new

[kahina_paper1.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 276c50a9b3eb46e64687e1ceacbf25a4beeca9b6..20686d252b697349ef0d6b9e868058bdb356bc74 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -348,7 +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
 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}
 %%$$ \exp \bigl(  \ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'}))- \ln(1- \exp(\ln(p(z)_{k})-ln(\ln(p(z)_{k}^{'})+\ln\sum_{i\neq j}^{n}\frac{1}{z_{k}-z_{j}})$$
 \begin{equation}
 \label{Log_H2}


@@ -394,7 +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
 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
 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


@@ -413,8 +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.
 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
 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