CUDA编程接口：共享存储器实现矩阵相乘

转自http://cuda.it168.com/a2011/1207/1285/000001285186.shtml

正如在前面的文章提到的，共享存储器应当比全局存储器更快，详细内容将在后续文章中介绍。任何用访问共享存储器取代访问全局存储器的机会应当被发掘，如下面的矩阵相乘例子展示的那样。 下面的代码是矩阵相乘的一个直接的实现，没有利用到共享存储器。每个线程读入A的一行和B的一列，然后计算C中对应的元素，如图1所示。这样，A读了B.width次，B读了A.height次。

// Matrices are stored in row-major order:
// M(row, col) = *(M.elements + row * M.width + col) 
typedef struct {
int width;
int height;
float* elements; } Matrix; 
// Thread block size 
#define BLOCK_SIZE 16 
// Forward declaration of the matrix multiplication kernel 
__global__ void MatMulKernel(const Matrix, const Matrix, Matrix); 
// Matrix multiplication - Host code 
// Matrix dimensions are assumed to be multiples of BLOCK_SIZE 
void MatMul(const Matrix A, const Matrix B, Matrix C) { 
   // Load A and B to device memory
   Matrix d_A; d_A.width = A.width;
   d_A.height = A.height;
   size_t size = A.width * A.height * sizeof(float);
   cudaMalloc((void**)&d_A.elements, size); 
   cudaMemcpy(d_A.elements, A.elements, size, 
   cudaMemcpyHostToDevice);
   Matrix d_B; 
   d_B.width = B.width;
   d_B.height = B.height; size = B.width * B.height * sizeof(float);
   cudaMalloc((void**)&d_B.elements, size);
   cudaMemcpy(d_B.elements, B.elements, size, cudaMemcpyHostToDevice);
   // Allocate C in device memory Matrix d_C; 
   d_C.width = C.width;
   d_C.height = C.height;
   size = C.width * C.height * sizeof(float); 
   cudaMalloc((void**)&d_C.elements, size);
   // Invoke kernel dim3 dimBlock(BLOCK_SIZE, BLOCK_SIZE);
   dim3 dimGrid(B.width / dimBlock.x, A.height / dimBlock.y);
   MatMulKernel<<>>(d_A, d_B, d_C);
   // Read C from device memory cudaMemcpy(C.elements, Cd.elements, size, cudaMemcpyDeviceToHost);
   // Free device memory cudaFree(d_A.elements); cudaFree(d_B.elements);
   cudaFree(d_C.elements);
}
// Matrix multiplication kernel called by MatMul() 
__global__ void MatMulKernel(Matrix A, Matrix B, Matrix C) 
{ 
   // Each thread computes one element of C
   // by accumulating results into Cvalue float Cvalue = 0;
   int row = blockIdx.y * blockDim.y + threadIdx.y;
   int col = blockIdx.x * blockDim.x + threadIdx.x;
   for (int e = 0; e < A.width; ++e) Cvalue += A.elements[row * A.width + e] * B.elements[e * B.width + col];
   C.elements[row * C.width + col] = Cvalue;
}

　

　　▲图1 没有使用共享存储器的矩阵相乘

下面的例子代码利用了共享存储器实现矩阵相乘。本实现中，每个线程块负责计算一个小方阵Csub，Csub是C的一部分，而块内的每个线程计算Csub的一个元素。如图2所示。Csub等于两个长方形矩阵的乘积：A的子矩阵尺寸是(A.width,block_size)，行索引与Csub相同，B的子矩阵的尺寸是(block_size,A.width)，列索引与Csub相同。为了满足设备的资源，两个长方形的子矩阵分割为尺寸为block_size的方阵，Csub是这些方阵积的和。每次乘法的计算是这样的，首先从全局存储器中将二个对应的方阵载入共享存储器中，载入的方式是一个线程载入一个矩阵元素，然后一个线程计算乘积的一个元素。每个线程积累每次乘法的结果并写入寄存器中，结束后，再写入全局存储器。

　　采用这种将计算分块的方式，利用了快速的共享存储器，节约了许多全局存储器带宽，因为在全局存储器中，A只被读了(B.width/block_size)次同时B读了(A.height/block_size)次。

　　前面代码中的Matrix 类型增加了一个stride域，这样子矩阵能够用同样的类型有效表示。__device__函数(相关阅读的文章中提及)用于读写元素和从矩阵中建立子矩阵。

