本文考慮了稀疏線性系統快速求解的一個非常重要的問題。
自2015年11月，cuSPARSE庫提供的例程基於LU分解稀疏線性系統的解，特別是
cusparse<t>csrilu02
和
cusparse<t>csrsv2_solve
此外，cuSPARSE提供
cusparse<t>csrcolor
哪個工具nts 圖着色。用於不完全LU-因式分解利用圖着色的在
Graph Coloring: More Parallelism for Incomplete-LU Factorization
和
M.Naumov, P.Castonguay, J. Cohen, 「Parallel Graph Coloring with Applications to the incomplete-LU factorization on the GPU」, NVIDIA Research Technical Report, May, 2015進行說明。
這個想法是將圖着色算法應用到與系統的係數矩陣相關聯的行依賴圖中，然後對系統方程進行相應的重新排序，這樣LU分解例程可以提取更多的並行性。
下面，請使用上述想法找到一個完全樣例：
#include <stdio.h>
#include <stdlib.h>
#include <iostream>
#include <assert.h>
#include "Utilities.cuh"
#include <cuda_runtime.h>
#include <cusparse_v2.h>
#define BLOCKSIZE 256
/**************************/
/* SETTING UP THE PROBLEM */
/**************************/
void setUpTheProblem(double **h_A_dense, double **h_x_dense, double **d_A_dense, double **d_x_dense, const int N) {
// --- Host side dense matrix
h_A_dense[0] = (double*)calloc(N * N, sizeof(*h_A_dense));
// --- Column-major ordering
h_A_dense[0][0] = 0.4612f; h_A_dense[0][4] = -0.0006f; h_A_dense[0][8] = 0.f; h_A_dense[0][12] = 0.0f;
h_A_dense[0][1] = -0.0006f; h_A_dense[0][5] = 0.f; h_A_dense[0][9] = 0.0723f; h_A_dense[0][13] = 0.04f;
h_A_dense[0][2] = 0.3566f; h_A_dense[0][6] = 0.0723f; h_A_dense[0][10] = 0.f; h_A_dense[0][14] = 0.0f;
h_A_dense[0][3] = 0.0f; h_A_dense[0][7] = 0.0f; h_A_dense[0][11] = 1.0f; h_A_dense[0][15] = 0.1f;
h_x_dense[0] = (double *)malloc(N * sizeof(double));
h_x_dense[0][0] = 100.0; h_x_dense[0][1] = 200.0; h_x_dense[0][2] = 400.0; h_x_dense[0][3] = 500.0;
// --- Create device arrays and copy host arrays to them
gpuErrchk(cudaMalloc(&d_A_dense[0], N * N * sizeof(double)));
gpuErrchk(cudaMemcpy(d_A_dense[0], h_A_dense[0], N * N * sizeof(double), cudaMemcpyHostToDevice));
gpuErrchk(cudaMalloc(&d_x_dense[0], N * sizeof(double)));
gpuErrchk(cudaMemcpy(d_x_dense[0], h_x_dense[0], N * sizeof(double), cudaMemcpyHostToDevice));
}
/************************/
/* FROM DENSE TO SPARSE */
/************************/
void fromDenseToSparse(const cusparseHandle_t handle, double *d_A_dense, double **d_A, int **d_A_RowIndices, int **d_A_ColIndices, int *nnz,
cusparseMatDescr_t *descrA, const int N) {
cusparseSafeCall(cusparseCreateMatDescr(&descrA[0]));
cusparseSafeCall(cusparseSetMatType (descrA[0], CUSPARSE_MATRIX_TYPE_GENERAL));
cusparseSafeCall(cusparseSetMatIndexBase(descrA[0], CUSPARSE_INDEX_BASE_ZERO));
nnz[0] = 0; // --- Number of nonzero elements in dense matrix
const int lda = N; // --- Leading dimension of dense matrix
// --- Device side number of nonzero elements per row
int *d_nnzPerVector; gpuErrchk(cudaMalloc(&d_nnzPerVector, N * sizeof(int)));
cusparseSafeCall(cusparseDnnz(handle, CUSPARSE_DIRECTION_ROW, N, N, descrA[0], d_A_dense, lda, d_nnzPerVector, &nnz[0]));
// --- Host side number of nonzero elements per row
int *h_nnzPerVector = (int *)malloc(N * sizeof(int));
gpuErrchk(cudaMemcpy(h_nnzPerVector, d_nnzPerVector, N * sizeof(int), cudaMemcpyDeviceToHost));
