X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/book_gpu.git/blobdiff_plain/3c87d3518228d3b7bcfbd0a08cb79682f05132ee..0cd5e33087ba2395b349cfca863c4c73948029ab:/BookGPU/Chapters/chapter15/ch15.tex diff --git a/BookGPU/Chapters/chapter15/ch15.tex b/BookGPU/Chapters/chapter15/ch15.tex index 1eca6ec..9b0bf27 100644 --- a/BookGPU/Chapters/chapter15/ch15.tex +++ b/BookGPU/Chapters/chapter15/ch15.tex @@ -5,10 +5,9 @@ \chapterauthor{Stan Scott}{School of Electronics, Electrical Engineering \& Computer Science, The Queen's University of Belfast} -\newcommand{\fixme}[1]{{\bf #1}} +%\newcommand{\fixme}[1]{{\bf #1}} -\chapter{Numerical validation and performance optimization on GPUs of -an application in atomic physics} +\chapter[Numerical validation and performance optimization on GPUs in atomic physics]{Numerical validation and performance optimization on GPUs of an application in atomic physics} \label{chapter15} \section{Introduction}\label{ch15:intro} @@ -22,11 +21,11 @@ GPUs, applications should be coarse-grained and have a high arithmetic intensity ($i.e.$ the ratio of arithmetic operations to memory operations). Another important aspect of GPU programming is that floating-point -operations are preferably performed in single precision, if the +operations are preferably performed in single precision\index{precision!single precision}, if the validity of results is not impacted by that format. The GPU compute power for floating-point operations is indeed greater in -single precision than in double precision. -The peak performance ratio between single precision and double +single precision\index{precision!single precision} than in double precision\index{precision!double precision}. +The peak performance ratio between single precision\index{precision!single precision} and double precision varies for example for NVIDIA GPUs from $12$ for the first Tesla GPUs (C1060), to $2$ for the Fermi GPUs (C2050 and C2070) @@ -34,7 +33,7 @@ and to $3$ for the latest Kepler architecture (K20/K20X). As far as AMD GPUs are concerned, the latest AMD GPU (Tahiti HD 7970) presents a ratio of $4$. Moreover, GPU internal memory accesses and CPU-GPU data transfers are -faster in single precision than in double precision, +faster in single precision\index{precision!single precision} than in double precision\index{precision!double precision}, because of the different format lengths. This chapter describes the deployment on GPUs of PROP, a program of the @@ -66,7 +65,7 @@ For matrix products the GPU performance gain over CPU increases indeed with the matrix size, since the CPU-GPU transfer overhead becomes less significant and since CPUs are still more efficient for fine computation grains. -Then, using HMPP\footnote{ +Then, using HMPP\index{HMPP}\footnote{ HMPP or {\em CAPS compiler}, see: \url{www.caps-entreprise.com/hmpp.html}}, a commercial hybrid and parallel compiler, CAPS @@ -79,8 +78,8 @@ significant acceleration. The work described in this chapter, which is based on a study presented in \cite{PF_PDSEC2011}, aims at improving PROP performance on GPUs by exploring two directions. First, because the original version of PROP is written -in double precision, -we study the numerical stability of PROP in single precision. +in double precision\index{precision!double precision}, +we study the numerical stability of PROP in single precision\index{precision!single precision}. Second, we deploy the whole computation code of PROP on GPUs to avoid the overhead generated by @@ -292,6 +291,22 @@ for the next evaluation. %% \end{algorithmic} %% \end{algorithm} +\begin{algorithm} +\caption{\label{prop-algo}PROP algorithm} +%\begin{algorithmic} +\For{all scattering energies} { + \For{all sectors}{ + Read amplitude arrays\; + Read correction data\; + Construct local $R$-matrices\; + From $\Re^{I}$ and local $R$-matrices, compute $\Re^{O}$\; + $\Re^{O}$ becomes $\Re^{I}$ for the next sector\; + } + Compute physical $R$-Matrix \; +} +%\end{algorithmic} +\end{algorithm} + On the first sector, there is no input $R$-matrix yet. To bootstrap the propagation, the first output $R$-matrix is constructed using only @@ -310,7 +325,7 @@ depending on the computation performed: The serial version of PROP is implemented in Fortran~90 and uses -for linear algebra operations BLAS and LAPACK routines +for linear algebra operations BLAS\index{BLAS} and LAPACK\index{LAPACK} routines which are fully optimized for x86 architecture. This program @@ -359,20 +374,20 @@ copies fall from 22 to 5, matrix multiplications from 4 to~1 and calls to a linear equation solver from 2 to 1. To implement this version, CAPS -used HMPP, a +used HMPP\index{HMPP}, a commercial hybrid and parallel compiler, -based on