この投稿では、スパース線形システムの高速解法という非常に重要な問題について考察します。
2015 年 11 月の時点で、cuSPARSEライブラリは LU 分解に基づく疎な線形システムを解くためのルーチンを提供しています。
cusparse<t>csrilu02
と
cusparse<t>csrsv2_solve
さらにcuSPARSE、
cusparse<t>csrcolor
グラフの色付けを実装します。不完全な LU 因数分解に対するグラフの色付けの使用については、
グラフの色分け: 不完全 LU 因数分解の並列性を高める
と
M.Naumov、P.Castonguay、J. Cohen、「GPU での不完全 LU 因数分解への適用を伴う並列グラフの色付け」、NVIDIA Research Technical Report、2015 年 5 月。
アイデアは、システムの係数行列に関連付けられた行依存グラフにグラフの色付けアルゴリズムを適用し、それに応じてシステム方程式を並べ替えて、LU 分解ルーチンがより多くの並列処理を抽出できるようにすることです。
以下に、上記のアイデアを使用して完全に機能する例を示します。
#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);
}