printf("Number of nonzero elements in dense matrix = %i\n\n", nnz[0]);
for (int i = 0; i < N; ++i) printf("Number of nonzero elements in row %i = %i \n", i, h_nnzPerVector[i]);
printf("\n");
// --- Device side sparse matrix
gpuErrchk(cudaMalloc(&d_A[0], nnz[0] * sizeof(double)));
gpuErrchk(cudaMalloc(&d_A_RowIndices[0], (N + 1) * sizeof(int)));
gpuErrchk(cudaMalloc(&d_A_ColIndices[0], nnz[0] * sizeof(int)));
cusparseSafeCall(cusparseDdense2csr(handle, N, N, descrA[0], d_A_dense, lda, d_nnzPerVector, d_A[0], d_A_RowIndices[0], d_A_ColIndices[0]));
// --- Host side sparse matrix
double *h_A = (double *)malloc(nnz[0] * sizeof(double));
int *h_A_RowIndices = (int *)malloc((N + 1) * sizeof(*h_A_RowIndices));
int *h_A_ColIndices = (int *)malloc(nnz[0] * sizeof(*h_A_ColIndices));
gpuErrchk(cudaMemcpy(h_A, d_A[0], nnz[0] * sizeof(double), cudaMemcpyDeviceToHost));
gpuErrchk(cudaMemcpy(h_A_RowIndices, d_A_RowIndices[0], (N + 1) * sizeof(int), cudaMemcpyDeviceToHost));
gpuErrchk(cudaMemcpy(h_A_ColIndices, d_A_ColIndices[0], nnz[0] * sizeof(int), cudaMemcpyDeviceToHost));
printf("\nOriginal matrix in CSR format\n\n");
for (int i = 0; i < nnz[0]; ++i) printf("A[%i] = %f ", i, h_A[i]); printf("\n");
printf("\n");
for (int i = 0; i < (N + 1); ++i) printf("h_A_RowIndices[%i] = %i \n", i, h_A_RowIndices[i]); printf("\n");
for (int i = 0; i < nnz[0]; ++i) printf("h_A_ColIndices[%i] = %i \n", i, h_A_ColIndices[i]);
}
/******************/
/* GRAPH COLORING */
/******************/
__global__ void setRowIndices(int *d_B_RowIndices, const int N) {
const int tid = threadIdx.x + blockDim.x * blockIdx.x;
if (tid == N) d_B_RowIndices[tid] = N;
else if (tid < N) d_B_RowIndices[tid] = tid;
}
__global__ void setB(double *d_B, const int N) {
const int tid = threadIdx.x + blockDim.x * blockIdx.x;
if (tid < N) d_B[tid] = 1.f;
}
void graphColoring(const cusparseHandle_t handle, const int nnz, const cusparseMatDescr_t descrA, const double fractionToColor, double *d_A,
const int *d_A_RowIndices, const int *d_A_ColIndices, double **d_B, int **d_B_RowIndices, int **d_B_ColIndices,
cusparseMatDescr_t *descrB, const int N) {
cusparseColorInfo_t info; cusparseSafeCall(cusparseCreateColorInfo(&info));
int ncolors;
int *d_coloring; gpuErrchk(cudaMalloc(&d_coloring, N * sizeof(double)));
gpuErrchk(cudaMalloc(&d_B_ColIndices[0], N * sizeof(double)));
cusparseSafeCall(cusparseDcsrcolor(handle, N, nnz, descrA, d_A, d_A_RowIndices, d_A_ColIndices, &fractionToColor, &ncolors, d_coloring,
d_B_ColIndices[0], info));
int *h_coloring = (int *)malloc(N * sizeof(double));
int *h_B_ColIndices = (int *)malloc(N * sizeof(double));
gpuErrchk(cudaMemcpy(h_coloring, d_coloring, N * sizeof(double), cudaMemcpyDeviceToHost));
gpuErrchk(cudaMemcpy(h_B_ColIndices, d_B_ColIndices[0], N * sizeof(double), cudaMemcpyDeviceToHost));
for (int i = 0; i < N; i++) printf("h_coloring = %i; h_B_ColIndices = %i\n", h_coloring[i], h_B_ColIndices[i]);
gpuErrchk(cudaMalloc(&d_B_RowIndices[0], (N + 1) * sizeof(int)));
int *h_B_RowIndices = (int *)malloc((N + 1) * sizeof(double));
