如何编写可以与本征竞争的矩阵矩阵积
How to write a matrix matrix product that can compete with Eigen?
下面是比较 Eigen 和 For Loop 执行矩阵-矩阵乘积所花费时间的C++实现。For 循环已经过优化，可最大程度地减少缓存未命中。for 循环最初比特征循环快，但最终变慢（对于 500 x 500 矩阵，最多为 2 倍）。我还应该怎么做才能与本征竞争？阻塞是本征性能更好的原因吗？如果是这样，我应该如何向 for 循环添加阻塞？
#include<iostream>
#include<Eigen/Dense>
#include<ctime>
int main(int argc, char* argv[]) {
srand(time(NULL));
// Input the size of the matrix from the user
int N = atoi(argv[1]);
int M = N*N;
// The matrices stored as row-wise vectors
double a[M];
double b[M];
double c[M];
// Initializing Eigen Matrices
Eigen::MatrixXd a_E = Eigen::MatrixXd::Random(N,N);
Eigen::MatrixXd b_E = Eigen::MatrixXd::Random(N,N);
Eigen::MatrixXd c_E(N,N);
double CPS = CLOCKS_PER_SEC;
clock_t start, end;
// Matrix vector product by Eigen
start = clock();
c_E = a_E*b_E;
end = clock();
std::cout << "nTime taken by Eigen is: " << (end-start)/CPS << "n";
// Initializing For loop vectors
int count = 0;
for (int j=0; j<N; ++j) {
for (int k=0; k<N; ++k) {
a[count] = a_E(j,k);
b[count] = b_E(j,k);
++count;
}
}
// Matrix vector product by For loop
start = clock();
count = 0;
int count1, count2;
for (int j=0; j<N; ++j) {
count1 = j*N;
for (int k=0; k<N; ++k) {
c[count] = a[count1]*b[k];
++count;
}
}
for (int j=0; j<N; ++j) {
count2 = N;
for (int l=1; l<N; ++l) {
count = j*N;
count1 = count+l;
for (int k=0; k<N; ++k) {
c[count]+=a[count1]*b[count2];
++count;
++count2;
}
}
}
end = clock();
std::cout << "nTime taken by for-loop is: " << (end-start)/CPS << "n";
}
没有必要神秘地解释如何实现矩阵-矩阵乘积的高性能实现。 事实上，我们需要更多的人了解它，以面对高性能计算的未来挑战。 为了进入这个主题，阅读 BLIS：快速实例化 BLAS 功能的框架是一个很好的起点。
因此，为了揭开神秘面纱并回答这个问题（如何编写可以与 Eigen 竞争的矩阵矩阵产品），我将 ggael 发布的代码扩展到总共 400 行。 我刚刚在AVX机器（英特尔（R）酷睿（TM）i5-3470 CPU @ 3.20GHz）上对其进行了测试。 这里有一些结果：
g++-5.3 -O3 -DNDEBUG -std=c++11 -mavx -m64 -I ../eigen.3.2.8/ gemm.cc -lrt
lehn@heim:~/work/test_eigen$ ./a.out 500
Time taken by Eigen is: 0.0190425
Time taken by for-loop is: 0.0121688
lehn@heim:~/work/test_eigen$ ./a.out 1000
Time taken by Eigen is: 0.147991
Time taken by for-loop is: 0.0959097
lehn@heim:~/work/test_eigen$ ./a.out 1500
Time taken by Eigen is: 0.492858
Time taken by for-loop is: 0.322442
lehn@heim:~/work/test_eigen$ ./a.out 5000
Time taken by Eigen is: 18.3666
Time taken by for-loop is: 12.1023
如果您有 FMA，则可以使用
g++-5.3 -O3 -DNDEBUG -std=c++11 -mfma -m64 -I ../eigen.3.2.8/ -DHAVE_FMA gemm.cc -lrt
如果您还想使用 openMP 进行多线程处理，也可以使用 -fopenmp
这里基于BLIS论文思想的完整代码。 它是自包含的，除了它需要完整的特征源文件，如 ggael 已经指出的那样：
#include<iostream>
#include<Eigen/Dense>
#include<bench/BenchTimer.h>
#if defined(_OPENMP)
#include <omp.h>
#endif
//-- malloc with alignment --------------------------------------------------------
void *
malloc_(std::size_t alignment, std::size_t size)
{
alignment = std::max(alignment, alignof(void *));
size += alignment;
