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\begin{document}

\begin{frontmatter}

\title{Efficient high degree polynomial root finding using GPU}

%% Group authors per affiliation:
%\author{Elsevier\fnref{myfootnote}}
%\address{Radarweg 29, Amsterdam}
%\fntext[myfootnote]{Since 1880.}

%% or include affiliations in footnotes:
\author[mymainaddress]{Kahina Ghidouche}
%%\ead[url]{kahina.ghidouche@univ-bejaia.dz}
\cortext[mycorrespondingauthor]{Corresponding author}
\ead{kahina.ghidouche@univ-bejaia.dz}

\author[mysecondaryaddress]{Raphaël Couturier\corref{mycorrespondingauthor}}
%%\cortext[mycorrespondingauthor]{Corresponding author}
\ead{raphael.couturier@univ-fcomte.fr}

\author[mymainaddress]{Abderrahmane Sider}
%%\cortext[mycorrespondingauthor]{Corresponding author}
\ead{ar.sider@univ-bejaia.dz}

\address[mymainaddress]{Laboratoire LIMED, Faculté des sciences
  exactes, Université de Bejaia, 06000, Algeria}
\address[mysecondaryaddress]{FEMTO-ST Institute, University of
  Bourgogne Franche-Comte, France }

\begin{abstract}
Polynomials are mathematical algebraic structures that play a great
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
GPU architectures. The main result of this
work is to be able to solve high degree polynomials (up
to 1,000,000)  efficiently. We also compare the results with a
sequential implementation and the Durand-Kerner method on full and
sparse polynomials.
\end{abstract}

\begin{keyword}
Polynomial root finding, Iterative methods, Ehrlich-Aberth, Durand-Kerner, GPU
\end{keyword}

\end{frontmatter}

\linenumbers

\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 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}

The problem of finding a root is equivalent to that of solving a fixed-point problem. To see this, consider the fixed-point problem of finding the $n$-dimensional
vector $x$ such that :
\begin{center}
$x=g(x)$
\end{center}
where $g : C^{n}\longrightarrow C^{n}$. Usually, we can easily
rewrite this fixed-point problem as a root-finding problem by
setting $f(x) = x-g(x)$ and likewise we can recast the
root-finding problem into a fixed-point problem by setting :
\begin{center}
$g(x)= f(x)-x$.
\end{center}

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 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 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 other methods for the simultaneous
approximation of all the roots, starting with the Durand-Kerner (DK)
method:
%%\begin{center}
\begin{equation}
\label{DK}
 DK: z_i^{k+1}=z_{i}^{k}-\frac{P(z_i^{k})}{\prod_{i\neq j}(z_i^{k}-z_j^{k})},   i = 1, . . . , n,
\end{equation}
%%\end{center}
where $z_i^k$ is the $i^{th}$ root of the polynomial $p$ at the
iteration $k$.


This formula is mentioned for the first time by
Weiestrass~\cite{Weierstrass03} as part of the fundamental theorem
of Algebra and is rediscovered by Ilieff~\cite{Ilie50},
Docev~\cite{Docev62}, Durand~\cite{Durand60},
Kerner~\cite{Kerner66}. Another method discovered by
Borsch-Supan~\cite{ Borch-Supan63} and also described and brought
in the following form by Ehrlich~\cite{Ehrlich67} and
Aberth~\cite{Aberth73} uses a different iteration formula given as:
%%\begin{center}
\begin{equation}
\label{Eq:EA}
 EA: z_i^{k+1}=z_i^{k}-\frac{1}{{\frac {P'(z_i^{k})} {P(z_i^{k})}}-{\sum_{i\neq j}\frac{1}{(z_i^{k}-z_j^{k})}}}, i = 1, . . . , n,
\end{equation}
%%\end{center}
where $p'(z)$ is the polynomial derivative of $p$ evaluated in the
point $z$.

Aberth, Ehrlich and Farmer-Loizou~\cite{Loizou83} have proved that
the Ehrlich-Aberth method (EA) has a cubic order of convergence for simple roots whereas the Durand-Kerner has a quadratic order of convergence.