setRowIndices<<<iDivUp(N + 1, BLOCKSIZE), BLOCKSIZE>>>(d_B_RowIndices[0], N);
gpuErrchk(cudaMemcpy(h_B_RowIndices, d_B_RowIndices[0], (N + 1) * sizeof(int), cudaMemcpyDeviceToHost));
printf("\n"); for (int i = 0; i <= N; i++) printf("h_B_RowIndices = %i\n", h_B_RowIndices[i]);
gpuErrchk(cudaMalloc(&d_B[0], N * sizeof(double)));
double *h_B = (double *)malloc(N * sizeof(double));
setB<<<iDivUp(N, BLOCKSIZE), BLOCKSIZE>>>(d_B[0], N);
gpuErrchk(cudaMemcpy(h_B, d_B[0], N * sizeof(double), cudaMemcpyDeviceToHost));
printf("\n"); for (int i = 0; i < N; i++) printf("h_B = %f\n", h_B[i]);
// --- Descriptor for sparse mutation matrix B
cusparseSafeCall(cusparseCreateMatDescr(&descrB[0]));
cusparseSafeCall(cusparseSetMatType (descrB[0], CUSPARSE_MATRIX_TYPE_GENERAL));
cusparseSafeCall(cusparseSetMatIndexBase(descrB[0], CUSPARSE_INDEX_BASE_ZERO));
}
/*************************/
/* MATRIX ROW REORDERING */
/*************************/
void matrixRowReordering(const cusparseHandle_t handle, int nnzA, int nnzB, int *nnzC, cusparseMatDescr_t descrA, cusparseMatDescr_t descrB,
cusparseMatDescr_t *descrC, double *d_A, int *d_A_RowIndices, int *d_A_ColIndices, double *d_B, int *d_B_RowIndices,
int *d_B_ColIndices, double **d_C, int **d_C_RowIndices, int **d_C_ColIndices, const int N) {
// --- Descriptor for sparse matrix C
cusparseSafeCall(cusparseCreateMatDescr(&descrC[0]));
cusparseSafeCall(cusparseSetMatType (descrC[0], CUSPARSE_MATRIX_TYPE_GENERAL));
cusparseSafeCall(cusparseSetMatIndexBase(descrC[0], CUSPARSE_INDEX_BASE_ZERO));
const int lda = N; // --- Leading dimension of dense matrix
// --- Device side sparse matrix
gpuErrchk(cudaMalloc(&d_C_RowIndices[0], (N + 1) * sizeof(int)));
// --- Host side sparse matrices
int *h_C_RowIndices = (int *)malloc((N + 1) * sizeof(int));
// --- Performing the matrix - matrix multiplication
int baseC;
int *nnzTotalDevHostPtr = &nnzC[0];
cusparseSafeCall(cusparseSetPointerMode(handle, CUSPARSE_POINTER_MODE_HOST));
cusparseSafeCall(cusparseXcsrgemmNnz(handle, CUSPARSE_OPERATION_NON_TRANSPOSE, CUSPARSE_OPERATION_NON_TRANSPOSE, N, N, N, descrB, nnzB,
d_B_RowIndices, d_B_ColIndices, descrA, nnzA, d_A_RowIndices, d_A_ColIndices, descrC[0], d_C_RowIndices[0],
nnzTotalDevHostPtr));
if (NULL != nnzTotalDevHostPtr) nnzC[0] = *nnzTotalDevHostPtr;
else {
gpuErrchk(cudaMemcpy(&nnzC[0], d_C_RowIndices + N, sizeof(int), cudaMemcpyDeviceToHost));
gpuErrchk(cudaMemcpy(&baseC, d_C_RowIndices, sizeof(int), cudaMemcpyDeviceToHost));
nnzC -= baseC;
}
gpuErrchk(cudaMalloc(&d_C_ColIndices[0], nnzC[0] * sizeof(int)));
gpuErrchk(cudaMalloc(&d_C[0], nnzC[0] * sizeof(double)));
double *h_C = (double *)malloc(nnzC[0] * sizeof(double));
int *h_C_ColIndices = (int *)malloc(nnzC[0] * sizeof(int));
cusparseSafeCall(cusparseDcsrgemm(handle, CUSPARSE_OPERATION_NON_TRANSPOSE, CUSPARSE_OPERATION_NON_TRANSPOSE, N, N, N, descrB, nnzB,
d_B, d_B_RowIndices, d_B_ColIndices, descrA, nnzA, d_A, d_A_RowIndices, d_A_ColIndices, descrC[0],
d_C[0], d_C_RowIndices[0], d_C_ColIndices[0]));
