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
-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
-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
@@ -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
-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}