From c90f146cdefa047c69990e5c0d1891ef6def1bc2 Mon Sep 17 00:00:00 2001 From: Kahina Date: Mon, 2 Nov 2015 11:44:07 +0100 Subject: [PATCH 01/16] MAJ eq:Aberth-Conv-Cond --- paper.tex | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/paper.tex b/paper.tex index c4fa445..c78df7b 100644 --- a/paper.tex +++ b/paper.tex @@ -312,8 +312,7 @@ The convergence condition determines the termination of the algorithm. It consis \begin{equation} \label{eq:Aberth-Conv-Cond} -\forall i \in -[1,n];\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}<\xi +\forall i \in [1,n];\vert\frac{z_{i}^{k}-z_{i}^{k-1}}{z_{i}^{k}}\vert<\xi \end{equation} -- 2.39.5 From 0cf063826e3d100cb00eaa4b104306a581de2ac3 Mon Sep 17 00:00:00 2001 From: Kahina Date: Mon, 2 Nov 2015 14:13:13 +0100 Subject: [PATCH 02/16] MAJ Mybibfile --- mybibfile.bib | 147 ++++++++++++++++++++++++++++++++++++-------------- paper.tex | 34 +++++++----- 2 files changed, 126 insertions(+), 55 deletions(-) diff --git a/mybibfile.bib b/mybibfile.bib index 2347318..d29bcb2 100644 --- a/mybibfile.bib +++ b/mybibfile.bib @@ -53,51 +53,116 @@ author = "K. Docev", }x -@Article{Durand60, - title = "Solution Numerique des Equations Algebriques, Vol. 1, Equations du Type F(x)=0, Racines d'une Polynome", - journal = "", - volume = "Vol.1", - number = "", - pages = "", - year = "1960", - author = "E. Durand", -}x +%@Article{Durand60, + % title = "Solution Numerique des Equations Algebriques, Vol. 1, Equations du Type F(x)=0, Racines d'une Polynome", + %journal = "", +% volume = "Vol.1", +% number = "", + % pages = "", + %year = "1960", + % author = "E. Durand", +%}x +@Book{Durand60, + author = "\'E. Durand", + publisher = "Masson, Paris", + title = "Solutions num\'eriques des \'equations alg\'ebriques. + {T}ome {I}: \'{E}quations du type {$F(x)=0$}; racines + d'un polyn\^ome", + year = "1960", +}x + +%@Article{Kerner66, + %title = "Ein Gesamtschritteverfahren zur Berechnung der Nullstellen von Polynomen", + % journal = "Numerische Mathematik", +% volume = "8", +% number = "3", +% pages = "290-294", + %year = "1966", + % author = "I. Kerner", +%}x + @Article{Kerner66, - title = "Ein Gesamtschritteverfahren zur Berechnung der Nullstellen von Polynomen", - journal = " ", - volume = "", - number = "8", - pages = "290-294", - year = "1966", - author = "I. Kerner", -}x + author = "Immo O. Kerner", + title = "{Ein Gesamtschrittverfahren zur Berechnung der + Nullstellen von Polynomen}. ({German}) [{A} Complete + Step Method for the Computation of Zeros of + Polynomials]", + journal = "Numerische Mathematik", + volume = "8", + number = "3", + pages = "290--294", + month = may, + year = "1966", + CODEN = "NUMMA7", + ISSN = "0029-599X (print), 0945-3245 (electronic)", + bibdate = "Mon Oct 18 01:28:20 MDT 1999", + bibsource = "http://www.math.utah.edu/pub/tex/bib/nummath.bib", + acknowledgement = "Nelson H. F. Beebe, University of Utah, Department + of Mathematics, 110 LCB, 155 S 1400 E RM 233, Salt Lake + City, UT 84112-0090, USA, Tel: +1 801 581 5254, FAX: +1 + 801 581 4148, e-mail: \path|beebe@math.utah.edu|, + \path|beebe@acm.org|, \path|beebe@computer.org| + (Internet), URL: + \path|http://www.math.utah.edu/~beebe/|", + fjournal = "Numerische Mathematik", + journal-url = "http://link.springer.com/journal/211", + language = "German", +} +%@Article{Borch-Supan63, +% title = "A posteriori error for the zeros of polynomials", + %journal = " Numerische Mathematik", + % volume = "5", +% number = "", +% pages = "380-398", + %year = "1963", + % author = "W. Borch-Supan", +%}x @Article{Borch-Supan63, - title = "A posteriori error for the zeros of polynomials", - journal = " ", - volume = "", - number = "5", - pages = "380-398", - year = "1963", - author = "W. Borch-Supan", -}x + author = "W. Boersch-Supan", + title = "A Posteriori Error Bounds for the Zeros of + Polynomials", + journal = "Numerische Mathematik", + volume = "5", + pages = "380--398", + year = "1963", + CODEN = "NUMMA7", + ISSN = "0029-599X", + bibdate = "Fri Jan 12 11:37:56 1996", + acknowledgement = "Jon Rokne, Department of Computer Science, The + University of Calgary, 2500 University Drive N.W., + Calgary, Alberta T2N 1N4, Canada", +} -@Article{Ehrlich67, - title = "A modified Newton method for polynomials", - journal = " Comm. Ass. Comput. Mach.", - volume = "", - number = "10", - pages = "107-108", - year = "1967", - author = "L.W. Ehrlich", -}x +%@Article{Ehrlich67, +% title = "A modified Newton method for polynomials", +% journal = " Comm. Ass. Comput. Mach.", +% volume = "10", +% number = "2", +% pages = "107-108", + %year = "1967", + % author = "L.W. Ehrlich", +%}x -@Article{Loizon83, +@Article{Ehrlich67, + title = "A modified Newton method for polynomials", + author = "Louis W. Ehrlich", + journal = "Commun. ACM", + year = "1967", + number = "2", + volume = "10", + bibdate = "2003-11-20", + bibsource = "DBLP, + http://dblp.uni-trier.de/db/journals/cacm/cacm10.html#Ehrlich67", + pages = "107--108", + URL = "http://doi.acm.org/10.1145/363067.363115", +} +@Article{Loizou83, title = "Higher-order iteration functions for simultaneously approximating polynomial zeros", journal = " Intern. J. Computer Math", - volume = "", - number = "14", + volume = "14", + number = "", pages = "45-58", year = "1983", author = "G. Loizon", @@ -106,8 +171,8 @@ @Article{Freeman89, title = " Calculating polynomial zeros on a local memory parallel computer", journal = " Parallel Computing", - volume = "", - number = "12", + volume = "12", + number = "", pages = "351-358", year = "1989", author = "T.L. Freeman", @@ -116,8 +181,8 @@ @Article{Freemanall90, title = " Asynchronous polynomial zero-finding algorithms", journal = " Parallel Computing", - volume = "", - number = "17", + volume = "17", + number = "", pages = "673-681", year = "1990", author = "T.L. Freeman AND R.K. Brankin", diff --git a/paper.tex b/paper.tex index c78df7b..4213238 100644 --- a/paper.tex +++ b/paper.tex @@ -493,32 +493,37 @@ read-only caches. \subsection{A 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} -%\LinesNumbered + \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)} -\KwOut {Z(The solution root's vector)} +\KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error tolerance threshold), P(Polynomial to solve),$\Delta z_{max}$ (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 the coefficients of the polynomial to solve\; Initialization of the solution vector $Z^{0}$\; +$\Delta z_{max}=0$\; + k=0\; -\While {$\Delta z_{max}\succ \epsilon$}{ +\While {$\Delta z_{max} > \varepsilon$}{ Let $\Delta z_{max}=0$\; \For{$j \gets 0 $ \KwTo $n$}{ -$ZPrec\left[j\right]=Z\left[j\right]$\; -$Z\left[j\right]=H\left(j,Z\right)$\; +$ZPrec\left[j\right]=Z\left[j\right]$;// save Z at the iteration k.\ + +$Z\left[j\right]=H\left(j,Z\right)$;//update Z with the iterative function.\ } +k=k+1\; \For{$i \gets 0 $ \KwTo $n-1$}{ -$c=\frac{\left|Z\left[i\right]-ZPrec\left[i\right]\right|}{Z\left[i\right]}$\; +$c= testConverge(\Delta z_{max},ZPrec\left[j\right],Z\left[j\right])$\; \If{$c > \Delta z_{max}$ }{ $\Delta z_{max}$=c\;} } + } \end{algorithm} @@ -555,8 +560,7 @@ Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth algorithm using C %\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)} +\KwIn{$Z^{0}$(Initial root's vector),$\varepsilon$ (error tolerance threshold), P(Polynomial to solve), $\Delta z_{max}$ (maximum value of stop condition)} \KwOut {Z(The solution root's vector)} @@ -565,12 +569,14 @@ tolerance threshold),P(Polynomial to solve)} Initialization of the coeffcients of the polynomial to solve\; Initialization of the solution vector $Z^{0}$\; Allocate and copy initial data to the GPU global memory\; - +k=0\; \While {$\Delta z_{max}\succ \epsilon$}{ Let $\Delta z_{max}=0$\; -$ kernel\_save(d\_z^{k-1})$\; -$ kernel\_update(d\_z^{k})$\; -$kernel\_testConverge(\Delta z_{max},d_z^{k},d_z^{k-1})$\; +$ kernel\_save(d\_Z^{k-1})$\; +k=k+1\; +$ kernel\_update(d\_Z^{k})$\; +$kernel\_testConverge(\Delta z_{max},d\_Z^{k},d\_Z^{k-1})$\; + } \end{algorithm} ~\\ -- 2.39.5 From 8f44fe5aa15f1d0795d9c1f5be461a09bc6ca4ca Mon Sep 17 00:00:00 2001 From: couturie Date: Mon, 2 Nov 2015 10:25:22 -0500 Subject: [PATCH 03/16] new --- paper.tex | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/paper.tex b/paper.tex index d9c3324..6507d22 100644 --- a/paper.tex +++ b/paper.tex @@ -229,10 +229,11 @@ and experimental study results. Finally, Section\ref{sec7} 6 concludes this paper and gives some hints for future research directions in this topic. -\section{The Sequential Aberth method} +\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 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 -- 2.39.5 From e74bf528b696246b88bb8a5542efc542fd09788a Mon Sep 17 00:00:00 2001 From: couturie Date: Mon, 2 