\relax \providecommand\hyper@newdestlabel[2]{} \providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} \HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined \global\let\oldcontentsline\contentsline \gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} \global\let\oldnewlabel\newlabel \gdef\newlabel#1#2{\newlabelxx{#1}#2} \gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} \AtEndDocument{\ifx\hyper@anchor\@undefined \let\contentsline\oldcontentsline \let\newlabel\oldnewlabel \fi} \fi} \global\let\hyper@last\relax \gdef\HyperFirstAtBeginDocument#1{#1} \providecommand*\HyPL@Entry[1]{} \bibstyle{elsarticle-num} \emailauthor{kahina.ghidouche@gmail.com}{Ghidouche Kahina\corref {mycorrespondingauthor}} \emailauthor{raphael.couturier@univ-fcomte.fr}{Couturier Raphael\corref {mycorrespondingauthor}} \emailauthor{ar.sider@univ-bejaia.dz}{Abderrahmane Sider\corref {mycorrespondingauthor}} \HyPL@Entry{0<>} \Newlabel{mycorrespondingauthor}{1} \Newlabel{myfootnote}{1} \Newlabel{mymainaddress}{a} \Newlabel{mysecondaryaddress}{b} \@writefile{toc}{\contentsline {section}{\numberline {1}Root finding problem}{1}{section.1}} \citation{Weierstrass03} \citation{Ilie50} \citation{Docev62} \citation{Durand60} \citation{Kerner66} \citation{Borch-Supan63} \citation{Ehrlich67} \citation{Aberth73} \citation{Loizon83} \citation{Freeman89} \citation{Loizon83} \citation{Freemanall90} \citation{Raphaelall01} \citation{CUDA10} \citation{Kahinall14} \@writefile{toc}{\contentsline {paragraph}{}{3}{section*.1}} \citation{Aberth73} \citation{Weierstrass03} \citation{Aberth73} \citation{Bini96} \citation{Ostrowski41} \@writefile{toc}{\contentsline {section}{\numberline {2}Aberth method}{4}{section.2}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Polynomials Initialization}{4}{subsection.2.1}} \citation{Karimall98} \@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Vector $Z^{(0)}$ Initialization}{5}{subsection.2.2}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Iterative Function $H_{i}$}{5}{subsection.2.3}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.4}Convergence condition}{5}{subsection.2.4}} \@writefile{toc}{\contentsline {section}{\numberline {3}Amelioration of Aberth method }{5}{section.3}} \citation{Karimall98} \citation{Mirankar68,Mirankar71} \citation{Schedler72} \citation{Winogard72} \citation{Benall68,Jana06,Janall99,Riceall06} \citation{Azad07,Gemignani07,Kalantari08,Skachek08,Zhancall08,Zhuall08} \citation{Bini04} \citation{Cosnard90,Freeman89,Freemanall90,,Jana99,Janall99} \citation{Freemanall90} \citation{Mirankar71} \@writefile{toc}{\contentsline {section}{\numberline {4}The implementation of simultaneous methods in a parallel computer}{6}{section.4}} \citation{Kahinall14} \citation{NVIDIA10} \@writefile{toc}{\contentsline {section}{\numberline {5}A parallel implementation of Aberth method}{8}{section.5}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Background on the GPU architecture}{8}{subsection.5.1}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Background on the CUDA Programming Model}{9}{subsection.5.2}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.3} The implementation of Aberth method on GPU}{10}{subsection.5.3}} \@writefile{toc}{\contentsline {subsubsection}{\numberline {5.3.1}A sequential Aberth algorithm}{10}{subsubsection.5.3.1}} \@writefile{loa}{\contentsline {algocf}{\numberline {1}{\ignorespaces Algorithm to find root