compiler directives like the new OpenACC standard\footnote{See: \url{www.openacc-standard.org}}. +based on compiler directives like the new OpenACC\index{OpenACC} standard\footnote{See: \url{www.openacc-standard.org}}. If the matrices are large enough (the limit sizes are experimental parameters), they are multiplied on the GPU, otherwise on the CPU. CAPS - used the MKL BLAS implementation on an Intel Xeon + used the MKL BLAS\index{BLAS} implementation on an Intel Xeon x5560 quad core CPU (2.8 GHz) -and the CUBLAS library (CUDA 2.2) on one Tesla C1060 GPU. +and the CUBLAS\index{CUBLAS} library (CUDA 2.2) on one Tesla C1060 GPU. On the large data set (see Table~\ref{data-sets}), CAPS obtained a speedup of 1.15 for the GPU version over the CPU one (with multi-threaded MKL calls on the four CPU cores). This limited gain in performance is mainly -due to the use of double precision computation +due to the use of double precision\index{precision!double precision} computation and to the small or medium sizes of most matrices. For these matrices, the computation gain on the GPU is indeed @@ -386,14 +401,14 @@ code to the GPU and therefore avoiding the intermediate data transfers between the host (CPU) and the GPU. We will also study the -stability of PROP in single precision because -single precision computation is faster on the GPU +stability of PROP in single precision\index{precision!single precision} because +single precision\index{precision!single precision} computation is faster on the GPU and CPU-GPU data transfers are twice as fast as those performed in -double precision. +double precision\index{precision!double precision}. -\section{Numerical validation of PROP in single precision} +\section{Numerical validation\index{numerical validation} of PROP in single precision\index{precision!single precision}} \label{single-precision} \begin{comment} @@ -419,13 +434,13 @@ double precision. \hline \end{tabular} \end{center} -\caption{\label{sp-distrib}Error distribution for medium case in single precision} +\caption{\label{sp-distrib}Error distribution for medium case in single precision\index{precision!single precision}} \end{table} \end{comment} Floating-point input data, computation and output data of PROP are -originally in double precision format. +originally in double precision\index{precision!double precision} format. PROP produces a standard $R$-matrix H-file \cite{FARM_2DRMP} and a collection of Rmat00X files (where X ranges from 0 to the number of scattering energies - 1) @@ -435,21 +450,21 @@ The H-file and the Rmat00X files are binary input files of the FARM program \c (last program of the 2DRMP suite). Their text equivalent are the prop.out and the prop00X.out files. -To study the validity of PROP results in single precision, +To study the validity of PROP results in single precision\index{precision!single precision}, first, reference results are - generated by running the serial version of PROP in double precision. + generated by running the serial version of PROP in double precision\index{precision!double precision}. Data used in the most costly computation parts are read from input files in -double precision format and then -cast to single precision format. -PROP results (input of FARM) are computed in single precision and written -into files in double precision. +double precision\index{precision!double precision} format and then +cast to single precision\index{precision!single precision} format. +PROP results (input of FARM) are computed in single precision\index{precision!single precision} and written +into files in double precision\index{precision!double precision}. \subsection{Medium case study} \begin{figure}[h] \begin{center} \includegraphics*[width=0.9\linewidth]{Chapters/chapter15/figures/error.pdf} -\caption{\label{fig:sp-distrib} Error distribution for medium case in single precision} +\caption{\label{fig:sp-distrib} Error distribution for medium case in single precision\index{precision!single precision}} \end{center} \end{figure} @@ -476,11 +491,11 @@ in the same prop00X.out file. Relative errors of approximately 5\% impact values the order of magnitude of which is at most 1.E2. For instance, the value 164 produced by the reference version of - PROP becomes 172 in the single precision version. + PROP becomes 172 in the single precision\index{precision!single precision} version. \end{itemize} -To study the impact of the single precision version of PROP on the +To study the impact of the single precision\index{precision!single precision} version of PROP on the FARM program, the cross-section results files corresponding to transitions @@ -504,25 +519,25 @@ values are impacted by low errors. For instance, the maximum value {1s2p} & 