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 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 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 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. 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.

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
hardware resources provided by GPU in order to offer a stronger
computing ability to the massive data computing.


Ghidouche and al~\cite{Kahinall14} proposed an implementation of the
Durand-Kerner method on GPU. Their main
result showed that a parallel CUDA implementation is about 10 times faster than
the sequential implementation on a single CPU for  sparse
polynomials of degree 48,000. 


In this paper, we focus on the implementation of the Ehrlich-Aberth
method for high degree polynomials on GPU. We propose an adaptation of
the exponential logarithm in order to be able to solve sparse and full
polynomial of degree up to $1,000,000$. The paper is organized as
follows. Initially, we recall the Ehrlich-Aberth method in
Section~\ref{sec1}. Improvements for the Ehrlich-Aberth method are
proposed in Section \ref{sec2}. Related work to the implementation of
simultaneous methods using a parallel approach is presented in Section
\ref{secStateofArt}.  In Section~\ref{sec5} we propose a parallel
implementation of the Ehrlich-Aberth method on GPU and discuss
it. Section~\ref{sec6} presents and investigates our implementation
and experimental study results. Finally, Section~\ref{sec7} concludes
this paper and gives some hints for future research directions in this
topic.

\section{Ehrlich-Aberth method}
\label{sec1}
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 

%\begin{equation}
%w_{i}(z)=\prod_{j=1,j \neq i}^{n} (z-x_{j})
%\end{equation}

%And let a rational function $R_{i}(z)$ be the correction term of the
%Weistrass method~\cite{Weierstrass03}

%\begin{equation}
%R_{i}(z)=\frac{p(z)}{w_{i}(z)} , i=1,2,...,n.
%\end{equation}

%Differentiating the rational function $R_{i}(z)$ and applying the
%Newton method, we have:

%\begin{equation}
%\frac{R_{i}(z)}{R_{i}^{'}(z)}= \frac{p(z)}{p^{'}(z)-p(z)\frac{w_{i}(z)}{w_{i}^{'}(z)}}= \frac{p(z)}{p^{'}(z)-p(z) \sum _{j=1,j \neq i}^{n}\frac{1}{z-x_{j}}}, i=1,2,...,n
%\end{equation}
%where R_{i}^{'}(z)is the rational function derivative of F evaluated in the point z 
%Substituting $x_{j}$ for $z_{j}$ we obtain the Aberth iteration method.% 


\subsection{Polynomials Initialization}
The initialization of a polynomial $p(z)$ is done by setting each of the $n$ complex coefficients $a_{i}$:

\begin{equation}
\label{eq:SimplePolynome}
  p(z)=\sum{a_{i}z^{n-i}} , a_{n} \neq 0,a_{0}=1, a_{i}\subset C
\end{equation}


\subsection{Vector $Z^{(0)}$ Initialization}
\label{sec:vec_initialization}
As for any iterative method, we need to choose $n$ initial guess points $z^{0}_{i}, i = 1, . . . , n.$
The initial guess is very important since the number of steps needed by the iterative method to reach
a given approximation strongly depends on it.
In~\cite{Aberth73} the Ehrlich-Aberth iteration is started by selecting $n$
equi-spaced points on a circle of center 0 and radius r, where r is
an upper bound to the moduli of the zeros. Later, Bini and al.~\cite{Bini96}
performed this choice by selecting complex numbers along different
circles which relies on the result of~\cite{Ostrowski41}.

\begin{equation}
\label{eq:radiusR}
%%\begin{align}
\sigma_{0}=\frac{u+v}{2};u=\frac{\sum_{i=1}^{n}u_{i}}{n.max_{i=1}^{n}u_{i}};
v=\frac{\sum_{i=0}^{n-1}v_{i}}{n.min_{i=0}^{n-1}v_{i}};\\
%%\end{align}
\end{equation}
Where:
\begin{equation}
u_{i}=2.|a_{i}|^{\frac{1}{i}};
v_{i}=\frac{|\frac{a_{n}}{a_{i}}|^{\frac{1}{n-i}}}{2}.
\end{equation}

\subsection{Iterative Function}
%The operator used by the Aberth method is corresponding to the
%following equation~\ref{Eq:EA} which will enable the convergence towards
%polynomial solutions, provided all the roots are distinct.

