suite

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 39536c619accd44ebe4ff4c130b2bcfb4a56ab2e..41e628a97bc3cd8695dd4bed1ffea352591b5be5 100644 (file)
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -56,15 +56,16 @@ generator (PRNG) needs  to verify some features to be used  by scientists. It is
 important  to  be  able  to  generate  pseudo-random  numbers  efficiently,  the
 generation  needs to  be reproducible  and a  PRNG needs  to satisfy  many usual
 statistical properties. Finally, from our point a view, it is essential to prove
 important  to  be  able  to  generate  pseudo-random  numbers  efficiently,  the
 generation  needs to  be reproducible  and a  PRNG needs  to satisfy  many usual
 statistical properties. Finally, from our point a view, it is essential to prove

-that a  PRNG is chaotic.  Devaney~\cite{Devaney} proposed  a common mathematical
-formulation  of chaotic  dynamical  systems. Concerning  the statistical  tests,
-TestU01the is  the best-known public-domain statistical testing  packages. So we
-use it for all our PRNGs, especially  the {\it BigCrush} which is based on the largest
-serie of tests.
+that  a PRNG  is  chaotic.  Concerning  the  statistical tests,  TestU01 is  the
+best-known public-domain statistical testing package.   So we use it for all our
+PRNGs, especially the {\it BigCrush}  which provides the largest serie of tests.
+Concerning  the  chaotic properties,  Devaney~\cite{Devaney}  proposed a  common
+mathematical formulation of chaotic dynamical systems.

 
 In a  previous work~\cite{bgw09:ip}  we have proposed  a new familly  of chaotic
 
 In a  previous work~\cite{bgw09:ip}  we have proposed  a new familly  of chaotic

-PRNG  based on  chaotic iterations  (IC).   In this  paper we  propose a  faster
-version which is also proven to be chaotic with the Devaney formulation.
+PRNG  based on  chaotic iterations  (IC). We  have proven  that these  PRNGs are
+chaotic in the Devaney's sense.  In this paper we propose a faster version which
+is also proven to be chaotic.

 
 Although graphics  processing units (GPU)  was initially designed  to accelerate
 the manipulation of  images, they are nowadays commonly  used in many scientific
 
 Although graphics  processing units (GPU)  was initially designed  to accelerate
 the manipulation of  images, they are nowadays commonly  used in many scientific


@@ -92,8 +93,13 @@ chaotic. Concerning  the speed  of generation, they  can generate  about 3200000
 random numbers per seconds on a GeForce 7800 GTX GPU (which is quite old now).
 
 In \cite{ZRKB10}, the authors propose  different versions of efficient GPU PRNGs
 random numbers per seconds on a GeForce 7800 GTX GPU (which is quite old now).
 
 In \cite{ZRKB10}, the authors propose  different versions of efficient GPU PRNGs

-based on  Lagged Fibonacci,  Hybrid Taus  or Hybrid Taus.  They have  used these
-PRNGs for Langevin simulations of biomolecules fully implemented on GPU.
+based on  Lagged Fibonacci, Hybrid  Taus or Hybrid  Taus.  They have  used these
+PRNGs   for  Langevin   simulations   of  biomolecules   fully  implemented   on
+GPU. Performance of  the GPU versions are far better than  those obtained with a
+CPU and these PRNGs succeed to pass the {\it BigCrush} test of TestU01. There is
+no mention that their PRNGs have chaos mathematical properties.
+
+To the best of our knowledge no GPU implementation have been proven to have chaotic properties.

 
 \section{Basic Recalls}
 \label{section:BASIC RECALLS}
 
 \section{Basic Recalls}
 \label{section:BASIC RECALLS}