2.3CUDA矩阵乘法
CPU 矩阵乘法
能相乘的两个矩阵，必须满足一个矩阵的行数和第二个矩阵的列数相同.
A(N*P) * B(P*M) = C(N*M). 其中P是行数，N是列数， 从宽高的角度来说，即 A的宽度和B的高度是相同的．C矩阵 = ha * wb.
其中C(i,j) = A矩阵中的i行和B矩阵中的j列进行点乘得到该点的值．
//C = A*B void MatrixMulCPU(float* _C,const float *_A,const float *_B,int _wa,int _ha,int _wb) { float sum = 0; for (int i = 0; i < _ha; ++i) { for (int j = 0; j < _wb; ++j) { sum = 0; for (int k = 0; k < _wa; ++k) {
//i*_wa得到当前对应的是A矩阵中的哪一行，+k对应当前行的哪一列．矩阵Ａ的stride是wa
//j对应当前矩阵Ｂ的哪一列，＋k*wb依次表示第０行的j列,第１行的j列...第wa-1行的j列．矩阵Ｂ的stride是wb sum += (float)_A[i*_wa+k]*(float)_B[k*_wb+ j]; } _C[i*_wb+j] = (float)sum; } } }
简单矩阵乘法
C(i,j) = sum { A(i,k)*B(k,j) } 0<=k < _wa;耦合程度很小，所以我们可以通过划分区域的方法，让每个线程负责一个区域。
怎么划分呢？首先最初的想法是让每一个线程计算一个C(i,j)，那么估算一下，应该需要height_c*width_c，也就是ha*wb个线程。进一步，我们将矩阵按一个大方格Block划分，如果一个方格Block大小是16*16，那么矩阵80*48的可以表示为5(*16) * 3(*16)，即5*3个大格子(Grid)，所以grid的划分自然就是(height_c/16) *(width_c/16)个线程了。
怎么划分呢？首先最初的想法是让每一个线程计算一个C(i,j)，那么估算一下，应该需要height_c*width_c，也就是ha*wb个线程。进一步，我们将矩阵按一个大方格Block划分，如果一个方格Block大小是16*16，那么矩阵80*48的可以表示为5(*16) * 3(*16)，即5*3个大格子(Grid)，所以grid的划分自然就是(height_c/16) *(width_c/16)个线程了。
好了，划分完后，内核代码如下： 这个kernel的代码只是把外层两个循环变成
计算版本0：
__global__ void matrix_kernel_0(float* _C,const float* _A,const float *_B,int _wa,int _wb) { float sum = 0; //找出该线程所在的行列 int row = blockIdx.y*blockDim.y + threadIdx.y; // X 对应矩阵row, Y对应举证col int col = blockIdx.x*blockDim.x + threadIdx.x; //线程Thread(row,col)负责计算C(row,col) for (int i = 0; i < _wa; ++i) { sum += _A[row*_wa + i]*_B[i*_wb + col]; } _C[row*_wb + col] = sum; }
这个是Heterogeneous Parallel Programming lab3:Basic Matrix Matrix Multiplication的代码：
#include <wb.h> #define wbCheck(stmt) \ do { \ cudaError_t err = stmt; \ if (err != cudaSuccess) { \ wbLog(ERROR, "Failed to run stmt ", #stmt); \ wbLog(ERROR, "Got CUDA error ... ", cudaGetErrorString(err)); \ return -1; \ } \ } while (0) #if 0 //This is C verison matrixMUl //C = A*B void MatrixMulCPU(float* _C,const float *_A,const float *_B,int _wa,int _ha,int _wb) { float sum = 0; for (int i = 0; i < _ha; ++i) { for (int j = 0; j < _wb; ++j) { sum = 0; for (int k = 0; k < _wa; ++k) { sum += (float)_A[i*_wa+k]*(float)_B[k*_wb+ j]; } _C[i*_wb+j] = (float)sum; } } } #endif // Compute C = A * B , Matrix C = hA * wB = rowA * columnB __global__ void matrixMultiply(float *A, float *B, float *C, int numARows, int numAColumns, int numBRows, int numBColumns, int numCRows, int numCColumns) { //@@ Insert code to implement matrix multiplication here float sum = 0.0f; int row = blockIdx.y*blockDim.y + threadIdx.y; int col = blockIdx.x*blockDim.x + threadIdx.x; if(row < numCRows && col < numCColumns){ for (int i = 0; i < numAColumns; ++i) { sum += A[row*numAColumns + i] * B[i*numBColumns + col]; } C[row*numBColumns + col] = sum; } printf("C = %f\n",C[row*numBColumns + col]); } int main(int argc, char **argv) { wbArg_t args; float *hostA; // The A matrix float *hostB; // The B matrix float *hostC; // The output C matrix float *deviceA; float *deviceB; float *deviceC; int numARows; // number of rows in the matrix A int numAColumns; // number of columns in the matrix A int numBRows; // number of rows in the matrix B int numBColumns; // number of columns in the matrix B int