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。

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
",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;
}