Nov 2015 10:38:46 -0500 Subject: [PATCH 04/16] new --- paper.tex | 7 +++++-- 1 file changed, 5 insertions(+), 2 deletions(-) diff --git a/paper.tex b/paper.tex index 3be1eea..76b626e 100644 --- a/paper.tex +++ b/paper.tex @@ -269,7 +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 $n$ @@ -303,7 +303,10 @@ 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: -- 2.39.5 From 7b00f9048df407936aee5458d1d219bbe59844ba Mon Sep 17 00:00:00 2001 From: couturie Date: Mon, 2 Nov 2015 10:50:32 -0500 Subject: [PATCH 05/16] new --- paper.tex | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/paper.tex b/paper.tex index 76b626e..276c50a 100644 --- a/paper.tex +++ b/paper.tex @@ -308,7 +308,7 @@ 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} @@ -319,7 +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 -- 2.39.5 From 5771bd90bdc89ce5ca4a1947aca277e3101ea508 Mon Sep 17 00:00:00 2001 From: couturie Date: Mon, 2 Nov 2015 11:09:31 -0500 Subject: [PATCH 06/16] new --- paper.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/paper.tex b/paper.tex index 276c50a..c4fa445 100644 --- 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 -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} -- 2.39.5 From 1fc948028049feeeacad3808eedfb3b2acb32775 Mon Sep 17 00:00:00 2001 From: couturie Date: Mon, 2 Nov 2015 13:53:29 -0500 Subject: [PATCH 07/16] new --- paper.tex | 9 ++++++--- 1 file changed, 6 insertions(+), 3 deletions(-) diff --git a/paper.tex b/paper.tex index c4fa445..20686d2 100644 --- a/paper.tex +++ b/paper.tex @@ -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 -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 @@ -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. -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 -- 2.39.5 From 165e7b481ee54e4e3f0be7828ca7b071aaa93888 Mon Sep 17 00:00:00 2001 From: couturie Date: Mon, 2 Nov 2015 14:13:41 -0500 Subject: [PATCH 08/16] new --- paper.tex | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) diff --git a/paper.tex b/paper.tex index e4873d7..cf8c180 100644 --- a/paper.tex +++ b/paper.tex @@ -423,13 +423,15 @@ 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 48000. In this paper we present a parallel implementation of Ehlisch-Aberth method on -GPUs, which details are discussed in the sequel. +polynomials of 48000. +%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. First, the hardware and software of the GPUs are presented. Then, a CUDA parallel Ehrlich-Aberth method are presented. +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 -- 2.39.5 From db8153d47e6b58c304ac2689b6d338e8023ff6e1 Mon Sep 17 00:00:00 2001 From: Kahina Date: Tue, 3 Nov 2015 05:49:29 +0100 Subject: [PATCH 09/16] MAJ Mybib --- mybibfile.bib | 208 +++++++++++++++++++++++++------------------------- paper.tex | 29 ++----- 2 files changed, 112 insertions(+), 125 deletions(-) diff --git a/mybibfile.bib b/mybibfile.bib index d29bcb2..cd9bf5a 100644 --- a/mybibfile.bib +++ b/mybibfile.bib @@ -31,37 +31,28 @@ }x - @Article{Ilie50, title = "On the approximations of Newton", journal = "Annual Sofia Univ", - volume = "", - number = "46", + volume = "46", + number = "", pages = "167--171", year = "1950", doi = "10.1016/0003-4916(63)90068-X", author = "L. Ilieff", }x + @Article{Docev62, title = "An alternative method of Newton for simultaneous calculation of all the roots of a given algebraic equation", journal = "Phys. Math. J", - volume = "", - number = "5", + volume = "5", + number = "", pages = "136-139", year = "1962", author = "K. Docev", }x -%@Article{Durand60, - % title = "Solution Numerique des Equations Algebriques, Vol. 1, Equations du Type F(x)=0, Racines d'une Polynome", - %journal = "", -% volume = "Vol.1", -% number = "", - % pages = "", - %year = "1960", - % author = "E. Durand", -%}x @Book{Durand60, author = "\'E. Durand", publisher = "Masson, Paris", @@ -71,17 +62,6 @@ year = "1960", }x -%@Article{Kerner66, - %title = "Ein Gesamtschritteverfahren zur Berechnung der Nullstellen von Polynomen", - % journal = "Numerische Mathematik", -% volume = "8", -% number = "3", -% pages = "290-294", - %year = "1966", - % author = "I. Kerner", -%}x - - @Article{Kerner66, author = "Immo O. Kerner", title = "{Ein Gesamtschrittverfahren zur Berechnung der @@ -109,15 +89,6 @@ journal-url = "http://link.springer.com/journal/211", language = "German", } -%@Article{Borch-Supan63, -% title = "A posteriori error for the zeros of polynomials", - %journal = " Numerische Mathematik", - % volume = "5", -% number = "", -% pages = "380-398", - %year = "1963", - % author = "W. Borch-Supan", -%}x @Article{Borch-Supan63, author = "W. Boersch-Supan", @@ -135,16 +106,6 @@ Calgary, Alberta T2N 1N4, Canada", } -%@Article{Ehrlich67, -% title = "A modified Newton method for polynomials", -% journal = " Comm. Ass. Comput. Mach.", -% volume = "10", -% number = "2", -% pages = "107-108", - %year = "1967", - % author = "L.W. Ehrlich", -%}x - @Article{Ehrlich67, title = "A modified Newton method for polynomials", author = "Louis W. Ehrlich", @@ -343,26 +304,62 @@ OPTannote = {•} year = "2006", author = "PK. Jana", }x -@Article{Kalantari08, - title = " Polynomial root finding and polynomiography.", - journal = " World Scientifict,New Jersey", - volume = "", - number = "", - pages = "", - year = "", - author = "B. Kalantari", -}x -@Article{Gemignani07, - title = " Structured matrix methods for polynomial root finding.", - journal = " n: Proc of the 2007 Intl symposium on symbolic and algebraic computation", - volume = "", - number = "", - pages = "175-180", - year = "2007", - author = "L. Gemignani", -}x +@Book{Kalantari08, +ALTauthor = {B. Kalantari}, +title = {Polynomial root finding and polynomiography.}, +publisher = {World Scientifict,New Jersey}, +year = {2008}, +OPTkey = {•}, +OPTvolume = {•}, +OPTnumber = {•}, +OPTseries = {•}, +OPTaddress = {•}, +OPTmonth = {December}, +OPTnote = {•}, +OPTannote = {•} +} + + +@InProceedings{Gemignani07, + author = "Luca Gemignani", + title = "Structured matrix methods for polynomial + root-finding", + editor = "C. W. Brown", + booktitle = "Proceedings of the 2007 International Symposium on + Symbolic and Algebraic Computation, July 29--August 1, + 2007, University of Waterloo, Waterloo, Ontario, + Canada", + publisher = "ACM Press", + address = "pub-ACM:adr", + ISBN = "1-59593-743-9 (print), 1-59593-742-0 (CD-ROM)", + isbn-13 = "978-1-59593-743-8 (print), 978-1-59593-742-1 + (CD-ROM)", + pages = "175--180", + year = "2007", + doi = "http://doi.acm.org/10.1145/1277548.1277573", + bibdate = "Fri Jun 20 08:46:50 MDT 2008", + bibsource = "http://portal.acm.org/; + http://www.math.utah.edu/pub/tex/bib/issac.bib", + abstract = "In this paper we discuss the use of structured matrix + methods for the numerical approximation of the zeros of + a univariate polynomial. In particular, it is shown + that root-finding algorithms based on floating-point + eigenvalue computation can benefit from the structure + of the matrix problem to reduce their complexity and + memory requirements by an order of magnitude.", + acknowledgement = "Nelson H. F. Beebe, University of Utah, Department + of Mathematics, 110 LCB, 155 S 1400 E RM 233, Salt Lake + City, UT 84112-0090, USA, Tel: +1 801 581 5254, FAX: +1 + 801 581 4148, e-mail: \path|beebe@math.utah.edu|, + \path|beebe@acm.org|, \path|beebe@computer.org| + (Internet), URL: + \path|http://www.math.utah.edu/~beebe/|", + keywords = "complexity; eigenvalue computation; polynomial + root-finding; rank-structured matrices", + doi-url = "http://dx.doi.org/10.1145/1277548.1277573", +} @Article{Skachek08, @@ -375,25 +372,20 @@ OPTannote = {•} author = "V. Skachek", }x -@BOOK{Skachek008, - AUTHOR = {V. Skachek}, - editor = {}, - TITLE = {Probabilistic algorithm for finding roots of linearized polynomials}, - PUBLISHER = {codes and cryptography. Kluwer}, - YEAR = {2008}, - volume = {}, - number = {}, - series = {}, - address = {}, - edition = {Design}, - month = {}, - note = {}, - abstract = {}, - isbn = {}, - price = {}, - keywords = {}, - source = {}, -}x +@Article{Skachek008, + title = "Probabilistic algorithm for finding roots of + linearized polynomials", + author = "Vitaly Skachek and Ron M. Roth", + journal = "Des. Codes Cryptography", + year = "2008", + number = "1", + volume = "46", + bibdate = "2008-03-11", + bibsource = "DBLP, + http://dblp.uni-trier.de/db/journals/dcc/dcc46.html#SkachekR08", + pages = "17--23", + URL = "http://dx.doi.org/10.1007/s10623-007-9125-y", +} @Article{Zhancall08, title = " A constrained learning algorithm for finding multiple real roots of polynomial", @@ -406,15 +398,24 @@ OPTannote = {•} }x -@Article{Zhuall08, - title = " an adaptive algorithm finding multiple roots of polynomials", - journal = " Lect Notes Comput Sci ", - volume = "", - number = "5262", - pages = "674-681", - year = "2008", - author = "W. Zhu AND w. Zeng AND D. Lin", -}x +@InProceedings{Zhuall08, + title = "An Adaptive Algorithm Finding Multiple Roots of Polynomials", + author = "Wei Zhu and Zhe-zhao Zeng and Dong-mei Lin", + bibdate = "2008-09-25", + bibsource = "DBLP, + http://dblp.uni-trier.de/db/conf/isnn/isnn2008-2.html#ZhuZL08", + booktitle = "ISNN (2)", + publisher = "Springer", + year = "2008", + volume = "5264", + editor = "Fuchun Sun and Jianwei Zhang 0001 and Ying Tan and + Jinde Cao and Wen Yu 0001", + ISBN = "978-3-540-87733-2", + pages = "674--681", + series = "Lecture Notes in Computer Science", + URL = "http://dx.doi.org/10.1007/978-3-540-87734-9_77", +} + @Article{Azad07, title = " The performance of synchronous parallel polynomial root extraction on a ring multicomputer", journal = " Clust Comput ", @@ -431,8 +432,8 @@ OPTannote = {•} @Article{Bini04, title = " Inverse power and Durand Kerner iterations for univariate polynomial root finding", journal = " Comput Math Appl ", - volume = "", - number = "47", + volume = "47", + number = "", pages = "447-459", year = "2004", author = "DA. Bini AND L. Gemignani", @@ -457,17 +458,18 @@ OPTannote = {•} year = "1903", author = "K. Weierstrass", }x +@Manual{NVIDIA10, +title = {NVIDIA CUDA C Programming Guide}, +OPTkey = {•}, +OPTauthor = {NVIDIA Corporation}, +OPTorganization = {Design Guide}, +OPTaddress = {•}, +OPTedition = {•}, +OPTmonth = {march}, +OPTyear = {2015}, +OPTnote = {•}, +OPTannote = {•} +} -@BOOK{NVIDIA10, - AUTHOR = {NVIDIA}, - editor = {Design Guide}, - TITLE = {NVIDIA CUDA C Programming Guide}, - PUBLISHER = {PG}, - YEAR = {2015}, - volume = {7}, - number = {02829}, - series = {001}, - month = {march}, -}x diff --git a/paper.tex b/paper.tex index 43bbf1e..30fe880 100644 --- a/paper.tex +++ b/paper.tex @@ -172,7 +172,7 @@ Aberth~\cite{Aberth73} uses a different iteration formula given as: where $P'(z)$ is the polynomial derivative of $P$ evaluated in the point $z$. -Aberth, Ehrlich and Farmer-Loizou~\cite{Loizon83} have proved that +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. @@ -186,7 +186,7 @@ time. 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{Loizon83}, on a 8-processor linear +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, Freeman and Bane~\cite{Freemanall90} considered asynchronous @@ -339,7 +339,7 @@ propose to use the logarithm and the exponential of a complex in order to comput \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{defexpcomplex} + & =\exp(x).\cos(y)+i.\exp(x).\sin(y)\label{defexpcomplex1} \end{align} %%\end{equation} @@ -651,12 +651,6 @@ All experimental results obtained from the simulations are made in double precis \subsection{The execution time in seconds of Ehrlich-Aberth algorithm on CPU OpenMP (1 core, 4 cores) vs. on a Tesla GPU} -%\begin{figure}[H] -%\centering - % \includegraphics[width=0.8\textwidth]{figures/Compar_EA_algorithm_CPU_GPU} -%\caption{The execution time in seconds of Ehrlich-Aberth algorithm on CPU core vs. on a Tesla GPU} -%\label{fig:01} -%\end{figure} \begin{figure}[H] \centering @@ -677,7 +671,7 @@ For that, we notice that the maximum number of threads per block for the Nvidia \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:01} +\label{fig:02} \end{figure} The figure 2 show that, the best execution time for both sparse and full polynomial are given when the threads number varies between 64 and 256 threads per bloc. We notice that with small polynomials the best number of threads per block is 64, Whereas, the large polynomials the best number of threads per block is 256. However,In the following experiments we specify that the number of thread by block is 256. @@ -689,7 +683,7 @@ In this experiment we report the performance of log.exp solution describe in ~\r \centering \includegraphics[width=0.8\textwidth]{figures/sparse_full_explog} \caption{The impact of exp-log solution to compute very high degrees of polynomial.