numCRows; // number of rows in the matrix C (you have to set this) int numCColumns; // number of columns in the matrix C (you have to set this) args = wbArg_read(argc, argv); wbTime_start(Generic, "Importing data and creating memory on host"); hostA = ( float * )wbImport(wbArg_getInputFile(args, 0), &numARows, &numAColumns); hostB = ( float * )wbImport(wbArg_getInputFile(args, 1), &numBRows, &numBColumns); //@@ Set numCRows and numCColumns numCRows = 0; numCColumns = 0; if(numAColumns != numBRows){ wbLog(TRACE, "numAColumns != numBRows, Break "); return 1; } numCRows = numARows; numCColumns = numBColumns; unsigned int A_size = numARows * numAColumns * sizeof(float); unsigned int B_size = numBRows * numBColumns * sizeof(float); unsigned int C_size = numCRows * numCColumns * sizeof(float); //@@ Allocate the hostC matrix hostC = ( float * )malloc(C_size); wbTime_stop(Generic, "Importing data and creating memory on host"); wbLog(TRACE, "The dimensions of A are ", numARows, " x ", numAColumns); wbLog(TRACE, "The dimensions of B are ", numBRows, " x ", numBColumns); wbTime_start(GPU, "Allocating GPU memory."); //@@ Allocate GPU memory here wbCheck(cudaMalloc((void**)&deviceA, A_size)); wbCheck(cudaMalloc((void**)&deviceB, B_size)); wbCheck(cudaMalloc((void**)&deviceC, C_size)); wbTime_stop(GPU, "Allocating GPU memory."); wbTime_start(GPU, "Copying input memory to the GPU."); //@@ Copy memory to the GPU here wbCheck(cudaMemcpy(deviceA, hostA, A_size, cudaMemcpyHostToDevice)); wbCheck(cudaMemcpy(deviceB, hostB, B_size, cudaMemcpyHostToDevice)); wbCheck(cudaMemcpy(deviceC, hostC, C_size, cudaMemcpyHostToDevice)); wbTime_stop(GPU, "Copying input memory to the GPU."); //@@ Initialize the grid and block dimensions here dim3 DimGrid((numCColumns-1)/16+1, (numCRows-1)/16+1, 1); dim3 DimBlock(16, 16, 1); wbTime_start(Compute, "Performing CUDA computation"); //@@ Launch the GPU Kernel here matrixMultiply<<< DimGrid, DimBlock >>>(deviceA, deviceB, deviceC, numARows, numAColumns, numBRows, numBColumns, numCRows, numCColumns); cudaDeviceSynchronize(); wbTime_stop(Compute, "Performing CUDA computation"); wbTime_start(Copy, "Copying output memory to the CPU"); //@@ Copy the GPU memory back to the CPU here //@@ Copy the GPU memory back to the CPU here wbCheck(cudaMemcpy(hostC, deviceC, C_size, cudaMemcpyDeviceToHost)); wbTime_stop(Copy, "Copying output memory to the CPU"); wbTime_start(GPU, "Freeing GPU Memory"); //@@ Free the GPU memory here wbCheck(cudaFree(deviceA)); wbCheck(cudaFree(deviceB)); wbCheck(cudaFree(deviceC)); wbTime_stop(GPU, "Freeing GPU Memory"); wbSolution(args, hostC, numCRows, numCColumns); free(hostA); free(hostB); free(hostC); return 0; }
使用tile来划分矩阵乘法
另外一种思路，我们不让每一个线程完整计算一个C(i,j)，通过C(i,j) = sum { A(i,k)*B(k,j) }发现，我们还可以再细度划分：
Csub(i,j) = sum{A(i,ksub+offsetA)*B(ksub+offsetB,j)} 0<=ksub < blockSize
C(i,j) = sum{Csub(i,j)}
就是把矩阵分成n*n个大的子块，然后每一个block负责计算子块i 和 子块j的子乘积，计算完毕后加起来则可。这里主要引入shared Memory来提高程序效率．
计算矩阵我们
