MAJ Mybibfile

diff --git a/mybibfile.bib b/mybibfile.bib
index 234731893a2ff0f829604900819750bf26a23b85..d29bcb2720a6ef8389953c781b5a24752f482a72 100644 (file)
--- a/mybibfile.bib
+++ b/mybibfile.bib
@@ -53,51 +53,116 @@
   author =  "K. Docev",
 }x
 
   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,
 
 @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,
 
 @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",
   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",
   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",
 @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",
   pages =   "351-358",
   year =    "1989",
   author =  "T.L. Freeman",


@@ -116,8 +181,8 @@
 @Article{Freemanall90,
   title =   " Asynchronous polynomial zero-finding algorithms",
   journal = "  Parallel Computing",
 @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",
   pages =   "673-681",
   year =    "1990",
   author =  "T.L. Freeman AND R.K. Brankin",

diff --git a/paper.tex b/paper.tex
index c78df7b0f8e1304ef642b6471ee068440e90627c..4213238dc15bba747cab568e22b4b4baecab2c2e 100644 (file)
--- 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} :
 
 \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}
 \begin{algorithm}[H]
 \label{alg1-seq}

-%\LinesNumbered
+

 \caption{A sequential algorithm to find roots with the Ehrlich-Aberth method}
 
 \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 $Z^{0}$\;
 
 \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 $n$}{
  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 $n-1$}{
 
 \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\;}
 }
 \If{$c > \Delta z_{max}$ }{
 $\Delta z_{max}$=c\;}
 }

+

 }
 \end{algorithm}
 
 }
 \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}
 
 %\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)}
 
 
 \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\;
 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 $\Delta z_{max}=0$\;
 \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}
 ~\\ 
 }
 \end{algorithm}
 ~\\