我编写了一个使用 GPU 进行正向求解的代码（从 A = LU 获取 U 矩阵，其中 U 的对角线条目未设置为单位）。我的代码适用于尺寸小于 128x128 的矩阵，这是块大小的一个因素，即 16。我很困惑为什么更大的矩阵会给我错误的结果。最后一行块出现问题。
------------ gaussian.h --------
// Thread block size
#define BLOCK_SIZE 16 //for forward solve A = LU
#define BLOCK_SIZE2 32 //for finding the pivot factor
// A[i,j] = A[i,j] - A[k,j] / A[k,k] * A[i,k]
// pivot factor = A[k,j] / A[k,k]
///////////////////////////////////////// /////////////////////////////////
------------ gaussian.cu --------
// includes, system
#include <stdlib.h>
#include <stdio.h>
#include <string.h>
#include <math.h>
#include <sys/time.h>
#include <cuda.h>
#include "gaussian_kernel.cu"
#define OUTPUT
void runTest(int argc, char** argv);
double gettime() {
struct timeval t;
gettimeofday(&t,NULL);
return t.tv_sec+t.tv_usec*1e-6;
}
int main(int argc, char** argv)
{
runTest(argc, argv);
}
void runTest(int argc, char** argv)
{
int dim;
if (argc == 2)
{
dim = atoi(argv[1]);
}
else{
printf("Wrong Usage\n");
exit(1);
}
// Note: A is a square matrix of size NxN where N % BLOCK_SIZE = 0
// Every row of A has a pivot column
// It is known that all the arithmetic operations involved for
// Gaussian Elimination are of type int (not float)
// allocate host memory for matrix A augmented with vector b (Ax=b)
unsigned int size_A = dim * (dim + BLOCK_SIZE);
unsigned int mem_size_A = sizeof(int) * size_A;
int* h_A = (int*) malloc(mem_size_A);
// initialize host memory, generate a test case such as below
// augmented matrix A with padding to make the size evenly divisible by BLOCK_SIZE
// ----A----- | b --padding--
// 1 1 1 1 .. | 0 * * * ..
// 1 2 2 2 .. | 1 * * * ..
// 1 2 3 3 .. | 2 * * * ..
// 1 2 3 4 .. | 3 * * * ..
// .......... | ...
// * means uninitialized entry
// A is size NxN
// Augmented matrix with padding is size Nx(N+BLOCK_SIZE)
int dimRow = dim;
int dimCol = dim + BLOCK_SIZE;
for(int i = 0; i < dim; i++)
{
h_A[(i + 1) * dimCol - BLOCK_SIZE] = i; // b vector stored in (N+1)th column
// of augmented matrix
for (int j = 0; j < dimRow - i; j++)
{
h_A[j + i + i * dimCol] = i + 1; // A[i][j] entry
h_A[j * dimCol + i + i * dimCol] = i + 1; // A[j][i] entry
}
}
//display the test case [ A | b ]
for ( int m = 0 ; m < dimRow; m++)
{
for ( int n = 0 ; n <= dimRow; n++)
//for ( int n = 0 ; n < dimCol; n++)
{
printf("%d ", h_A[m * dimCol + n]);
}
printf("\n");
}
// allocate device memory for the augmented matrix A
int* d_A;
cudaMalloc(&d_A, mem_size_A);
// start timer
double timer1 = gettime();
// copy host memory to device
cudaMemcpy(d_A, h_A, mem_size_A, cudaMemcpyHostToDevice);
// setup execution parameters for obtaining the pivot factor
int gridP = (dimRow / BLOCK_SIZE2) + ( (dimRow % BLOCK_SIZE2) == 0 ? 0 : 1 );
// add 1 if dimRow/BLOCK_SIZE2 is not evenly divisible
// setup execution parameters for the forward solve (Gaussian Elimination)
dim3 threads(BLOCK_SIZE, BLOCK_SIZE);
dim3 grid(dimCol / threads.x, dimRow / threads.y);
// execute the kernel
for ( int i = 0 ; i < dim; i++)
{
Gaussian_Pivot_CUDA<<< gridP, BLOCK_SIZE2 >>>(d_A, dimCol, i);
Gaussian_Elim_CUDA<<< grid, threads >>>(d_A, dimCol, i);
}
// copy result from device to host
cudaMemcpy(h_A, d_A, mem_size_A, cudaMemcpyDeviceToHost);
// stop timer
double timer2 = gettime();
printf("GPU time = %lf\n",(timer2-timer1)*1000);
// Back substitution
int X[dimRow];
for (int row = dimRow - 1; row >= 0; row--)
{
X[row] = h_A[(row + 1) * dimCol - BLOCK_SIZE]; // b[row] entry
for (int col = dimRow - 1; col > row; col--)
{
X[row] -= h_A[row * dimCol + col] * X[col];
}
X[row] /= h_A[row * dimCol + row];
printf("X[%d] = %d\t",row,X[row]);
}
printf("\n");
#ifdef OUTPUT
// result of Gaussian Elimination
// ----A----- | b --padding--
// 1 1 1 1 .. | 0 * * * ..
// 0 1 1 1 .. | 1 * * * ..
// 0 0 1 1 .. | 1 * * * ..
// 0 0 0 1 .. | 1 * * * ..
// .......... | ...
// * means garbage entry
for ( int m = 0 ; m < dimRow; m++)
{
for ( int n = 0 ; n <= dimRow; n++)
{
printf("%d ", h_A[m * dimCol + n]);
}
printf("\n");
}
#endif
free(h_A);
cudaFree(d_A);
}
///////////////////////////////////////// /////////////////////////
------------ gaussian_kernel.cu --------
#include "gaussian.h"
__global__
void Gaussian_Pivot_CUDA(int* A, int widthA, int Pcol)
{
// find the pivot factor
// A[i,j] = A[i,j] - A[k,j] / A[k,k] * A[i,k]
// pivot factor = A[k,j] / A[k,k]
int bx = blockIdx.x;
int tx = threadIdx.x;
int index_row = BLOCK_SIZE2 * bx + tx;
int index_rowA = index_row * widthA;
__shared__ int A_RowRow;
if (tx == 0)
{
A_RowRow = A[Pcol * widthA + Pcol]; // get A[k,k] where k is the pivot column
// for this problem pivot column = pivot row
}
__syncthreads();
A[index_rowA - 1] = A[index_rowA + Pcol] / A_RowRow;
// INVALID WRITE HERE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
// store pivot factor at A[dimCol - 1][row]
}
__global__
void Gaussian_Elim_CUDA(int* A, int widthA, int Pcol)
{
// Block index
int bx = blockIdx.x;
int by = blockIdx.y;
// Thread index
int tx = threadIdx.x;
int ty = threadIdx.y;
int index_col = BLOCK_SIZE * bx + tx;
int index_row = BLOCK_SIZE * by + ty;
int index = widthA * index_row + index_col;
// d_A[index] "=" d_A[index_row][index_col]
// store pivot factor for rows that we are working on in this block
__shared__ int pivotFactor[BLOCK_SIZE];
if ( ty == 0 ) // use ty instead of tx for coalesced accesses
{
pivotFactor[tx] = A[(index_row + tx) * widthA - 1];
// INVALID WRITE HERE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
}
__syncthreads();
// implement the gaussian elimination for the current row
if ( index_row > Pcol )
{
A[index] -= A[Pcol * widthA + index_col] * pivotFactor[ty];
// A[i,j] = A[i,j] - A[k,j] / A[k,k] * A[i,k]
// pivot factor = A[k,j] / A[k,k]
}
}