Here we give a second form of the iterative function used by the Ehrlich-Aberth method: 

\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=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
Ehrlich-Aberth method and find the roots of high degree 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 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];\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}
With high degree polynomial, the Ehrlich-Aberth method implementation,
as well as the Durand-Kerner implementation, 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 $|z|$ stands for the modolus of a complex $z$. Indeed, the limited number in the
mantissa of floating points representations makes the computation of $p(z)$ wrong when z
is large. For example $(10^{50}) +1+ (- 10^{50})$ will give the wrong result
of $0$ instead of $1$. Consequently, we can not compute the roots
for large degrees. This problem was discussed earlier in
~\cite{Karimall98} for the Durand-Kerner method. The authors
propose to use the logarithm and the exponential of a complex in order to compute the power at a high exponent.

\begin{equation}
\label{deflncomplex}
 \forall(x,y)\in R^{*2}; \ln (x+i.y)=\ln(x^{2}+y^{2})
2+i.\arcsin(y\sqrt{x^{2}+y^{2}})_{\left] -\pi, \pi\right[ }
\end{equation}
%%\begin{equation}
\begin{align}
\label{defexpcomplex}
 \forall(x,y)\in R^{*2}; \exp(x+i.y) & = \exp(x).\exp(i.y)\\
                                     & =\exp(x).\cos(y)+i.\exp(x).\sin(y)\label{defexpcomplex1}
\end{align}
%%\end{equation}

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 degrees can be looked for successfully~\cite{Karimall98}.

Applying this solution for the iteration function Eq.~\ref{Eq:Hi} of
Ehrlich-Aberth method, we obtain the following 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),
\end{equation}

where:

\begin{equation}
\label{Log_H1}
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}

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{equation}

%R = \exp( \log(DBL\_MAX) / (2*n) )
%\end{equation}
 where \verb=DBL_MAX= stands for the maximum representable \verb=double= value.

\section{Implementation of simultaneous methods in a parallel computer}
\label{secStateofArt}   
The main problem of simultaneous methods is that the 
time needed for convergence is increased when we increase
the degree of the polynomial. The parallelization of these
algorithms is expected to improve the convergence time.
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 in Miranker
~\cite{Mirankar68,Mirankar71}, Schedler~\cite{Schedler72} and
Winograd~\cite{Winogard72}. The second approach aims at reducing the
computation time per iteration, as reported
in~\cite{Benall68,Jana06,Janall99,Riceall06}. 

There are many schemes for the simultaneous approximation of all roots of a given
polynomial. Several works on different methods and issues of root
finding have been reported in~\cite{Azad07, Gemignani07, Kalantari08,
  Zhancall08, Zhuall08}. However, the Durand-Kerner and the Ehrlich-Aberth methods are the most practical choices among
them~\cite{Bini04}. These two methods have been extensively
studied for parallelization due to their intrinsic parallelism, i.e. the
computations involved in both methods have some inherent
parallelism that can be suitably exploited by SIMD machines.
Moreover, they have fast a rate of convergence (quadratic for the
Durand-Kerner and cubic for the Ehrlich-Aberth). Various parallel
algorithms reported for these methods can be found
in~\cite{Cosnard90, Freeman89,Freemanall90,Jana99,Janall99}.
Freeman and Bane~\cite{Freemanall90} presented two parallel
algorithms on a local memory MIMD computer with the compute-to
communication time ratio O(n). However, their algorithms require
each processor to communicate its current approximation to all
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 the 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
practical for solving large degree polynomials.

%Until very recently, the literature did not mention implementations
%able to compute the roots of large degree polynomials (higher then
%1000) and within small or at least tractable times.

Finding polynomial roots rapidly and accurately is the main objective of our work. 
With the advent of CUDA (Compute Unified Device
Architecture), finding the roots of polynomials receives a new attention because of the new possibilities to solve higher degree polynomials in less time. 
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
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 $1,000,000$).