double *h_C_dense = (double*)malloc(N * N * sizeof(double));
double *d_C_dense; gpuErrchk(cudaMalloc(&d_C_dense, N * N * sizeof(double)));
cusparseSafeCall(cusparseDcsr2dense(handle, N, N, descrC[0], d_C[0], d_C_RowIndices[0], d_C_ColIndices[0], d_C_dense, N));
gpuErrchk(cudaMemcpy(h_C , d_C[0], nnzC[0] * sizeof(double), cudaMemcpyDeviceToHost));
gpuErrchk(cudaMemcpy(h_C_RowIndices, d_C_RowIndices[0], (N + 1) * sizeof(int), cudaMemcpyDeviceToHost));
gpuErrchk(cudaMemcpy(h_C_ColIndices, d_C_ColIndices[0], nnzC[0] * sizeof(int), cudaMemcpyDeviceToHost));
printf("\nResult matrix C in CSR format\n\n");
for (int i = 0; i < nnzC[0]; ++i) printf("C[%i] = %f ", i, h_C[i]); printf("\n");
printf("\n");
for (int i = 0; i < (N + 1); ++i) printf("h_C_RowIndices[%i] = %i \n", i, h_C_RowIndices[i]); printf("\n");
printf("\n");
for (int i = 0; i < nnzC[0]; ++i) printf("h_C_ColIndices[%i] = %i \n", i, h_C_ColIndices[i]);
gpuErrchk(cudaMemcpy(h_C_dense, d_C_dense, N * N * sizeof(double), cudaMemcpyDeviceToHost));
for (int j = 0; j < N; j++) {
for (int i = 0; i < N; i++)
printf("%f \t", h_C_dense[i * N + j]);
printf("\n");
}
}
/******************/
/* ROW REORDERING */
/******************/
void rowReordering(const cusparseHandle_t handle, int nnzA, cusparseMatDescr_t descrB, double *d_B, int *d_B_RowIndices, int *d_B_ColIndices,
double *d_x_dense, double **d_y_dense, const int N) {
gpuErrchk(cudaMalloc(&d_y_dense[0], N * sizeof(double)));
const double alpha = 1.;
const double beta = 0.;
cusparseSafeCall(cusparseDcsrmv(handle, CUSPARSE_OPERATION_NON_TRANSPOSE, N, N, nnzA, &alpha, descrB, d_B, d_B_RowIndices, d_B_ColIndices, d_x_dense,
&beta, d_y_dense[0]));
double *h_y_dense = (double*)malloc(N * sizeof(double));
gpuErrchk(cudaMemcpy(h_y_dense, d_y_dense[0], N * sizeof(double), cudaMemcpyDeviceToHost));
printf("\nResult vector\n\n");
for (int i = 0; i < N; ++i) printf("h_y[%i] = %f ", i, h_y_dense[i]); printf("\n");
}
/*****************************/
/* SOLVING THE LINEAR SYSTEM */
/*****************************/
void LUDecomposition(const cusparseHandle_t handle, int nnzC, cusparseMatDescr_t descrC, double *d_C, int *d_C_RowIndices, int *d_C_ColIndices,
double *d_x_dense, double **d_y_dense, const int N) {
/******************************************/
/* STEP 1: CREATE DESCRIPTORS FOR L AND U */
/******************************************/
cusparseMatDescr_t descr_L = 0;
cusparseSafeCall(cusparseCreateMatDescr (&descr_L));
cusparseSafeCall(cusparseSetMatIndexBase(descr_L, CUSPARSE_INDEX_BASE_ZERO));
cusparseSafeCall(cusparseSetMatType (descr_L, CUSPARSE_MATRIX_TYPE_GENERAL));
cusparseSafeCall(cusparseSetMatFillMode (descr_L, CUSPARSE_FILL_MODE_LOWER));
cusparseSafeCall(cusparseSetMatDiagType (descr_L, CUSPARSE_DIAG_TYPE_UNIT));
cusparseMatDescr_t descr_U = 0;
cusparseSafeCall(cusparseCreateMatDescr (&descr_U));
cusparseSafeCall(cusparseSetMatIndexBase(descr_U, CUSPARSE_INDEX_BASE_ZERO));
cusparseSafeCall(cusparseSetMatType (descr_U, CUSPARSE_MATRIX_TYPE_GENERAL));