void *ptr = std::malloc(size);
void *ptr2 = (void *)(((uintptr_t)ptr + alignment) & ~(alignment-1));
void **vp = (void**) ptr2 - 1;
*vp = ptr;
return ptr2;
}
void
free_(void *ptr)
{
std::free(*((void**)ptr-1));
}
//-- Config --------------------------------------------------------------------
// SIMD-Register width in bits
// SSE: 128
// AVX/FMA: 256
// AVX-512: 512
#ifndef SIMD_REGISTER_WIDTH
#define SIMD_REGISTER_WIDTH 256
#endif
#ifdef HAVE_FMA
# ifndef BS_D_MR
# define BS_D_MR 4
# endif
# ifndef BS_D_NR
# define BS_D_NR 12
# endif
# ifndef BS_D_MC
# define BS_D_MC 256
# endif
# ifndef BS_D_KC
# define BS_D_KC 512
# endif
# ifndef BS_D_NC
# define BS_D_NC 4092
# endif
#endif
#ifndef BS_D_MR
#define BS_D_MR 4
#endif
#ifndef BS_D_NR
#define BS_D_NR 8
#endif
#ifndef BS_D_MC
#define BS_D_MC 256
#endif
#ifndef BS_D_KC
#define BS_D_KC 256
#endif
#ifndef BS_D_NC
#define BS_D_NC 4096
#endif
template <typename T>
struct BlockSize
{
static constexpr int MC = 64;
static constexpr int KC = 64;
static constexpr int NC = 256;
static constexpr int MR = 8;
static constexpr int NR = 8;
static constexpr int rwidth = 0;
static constexpr int align = alignof(T);
static constexpr int vlen = 0;
static_assert(MC>0 && KC>0 && NC>0 && MR>0 && NR>0, "Invalid block size.");
static_assert(MC % MR == 0, "MC must be a multiple of MR.");
static_assert(NC % NR == 0, "NC must be a multiple of NR.");
};
template <>
struct BlockSize<double>
{
static constexpr int MC = BS_D_MC;
static constexpr int KC = BS_D_KC;
static constexpr int NC = BS_D_NC;
static constexpr int MR = BS_D_MR;
static constexpr int NR = BS_D_NR;
static constexpr int rwidth = SIMD_REGISTER_WIDTH;
static constexpr int align = rwidth / 8;
static constexpr int vlen = rwidth / (8*sizeof(double));
static_assert(MC>0 && KC>0 && NC>0 && MR>0 && NR>0, "Invalid block size.");
static_assert(MC % MR == 0, "MC must be a multiple of MR.");
static_assert(NC % NR == 0, "NC must be a multiple of NR.");
static_assert(rwidth % sizeof(double) == 0, "SIMD register width not sane.");
};
//-- aux routines --------------------------------------------------------------
template <typename Index, typename Alpha, typename TX, typename TY>
void
geaxpy(Index m, Index n,
const Alpha &alpha,
const TX *X, Index incRowX, Index incColX,
TY *Y, Index incRowY, Index incColY)
{
for (Index j=0; j<n; ++j) {
for (Index i=0; i<m; ++i) {
Y[i*incRowY+j*incColY] += alpha*X[i*incRowX+j*incColX];
}
}
}
template <typename Index, typename Alpha, typename TX>
void
gescal(Index m, Index n,
const Alpha &alpha,
TX *X, Index incRowX, Index incColX)
{
if (alpha!=Alpha(0)) {
for (Index j=0; j<n; ++j) {
for (Index i=0; i<m; ++i) {
X[i*incRowX+j*incColX] *= alpha;
}
}
} else {
for (Index j=0; j<n; ++j) {
for (Index i=0; i<m; ++i) {
X[i*incRowX+j*incColX] = Alpha(0);
}
