methods for simultaneous
+the Muller method, and several other methods for the simultaneous
approximation of all the roots, starting with the Durand-Kerner (DK)
method:
%%\begin{center}
the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of convergence.
-Iterative methods raise several problem when implemented e.g.
-specific sizes of numbers must be used to deal with this
-difficulty. Moreover, the convergence time of iterative methods
+Moreover, the convergence times of iterative methods
drastically increases like the degrees of high polynomials. It is expected that the
-parallelization of these algorithms will improve the convergence
-time.
+parallelization of these algorithms will reduce the execution times.
Many authors have dealt with the parallelization of
simultaneous methods, i.e. that find all the zeros simultaneously.
Freeman~\cite{Freeman89} implemented and compared DK, EA and another method of the fourth order proposed
-by Farmer and Loizou~\cite{Loizou83}, on a 8-processor linear
-chain, for polynomials of degree up to 8. The third method often
-diverges, but the first two methods have speed-up equal to 5.5. Later,
+by Farmer and Loizou~\cite{Loizou83}, on an 8-processor linear
+chain, for polynomials of degree 8. The third method often
+diverges, but the first two methods have speed-ups equal to 5.5. Later,
Freeman and Bane~\cite{Freemanall90} considered asynchronous
algorithms, in which each processor continues to update its
-approximations even though the latest values of other $z_i^{k}$
-have not been received from the other processors, in contrast with synchronous algorithms where it would wait those values before making a new iteration.
+approximations even though the latest values of other roots
+have not yet been received from the other processors. In contrast,
+synchronous algorithms wait the computation of all roots at a given
+iterations before making a new one.
Couturier and al.~\cite{Raphaelall01} proposed two methods of parallelization for
a shared memory architecture and for distributed memory one. They were able to
compute the roots of sparse polynomials of degree 10,000 in 430 seconds with only 8
-personal computers and 2 communications per iteration. Comparing to the sequential implementation
-where it takes up to 3,300 seconds to obtain the same results, the authors show an interesting speedup.
+personal computers and 2 communications per iteration. Compared to sequential implementations
+where it takes up to 3,300 seconds to obtain the same results, the
+authors' work experiment show an interesting speedup.
-Very few works had been performed since this last work until the appearing of
+Few works have been conducted after those works until the appearance of
the Compute Unified Device Architecture (CUDA)~\cite{CUDA10}, a
parallel computing platform and a programming model invented by
NVIDIA. The computing power of GPUs (Graphics Processing Unit) has exceeded that of CPUs. However, CUDA adopts a totally new computing architecture to use the
\section{Ehrlich-Aberth method}
\label{sec1}
-A cubically convergent iteration method for finding zeros of
-polynomials was proposed by O. Aberth~\cite{Aberth73}. The Ehrlich-Aberth method contain 4 main steps, presented in the following.
+A cubically convergent iteration method to find zeros of
+polynomials was proposed by O. Aberth~\cite{Aberth73}. The
+Ehrlich-Aberth method contains 4 main steps, presented in what
+follows.
+
%The Aberth method is a purely algebraic derivation.
%To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors
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(
-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:
\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}
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
+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.
\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
$z^{k}_{i}$ to compute the new values $z^{k+1}_{i}$, that is :
\begin{equation}
\label{eq:Aberth-H-GS}
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=1,j\neq i}^{j=n}{\frac{1}{(z_{i}^{k}-z_{j}^{k})}})}, i=1,. . . .,n
+{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}
Using Eq.~\ref{eq:Aberth-H-GS} to update the vector solution
%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}
-\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (error tolerance
+\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 {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}
\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 {$\Delta z_{max} > \epsilon$}{
- 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{enumerate}
~\\
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 6 of
+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
-8, Algorithm~\ref{alg2-cuda}).
+10, Algorithm~\ref{alg2-cuda}).
-The second kernel executes the iterative function $H$ and updates
-Z, according to Algorithm~\ref{alg3-update}. We notice that the
+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.
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
-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
%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}
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