%% \section {A CUDA parallel Ehrlich-Aberth method}
%% In the following, we describe the parallel implementation of Ehrlich-Aberth method on GPU 
%% for solving high degree polynomials (up to $1,000,000$). First, the hardware and software of the GPUs are presented. Then, the CUDA parallel Ehrlich-Aberth method is presented.

%% \subsection{Background on the GPU architecture}
%% A GPU is viewed as an accelerator for the data-parallel and
%% intensive arithmetic computations. It draws its computing power
%% from the parallel nature of its hardware and software
%% architectures. A GPU is composed of hundreds of Streaming
%% Processors (SPs) organized in several blocks called Streaming
%% Multiprocessors (SMs). It also has a memory hierarchy. It has a
%% private read-write local memory per SP, fast shared memory and
%% read-only constant and texture caches per SM and a read-write
%% global memory shared by all its SPs~\cite{NVIDIA10}.

%% On a CPU equipped with a GPU, all the data-parallel and intensive
%% functions of an application running on the CPU are off-loaded onto
%% the GPU in order to accelerate their computations. A similar
%% data-parallel function is executed on a GPU as a kernel by
%% thousands or even millions of parallel threads, grouped together
%% as a grid of thread blocks. Therefore, each SM of the GPU executes
%% one or more thread blocks in SIMD fashion (Single  Instruction,
%% Multiple Data) and in turn each SP of a GPU SM runs one or more
%% threads within a block in SIMT fashion (Single Instruction,
%% Multiple threads). Indeed at any given clock cycle, the threads
%% execute the same instruction of a kernel, but each of them
%% operates on different data.
%%  GPUs only work on data filled in their
%% global memories and the final results of their kernel executions
%% must be communicated to their CPUs. Hence, the data must be
%% transferred in and out of the GPU. However, the speed of memory
%% copy between the GPU and the CPU is slower than the memory
%% bandwidths of the GPU memories and, thus, it dramatically affects
%% the performances of GPU computations. Accordingly, it is necessary
%% to limit as much as possible, data transfers between the GPU and its CPU during the
%% computations.
%% \subsection{Background on the CUDA Programming Model}

%% The CUDA programming model is similar in style to a single program
%% multiple-data (SPMD) software model. The GPU is viewed as a
%% coprocessor that executes data-parallel kernel functions. CUDA
%% provides three key abstractions, a hierarchy of thread groups,
%% shared memories, and barrier synchronization. Threads have a three
%% level hierarchy. A grid is a set of thread blocks that execute a
%% kernel function. Each grid consists of blocks of threads. Each
%% block is composed of hundreds of threads. Threads within one block
%% can share data using shared memory and can be synchronized at a
%% barrier. All threads within a block are executed concurrently on a
%% multithreaded architecture.The programmer specifies the number of
%% threads per block, and the number of blocks per grid. A thread in
%% the CUDA programming language is much lighter weight than a thread
%% in traditional operating systems. A thread in CUDA typically
%% processes one data element at a time. The CUDA programming model
%% has two shared read-write memory spaces, the shared memory space
%% and the global memory space. The shared memory is local to a block
%% and the global memory space is accessible by all blocks. CUDA also
%% provides two read-only memory spaces, the constant space and the
%% texture space, which reside in external DRAM, and are accessed via
%% read-only caches.

\section{ Implementation of the Ehrlich-Aberth method on GPU}
\label{sec5}
%%\subsection{A CUDA implementation of the Aberth's method }
%%\subsection{A GPU implementation of the Aberth's method }



%% \subsection{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}

%% \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),$Pu$ (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)}

%% \BlankLine

%% Initialization of $P$\;
%% Initialization of $Pu$\;
%% Initialization of the solution vector $Z^{0}$\;
%% $\Delta z_{max}=0$\;
%%  k=0\;

%% \While {$\Delta z_{max} > \varepsilon$}{
%%  Let $\Delta z_{max}=0$\;
%% \For{$j \gets 0 $ \KwTo $n$}{
%% $ZPrec\left[j\right]=Z\left[j\right]$;// save Z at the iteration k.\

