X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/kahina_paper1.git/blobdiff_plain/5870d4e72db0c954d03231f1d807de7aee47ff74..1fc948028049feeeacad3808eedfb3b2acb32775:/paper.tex

diff --git a/paper.tex b/paper.tex
index f66bf99..20686d2 100644
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+++ b/paper.tex
@@ -232,7 +232,8 @@ topic.
 \section{The Sequential Ehrlich-Aberth method}
 \label{sec1}
 A cubically convergent iteration method for finding zeros of
-polynomials was proposed by O. Aberth~\cite{Aberth73}. In the fellowing we present the main stages of the running of the Ehrlich-Aberth method.
+polynomials was proposed by O. Aberth~\cite{Aberth73}. In the
+following we present the main stages of our implementation the Ehrlich-Aberth method.
 %The Aberth method is a purely algebraic derivation. 
 %To illustrate the derivation, we let $w_{i}(z)$ be the product of linear factors 
 
@@ -268,7 +269,7 @@ The initialization of a polynomial p(z) is done by setting each of the $n$ compl
 
 \subsection{Vector $z^{(0)}$ Initialization}
 
-Like for any iterative method, we need to choose $n$ initial guess points $z^{(0)}_{i}, i = 1, . . . , n.$
+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 $n$
@@ -302,9 +303,12 @@ Here we give a second form of the iterative function used by Ehrlich-Aberth meth
 EA2: z^{k+1}=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=0,. . . .,n
 \end{equation}
-we notice that the function iterative in Eq.~\ref{Eq:Hi} it the same those presented in Eq.~\ref{Eq:EA}, but we prefer used the last one seen the advantage of its use to improve the Ehrlich-Aberth method and resolve very high degrees polynomials. More detail in the section ~\ref{sec2}.    
+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 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}
@@ -315,7 +319,8 @@ The convergence condition determines the termination of the algorithm. It consis
 
 \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
@@ -343,7 +348,7 @@ 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
-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}
@@ -389,7 +394,7 @@ There are many schemes for the simultaneous approximation of all roots of a give
 polynomial. Several works on different methods and issues of root
 finding have been reported in~\cite{Azad07, Gemignani07, Kalantari08, Skachek08, Zhancall08, Zhuall08}. However, Durand-Kerner and Ehrlich-Aberth methods are the most practical choices among
 them~\cite{Bini04}. These two methods have been extensively
-studied for parallelization due to their intrinsics, i.e. the
+studied for parallelization due to their intrinsics parallelism, i.e. the
 computations involved in both methods has some inherent
 parallelism that can be suitably exploited by SIMD machines.
 Moreover, they have fast rate of convergence (quadratic for the
@@ -408,8 +413,11 @@ 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.
-Until very recently, the literature doen 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. 
+%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