} -\label{fig:01} +\label{fig:03} \end{figure} The figure 3, show a comparison between the execution time of the Ehrlich-Aberth algorithm applying exp.log solution and the execution time of the Ehrlich-Aberth algorithm without applying exp.log solution, with full and sparse polynomials degrees. We can see that the execution time for the both algorithms are the same while the full polynomials degrees are less than 4000 and full polynomials are less than 150,000. After,we show clearly that the classical version of Ehrlich-Aberth algorithm (without applying log.exp) stop to converge and can not solving any polynomial sparse or full. In counterpart, the new version of Ehrlich-Aberth algorithm (applying log.exp solution) can solve very high and large full polynomial exceed 100,000 degrees. @@ -697,16 +691,7 @@ The figure 3, show a comparison between the execution time of the Ehrlich-Aberth 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 log.exp 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 . -%\begin{figure}[H] -\%centering - %\includegraphics[width=0.8\textwidth]{figures/log_exp_Sparse} -%\caption{The impact of exp-log solution to compute very high degrees of polynomial.} -%\label{fig:01} -%\end{figure} - -%we report the performances of the exp.log for the Ehrlich-Aberth algorithm for solving very high degree of polynomial. - \subsection{A comparative study between Ehrlich-Aberth algorithm and Durand-kerner algorithm} In this part, we are interesting to compare the simultaneous methods, Ehrlich-Aberth and Durand-Kerner in parallel computer using GPU. We took into account the execution time, the number of iteration and the polynomial's size. for the both sparse and full polynomials. @@ -714,7 +699,7 @@ In this part, we are interesting to compare the simultaneous methods, Ehrlich-Ab \centering \includegraphics[width=0.8\textwidth]{figures/EA_DK} \caption{The execution time of Ehrlich-Aberth versus Durand-Kerner algorithm on GPU} -\label{fig:01} +\label{fig:04} \end{figure} This figure show the execution time of the both algorithm EA and DK with sparse polynomial degrees ranging from 1000 to 1000000. We can see that the Ehrlich-Aberth algorithm are faster than Durand-Kerner algorithm, with an average of 25 times as fast. Then, when degrees of polynomial exceed 500000 the execution time with EA is of the order 100 whereas DK passes in the order 1000. %with double precision not exceed $10^{-5}$. @@ -723,7 +708,7 @@ This figure show the execution time of the both algorithm EA and DK with sparse \centering \includegraphics[width=0.8\textwidth]{figures/EA_DK_nbr} \caption{The iteration number of Ehrlich-Aberth versus Durand-Kerner algorithm} -\label{fig:01} +\label{fig:05} \end{figure} %\subsubsection{The execution time of Ehrlich-Aberth algorithm on OpenMP(1 core, 4 cores) vs. on a Tesla GPU} -- 2.39.5 From 9213a919b9009602014fa0b33a14063f019562dc Mon Sep 17 00:00:00 2001 From: Kahina Date: Tue, 3 Nov 2015 06:20:51 +0100 Subject: [PATCH 10/16] new bib --- mybibfile.bib | 8 ++++---- paper.tex | 2 +- 2 files changed, 5 insertions(+), 5 deletions(-) diff --git a/mybibfile.bib b/mybibfile.bib index cd9bf5a..7e943e1 100644 --- a/mybibfile.bib +++ b/mybibfile.bib @@ -307,9 +307,9 @@ OPTannote = {•} @Book{Kalantari08, -ALTauthor = {B. Kalantari}, -title = {Polynomial root finding and polynomiography.}, -publisher = {World Scientifict,New Jersey}, +author = {B. Kalantari}, +title = {Polynomial root finding and polynomiography}, +publisher = {World Scientifict}, year = {2008}, OPTkey = {•}, OPTvolume = {•}, @@ -363,7 +363,7 @@ OPTannote = {•} @Article{Skachek08, - title = " Structured matrix methods for polynomial root finding.", + title = " Structured matrix methods for polynomial root finding", journal = " n: Proc of the 2007 Intl symposium on symbolic and algebraic computation", volume = "", number = "", diff --git a/paper.tex b/paper.tex index 30fe880..d9b3e5e 100644 --- a/paper.tex +++ b/paper.tex @@ -391,7 +391,7 @@ 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, Skachek08, Zhancall08, Zhuall08}. However, Durand-Kerner and Ehrlich-Aberth methods are the most practical choices among +finding have been reported in~\cite{Azad07, Gemignani07, Kalantari08, 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 parallelism, i.e. the computations involved in both methods has some inherent -- 2.39.5 From 0ecc70e5cfcddd4891a106344c71d6244677044d Mon Sep 17 00:00:00 2001 From: Kahina Date: Tue, 3 Nov 2015 06:29:47 +0100 Subject: [PATCH 11/16] Biblio --- mybibfile.bib | 23 +++++++++++------------ 1 file changed, 11 insertions(+), 12 deletions(-) diff --git a/mybibfile.bib b/mybibfile.bib index 7e943e1..308a403 100644 --- a/mybibfile.bib +++ b/mybibfile.bib @@ -126,7 +126,7 @@ number = "", pages = "45-58", year = "1983", - author = "G. Loizon", + author = "G. Loizou", }x @Article{Freeman89, @@ -321,6 +321,16 @@ OPTnote = {•}, OPTannote = {•} } +Article{Skachek08, + title = " Structured matrix methods for polynomial root finding", + journal = " n: Proc of the 2007 Intl symposium on symbolic and algebraic computation", + volume = "", + number = "", + pages = "175-180", + year = "2008", + author = "V. Skachek", +}x + @InProceedings{Gemignani07, author = "Luca Gemignani", @@ -361,17 +371,6 @@ OPTannote = {•} doi-url = "http://dx.doi.org/10.1145/1277548.1277573", } - -@Article{Skachek08, - title = " Structured matrix methods for polynomial root finding", - journal = " n: Proc of the 2007 Intl symposium on symbolic and algebraic computation", - volume = "", - number = "", - pages = "175-180", - year = "2008", - author = "V. Skachek", -}x - @Article{Skachek008, title = "Probabilistic algorithm for finding roots of linearized polynomials", -- 2.39.5 From c4044fa1a2cf8748ba82d3ff09d72252a42ef489 Mon Sep 17 00:00:00 2001 From: Kahina Date: Tue, 3 Nov 2015 06:31:47 +0100 Subject: [PATCH 12/16] BIBLIO --- mybibfile.bib | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/mybibfile.bib b/mybibfile.bib index 308a403..d5dfdc9 100644 --- a/mybibfile.bib +++ b/mybibfile.bib @@ -123,7 +123,7 @@ title = "Higher-order iteration functions for simultaneously approximating polynomial zeros", journal = " Intern. J. Computer Math", volume = "14", - number = "", + number = "1", pages = "45-58", year = "1983", author = "G. Loizou", -- 2.39.5 From 35a5c9f528a97942bbfdc3ec764d654a2192dc38 Mon Sep 17 00:00:00 2001 From: couturie Date: Tue, 3 Nov 2015 10:05:10 -0500 Subject: [PATCH 13/16] correct --- paper.tex | 16 ++++++++++------ 1 file changed, 10 insertions(+), 6 deletions(-) diff --git a/paper.tex b/paper.tex index d9b3e5e..8506d85 100644 --- a/paper.tex +++ b/paper.tex @@ -489,14 +489,14 @@ 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{ The implementation of Ehrlich-Aberth method on GPU} +\section{ Implementation of 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{A sequential Ehrlich-Aberth algorithm} +\subsection{Sequential Ehrlich-Aberth algorithm} The main steps of Ehrlich-Aberth method are shown in Algorithm.~\ref{alg1-seq} : %\LinesNumbered \begin{algorithm}[H] @@ -504,8 +504,10 @@ The main steps of Ehrlich-Aberth method are shown in Algorithm.~\ref{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),$\Delta z_{max}$ (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)} +\KwIn{$Z^{0}$ (Initial root's vector), $\varepsilon$ (error tolerance + threshold), P (Polynomial to solve), $\Delta z_{max}$ (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 @@ -546,13 +548,15 @@ With the Gauss-Seidel iteration, we have: 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} %%Here a finiched my revision %% -Using Equation.~\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 its ancestor (for solving linear systems of equations), it uses the most fresh computed roots $z^{k+1}_{i}$. +Using Equation.~\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 most fresh computed roots $z^{k+1}_{i}$. The $4^{th}$ step of the algorithm checks the convergence condition using Equation.~\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. -\subsection{A Parallel implementation with CUDA } +\subsection{Parallel implementation with CUDA } 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 schduler 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. It's up to the programmer, to describe the execution context, that is the size of the Grid, the number of blocks and the number of threads per block upon the call of a given kernel, according to a special syntax defined by CUDA. -- 2.39.5 From 6dcd06b01e947fd9dba5e4a45b2e1ac19835e5db Mon Sep 17 00:00:00 2001 From: couturie Date: Tue, 3 Nov 2015 12:05:24 -0500 Subject: [PATCH 14/16] new --- paper.tex | 15 ++++++++++----- 1 file changed, 10 insertions(+), 5 deletions(-) diff --git a/paper.tex b/paper.tex index 8506d85..52fe61c 100644 --- a/paper.tex +++ b/paper.tex @@ -558,9 +558,14 @@ Both steps 3 and 4 use 1 thread to compute all the $n$ roots on CPU, which is ve \subsection{Parallel implementation with CUDA } 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 schduler 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. It's up to the programmer, to describe the execution context, that