// Matrices are stored in row-major order:
// M(row, col) = *(M.elements + row * M.stride + col)
typedef struct {
  int width;
  int height; int stride;
  float* elements;
}
Matrix; 
// Get a matrix element 
__device__ float GetElement(const Matrix A, int row, int col) { 
  return A.elements[row * A.stride + col];
} 
// Set a matrix element 
__device__ void SetElement(Matrix A, int row, int col, float value) {
A.elements[row * A.stride + col] = value;
} 
// Get the BLOCK_SIZExBLOCK_SIZE sub-matrix Asub of A that is
// located col sub-matrices to the right and row sub-matrices down 
// from the upper-left corner of A 
__device__ Matrix GetSubMatrix(Matrix A, int row, int col) {
  Matrix Asub;
  Asub.width = BLOCK_SIZE; 
  Asub.height = BLOCK_SIZE;
  Asub.stride = A.stride;
  Asub.elements = &A.elements[A.stride * BLOCK_SIZE * row + BLOCK_SIZE * col];
  return Asub;
} 
// Thread block size #define BLOCK_SIZE 16 
// Forward declaration of the matrix multiplication kernel 
__global__ void MatMulKernel(const Matrix, const Matrix, Matrix); 
// Matrix multiplication - Host code 
// Matrix dimensions are assumed to be multiples of BLOCK_SIZE
void MatMul(const Matrix A, const Matrix B, Matrix C) {
   // Load A and B to device memory Matrix d_A; 
   d_A.width = d_A.stride = A.width;
   d_A.height = A.height;
   size_t size = A.width * A.height * sizeof(float);
   cudaMalloc((void**)&d_A.elements, size);
   cudaMemcpy(d_A.elements, A.elements, size, cudaMemcpyHostToDevice);
   Matrix d_B;
   d_B.width = d_B.stride = B.width;
   d_B.height = B.height;
   size = B.width * B.height * sizeof(float);   
   cudaMalloc((void**)&d_B.elements, size);
   cudaMemcpy(d_B.elements, B.elements, size,cudaMemcpyHostToDevice);
   // Allocate C in device memory Matrix d_C;
   d_C.width = d_C.stride = C.width;
   d_C.height = C.height;
   size = C.width * C.height * sizeof(float); cudaMalloc((void**)&d_C.elements, size);
   // Invoke kernel dim3 dimBlock(BLOCK_SIZE, BLOCK_SIZE);
   dim3 dimGrid(B.width / dimBlock.x, A.height / dimBlock.y);
   MatMulKernel<<>>(d_A, d_B, d_C);
   // Read C from device memory cudaMemcpy(C.elements, d_C.elements, size, cudaMemcpyDeviceToHost);
   // Free device memory cudaFree(d_A.elements); cudaFree(d_B.elements);
   cudaFree(d_C.elements);
}
// Matrix multiplication kernel called by MatMul() 
__global__ void MatMulKernel(Matrix A, Matrix B, Matrix C) {
   // Block row and column 
   int blockRow = blockIdx.y;
   int blockCol = blockIdx.x;
   // Each thread block computes one sub-matrix Csub of C
   Matrix Csub = GetSubMatrix(C, blockRow, blockCol);
   // Each thread computes one element of Csub 
   // by accumulating results into Cvalue
   float Cvalue = 0;
   // Thread row and column within Csub
   int row = threadIdx.y;
   int col = threadIdx.x;
   // Loop over all the sub-matrices of A and B that are
   // required to compute Csub
   // Multiply each pair of sub-matrices together
   // and accumulate the results
   for (int m = 0; m < (A.width / BLOCK_SIZE); ++m) {
   // Get sub-matrix Asub of A
   Matrix Asub = GetSubMatrix(A, blockRow, m);
   // Get sub-matrix Bsub of B Matrix Bsub = GetSubMatrix(B, m, blockCol);
   // Shared memory used to store Asub and Bsub respectively
   __shared__ float As[BLOCK_SIZE][BLOCK_SIZE];
   __shared__ float Bs[BLOCK_SIZE][BLOCK_SIZE];
   // Load Asub and Bsub from device memory to shared memory
   // Each thread loads one element of each sub-matrix
   As[row][col] = GetElement(Asub, row, col);
   Bs[row][col] = GetElement(Bsub, row, col);
   // Synchronize to make sure the sub-matrices are loaded
   // before starting the computation 
   __syncthreads();
   // Multiply Asub and Bsub together
   for (int e = 0; e < BLOCK_SIZE; ++e)
   Cvalue += As[row][e] * Bs[e][col];
   // Synchronize to make sure that the preceding
   // computation is done before loading two new
   // sub-matrices of A and B in the next iteration
   __syncthreads(); }
   // Write Csub to device memory
   // Each thread writes one element
   SetElement(Csub, row, col, Cvalue); 
}

　

　　▲图2 使用共享存储器的矩阵相乘