}
}
}
//-- Micro Kernel --------------------------------------------------------------
template <typename Index, typename T>
typename std::enable_if<BlockSize<T>::vlen != 0,
void>::type
ugemm(Index kc, T alpha, const T *A, const T *B, T beta,
T *C, Index incRowC, Index incColC)
{
typedef T vx __attribute__((vector_size (BlockSize<T>::rwidth/8)));
static constexpr Index vlen = BlockSize<T>::vlen;
static constexpr Index MR = BlockSize<T>::MR;
static constexpr Index NR = BlockSize<T>::NR/vlen;
A = (const T*) __builtin_assume_aligned (A, BlockSize<T>::align);
B = (const T*) __builtin_assume_aligned (B, BlockSize<T>::align);
vx P[MR*NR] = {};
for (Index l=0; l<kc; ++l) {
const vx *b = (const vx *)B;
for (Index i=0; i<MR; ++i) {
for (Index j=0; j<NR; ++j) {
P[i*NR+j] += A[i]*b[j];
}
}
A += MR;
B += vlen*NR;
}
if (alpha!=T(1)) {
for (Index i=0; i<MR; ++i) {
for (Index j=0; j<NR; ++j) {
P[i*NR+j] *= alpha;
}
}
}
if (beta!=T(0)) {
for (Index i=0; i<MR; ++i) {
for (Index j=0; j<NR; ++j) {
const T *p = (const T *) &P[i*NR+j];
for (Index j1=0; j1<vlen; ++j1) {
C[i*incRowC+(j*vlen+j1)*incColC] *= beta;
C[i*incRowC+(j*vlen+j1)*incColC] += p[j1];
}
}
}
} else {
for (Index i=0; i<MR; ++i) {
for (Index j=0; j<NR; ++j) {
const T *p = (const T *) &P[i*NR+j];
for (Index j1=0; j1<vlen; ++j1) {
C[i*incRowC+(j*vlen+j1)*incColC] = p[j1];
}
}
}
}
}
//-- Macro Kernel --------------------------------------------------------------
template <typename Index, typename T, typename Beta, typename TC>
void
mgemm(Index mc, Index nc, Index kc,
T alpha,
const T *A, const T *B,
Beta beta,
TC *C, Index incRowC, Index incColC)
{
const Index MR = BlockSize<T>::MR;
const Index NR = BlockSize<T>::NR;
const Index mp = (mc+MR-1) / MR;
const Index np = (nc+NR-1) / NR;
const Index mr_ = mc % MR;
const Index nr_ = nc % NR;
T C_[MR*NR];
#pragma omp parallel for
for (Index j=0; j<np; ++j) {
const Index nr = (j!=np-1 || nr_==0) ? NR : nr_;
for (Index i=0; i<mp; ++i) {
const Index mr = (i!=mp-1 || mr_==0) ? MR : mr_;
if (mr==MR && nr==NR) {
ugemm(kc, alpha,
&A[i*kc*MR], &B[j*kc*NR],
beta,
&C[i*MR*incRowC+j*NR*incColC],
incRowC, incColC);
} else {
ugemm(kc, alpha,
&A[i*kc*MR], &B[j*kc*NR],
T(0),
C_, Index(1), MR);
gescal(mr, nr, beta,
&C[i*MR*incRowC+j*NR*incColC],
incRowC, incColC);
geaxpy(mr, nr, T(1), C_, Index(1), MR,
&C[i*MR*incRowC+j*NR*incColC],
incRowC, incColC);
}
}
}
}
//-- Packing blocks ------------------------------------------------------------
template <typename Index, typename TA, typename T>
void
pack_A(Index mc, Index kc,
const TA *A, Index incRowA, Index incColA,
T *p)
{
Index MR = BlockSize<T>::MR;
Index mp = (mc+MR-1) / MR;
for (Index j=0; j<kc; ++j) {
for (Index l=0; l<mp; ++l) {
for (Index i0=0; i0<MR; ++i0) {
Index i = l*MR + i0;
Index nu = l*MR*kc + j*MR + i0;