__global__ void matrix_kernel_1(float* _C,const float* _A,const float *_B,int _wa,int _wb) //_wa是A矩阵的宽度，_wb是矩阵B的宽度 { int bx = blockIdx.x; //Block X的当前位置 int by = blockIdx.y; //Block y的当前位置 int tx = threadIdx.x; int ty = threadIdx.y; //该block要处理的A ,A的取值方向是X轴方向， B的取值方向是Y轴方向 int aBegin = _wa*(by*BLOCK_SIZE);//A(0,by) //在矩阵A上每个block的首地址 int aEnd = aBegin + _wa - 1; // int aStep = BLOCK_SIZE;//offsetA //因为A是横向取值，所以step是blocksize int bBegin = BLOCK_SIZE*bx;//B(bx,0) //矩阵B的首地址 int bStep = BLOCK_SIZE*_wb;//offsetB //因为B是纵向取值，所以step是blocksize*_wb. float cSub = 0;
//每一个线程计算一个像素点，分成wa/block 次来计算，每次计算一段A(sub) * B(sub)，最后累加得到C的结果．
//假设矩阵都是n*n的，那么旧的basicMatrix每个线程都需要执行2n次globalMemory的访问，这里用到sharedMemory只需要执行2n／blocksize，每个线程可以提高blocksize倍，
／／每个block里面的thread都是通过读取sharedMemory来执行计算的，速度会非常快． for (int a = aBegin,b = bBegin; a <= aEnd; a += aStep,b += bStep) { __shared__ float As[BLOCK_SIZE][BLOCK_SIZE]; __shared__ float Bs[BLOCK_SIZE][BLOCK_SIZE]; //每个线程负责一个元素拷贝,我们以block为单位来分析。假设blocksize=16, 一个block里面有 16*16个线程。
//每个block 可以填满需要用到的 As, 和Bs大小的矩阵。这里就是矩阵A里面的16*16的数据可以填满，保存在sharedMemory中。同样B矩阵也是。 As[ty][tx] = _A[a + _wa*ty + tx]; Bs[ty][tx] = _B[b + _wb*ty + tx]; __syncthreads(); //同步使得矩阵Ａ，和矩阵Ｂ的第一个tile*tile的数据保存在As和Bs里，供下面的计算使用． //每个线程负责计算一个子块i 和 子块j的子乘积宽度是block_size,执行到wa/block_size次，累加可得到Ｃ的值 for (int k = 0; k < BLOCK_SIZE; ++k) { cSub += As[ty][k]*Bs[k][tx]; } __syncthreads(); } //全局地址，向全局寄存器写回去 //一个线程负责一个元素，一个block负责一个子块 int cIndex = (by*BLOCK_SIZE + ty)*_wb + (bx*BLOCK_SIZE + tx); _C[cIndex] = cSub; }
二维矩阵的索引问题:
假设有一个32*48的矩阵，x的范围[0,47], y的范围是[0,31]。 在代码中是以一个二维数组保存，内存是连续的。目前我们要找到point(23,23)的索引：
一维指针指向的应该是: 23*48 + 23.
但是我们同样也可以用grid 和block 来划分这个矩阵:
grid(3,2) (2是列，即X维度，2是行，Y维度), block(16,16) 。 grid X的范围是是[0,2], Y的范围是[0,1]. 同理适用于block 我们也可以用下面的方式找到point(23,23)的索引：
point(23,23) 对应grid的坐标是（1,1） 对应的block坐标是(7,7)。block的宽度是16。
point.y = (by*Block_size + ty) = 1*16+7 =23
point.x = (bx*Block_size + tx)= 1*16+7 =23
point(x,y)的索引位置是： y * width + x
这个是Heterogeneous Parallel Programming lab4:Basic Matrix Matrix Multiplication的代码：
#include <wb.h> #define wbCheck(stmt) \ do { \ cudaError_t err = stmt; \ if (err != cudaSuccess) { \ wbLog(ERROR, "Failed to run stmt ", #stmt); \ wbLog(ERROR, "Got CUDA error ... ", cudaGetErrorString(err)); \ return -1; \ } \ } while (0) #define TILE_WIDTH 32 //block size ,each thread to calucate each block // Compute C = A * B __global__ void matrixMultiplyShared(float *A, float *B, float *C, int numARows, int numAColumns, int numBRows, int numBColumns, int numCRows, int numCColumns) { //@@ Insert code to implement matrix multiplication here //@@ You have to use shared memory for this MP __shared__ float sharedM[TILE_WIDTH][TILE_WIDTH]; __shared__ float sharedN[TILE_WIDTH][TILE_WIDTH]; int bx = blockIdx.x; int by = blockIdx.y; int tx = threadIdx.x; int ty = threadIdx.y; int row = by*TILE_WIDTH + ty; int col = bx*TILE_WIDTH + tx; float v = 0.0; for (int i = 0; i < (int)(ceil((float)numAColumns/TILE_WIDTH)); i++) { if (i*TILE_WIDTH + tx < numAColumns && row < numARows) sharedM[ty][tx] = A[row*numAColumns + i*TILE_WIDTH + tx]; else sharedM[ty][tx] = 0.0; if (i*TILE_WIDTH + ty < numBRows && col < numBColumns) sharedN[ty][tx] = B[(i*TILE_WIDTH + ty)*numBColumns + col]; else sharedN[ty][tx] = 0.0; __syncthreads(); for(int j = 0; j < TILE_WIDTH; j++) v += sharedM[ty][j] * sharedN[j][tx]; __syncthreads(); } if (row < numCRows && col < numCColumns) C[row*numCColumns + col] = v; } int main(int argc, char **argv) { wbArg_t args; float *hostA; // The A matrix float *hostB; // The B matrix float *hostC; // The output C matrix float *deviceA; float *deviceB; float *deviceC; int numARows; // number of rows in the matrix A int numAColumns; // number of columns in the matrix A int numBRows; // number of rows in the matrix B int numBColumns; // number of columns in the matrix B int numCRows; // number of rows in the matrix C (you have to set this) int numCColumns; // number of columns in the matrix C (you have to set this) args = wbArg_read(argc, argv); wbTime_start(Generic, "Importing data and creating memory on host"); hostA = ( float * )wbImport(wbArg_getInputFile(args, 0), &numARows, &numAColumns); hostB = ( float * )wbImport(wbArg_getInputFile(args, 1), &numBRows, &numBColumns); //@@ Set numCRows and numCColumns numCRows = 0; numCColumns = 0; if(numAColumns != numBRows){ wbLog(TRACE, "numAColumns != numBRows, Break "); return 1; } numCRows = numARows; numCColumns = numBColumns; unsigned int A_size = numARows * numAColumns * sizeof(float); unsigned int B_size = numBRows * numBColumns * sizeof(float); unsigned int C_size = numCRows * numCColumns * sizeof(float); //@@ Allocate the hostC matrix hostC = ( float * )malloc(C_size); wbTime_stop(Generic, "Importing data and creating memory on host"); wbLog(TRACE, "The dimensions of A are ", numARows, " x ", numAColumns); wbLog(TRACE, "The dimensions of B are ", numBRows, " x ", numBColumns); wbTime_start(GPU, "Allocating GPU memory."); //@@ Allocate GPU memory here wbCheck(cudaMalloc((void**)&deviceA, A_size)); wbCheck(cudaMalloc((void**)&deviceB, B_size)); wbCheck(cudaMalloc((void**)&deviceC, C_size)); wbTime_stop(GPU, "Allocating GPU memory."); wbTime_start(GPU, "Copying input memory to the GPU."); //@@ Copy memory to the GPU here wbCheck(cudaMemcpy(deviceA, hostA, A_size, cudaMemcpyHostToDevice)); wbCheck(cudaMemcpy(deviceB, hostB, B_size, cudaMemcpyHostToDevice)); wbCheck(cudaMemcpy(deviceC, hostC, C_size, cudaMemcpyHostToDevice)); wbTime_stop(GPU, "Copying input memory to the GPU."); //@@ Initialize the grid and block dimensions here dim3 DimGrid(ceil(numCColumns / 32.0), ceil(numCRows / 32.0), 1); dim3 DimBlock(TILE_WIDTH, TILE_WIDTH, 1); wbTime_start(Compute, "Performing CUDA computation"); //@@ Launch the GPU Kernel here matrixMultiplyShared<<< DimGrid, DimBlock >>>(deviceA, deviceB, deviceC, numARows, numAColumns, numBRows, numBColumns, numCRows, numCColumns); cudaDeviceSynchronize(); wbTime_stop(Compute, "Performing CUDA computation"); wbTime_start(Copy, "Copying output memory to the CPU"); //@@ Copy the GPU memory back to the CPU here //@@ Copy the GPU memory back to the CPU here wbCheck(cudaMemcpy(hostC, deviceC, C_size, cudaMemcpyDeviceToHost)); wbTime_stop(Copy, "Copying output memory to the CPU"); wbTime_start(GPU, "Freeing GPU Memory"); //@@ Free the GPU memory here wbCheck(cudaFree(deviceA)); wbCheck(cudaFree(deviceB)); wbCheck(cudaFree(deviceC)); wbTime_stop(GPU, "Freeing GPU Memory"); wbSolution(args, hostC, numCRows, numCColumns); free(hostA); free(hostB); free(hostC); return 0; }
