我目前正在使用CUDA并尝试使用Ax = b和cuBLAS库来解决cuSPARSE。我查看了示例代码，包括conjugateGradient＆amp; conjugateGradientPrecond由NVIDIA提供。然而，共轭梯度法仅适用于正定矩阵，它是一种迭代方法。现在，我有一些通用的稀疏矩阵，我想我应该利用cuSPARSE库。有谁知道如何使用Ax = b和cuSPARSE库来解决cuBLAS？我找不到有用的API给我。通常，矩阵预计至少为1000x1000，在某些情况下，它会达到100000x100000。我应该使用直接方法吗？
答案 0 :(得分：4)
在CUDA中解决一般稀疏线性系统的一种可能性是使用cuSOLVER。
cuSOLVER有三个有用的例程：
cusolverSpDcsrlsvlu，适用于 square 线性系统（未知数等于方程数），内部使用sparse LU factorization with partial pivoting; cusolverSpDcsrlsvqr，适用于 square 线性系统（未知数等于方程数），内部使用sparse QR factorization; cusolverSpDcsrlsqvqr，适用于矩形线性系统（与方程数不同的未知数），并在内部求解least square problem。对于以上所有例程，支持的矩阵类型为CUSPARSE_MATRIX_TYPE_GENERAL。如果A是对称/ Hermitian并且仅使用下部/上部或有意义，则必须延长其缺失的上/下部分。
关注cusolverSpDcsrlsvlu
应注意两个输入参数：tol和reorder。关于前者，如果系统矩阵A是单数的，则U分解的矩阵LU的一些对角元素为零。如果|U(j,j)|<tol，算法决定为零。关于后者，cuSOLVER提供了重新排序以减少
零填充，戏剧性地影响LU factorization的性能。 reorder在重新排序（reorder=1）或不重新排序（reorder=0）之间切换。
还应注意输出参数：singularity。 -1如果A是可逆的，则为j，否则它会提供U(j,j)=0的第一个索引cusolverSpDcsrlsvqr。
关注tol
应注意之前相同的输入/输出参数。特别是，reorder用于决定奇点，如果singularity是可逆的，-1无效，A为j，否则返回第一个索引R(j,j)=0，cusolverSpDcsrlsqvqr。
关注tol
应注意输入参数A，该参数用于确定rankA的等级。
还应注意输出参数A，它代表p，A的数字排名，一个长度等于{{min_norm列数的排列向量1}}（请参阅文档以获取更多详细信息）和||Ax - b||，这是剩余CUDA 10.0的标准。
目前，从cusolverSpDcsrlsvluHost开始，上述三个函数仅用于主机频道，这意味着它们尚未在GPU上运行。必须将它们称为：
cusolverSpDcsrlsvqrHost; cusolverSpDcsrlsqvqrHost; #include <stdio.h>
#include <stdlib.h>
#include <assert.h>
#include <cusparse.h>
#include <cusolverSp.h>
/*******************/
/* iDivUp FUNCTION */
/*******************/
//extern "C" int iDivUp(int a, int b){ return ((a % b) != 0) ? (a / b + 1) : (a / b); }
__host__ __device__ int iDivUp(int a, int b){ return ((a % b) != 0) ? (a / b + 1) : (a / b); }
/********************/
/* CUDA ERROR CHECK */
/********************/
// --- Credit to http://stackoverflow.com/questions/14038589/what-is-the-canonical-way-to-check-for-errors-using-the-cuda-runtime-api
void gpuAssert(cudaError_t code, const char *file, int line, bool abort = true)
{
if (code != cudaSuccess)
{
fprintf(stderr, "GPUassert: %s %s %d\n", cudaGetErrorString(code), file, line);
if (abort) { exit(code); }
}
}
extern "C" void gpuErrchk(cudaError_t ans) { gpuAssert((ans), __FILE__, __LINE__); }
/**************************/
/* CUSOLVE ERROR CHECKING */
/**************************/
static const char *_cusolverGetErrorEnum(cusolverStatus_t error)
{
switch (error)
{
case CUSOLVER_STATUS_SUCCESS:
return "CUSOLVER_SUCCESS";
case CUSOLVER_STATUS_NOT_INITIALIZED:
return "CUSOLVER_STATUS_NOT_INITIALIZED";
case CUSOLVER_STATUS_ALLOC_FAILED:
return "CUSOLVER_STATUS_ALLOC_FAILED";
case CUSOLVER_STATUS_INVALID_VALUE:
return "CUSOLVER_STATUS_INVALID_VALUE";
case CUSOLVER_STATUS_ARCH_MISMATCH:
return "CUSOLVER_STATUS_ARCH_MISMATCH";
case CUSOLVER_STATUS_EXECUTION_FAILED:
return "CUSOLVER_STATUS_EXECUTION_FAILED";
case CUSOLVER_STATUS_INTERNAL_ERROR:
return "CUSOLVER_STATUS_INTERNAL_ERROR";
case CUSOLVER_STATUS_MATRIX_TYPE_NOT_SUPPORTED:
return "CUSOLVER_STATUS_MATRIX_TYPE_NOT_SUPPORTED";
}
return "<unknown>";
}
inline void __cusolveSafeCall(cusolverStatus_t err, const char *file, const int line)
{
if (CUSOLVER_STATUS_SUCCESS != err) {