cusparseSafeCall(cusparseSetMatFillMode (descr_U, CUSPARSE_FILL_MODE_UPPER));
cusparseSafeCall(cusparseSetMatDiagType (descr_U, CUSPARSE_DIAG_TYPE_NON_UNIT));
/**************************************************************************************************/
/* STEP 2: QUERY HOW MUCH MEMORY USED IN LU FACTORIZATION AND THE TWO FOLLOWING SYSTEM INVERSIONS */
/**************************************************************************************************/
csrilu02Info_t info_C = 0; cusparseSafeCall(cusparseCreateCsrilu02Info (&info_C));
csrsv2Info_t info_L = 0; cusparseSafeCall(cusparseCreateCsrsv2Info (&info_L));
csrsv2Info_t info_U = 0; cusparseSafeCall(cusparseCreateCsrsv2Info (&info_U));
int pBufferSize_M, pBufferSize_L, pBufferSize_U;
cusparseSafeCall(cusparseDcsrilu02_bufferSize(handle, N, nnzC, descrC, d_C, d_C_RowIndices, d_C_ColIndices, info_C, &pBufferSize_M));
cusparseSafeCall(cusparseDcsrsv2_bufferSize (handle, CUSPARSE_OPERATION_NON_TRANSPOSE, N, nnzC, descr_L, d_C, d_C_RowIndices, d_C_ColIndices, info_L, &pBufferSize_L));
cusparseSafeCall(cusparseDcsrsv2_bufferSize (handle, CUSPARSE_OPERATION_NON_TRANSPOSE, N, nnzC, descr_U, d_C, d_C_RowIndices, d_C_ColIndices, info_U, &pBufferSize_U));
int pBufferSize = max(pBufferSize_M, max(pBufferSize_L, pBufferSize_U));
void *pBuffer = 0; gpuErrchk(cudaMalloc((void**)&pBuffer, pBufferSize));
/************************************************************************************************/
/* STEP 3: ANALYZE THE THREE PROBLEMS: LU FACTORIZATION AND THE TWO FOLLOWING SYSTEM INVERSIONS */
/************************************************************************************************/
int structural_zero;
cusparseSafeCall(cusparseDcsrilu02_analysis(handle, N, nnzC, descrC, d_C, d_C_RowIndices, d_C_ColIndices, info_C, CUSPARSE_SOLVE_POLICY_NO_LEVEL, pBuffer));
cusparseStatus_t status = cusparseXcsrilu02_zeroPivot(handle, info_C, &structural_zero);
if (CUSPARSE_STATUS_ZERO_PIVOT == status){ printf("A(%d,%d) is missing\n", structural_zero, structural_zero); }
cusparseSafeCall(cusparseDcsrsv2_analysis(handle, CUSPARSE_OPERATION_NON_TRANSPOSE, N, nnzC, descr_L, d_C, d_C_RowIndices, d_C_ColIndices, info_L, CUSPARSE_SOLVE_POLICY_NO_LEVEL, pBuffer));
cusparseSafeCall(cusparseDcsrsv2_analysis(handle, CUSPARSE_OPERATION_NON_TRANSPOSE, N, nnzC, descr_U, d_C, d_C_RowIndices, d_C_ColIndices, info_U, CUSPARSE_SOLVE_POLICY_USE_LEVEL, pBuffer));
/************************************/
/* STEP 4: FACTORIZATION: A = L * U */
/************************************/
int numerical_zero;
cusparseSafeCall(cusparseDcsrilu02(handle, N, nnzC, descrC, d_C, d_C_RowIndices, d_C_ColIndices, info_C, CUSPARSE_SOLVE_POLICY_NO_LEVEL, pBuffer));
status = cusparseXcsrilu02_zeroPivot(handle, info_C, &numerical_zero);
if (CUSPARSE_STATUS_ZERO_PIVOT == status){ printf("U(%d,%d) is zero\n", numerical_zero, numerical_zero); }
/*********************/
/* STEP 5: L * z = x */
/*********************/
// --- Allocating the intermediate result vector