0.08 & {1s4s} & 0.20 \\ \hline {1s3s} & 0.17 &2p4d & 1.60 \\ \hline \end{tabular} -\caption{\label{sp-farm}Impact on FARM of the single precision version of PROP} +\caption{\label{sp-farm}Impact on FARM of the single precision\index{precision!single precision} version of PROP} \end{center} \end{table} To examine in more detail the impact of PROP on FARM, cross sections above the ionization threshold (1 Ryd) are compared in single and -double precision for +double precision\index{precision!double precision} for transitions amongst the 1s, \dots 4s, 2p, \dots 4p, 3d, 4d target states. This comparison is carried out by generating 45 plots. In all the - plots, results in single and double precision match except for few + plots, results in single and double precision\index{precision!double precision} match except for few scattering energies which are very close to pseudo-state thresholds. For example Fig.~\ref{1s2p} and \ref{1s4d} present the scattering energies corresponding to the -{1s2p} and {1s4d} cross-sections computed in single and double precision. For some cross-sections, +{1s2p} and {1s4d} cross-sections computed in single and double precision\index{precision!double precision}. For some cross-sections, increasing a threshold parameter from 1.E-4 to 1.E-3 in the FARM program results in energies close to threshold being avoided and therefore -the cross-sections in double and single precision match more +the cross-sections in double and single precision\index{precision!single precision} match more accurately. This is the case for instance for cross-section 1s2p (see Fig.~\ref{1s2p3}). However for other cross-sections (such as 1s4d) some problematic energies remain even if the @@ -559,12 +574,12 @@ threshold parameter would be required for such cross-sections. \end{figure} As a conclusion, the medium case study shows that the execution of -PROP in single precision leads to a few inexact scattering energies to +PROP in single precision\index{precision!single precision} leads to a few inexact scattering energies to be computed by the FARM program for some cross-sections. Thanks to a suitable threshold parameter in the FARM program these problematic energies may possibly be skipped. Instead of investigating deeper the choice of such a parameter for the medium case, we analyze the -single precision computation in a more +single precision\index{precision!single precision} computation in a more realistic case in Sect.~\ref{huge}. \begin{comment} The conclusion of the medium case study is that running PROP in single @@ -572,7 +587,7 @@ precision gives relatively stable results provided that suitable parameter values are used in the FARM program in order to skip the problematic energies that are too close to the pseudo-state thresholds. To verify if this conclusion is still valid with a larger -data set, the single precision computation is analyzed in a more +data set, the single precision\index{precision!single precision} computation is analyzed in a more realistic case in Sect.~\ref{huge}. \end{comment} @@ -592,7 +607,7 @@ realistic case in Sect.~\ref{huge}. \end{figure} We study here the impact on FARM of the PROP program run in -single precision for the huge case (see Table~\ref{data-sets}). +single precision\index{precision!single precision} for the huge case (see Table~\ref{data-sets}). The cross-sections corresponding to all atomic target states 1s \dots 7i are explored, which @@ -602,24 +617,24 @@ It should be noted that in this case, over the same energy range above the ioniz As expected, all the plots exhibit large differences between single and double precision cross-sections. For example Fig.~\ref{1s2pHT} and \ref{1s4dHT} present the 1s2p and 1s4d cross-sections computed in -single and double precision for the huge case. -We can conclude that PROP in single precision gives invalid results +single and double precision\index{precision!double precision} for the huge case. +We can conclude that PROP in single precision\index{precision!single precision} gives invalid results for realistic simulation cases above the ionization threshold. Therefore the deployment of PROP on GPU, described in Sect.~\ref{gpu-implem}, -has been carried out in double precision. +has been carried out in double precision\index{precision!double precision}. \section{Towards a complete deployment of PROP on GPUs} \label{gpu-implem} We now detail how PROP has been progressively deployed on -GPUs in double precision in order to avoid the +GPUs in double precision\index{precision!double precision} in order to avoid the expensive memory transfers between the host and the GPU. Different versions with successive improvements and optimizations are presented. We use CUDA~\cite{CUDA_ProgGuide} for GPU programming, as well as the -CUBLAS~\cite{CUBLAS} +CUBLAS\index{CUBLAS}~\cite{CUBLAS} and MAGMA \cite{MAGMA} libraries for linear algebra operations. -Since PROP is written in Fortran 90, {\em wrappers} in C are used to -enable calls to CUDA kernels from PROP routines. +Since PROP is written in Fortran 90, {\em wrappers\index{wrapper}} in C are used to +enable calls to CUDA kernels from PROP routines. \subsection{Computing the output $R$-matrix on GPU} @@ -647,7 +662,7 @@ Fig.