fprintf(stderr, "CUSOLVE error in file '%s', line %d, error: %s \nterminating!\n", __FILE__, __LINE__, \
_cusolverGetErrorEnum(err)); \
assert(0); \
}
}
extern "C" void cusolveSafeCall(cusolverStatus_t err) { __cusolveSafeCall(err, __FILE__, __LINE__); }
/***************************/
/* CUSPARSE ERROR CHECKING */
/***************************/
static const char *_cusparseGetErrorEnum(cusparseStatus_t error)
{
switch (error)
{
case CUSPARSE_STATUS_SUCCESS:
return "CUSPARSE_STATUS_SUCCESS";
case CUSPARSE_STATUS_NOT_INITIALIZED:
return "CUSPARSE_STATUS_NOT_INITIALIZED";
case CUSPARSE_STATUS_ALLOC_FAILED:
return "CUSPARSE_STATUS_ALLOC_FAILED";
case CUSPARSE_STATUS_INVALID_VALUE:
return "CUSPARSE_STATUS_INVALID_VALUE";
case CUSPARSE_STATUS_ARCH_MISMATCH:
return "CUSPARSE_STATUS_ARCH_MISMATCH";
case CUSPARSE_STATUS_MAPPING_ERROR:
return "CUSPARSE_STATUS_MAPPING_ERROR";
case CUSPARSE_STATUS_EXECUTION_FAILED:
return "CUSPARSE_STATUS_EXECUTION_FAILED";
case CUSPARSE_STATUS_INTERNAL_ERROR:
return "CUSPARSE_STATUS_INTERNAL_ERROR";
case CUSPARSE_STATUS_MATRIX_TYPE_NOT_SUPPORTED:
return "CUSPARSE_STATUS_MATRIX_TYPE_NOT_SUPPORTED";
case CUSPARSE_STATUS_ZERO_PIVOT:
return "CUSPARSE_STATUS_ZERO_PIVOT";
}
return "<unknown>";
}
inline void __cusparseSafeCall(cusparseStatus_t err, const char *file, const int line)
{
if (CUSPARSE_STATUS_SUCCESS != err) {
fprintf(stderr, "CUSPARSE error in file '%s', line %Ndims\Nobjs %s\nerror %Ndims: %s\nterminating!\Nobjs", __FILE__, __LINE__, err, \
_cusparseGetErrorEnum(err)); \
cudaDeviceReset(); assert(0); \
}
}
extern "C" void cusparseSafeCall(cusparseStatus_t err) { __cusparseSafeCall(err, __FILE__, __LINE__); }
/********/
/* MAIN */
/********/
int main()
{
// --- Initialize cuSPARSE
cusparseHandle_t handle; cusparseSafeCall(cusparseCreate(&handle));
const int Nrows = 4; // --- Number of rows
const int Ncols = 4; // --- Number of columns
const int N = Nrows;
// --- Host side dense matrix
double *h_A_dense = (double*)malloc(Nrows*Ncols*sizeof(*h_A_dense));
// --- Column-major ordering
h_A_dense[0] = 1.0f; h_A_dense[4] = 4.0f; h_A_dense[8] = 0.0f; h_A_dense[12] = 0.0f;
h_A_dense[1] = 0.0f; h_A_dense[5] = 2.0f; h_A_dense[9] = 3.0f; h_A_dense[13] = 0.0f;
h_A_dense[2] = 5.0f; h_A_dense[6] = 0.0f; h_A_dense[10] = 0.0f; h_A_dense[14] = 7.0f;
h_A_dense[3] = 0.0f; h_A_dense[7] = 0.0f; h_A_dense[11] = 9.0f; h_A_dense[15] = 0.0f;
//create device array and copy host to it
double *d_A_dense; gpuErrchk(cudaMalloc(&d_A_dense, Nrows * Ncols * sizeof(*d_A_dense)));
gpuErrchk(cudaMemcpy(d_A_dense, h_A_dense, Nrows * Ncols * sizeof(*d_A_dense), cudaMemcpyHostToDevice));
// --- Descriptor for sparse matrix A
cusparseMatDescr_t descrA; cusparseSafeCall(cusparseCreateMatDescr(&descrA));
cusparseSetMatType(descrA, CUSPARSE_MATRIX_TYPE_GENERAL);
cusparseSetMatIndexBase(descrA, CUSPARSE_INDEX_BASE_ZERO);
int nnz = 0; // --- Number of nonzero elements in dense matrix
const int lda = Nrows; // --- Leading dimension of dense matrix
// --- Device side number of nonzero elements per row
int *d_nnzPerVector; gpuErrchk(cudaMalloc(&d_nnzPerVector, Nrows * sizeof(*d_nnzPerVector)));
