\chapterauthor{Raphaël Couturier}{Femto-ST Institute, University of Franche-Comte}
-\chapter{Introduction to CUDA}
+\chapter{Introduction to Cuda}
\label{chapter2}
-\section{Introduction}\label{intro}
-In this chapter we give some simple examples on CUDA programming. The goal is
-not to provide an exhaustive presentation of all the functionalities of CUDA but
-rather giving some basic elements. Of course, readers that do not know CUDA are
-invited to read other books that are specialized on CUDA programming (for
+\section{Introduction}
+\label{ch2:intro}
+
+In this chapter we give some simple examples of Cuda programming. The goal is
+not to provide an exhaustive presentation of all the functionalities of Cuda but
+rather to give some basic elements. Of course, readers that do not know Cuda are
+invited to read other books that are specialized on Cuda programming (for
example: \cite{ch2:Sanders:2010:CEI}).
\section{First example}
+\label{ch2:1ex}
-This first example is intented to show how to build a very simple example with
-CUDA. The goal of this example is to performed the sum of two arrays and
-putting the result into a third array. A cuda program consists in a C code
-which calls CUDA kernels that are executed on a GPU. The listing of this code is
-in Listing~\ref{ch2:lst:ex1}.
+This first example is intented to show how to build a very simple program with
+Cuda. Its goal is to perform the sum of two arrays and put the result into a
+third array. A Cuda program consists in a C code which calls Cuda kernels that
+are executed on a GPU. The listing of this code is in Listing~\ref{ch2:lst:ex1}.
As GPUs have their own memory, the first step consists in allocating memory on
-the GPU. A call to \texttt{cudaMalloc} allows to allocate memory on the GPU. The
-first parameter of this function is a pointer on a memory on the device
-(i.e. the GPU). In this example, \texttt{d\_} is added on each variable allocated
-on the GPU meaning this variable is on the GPU. The second parameter represents
-the size of the allocated variables, this size is in bits.
+the GPU. A call to \texttt{cudaMalloc}\index{Cuda~functions!cudaMalloc} allows
+to allocate memory on the GPU. The first parameter of this function is a pointer
+on a memory on the device (i.e. the GPU). In this example, \texttt{d\_} is added
+on each variable allocated on the GPU, meaning this variable is on the GPU. The
+second parameter represents the size of the allocated variables, this size is expressed in
+bits.
In this example, we want to compare the execution time of the additions of two
arrays in CPU and GPU. So for both these operations, a timer is created to
-measure the time. CUDA proposes to manipulate timers quick easily. The first
-step is to create the timer, then to start it and at the end to stop it. For
-each of these operations a dedicated functions is used.
+measure the time. Cuda proposes to manipulate timers quite easily. The first
+step is to create the timer\index{Cuda~functions!timer}, then to start it and at
+the end to stop it. For each of these operations a dedicated function is used.
-In order to compute the same sum with a GPU, the first step consits in
-transferring the data from the CPU (considered as the host with CUDA) to the GPU
-(considered as the device with CUDA). A call to \texttt{cudaMalloc} allows to
+In order to compute the same sum with a GPU, the first step consists in
+transferring the data from the CPU (considered as the host with Cuda) to the GPU
+(considered as the device with Cuda). A call to \texttt{cudaMemcpy} allows to
copy the content of an array allocated in the host to the device when the fourth
-parameter is set to \texttt{cudaMemcpyHostToDevice}. The first parameter of the
-function is the destination array, the second is the source array and the third
-is the number of elements to copy (exprimed in bytes).
-
-Now that the GPU contains the data needed to perform the addition. In sequential
-such addition is achieved out with a loop on all the elements. With a GPU, it
-is possible to perform the addition of all elements of the arrays in parallel
-(if the number of blocks and threads per blocks is sufficient). In
-Listing\ref{ch2:lst:ex1} at the beginning, a simple kernel,
+parameter is set
+to \texttt{cudaMemcpyHostToDevice}\index{Cuda~functions!cudaMemcpy}. The first
+parameter of the function is the destination array, the second is the
+source array and the third is the number of elements to copy (expressed in
+bytes).
+
+Now the GPU contains the data needed to perform the addition. In sequential
+programming, such addition is achieved out with a loop on all the elements.