%% $Z\left[j\right]=H\left(j, Z, P, Pu\right)$;//update Z with the iterative function.\
%% }
%% k=k+1\;

%% \For{$i \gets 0 $ \KwTo $n-1$}{
%% $c= testConverge(\Delta z_{max},ZPrec\left[j\right],Z\left[j\right])$\;
%% \If{$c > \Delta z_{max}$ }{
%% $\Delta z_{max}$=c\;}
%% }

%% }
%% \end{algorithm}

%% ~\\ 
%% In this sequential algorithm, one CPU thread  executes all the steps. Let us look to the $3^{rd}$ step i.e. the execution of the iterative function, 2 sub-steps are needed. The first sub-step \textit{save}s the solution vector of the previous iteration, the second sub-step \textit{update}s or computes the new values of the roots vector.

\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
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}
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:
%\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}
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}

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 freshest computed roots $z^{k+1}_{i}$.

%The $4^{th}$ step of the algorithm checks the convergence condition using Eq.~\ref{eq:Aberth-Conv-Cond}.
%Both steps 3 and 4 use 1 thread to compute all the $n$ roots on CPU, which is very harmful for performance in case of the large degree polynomials.



%On the CPU,  both steps 3 and 4 contain the loop \verb=for= and a single thread executes all the instructions in the loop $n$ times. In this subsection, we explain how the GPU architecture can compute this loop and reduce the execution time.
%In the GPU, the scheduler assigns the execution of this loop to a
%group of threads organised as a grid of blocks with block containing a
%number of threads. All threads within a block are executed
%concurrently in parallel. The instructions run on the GPU are grouped
%in special function called kernels. With CUDA, a programmer must
%describe the kernel execution context:  the size of the Grid, the number of blocks and the number of threads per block.

%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 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
  threshold), P (Polynomial to solve), Pu (Derivative of P), $n$ (Polynomial degrees), $\Delta z_{max}$ (Maximum value of stop condition)}

\KwOut {$Z$ (Solution root's vector), $ZPrec$ (Previous solution root's vector)}

\BlankLine

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

}
\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 7 of
Algorithm~\ref{alg2-cuda} consists in saving the vector of
polynomial roots 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. 

\begin{algorithm}[H]
\label{alg3-update}
%\LinesNumbered
\caption{Kernel update}

\eIf{$(\left|Z\right|<= R)$}{
$kernel\_update(Z,P,Pu)$\;}
{
$kernel\_update\_ExpoLog(Z,P,Pu)$\;
}
\end{algorithm}

If the modulus
of the current complex is less than a given value called the
radius i.e. ($ |z^{k}_{i}|<= R$), then the classical form of the EA
function Eq.~\ref{Eq:Hi} is executed else the EA.EL
function Eq.~\ref{Log_H2} is executed.
(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. 

The kernel terminates its computations when all the roots have
converged. It should be noticed that, as blocks of threads are
scheduled automatically by the GPU, we have absolutely no control on
the order of the blocks. Consequently, our algorithm is executed more
or less with the asynchronous iteration model, where blocks of roots
are updated in a non deterministic way. As the Durand-Kerner method
has been proved to converge with asynchronous iterations, we think it
is similar with the Ehrlich-Aberth method, but we did not try to prove
this in that paper. Another consequence of that, is that several
executions of our algorithm with the same polynomial do not
necessarily give the same result (but roots have the same accuracy)
and the same number of iterations (even if the variation is not very
significant).





%%HIER END MY REVISIONS (SIDER)
\section{Experimental study}
\label{sec6}
%\subsection{Definition of the used polynomials }
We study two categories of polynomials: sparse polynomials and the full polynomials.\\
{\it A sparse polynomial} is a polynomial for which only some
coefficients are not null. In this paper, we consider sparse polynomials for which the roots are distributed on 2 distinct circles:
\begin{equation}
        \forall \alpha_{1} \alpha_{2} \in C,\forall n_{1},n_{2} \in N^{*}; P(z)= (z^{n_{1}}-\alpha_{1})(z^{n_{2}}-\alpha_{2})
\end{equation}\noindent
{\it A full polynomial} is, in contrast, a polynomial for which
all the coefficients are not null. A full polynomial is defined by:
%%\begin{equation}
        %%\forall \alpha_{i} \in C,\forall n_{i}\in N^{*}; P(z)= \sum^{n}_{i=1}(z^{n^{i}}.a_{i})
%%\end{equation}

\begin{equation}
     {\Large \forall a_{i} \in C, i\in N;  p(x)=\sum^{n}_{i=0} a_{i}.x^{i}} 
\end{equation}
%With this form, we can have until \textit{n} non zero terms whereas the sparse ones have just two non zero terms.