polynomial with Aberth method}}{10}{algocf.1}} \@writefile{toc}{\contentsline {paragraph}{The execution time}{11}{section*.2}} \@writefile{toc}{\contentsline {subsubsection}{\numberline {5.3.2}Parallelize the steps on GPU }{12}{subsubsection.5.3.2}} \@writefile{loa}{\contentsline {algocf}{\numberline {2}{\ignorespaces Algorithm to find root polynomial with Aberth method}}{13}{algocf.2}} \@writefile{loa}{\contentsline {algocf}{\numberline {3}{\ignorespaces A global Algorithm for the iterative function}}{14}{algocf.3}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.4}Experimental study}{14}{subsection.5.4}} \@writefile{toc}{\contentsline {subsubsection}{\numberline {5.4.1}Definition of the polynomial used}{14}{subsubsection.5.4.1}} \bibdata{mybibfile} \@writefile{toc}{\contentsline {subsubsection}{\numberline {5.4.2}The study condition}{15}{subsubsection.5.4.2}} \@writefile{toc}{\contentsline {subsubsection}{\numberline {5.4.3}Comparative study}{15}{subsubsection.5.4.3}} \@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces the convergence of Aberth algorithm}}{15}{table.1}} \newlabel{tab:theConvergenceOfAberthAlgorithm}{{1}{15}{the convergence of Aberth algorithm}{table.1}{}} \@writefile{toc}{\contentsline {paragraph}{Aberth algorithm on CPU and GPU}{15}{table.1}} \@writefile{toc}{\contentsline {paragraph}{The impact of the thread's number into the convergence of Aberth algorithm}{15}{table.2}} \bibcite{Weierstrass03}{{1}{}{{}}{{}}} \bibcite{Ilie50}{{2}{}{{}}{{}}} \bibcite{Docev62}{{3}{}{{}}{{}}} \bibcite{Durand60}{{4}{}{{}}{{}}} \@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces The impact of the thread's number into the convergence of Aberth algorithm}}{16}{table.2}} \newlabel{tab:Theimpactofthethread'snumberintotheconvergenceofAberthalgorithm}{{2}{16}{The impact of the thread's number into the convergence of Aberth algorithm}{table.2}{}} \@writefile{lot}{\contentsline {table}{\numberline {3}{\ignorespaces Aberth algorithm compare to Durand-Kerner algorithm}}{16}{table.3}} \newlabel{tab:AberthAlgorithCompareToDurandKernerAlgorithm}{{3}{16}{Aberth algorithm compare to Durand-Kerner algorithm}{table.3}{}} \@writefile{toc}{\contentsline {paragraph}{A comparative study between Aberth and Durand-kerner algorithm}{16}{table.3}} \bibcite{Kerner66}{{5}{}{{}}{{}}} \bibcite{Borch-Supan63}{{6}{}{{}}{{}}} \bibcite{Ehrlich67}{{7}{}{{}}{{}}} \bibcite{Aberth73}{{8}{}{{}}{{}}} \bibcite{Loizon83}{{9}{}{{}}{{}}} \bibcite{Freeman89}{{10}{}{{}}{{}}} \bibcite{Freemanall90}{{11}{}{{}}{{}}} \bibcite{Raphaelall01}{{12}{}{{}}{{}}} \bibcite{CUDA10}{{13}{}{{}}{{}}} \bibcite{Kahinall14}{{14}{}{{}}{{}}} \bibcite{Bini96}{{15}{}{{}}{{}}} \bibcite{Ostrowski41}{{16}{}{{}}{{}}} \bibcite{Karimall98}{{17}{}{{}}{{}}} \bibcite{Mirankar68}{{18}{}{{}}{{}}} \bibcite{Mirankar71}{{19}{}{{}}{{}}} \bibcite{Schedler72}{{20}{}{{}}{{}}} \bibcite{Winogard72}{{21}{}{{}}{{}}} \bibcite{Benall68}{{22}{}{{}}{{}}} \bibcite{Jana06}{{23}{}{{}}{{}}} \bibcite{Janall99}{{24}{}{{}}{{}}} \bibcite{Riceall06}{{25}{}{{}}{{}}} \bibcite{Azad07}{{26}{}{{}}{{}}} \bibcite{Gemignani07}{{27}{}{{}}{{}}} \bibcite{Kalantari08}{{28}{}{{}}{{}}} \bibcite{Skachek08}{{29}{}{{}}{{}}} \bibcite{Zhancall08}{{30}{}{{}}{{}}} \bibcite{Zhuall08}{{31}{}{{}}{{}}} \bibcite{Bini04}{{32}{}{{}}{{}}} \bibcite{Cosnard90}{{33}{}{{}}{{}}} \bibcite{Jana99}{{34}{}{{}}{{}}} \bibcite{NVIDIA10}{{35}{}{{}}{{}}} \providecommand\NAT@force@numbers{}\NAT@force@numbers