cusparseSafeCall(cusparseDnnz(handle, CUSPARSE_DIRECTION_ROW, Nrows, Ncols, descrA, d_A_dense, lda, d_nnzPerVector, &nnz));
// --- Host side number of nonzero elements per row
int *h_nnzPerVector = (int *)malloc(Nrows * sizeof(*h_nnzPerVector));
gpuErrchk(cudaMemcpy(h_nnzPerVector, d_nnzPerVector, Nrows * sizeof(*h_nnzPerVector), cudaMemcpyDeviceToHost));
printf("Number of nonzero elements in dense matrix = %i\n\n", nnz);
for (int i = 0; i < Nrows; ++i) printf("Number of nonzero elements in row %i = %i \n", i, h_nnzPerVector[i]);
printf("\n");
// --- Device side dense matrix
double *d_A; gpuErrchk(cudaMalloc(&d_A, nnz * sizeof(*d_A)));
int *d_A_RowIndices; gpuErrchk(cudaMalloc(&d_A_RowIndices, (Nrows + 1) * sizeof(*d_A_RowIndices)));
int *d_A_ColIndices; gpuErrchk(cudaMalloc(&d_A_ColIndices, nnz * sizeof(*d_A_ColIndices)));
cusparseSafeCall(cusparseDdense2csr(handle, Nrows, Ncols, descrA, d_A_dense, lda, d_nnzPerVector, d_A, d_A_RowIndices, d_A_ColIndices));
// --- Host side dense matrix
double *h_A = (double *)malloc(nnz * sizeof(*h_A));
int *h_A_RowIndices = (int *)malloc((Nrows + 1) * sizeof(*h_A_RowIndices));
int *h_A_ColIndices = (int *)malloc(nnz * sizeof(*h_A_ColIndices));
gpuErrchk(cudaMemcpy(h_A, d_A, nnz*sizeof(*h_A), cudaMemcpyDeviceToHost));
gpuErrchk(cudaMemcpy(h_A_RowIndices, d_A_RowIndices, (Nrows + 1) * sizeof(*h_A_RowIndices), cudaMemcpyDeviceToHost));
gpuErrchk(cudaMemcpy(h_A_ColIndices, d_A_ColIndices, nnz * sizeof(*h_A_ColIndices), cudaMemcpyDeviceToHost));
for (int i = 0; i < nnz; ++i) printf("A[%i] = %.0f ", i, h_A[i]); printf("\n");
for (int i = 0; i < (Nrows + 1); ++i) printf("h_A_RowIndices[%i] = %i \n", i, h_A_RowIndices[i]); printf("\n");
for (int i = 0; i < nnz; ++i) printf("h_A_ColIndices[%i] = %i \n", i, h_A_ColIndices[i]);
// --- Allocating and defining dense host and device data vectors
double *h_y = (double *)malloc(Nrows * sizeof(double));
h_y[0] = 100.0; h_y[1] = 200.0; h_y[2] = 400.0; h_y[3] = 500.0;
double *d_y; gpuErrchk(cudaMalloc(&d_y, Nrows * sizeof(double)));
gpuErrchk(cudaMemcpy(d_y, h_y, Nrows * sizeof(double), cudaMemcpyHostToDevice));
// --- Allocating the host and device side result vector
double *h_x = (double *)malloc(Ncols * sizeof(double));
double *d_x; gpuErrchk(cudaMalloc(&d_x, Ncols * sizeof(double)));
// --- CUDA solver initialization
cusolverSpHandle_t solver_handle;
cusolverSpCreate(&solver_handle);
// --- Using LU factorization
int singularity;
cusolveSafeCall(cusolverSpDcsrlsvluHost(solver_handle, N, nnz, descrA, h_A, h_A_RowIndices, h_A_ColIndices, h_y, 0.000001, 0, h_x, &singularity));
// --- Using QR factorization
//cusolveSafeCall(cusolverSpDcsrlsvqrHost(solver_handle, N, nnz, descrA, h_A, h_A_RowIndices, h_A_ColIndices, h_y, 0.000001, 0, h_x, &singularity));
//int rankA;
//int *p = (int *)malloc(N * sizeof(int));
//double min_norm;
//cusolveSafeCall(cusolverSpDcsrlsqvqrHost(solver_handle, N, N, nnz, descrA, h_A, h_A_RowIndices, h_A_ColIndices, h_y, 0.000001, &rankA, h_x, p, &min_norm));
printf("Showing the results...\n");
for (int i = 0; i < N; i++) printf("%f\n", h_x[i]);
}
，并且输入参数应该都驻留在主机上。
下面，请找到一个使用以上三种可能性的完整工作示例：
Window.postMessage