%\subsection{The study condition} 
%Two parameters are studied are
%the polynomial degree and the execution time of our program
%to converge on the solution. The polynomial degree allows us
%to validate that our algorithm is powerful with high degree
%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) are used. 


%\subsection{Comparative study}
%First, performances of the Ehrlich-Aberth method  of root finding polynomials
%implemented on CPUs and on GPUs are studied.

We performed a set of experiments on the sequential and the parallel algorithms, for both sparse and full polynomials of different sizes. We took into account the execution times, the  polynomial size and the number of threads per block performed by sum or each experiment on CPU and on GPU.

All experimental results obtained from the simulations are made in
double precision data, the convergence threshold of the methods is set
to $10^{-7}$.
%Since we were more interested in the comparison of the
%performance behaviors of Ehrlich-Aberth and Durand-Kerner methods on
%CPUs versus on GPUs.
The initialization values of the vector solution
of the methods are given in Section~\ref{sec:vec_initialization}.

\subsection{Comparison of execution times of the Ehrlich-Aberth method
  on a CPU with OpenMP (1 core and 4 cores) vs. on a Tesla GPU}

\begin{figure}[htbp]
\centering
  \includegraphics[width=0.8\textwidth]{figures/openMP-GPU}
\caption{Comparison of execution times  of the Ehrlich-Aberth method
  on a CPU with OpenMP (1 core, 4 cores) and on a Tesla GPU}
\label{fig:01}
\end{figure}
%%Figure 1 %%show a comparison of execution time between the parallel
%%and sequential version of the Ehrlich-Aberth algorithm with sparse
%%polynomial exceed 100000,

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 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
CPU (4 cores) implementation exceed 5,000s for 250,000 degrees
polynomials. On the other hand, the GPU implementation for the same
polynomials do not take more 100s. With the GPU
we can solve high degree polynomials very quickly up to degree 1,000,000. We can also notice that the GPU implementation are
 almost 40 times faster then the implementation on the CPU (4 cores).


 

%This reduction of time allows us to compute roots of polynomial of more important degree at the same time than with a CPU.
 
 %We notice that the convergence precision is a round $10^{-7}$ for the both implementation on CPU and GPU. Consequently, we can conclude that Ehrlich-Aberth on GPU are faster and accurately then CPU implementation.

\subsection{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
To optimize the performances of an algorithm on a GPU, it is necessary to maximize the use of cores GPU (maximize the number of threads executed in parallel). In fact, it is interesting to see the influence of the number of threads per block on the execution time of Ehrlich-Aberth algorithm. 
For that, we noticed that the maximum number of threads per block for
the Nvidia Tesla K40 GPU is 1,024, so we varied the number of threads
per block from 8 to 1,024. We took into account the execution time for
10 different sparse and full polynomials of degree 50,000 and of degree 500,000.

\begin{figure}[htbp]
\centering
  \includegraphics[width=0.8\textwidth]{figures/influence_nb_threads}
\caption{Influence of the number of threads on the execution times of different polynomials (sparse and full)}
\label{fig:02}
\end{figure}

Figure~\ref{fig:02} shows that, the best execution time for both
sparse and full polynomial are given when the threads number varies
between 64 and 256 threads per block. We notice that with small
polynomials the best number of threads per block is 64, whereas for large polynomials the best number of threads per block is
256. However, in the following experiments we specify that the number
of threads per block is 256.