+With a GPU, it is possible to perform the addition of all the elements of the
+two arrays in parallel (if the number of blocks and threads per blocks is
+sufficient). In Listing\ref{ch2:lst:ex1} at the beginning, a simple kernel,
called \texttt{addition} is defined to compute in parallel the summation of the
-two arrays. With CUDA, a kernel starts with the
-keyword \texttt{\_\_global\_\_} \index{CUDA~keywords!\_\_shared\_\_} which
+two arrays. With Cuda, a kernel starts with the
+keyword \texttt{\_\_global\_\_} \index{Cuda~keywords!\_\_shared\_\_} which
indicates that this kernel can be called from the C code. The first instruction
in this kernel is used to compute the variable \texttt{tid} which represents the
thread index. This thread index\index{thread index} is computed according to
-the values of the block index (it is a variable of CUDA
-called \texttt{blockIdx}\index{CUDA~keywords!blockIdx}). Blocks of threads can
-be decomposed into 1 dimension, 2 dimensions or 3 dimensions. According to the
-dimension of data manipulated, the appropriate dimension can be useful. In our
-example, only one dimension is used. Then using notation \texttt{.x} we can
-access to the first dimension (\texttt{.y} and \texttt{.z} allow respectively to
-access to the second and third dimension). The
-variable \texttt{blockDim}\index{CUDA~keywords!blockDim} gives the size of each
-block.
+the values of the block index
+(called \texttt{blockIdx} \index{Cuda~keywords!blockIdx} in Cuda) and of the
+thread index (called \texttt{blockIdx}\index{Cuda~keywords!threadIdx} in
+Cuda). Blocks of threads and thread indexes can be decomposed into 1 dimension,
+2 dimensions or 3 dimensions. According to the dimension of manipulated data,
+the appropriate dimension can be useful. In our example, only one dimension is
+used. Then using notation \texttt{.x} we can access to the first dimension
+(\texttt{.y} and \texttt{.z} respectively allow to access to the second and
+third dimension). The variable \texttt{blockDim}\index{Cuda~keywords!blockDim}
+gives the size of each block.
\lstinputlisting[label=ch2:lst:ex1,caption=A simple example]{Chapters/chapter2/ex1.cu}
\section{Second example: using CUBLAS}
+\label{ch2:2ex}
-The Basic Linear Algebra Subprograms (BLAS) allows programmer to use performant
-routines that are often used. Those routines are heavily used in many scientific
-applications and are very optimzed for vector operations, matrix-vector
+The Basic Linear Algebra Subprograms (BLAS) allows programmers to use efficient
+routines that are often required. Those routines are heavily used in many
+scientific applications and are optimized for vector operations, matrix-vector
operations and matrix-matrix
-operations~\cite{ch2:journals/ijhpca/Dongarra02}. Some of those operations seems
-to be easy to implement with CUDA. Nevertheless, as soon as a reduction is
-needed, implementing an efficient reduction routines with CUDA is far from being
-simple.
+operations~\cite{ch2:journals/ijhpca/Dongarra02}. Some of those operations seem
+to be easy to implement with Cuda. Nevertheless, as soon as a reduction is
+needed, implementing an efficient reduction routine with Cuda is far from being
+simple. Roughly speaking, a reduction operation\index{reduction~operation} is an
+operation which combines all the elements of an array and extracts a number
+computed with all the elements. For example, a sum, a maximum or a dot product
+are reduction operations.
In this second example, we consider that we have two vectors $A$ and $B$. First
-of all we want to compute the sum of both vectors in a vector $C$. Then we want
+of all, we want to compute the sum of both vectors in a vector $C$. Then we want
to compute the scalar product between $1/C$ and $1/A$. This is just an example
-which has not direct interest except to show how to program it with CUDA.
+which has no direct interest except to show how to program it with Cuda.
-Listing~\ref{ch2:lst:ex2} shows this example with CUDA. The first kernel for the
-addition of two arrays is exactly the same that the one described in the
+Listing~\ref{ch2:lst:ex2} shows this example with Cuda. The first kernel for the
+addition of two arrays is exactly the same as the one described in the
previous example.
-The kernel to compute the inverse of the elements of an array is very
+The kernel to compute the opposite of the elements of an array is very
simple. For each thread index, the inverse of the array replaces the initial
array.
In the main function, the beginning is very similar to the one in the previous
-example. First the number of elements is asked to the user. Then a call
-to \texttt{cublasCreate} allows to initialize the cublas library. It creates an
-handle. Then all the arrays are allocated in the host and the device, as in the
-previous example. Both arrays $A$ and $B$ are initialized. Then the CPU
+example. First, the user is askef to define the number of elements. Then a
+call to \texttt{cublasCreate} allows to initialize the cublas library. It
+creates a handle. Then all the arrays are allocated in the host and the device,
+as in the previous example. Both arrays $A$ and $B$ are initialized. The CPU
computation is performed and the time for this CPU computation is measured. In
order to compute the same result on the GPU, first of all, data from the CPU
need to be copied into the memory of the GPU. For that, it is possible to use
-cublas function \texttt{cublasSetVector}.