is the size of the Grid, the number of blocks and the number of threads per block upon the call of a given kernel, according to a special syntax defined by CUDA. +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. +%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. @@ -569,13 +574,13 @@ Algorithm~\ref{alg2-cuda} shows a sketch of the Ehrlich-Aberth algorithm using C %\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), $\Delta z_{max}$ (maximum value of stop condition)} -\KwOut {Z(The solution root's vector)} +\KwOut {Z (The solution root's vector)} \BlankLine -Initialization of the coeffcients of the polynomial to solve\; +Initialization of the coefficients of the polynomial to solve\; Initialization of the solution vector $Z^{0}$\; Allocate and copy initial data to the GPU global memory\; k=0\; -- 2.39.5 From 19499f903293ae4b6d3313414aef5a790c81cb34 Mon Sep 17 00:00:00 2001 From: couturie Date: Tue, 3 Nov 2015 12:18:40 -0500 Subject: [PATCH 15/16] correct --- paper.tex | 14 +++++++++++--- 1 file changed, 11 insertions(+), 3 deletions(-) diff --git a/paper.tex b/paper.tex index 52fe61c..0d2bbd6 100644 --- a/paper.tex +++ b/paper.tex @@ -597,7 +597,11 @@ $kernel\_testConverge(\Delta z_{max},d\_Z^{k},d\_Z^{k-1})$\; 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}). -The second kernel executes the iterative function $H$ and updates $z^{k}$, 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 logarithm computation of the power of a complex. +The second kernel executes the iterative function $H$ and updates +$z^{k}$, 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. \begin{algorithm}[H] \label{alg3-update} @@ -607,11 +611,15 @@ The second kernel executes the iterative function $H$ and updates $z^{k}$, accor \eIf{$(\left|Z^{(k)}\right|<= R)$}{ $kernel\_update(d\_z^{k})$\;} { -$kernel\_update\_Log(d\_z^{k})$\; +$kernel\_update\_ExpoLog(d\_z^{k})$\; } \end{algorithm} -The first form executes formula \ref{eq:SimplePolynome} if the modulus of the current complex is less than the a certain value called the radius i.e. ($ |z^{k}_{i}|<= R$), else the kernel executes formulas (Eq.~\ref{deflncomplex},Eq.~\ref{defexpcomplex}). The radius $R$ is evaluated as : +The first form executes formula \ref{eq:SimplePolynome} if the modulus +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. -- 2.39.5 From fdc11565a5688a74effe8d08b78b4a1853387f3a Mon Sep 17 00:00:00 2001 From: couturie Date: Tue, 3 Nov 2015 12:30:07 -0500 Subject: [PATCH 16/16] suppression explication gpu --- paper.tex | 142 ++++++++++++++++++++++++++++++------------------------ 1 file changed, 79 insertions(+), 63 deletions(-) diff --git a/paper.tex b/paper.tex index 0d2bbd6..694ae43 100644 --- a/paper.tex +++ b/paper.tex @@ -429,65 +429,65 @@ polynomials of 48000. %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 {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 Ehrlich-Aberth method on GPU} \label{sec5} @@ -619,14 +619,30 @@ The first form executes formula \ref{eq:SimplePolynome} if the modulus 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 : +(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. -The last kernel verifies the convergence of the roots after each update of $Z^{(k)}$, according to formula. We used the functions of the CUBLAS Library (CUDA Basic Linear Algebra Subroutines) to implement this kernel. +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. Many important remarks should be noticed. First, 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 in an asynchronous iterations way, where blocks +of roots are updated in a non deterministic way. As the Durand-Kerner +method has been proved to convergence 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 polynomials do no +give necessarily the same result with the same number of iterations +(even if the variation is not very significant). + + + + -The kernels terminate it computations when all the roots converge. Finally, the solution of the root finding problem is copied back from GPU global memory to CPU memory. We use the communication functions of CUDA for the memory allocation in the GPU \verb=(cudaMalloc())= and for data transfers from the CPU memory to the GPU memory \verb=(cudaMemcpyHostToDevice)= -or from GPU memory to CPU memory \verb=(cudaMemcpyDeviceToHost))=. %%HIER END MY REVISIONS (SIDER) \section{Experimental study} \label{sec6} -- 2.39.5