\subsection{Influence of exp-log solution to compute high degree polynomials}

In this experiment we report the performance of the exp-log solution described in Section~\ref{sec2} to compute high degree polynomials.   
\begin{figure}[htbp]
\centering
  \includegraphics[width=0.8\textwidth]{figures/sparse_full_explog}
\caption{The impact of exp-log solution to compute  high degree  polynomials}
\label{fig:03}
\end{figure}


Figure~\ref{fig:03} shows a comparison between the execution time of
the Ehrlich-Aberth method using the exp-log solution and the
execution time of the Ehrlich-Aberth method without this solution,
with full and sparse polynomials degrees. We can see that the
execution times for both algorithms are the same with full polynomials
degree inferior to 4,000 and sparse polynomials inferior to 150,000. We
also clearly show that the classical version (without exp-log) of
Ehrlich-Aberth algorithm does not converge after these degrees with
sparse and full polynomials. On the contrary, the new version of
the Ehrlich-Aberth algorithm with the exp-log solution can solve
high degree polynomials.

%in fact, when the modulus of the roots are up than \textit{R} given in ~\ref{R},this exceed the limited number in the mantissa of floating points representations and can not compute the iterative function given in ~\ref{eq:Aberth-H-GS} to obtain the root solution, who justify the divergence of the classical Ehrlich-Aberth algorithm. However, applying exp-log solution given in ~\ref{sec2} took into account the limit of floating using the iterative function in(Eq.~\ref{Log_H1},Eq.~\ref{Log_H2} and allows to solve a very large polynomials degrees . 




\subsection{Comparison of the Durand-Kerner and the Ehrlich-Aberth methods}

In this part, we  compare the Durand-Kerner and the Ehrlich-Aberth
methods on GPU. We took into account the execution times, the number of iterations and the polynomials size for both sparse and full polynomials.  

\begin{figure}[htbp]
\centering
  \includegraphics[width=0.8\textwidth]{figures/EA_DK}
\caption{Execution times of the  Durand-Kerner and the Ehrlich-Aberth methods on GPU}
\label{fig:04}
\end{figure}

Figure~\ref{fig:04} shows the execution times of both methods with
sparse polynomial degrees ranging from 1,000 to 1,000,000. We can see
that the Ehrlich-Aberth algorithm is faster than Durand-Kerner
algorithm, being on average 25 times faster. Then, when degrees of
polynomials exceed 500,000 the execution times with DK are very long.

%with double precision not exceed $10^{-5}$.

\begin{figure}[htbp]
\centering
  \includegraphics[width=0.8\textwidth]{figures/EA_DK_nbr}
\caption{The number of iterations to converge for the Ehrlich-Aberth
  and the Durand-Kerner methods}
\label{fig:05}
\end{figure}

Figure~\ref{fig:05} shows the evaluation of the number of iterations according
to the degree of polynomials for both EA and DK algorithms. We can see
that the number of iterations of DK is of order 100 while EA is of order
10. Indeed the computation of the derivative of P in the iterative function (Eq.~\ref{Eq:Hi}) executed by EA
allows the algorithm to converge faster. On the contrary, the
DK operator (Eq.~\ref{DK}) needs low operations, consequently low
execution times per iteration, but it needs more iterations to converge.




 \section{Conclusion and perspectives}
\label{sec7}
In this paper we have presented the parallel implementation
Ehrlich-Aberth method on GPU for the problem of finding roots
polynomial. Moreover, we have improved the classical Ehrlich-Aberth
method which suffers from overflow problems, the exp-log solution
applied to the iterative function allows to solve high degree
polynomials.

We have performed many experiments with the Ehrlich-Aberth method in
GPU. These experiments highlight that this method is more efficient in
GPU than all the other implementations. The improvement with
the exponential logarithm solution allows us to solve sparse and full
high degree polynomials up to 1,000,000 degree. Hence, it may be
possible to consider  using polynomial root finding methods in other
numerical applications on GPU.


In future works, we plan to investigate the possibility of using
several multiple GPUs simultaneously, either with a multi-GPU machine or
with a cluster of GPUs. It may also be interesting to study the
implementation of other root finding polynomial methods on GPU.



\bibliography{mybibfile}

\end{document}