-
-\lstinputlisting[label=ch2:lst:ex2,caption=A simple example]{Chapters/chapter2/ex2.cu}
-
+cublas function \texttt{cublasSetVector}. This function has several
+arguments. More precisely, the first argument represents the number of elements
+to transfer, the second arguments is the size of each element, the third element
+represents the source of the array to transfer (in the GPU), the fourth is an
+offset between each element of the source (usually this value is set to 1), the
+fifth is the destination (in the GPU) and the last is an offset between each
+element of the destination. Then we call the kernel \texttt{addition} which
+computes the sum of all elements of arrays $A$ and $B$. The \texttt{inverse}
+kernel is called twice, once to inverse elements of array $C$ and once for
+$A$. Finally, we call the function \texttt{cublasDdot} which computes the dot
+product of two vectors. To use this routine, we must specify the handle
+initialized by Cuda, the number of elements to consider, then each vector is
+followed by the offset between every element. After the GPU computation, it is
+possible to check that both computation produce the same result.
+
+\lstinputlisting[label=ch2:lst:ex2,caption=A simple example with cublas]{Chapters/chapter2/ex2.cu}
+
+\section{Third example: matrix-matrix multiplication}
+\label{ch2:3ex}
+
+
+
+Matrix-matrix multiplication is an operation which is quite easy to parallelize
+with a GPU. If we consider that a matrix is represented using a two dimensional
+array, $A[i][j]$ represents the element of the $i^{th}$ row and of the $j^{th}$
+column. In many cases, it is easier to manipulate a 1D array instead of a 2D
+array. With Cuda, even if it is possible to manipulate 2D arrays, in the
+following we present an example based on a 1D array. For the sake of simplicity,
+we consider we have a square matrix of size \texttt{size}. So with a 1D
+array, \texttt{A[i*size+j]} allows us to have access to the element of the
+$i^{th}$ row and of the $j^{th}$ column.
+
+With a sequential programming, the matrix multiplication is performed using
+three loops. We assume that $A$, $B$ represent two square matrices and the
+result of the multiplication of $A \times B$ is $C$. The
+element \texttt{C[i*size+j]} is computed as follows:
+\begin{equation}
+C[i*size+j]=\sum_{k=0}^{size-1} A[i*size+k]*B[k*size+j];
+\end{equation}
+
+In Listing~\ref{ch2:lst:ex3}, the CPU computation is performed using 3 loops,
+one for $i$, one for $j$ and one for $k$. In order to perform the same
+computation on a GPU, a naive solution consists in considering that the matrix
+$C$ is split into 2 dimensional blocks. The size of each block must be chosen
+such as the number of threads per block is inferior to $1,024$.
+
+
+In Listing~\ref{ch2:lst:ex3}, we consider that a block contains 16 threads in
+each dimension, the variable \texttt{width} is used for that. The
+variable \texttt{nbTh} represents the number of threads per block. So, to be able
+to compute the matrix-matrix product on a GPU, each block of threads is assigned
+to compute the result of the product for the elements of this block. The main
+part of the code is quite similar to the previous code. Arrays are allocated in
+the CPU and the GPU. Matrices $A$ and $B$ are randomly initialized. Then
+arrays are transferred inside the GPU memory with call to \texttt{cudaMemcpy}.
+So the first step for each thread of a block is to compute the corresponding row
+and column. With a 2 dimensional decomposition, \texttt{int i=
+blockIdx.y*blockDim.y+ threadIdx.y;} allows us to compute the corresponding line
+and \texttt{int j= blockIdx.x*blockDim.x+ threadIdx.x;} the corresponding
+column. Then each thread has to compute the sum of the product of the line of
+$A$ by the column of $B$. In order to use a register, the
+kernel \texttt{matmul} uses a variable called \texttt{sum} to compute the
+sum. Then the result is set into the matrix at the right place. The computation
+of CPU matrix-matrix multiplication is performed as described previously. A
+timer measures the time. In order to use 2 dimensional blocks, \texttt{dim3
+dimGrid(size/width,size/width);} allows us to create \texttt{size/width} blocks
+in each dimension. Likewise, \texttt{dim3 dimBlock(width,width);} is used to
+create \texttt{width} thread in each dimension. After that, the kernel for the
+matrix multiplication is called. At the end of the listing, the matrix $C$
+computed by the GPU is transferred back into the CPU and we check if both matrices
+C computed by the CPU and the GPU are identical with a precision of $10^{-4}$.
+
+
+With $1,024 \times 1,024$ matrices, on a C2070M Tesla card, this code takes
+$37.68$ms to perform the multiplication. With an Intel Xeon E31245 at $3.30$GHz, it
+takes $2465$ms without any parallelization (using only one core). Consequently
+the speed up between the CPU and GPU version is about $65$ which is very good
+regarding the difficulty of parallelizing this code.
+
+\lstinputlisting[label=ch2:lst:ex3,caption=simple Matrix-matrix multiplication with cuda]{Chapters/chapter2/ex3.cu}
+
+\section{Conclusion}
+In this chapter, three simple Cuda examples have been presented. They are
+quite simple. As we cannot present all the possibilities of the Cuda
+programming, interested readers are invited to consult Cuda programming
+introduction books if some issues regarding the Cuda programming are not clear.
\putbib[Chapters/chapter2/biblio]