~\ref{offdiagonal} for an off-diagonal sector. These copies, along with possible scalings or transpositions, are implemented as CUDA kernels which can be applied to two matrices of any size and starting at any offset. - Memory accesses are coalesced \cite{CUDA_ProgGuide} in order to + Memory accesses are coalesced\index{coalesced memory accesses} \cite{CUDA_ProgGuide} in order to provide the best performance for such memory-bound kernels. \item[Step 2] (``Local copies''):~data are copied from local $R$-matrices to temporary arrays ($U$, $V$) and to $\Re^{O}$. @@ -656,13 +671,13 @@ Fig.~\ref{offdiagonal} for an off-diagonal sector. is added to matrix $A$ (via a CUDA kernel) and zeroes are written in $\Re^{O}$ where required. \item[Step 3] (``Linear system solving''):~matrix $A$ is factorized - using the MAGMA DGETRF + using the MAGMA DGETRF\index{MAGMA functions!DGETRF} routine and the result is stored in-place. \item[Step 4] (``Linear system solving'' cont.):~the matrix system - of linear equations $AW$ = $V$ is solved using the MAGMA DGETRS + of linear equations $AW$ = $V$ is solved using the MAGMA DGETRS\index{MAGMA functions!DGETRS} routine. The solution is stored in matrix $V$. \item[Step 5] (``Output matrix product''):~matrix $U$ - is multiplied by matrix $V$ using the CUBLAS DGEMM + is multiplied by matrix $V$ using the CUBLAS\index{CUBLAS} DGEMM routine. The result is stored in a temporary matrix~$t$. \item[Step 6] (``Output add''):~$t$ is added to $\Re^{O}$ (CUDA kernel). @@ -696,11 +711,11 @@ presented in Fig.~\ref{amplitudes}, is a matrix product between one $i$ amplitude array and one transposed $j$ amplitude array which is performed by a single DGEMM -BLAS call. +BLAS\index{BLAS} call. In this version, hereafter referred to as GPU V2\label{gpuv2}, $i$ and $j$ amplitude arrays are transferred to the GPU memory and the required matrix multiplications are performed on -the GPU (via CUBLAS routines). +the GPU (via CUBLAS\index{CUBLAS} routines). The involved matrices having medium sizes (either $3066 \times 383$ or @@ -734,7 +749,7 @@ Moreover, scaling $j$ amplitude arrays is expected to be faster on the GPU than on the CPU, thanks to the massively parallel architecture of the GPU and thanks to its higher internal memory bandwidth. -\subsection{Using double-buffering to overlap I/O and computation} +\subsection{Using double-buffering\index{double-buffering} to overlap I/O and computation} \begin{figure}[t] \centering @@ -761,7 +776,7 @@ time for the first sectors. But this evaluation time grows linearly with the strip number, and rapidly exceeds the I/O time. -It is thus interesting to use a double-buffering technique to overlap the +It is thus interesting to use a double-buffering\index{double-buffering} technique to overlap the I/O time with the evaluation time: for each sector, the evaluation of sector $n$ is performed (on GPU) simultaneously with the reading of data for sector @@ -769,7 +784,7 @@ $n+1$ (on CPU). This requires the duplication in the CPU memory of all the data structures used for storing data read from I/O files for each sector. This version, hereafter referred to as GPU -V4\label{gpuv4}, uses POSIX threads. Two threads are +V4\label{gpuv4}, uses POSIX threads\index{POSIX threads}. Two threads are executed concurrently: an I/O thread that reads data from I/O files for each sector, and a computation thread, dedicated to the propagation of the global $R$-matrix, that performs successively for each sector @@ -782,9 +797,9 @@ the computation thread which are implemented through standard POSIX thread mechanisms. -\subsection{Matrix padding} +\subsection{Matrix padding\index{padding}} The CUBLAS DGEMM -performance and the MAGMA DGETRF/DGETRS +performance and the MAGMA DGETRF\index{MAGMA functions!DGETRF}/DGETRS\index{MAGMA functions!DGETRS} performance is reduced when the sizes (or the leading dimensions) of the matrix are not multiples of the inner blocking size \cite{NTD10a}. This inner blocking size can be 32 or 64, depending on the computation @@ -797,9 +812,9 @@ multiples of 64. This corresponds indeed to the optimal size for the matrix product on the Fermi architecture \cite{NTD10b}. And as far as linear system