From 8defa47ac283c6ce000d76eba1c1ad62b4a8bd38 Mon Sep 17 00:00:00 2001 From: couchot Date: Thu, 16 Jul 2015 08:33:49 +0200 Subject: [PATCH] 15 --- 15RairoGen.tex | 33 +++++++++++++++++---------------- main.pdf | Bin 1970982 -> 1992805 bytes 2 files changed, 17 insertions(+), 16 deletions(-) diff --git a/15RairoGen.tex b/15RairoGen.tex index 4218094..f2b9243 100644 --- a/15RairoGen.tex +++ b/15RairoGen.tex @@ -1,12 +1,12 @@ Au bout d'un nombre $b$ d'itérations, -si la fonction, notée $G_f_u$ (ou bien $G_f_g$) +si la fonction, notée $G_{f_u}$ (ou bien $G_{f_g}$) présentée au chapitre~\ref{chap:carachaos}, a de \og bonnes\fg{} propriétés chaotiques, le mot $x^b$ devrait \og sembler ne plus dépendre\fg{} de $x^0$. On peut penser à exploiter une de ces fonctions $G_f$ comme un générateur aléatoire. Enfin, un bon générateur aléatoire se doit de -fournir des nombres selon une \gls{distributionuniforme} +fournir des nombres selon une {distributionuniforme} La suite de ce document donnera, dans le cas où le graphe d'itérations est fortement connexe, une condition nécessaire est suffisante pour que @@ -14,10 +14,11 @@ cette propriété soit satisfaite. Cette section présente une application directe de la théorie développée ci-avant -à la génération de nombres pseudo aléatoires. On présente tout d'abord le générateur +à la génération de nombres pseudo aléatoires. +On présente tout d'abord le générateur basé sur des fonctions chaotiques (section~\ref{sub:prng:algo}), -puis comment intégrer la contrainte de \gls{distributionuniforme} -(cf. glossaire) de la sortie +puis comment intégrer la contrainte de distributionuniforme +de la sortie dans le choix de la fonction à itérer (section~\ref{sub:prng:unif}). L'approche est évaluée dans la dernière section. \JFC{plan à revoir} @@ -64,20 +65,20 @@ de nombres pseudo aléatoires Cet algorithme est utilisée dans notre générateur pour construire la longueur de la stratégie ainsi que les éléments qui la composent. Pratiquement, il retourne des entiers dans $\llbracket 1 ; l \rrbracket$ -selon une \gls{distributionuniforme} (cf. glossaire) et utilise +selon une distributionuniforme et utilise \textit{XORshift} qui est une classe de générateurs de nombres pseudo aléatoires très rapides conçus par George Marsaglia. % L'algorithme \textit{XORshift} exploite itérativement -% la fonction \og \gls{xor}\fg{} $\oplus$ (cf. glossaire) -% sur des nombres obtenus grâce à des \glspl{decalageDeBits} (cf. glossaire). +% la fonction \og {xor}\fg{} $\oplus$ (cf. glossaire) +% sur des nombres obtenus grâce à des pl{decalageDeBits} (cf. glossaire). L'algorithme \textit{XORshift} exploite itérativement l'opérateur $\oplus$ -sur des nombres obtenus grâce à des \glspl{decalageDeBits} (cf. glossaire). +sur des nombres obtenus grâce à des decalages de bits. Cet opérateur, défini dans $\Bool^{n}$, -applique la fonction \og \gls{xor} \fg{} (cf. glossaire) +applique la fonction \og xor \fg{} aux bits de même rang de ses deux opérandes (\og opération bit à bit \fg{}). Une instance de cette classe est donnée dans l'algorithme~\ref{XORshift} donné ci-dessous. @@ -117,8 +118,8 @@ si la propriété suivante est établie: $$\exists k \in \mathds{N}^\ast, \forall i,j \in \llbracket 1; n \rrbracket, M_{ij}^k>0.$$ On énonce enfin le théorème suivant liant les -\glspl{vecteurDeProbabilite} (cf. glossaire) -et les \glspl{chaineDeMarkov} (cf. glossaire): +vecteurDeProbabilite +et les chaineDeMarkov: @@ -126,11 +127,11 @@ et les \glspl{chaineDeMarkov} (cf. glossaire): Si $M$ est une matrice stochastique régulière, alors $M$ possède un unique vecteur stationnaire de probabilités $\pi$ ($\pi.M = \pi$). - De plus, si $\pi^0$ est un \gls{vecteurDeProbabilite} + De plus, si $\pi^0$ est un {vecteurDeProbabilite} et si on définit la suite $(\pi^{k})^{k \in \Nats}$ par $\pi^{k+1} = \pi^k.M $ pour $k = 0, 1,\dots$ - alors la \gls{chaineDeMarkov} $\pi^k$ + alors la {chaineDeMarkov} $\pi^k$ converge vers $\pi$ lorsque $k$ tend vers l'infini. \end{Theo} @@ -158,10 +159,10 @@ pour tout sommet de $\Gamma(g)$ et de $\Gamma(h)$, chaque arc sortant de ce sommet a, parmi l'ensemble des arcs sortant de ce sommet, une probabilité $1/2$ d’être celui qui sera traversé. En d'autres mots, $\Gamma(g)$ est le graphe orienté d'une chaîne de Markov. -Il est facile de vérifier que la \gls{matriceDeTransitions} (cf. glossaire) +Il est facile de vérifier que la {matriceDeTransitions} d'un tel processus est $M_g = \frac{1}{2} \check{M}_g$, -où $\check{M}_g$ est la \gls{matriceDAdjacence} (cf. glossaire) donnée en +où $\check{M}_g$ est la {matriceDAdjacence} donnée en figure~\ref{fig:g:incidence} (voir ci-après), et similairement pour $M_h$. \begin{figure}[h] diff --git a/main.pdf b/main.pdf index f9c13dc8874400b3b9835034d732185b874be013..022022b4e275b5745290dad62719e8c0d2a339c2 100644 GIT binary patch delta 125348 zcmZsC1y~$Svo7wii@Uq~;=zM!aQEP@K^BML78VU|!QI_8I0S-|;1=93`Tqa>=bU@* z^Gwh7Y|9HYy*G&LMbn#&;OVq;I!&1Oo63UO52EZVFy;M&N;kL)Z%Qm_TqaUo4u=4=9r3q1|)C3)dR8j9>uWC|6jojTEZ z3t=Ld6Pk=VnLzk`UXl-_d2pkuC2~4Zuezkn04?`_BJcA}SR8i()11W*@ zR?2APnSlG_gc*5K;)xz*+haMToeyCrmGuh0X+1exMID zH3pvyXkwxlt%d12g6b|A;BtTOJ%kTWloKnf>)JhVM_jqH1H 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