double *d_z_dense; gpuErrchk(cudaMalloc(&d_z_dense, N * sizeof(double)));
const double alpha = 1.;
cusparseSafeCall(cusparseDcsrsv2_solve(handle, CUSPARSE_OPERATION_NON_TRANSPOSE, N, nnzC, &alpha, descr_L, d_C, d_C_RowIndices, d_C_ColIndices, info_L, d_x_dense, d_z_dense, CUSPARSE_SOLVE_POLICY_NO_LEVEL, pBuffer));
/*********************/
/* STEP 5: U * y = z */
/*********************/
gpuErrchk(cudaMalloc(&d_y_dense[0], N * sizeof(double)));
cusparseSafeCall(cusparseDcsrsv2_solve(handle, CUSPARSE_OPERATION_NON_TRANSPOSE, N, nnzC, &alpha, descr_U, d_C, d_C_RowIndices, d_C_ColIndices, info_U, d_z_dense, d_y_dense[0], CUSPARSE_SOLVE_POLICY_USE_LEVEL, pBuffer));
double *h_y_dense = (double *)malloc(N * sizeof(double));
gpuErrchk(cudaMemcpy(h_y_dense, d_y_dense[0], N * sizeof(double), cudaMemcpyDeviceToHost));
printf("\n\nFinal result\n");
for (int k=0; k<N; k++) printf("x[%i] = %f\n", k, h_y_dense[k]);
}
/********/
/* MAIN */
/********/
int main()
{
// --- Initialize cuSPARSE
cusparseHandle_t handle; cusparseSafeCall(cusparseCreate(&handle));
/*************************************************/
/* SETTING UP THE ORIGINAL LINEAR SYSTEM PROBLEM */
/*************************************************/
const int N = 4; // --- Number of rows and columns
double *h_A_dense; double *h_x_dense;
double *d_A_dense; double *d_x_dense;
setUpTheProblem(&h_A_dense, &h_x_dense, &d_A_dense, &d_x_dense, N);
/************************/
/* FROM DENSE TO SPARSE */
/************************/
//--- Descriptor for sparse matrix A
cusparseMatDescr_t descrA;
int *d_A_RowIndices, *d_A_ColIndices;
double *d_A;
int nnzA;
fromDenseToSparse(handle, d_A_dense, &d_A, &d_A_RowIndices, &d_A_ColIndices, &nnzA, &descrA, N);
/******************/
/* GRAPH COLORING */
/******************/
const double fractionToColor = 0.95;
int *d_B_RowIndices, *d_B_ColIndices;
double *d_B;
int nnzB;
cusparseMatDescr_t descrB;
graphColoring(handle, nnzB, descrA, fractionToColor, d_A, d_A_RowIndices, d_A_ColIndices, &d_B, &d_B_RowIndices, &d_B_ColIndices, &descrB, N);
/*************************/
/* MATRIX ROW REORDERING */
/*************************/
int nnzC;
int *d_C_RowIndices, *d_C_ColIndices;
double *d_C;
cusparseMatDescr_t descrC;
matrixRowReordering(handle, nnzA, nnzB, &nnzC, descrA, descrB, &descrC, d_A, d_A_RowIndices, d_A_ColIndices, d_B, d_B_RowIndices, d_B_ColIndices,
&d_C, &d_C_RowIndices, &d_C_ColIndices, N);
/******************/
/* ROW REORDERING */
/******************/
double *d_y_dense;
rowReordering(handle, nnzA, descrB, d_B, d_B_RowIndices, d_B_ColIndices, d_x_dense, &d_y_dense, N);
/*****************************/
/* SOLVING THE LINEAR SYSTEM */
/*****************************/
double *d_xsol_dense;
LUDecomposition(handle, nnzC, descrC, d_C, d_C_RowIndices, d_C_ColIndices, d_y_dense, &d_xsol_dense, N);
}
您的系統是否有哪些CPU？你提到UMFPACK 5.6.2（似乎可用），但你的tar文件似乎有UMFPACK 5.2.0在裏面？這很重要嗎？另外，在你的'Makefile'中，你指定了'-G'開關並且不指定任何'-arch'開關。我會刪除'-G'開關併爲GTX Titan指定'-arch = sm_35'開關。 –
使用的CPU是：Intel酷睿i7-4770四核心處理器。 UMFPACK版本並不重要。 – yassine93