solving is concerned, all the matrices have sizes which are multiples of 383: we -therefore use padding to obtain multiples of 384 (which are +therefore use padding\index{padding} to obtain multiples of 384 (which are again multiples of 64). -It can be noticed that this padding has to be performed dynamically +It can be noticed that this padding\index{padding} has to be performed dynamically as the matrices increase in size during the propagation (when possible the maximum required storage space is however allocated only once in the @@ -897,7 +912,7 @@ one C2050 (Fermi) GPU, located at UPMC (Universit\'e Pierre et Marie Curie, Paris, France). As a remark, the execution times measured on the C2050 would be the same on the C2070 and on the C2075, the only difference between these GPUs -being their memory size and their TDP (Thermal Design Power). +being their memory size and their TDP (Thermal Design Power)\index{TDP (Thermal Design Power)}. We emphasize that the execution times correspond to the complete propagation for all six energies of the large case (see Table~\ref{data-sets}), that is to say to the complete execution of @@ -906,7 +921,7 @@ Since energies are independent, execution times for more energies (e.g. the huge case) should be proportional to those reported in Table~\ref{table:time}. -These tests, which have been performed with CUDA 3.2, CUBLAS 3.2 and +These tests, which have been performed with CUDA 3.2, CUBLAS\index{CUBLAS} 3.2 and MAGMA 0.2, show that the successive GPU versions of PROP offer increasing, and at the end interesting, speedups. @@ -919,21 +934,21 @@ reduced thanks to~V3, which also accelerates the computation of amplitude arrays thanks to the GPU. The -double-buffering technique implemented in V4 +double-buffering\index{double-buffering} technique implemented in V4 effectively enables the overlapping of I/O operations with computation, while the -padding implemented in V5 also improves the computation times. +padding\index{padding} implemented in V5 also improves the computation times. It -is noticed that the padding +is noticed that the padding\index{padding} does offer much more performance gain with, -for example, CUDA 3.1 and the MAGMA DGEMM~\cite{NTD10b}: the +for example, CUDA 3.1 and the MAGMA DGEMM\index{MAGMA functions!DGEMM}~\cite{NTD10b}: the speedup with respect to one CPU core was increased from 6.3 to 8.1 on C1060, and from 9.5 to 14.3 on C2050. -Indeed since CUBLAS 3.2 performance has been improved for non block multiple +Indeed since CUBLAS\index{CUBLAS} 3.2 performance has been improved for non block multiple matrix sizes through MAGMA code~\cite{NTD10a}. Although for all versions the C2050 (with its improved -double precision performance) offers up to almost +double precision\index{precision!double precision} performance) offers up to almost double speedup compared to the C1060, the performance obtained with both architectures justifies the use of the GPU for such an application. @@ -1009,33 +1024,32 @@ obtained for all operations, except for the CPU-GPU transfers and the linear system solving. The CPU-GPU transfers are mainly due to the $j$ amplitude arrays, and currently still correspond to minor times. When required, the -double-buffering technique may also be used to overlap such transfers -with computation on the GPU. +double-buffering\index{double-buffering} technique may also be used to overlap such transfers with computation on the GPU. -\section{Propagation of multiple concurrent energies on GPU} +\section{Propagation of multiple concurrent energies on GPU}\index{concurrent kernel execution} Finally, we present here an improvement that can benefit from the Fermi architecture, as well as from the newest Kepler architecture, both of which enable the concurrent execution of multiple -CUDA kernels, thus offering additional speedup on +CUDA kernels\index{concurrent kernel execution}, thus offering additional speedup on GPUs for small or medium computation grain kernels. In our case, the performance gain on the GPU is indeed limited since most matrices have small or medium sizes. By using multiple streams within one CUDA context~\cite{CUDA_ProgGuide}, we can propagate multiple energies -concurrently on the Fermi GPU. +concurrently\index{concurrent kernel execution} on the Fermi GPU. It can be noticed that all GPU computations for all streams are launched by the same host thread. We therefore rely here on the {\em legacy -API} of CUBLAS~\cite{CUBLAS} (like MAGMA) +API} of CUBLAS\index{CUBLAS}~\cite{CUBLAS} (like MAGMA) without thread safety problems. A {\em breadth first} issue order is used for kernel launchs \cite{CUDA_stream}: for a given GPU kernel, all kernel launchs are indeed issued together in the host thread, using one stream for each concurrent energy, in order to maximize concurrent kernel -execution. +execution\index{concurrent kernel execution}. Of course, the memory available on the GPU must be large enough to store all data structures required by each energy. Moreover, multiple streams are also used within the @@ -1080,7 +1094,7 @@ in order to enable concurrent executions among the required kernels. In order to have enough GPU memory to run two or three concurrent energies for the large case, we use one C2070 GPU (featuring 6GB of memory) -with one Intel X5650 hex-core CPU, CUDA 4.1 and CUBLAS 4.1, as +with one Intel X5650 hex-core CPU, CUDA 4.1 and CUBLAS\index{CUBLAS} 4.1, as well as the latest MAGMA release (version 1.3.0). Substantial changes have been required in the MAGMA calls with respect to the previous versions of PROP that were using MAGMA 0.2. @@ -1093,15 +1107,14 @@ concurrent energies or more. With the more realistic large case, the performance gain is lower mainly because of the increase in matrix sizes, which implies a better GPU usage with only one energy on the GPU. The concurrent execution of multiple -kernels is also limited by other operations on the +kernels\index{concurrent kernel execution} is also limited by other operations on the GPU \cite{CUDA_ProgGuide,CUDA_stream} and by the current MAGMA code which prevents concurrent MAGMA calls in different streams. Better speedups can be here expected on the latest Kepler GPUs which offer additional compute power, and whose {\em Hyper-Q} feature may help improve further the GPU utilization with concurrent energies. On the contrary, the same code on the C1060 shows no speedup - since the concurrent kernel launches are -serialized on this previous GPU architecture. + since the concurrent kernel launches are serialized on this previous GPU architecture. @@ -1111,16 +1124,15 @@ serialized on this previous GPU architecture. \section{Conclusion and future work} -\label{conclusion} - +\label{conclusion} In this chapter, we have presented our methodology and our results in the deployment on a GPU of an application (the PROP program) in atomic physics. We have started by studying the numerical stability of PROP using -single precision arithmetic. This has shown that PROP -using single precision, while relatively stable for some small cases, +single precision\index{precision!single precision} arithmetic. This has shown that PROP +using single precision\index{precision!single precision}, while relatively stable for some small cases, gives unsatisfactory results for realistic simulation cases above the ionization threshold where there is a significant density of pseudo-states. It is @@ -1129,25 +1141,25 @@ where the actual target states are less dense. This requires further investigation. We have -therefore deployed the PROP code in double precision on +therefore deployed the PROP code in double precision\index{precision!double precision} on a GPU, with successive improvements. The different GPU versions each offer increasing speedups over the CPU version. Compared to the single (respectively four) core(s) CPU version, the optimal GPU implementation gives a speedup of 8.2 (resp. 4.6) on one C1060 GPU, and a speedup of 15.9 (resp. 9.0) on one -C2050 GPU with improved double precision performance. +C2050 GPU with improved double precision\index{precision!double precision} performance. An additional gain of around 15\% can also be obtained on one Fermi GPU with large memory (C2070) thanks to concurrent kernel execution. Such speedups cannot be directly compared with the 1.15 speedup -obtained with the HMPP version, since in our tests the CPUs are -different and the CUBLAS versions are more recent. +obtained with the HMPP\index{HMPP} version, since in our tests the CPUs are +different and the CUBLAS\index{CUBLAS} versions are more recent. However, the programming effort required progressively to deploy PROP on GPUs clearly offers improved and interesting speedups for this -real-life application in double precision with varying-size matrices. +real-life application in double precision\index{precision!double precision} with varying-size matrices. We are currently working on a hybrid CPU-GPU version that spreads the @@ -1155,7 +1167,7 @@ computations of the independent energies on both the CPU and the GPU. This will enable multiple energy execution on the CPU, with one or several core(s) dedicated to each energy (via multi-threaded -BLAS libraries). Multiple +BLAS\index{BLAS} libraries). Multiple concurrent energies may also be propagated on each Fermi GPU. By merging this work with the current MPI PROP program, we will obtain a scalable hybrid CPU-GPU version.