p[nu] = (i<mc) ? A[i*incRowA+j*incColA]
: T(0);
}
}
}
}
template <typename Index, typename TB, typename T>
void
pack_B(Index kc, Index nc,
const TB *B, Index incRowB, Index incColB,
T *p)
{
Index NR = BlockSize<T>::NR;
Index np = (nc+NR-1) / NR;
for (Index l=0; l<np; ++l) {
for (Index j0=0; j0<NR; ++j0) {
for (Index i=0; i<kc; ++i) {
Index j = l*NR+j0;
Index nu = l*NR*kc + i*NR + j0;
p[nu] = (j<nc) ? B[i*incRowB+j*incColB]
: T(0);
}
}
}
}
//-- Frame routine -------------------------------------------------------------
template <typename Index, typename Alpha,
typename TA, typename TB,
typename Beta,
typename TC>
void
gemm(Index m, Index n, Index k,
Alpha alpha,
const TA *A, Index incRowA, Index incColA,
const TB *B, Index incRowB, Index incColB,
Beta beta,
TC *C, Index incRowC, Index incColC)
{
typedef typename std::common_type<Alpha, TA, TB>::type T;
const Index MC = BlockSize<T>::MC;
const Index NC = BlockSize<T>::NC;
const Index MR = BlockSize<T>::MR;
const Index NR = BlockSize<T>::NR;
const Index KC = BlockSize<T>::KC;
const Index mb = (m+MC-1) / MC;
const Index nb = (n+NC-1) / NC;
const Index kb = (k+KC-1) / KC;
const Index mc_ = m % MC;
const Index nc_ = n % NC;
const Index kc_ = k % KC;
T *A_ = (T*) malloc_(BlockSize<T>::align, sizeof(T)*(MC*KC+MR));
T *B_ = (T*) malloc_(BlockSize<T>::align, sizeof(T)*(KC*NC+NR));
if (alpha==Alpha(0) || k==0) {
gescal(m, n, beta, C, incRowC, incColC);
return;
}
for (Index j=0; j<nb; ++j) {
Index nc = (j!=nb-1 || nc_==0) ? NC : nc_;
for (Index l=0; l<kb; ++l) {
Index kc = (l!=kb-1 || kc_==0) ? KC : kc_;
Beta beta_ = (l==0) ? beta : Beta(1);
pack_B(kc, nc,
&B[l*KC*incRowB+j*NC*incColB],
incRowB, incColB,
B_);
for (Index i=0; i<mb; ++i) {
Index mc = (i!=mb-1 || mc_==0) ? MC : mc_;
pack_A(mc, kc,
&A[i*MC*incRowA+l*KC*incColA],
incRowA, incColA,
A_);
mgemm(mc, nc, kc,
T(alpha), A_, B_, beta_,
&C[i*MC*incRowC+j*NC*incColC],
incRowC, incColC);
}
}
}
free_(A_);
free_(B_);
}
//------------------------------------------------------------------------------
void myprod(double *c, const double* a, const double* b, int N) {
gemm(N, N, N, 1.0, a, 1, N, b, 1, N, 0.0, c, 1, N);
}
int main(int argc, char* argv[]) {
int N = atoi(argv[1]);
int tries = 4;
int rep = std::max<int>(1,10000000/N/N/N);
Eigen::MatrixXd a_E = Eigen::MatrixXd::Random(N,N);
Eigen::MatrixXd b_E = Eigen::MatrixXd::Random(N,N);
Eigen::MatrixXd c_E(N,N);
Eigen::BenchTimer t1, t2;
BENCH(t1, tries, rep, c_E.noalias() = a_E*b_E );
BENCH(t2, tries, rep, myprod(c_E.data(), a_E.data(), b_E.data(), N));
std::cout << "Time taken by Eigen is: " << t1.best() << "n";
std::cout << "Time taken by for-loop is: " << t2.best() << "nn";
}
编译器已经很好地对代码进行了矢量化处理。提高性能的关键是分层阻塞，以优化寄存器和不同级别的缓存的使用。部分循环展开对于改善指令流水线也至关重要。达到 Eigen 产品的性能需要大量的努力和调整。
还应该注意的是，您的基准测试略有偏差，并不完全可靠。这是一个更可靠的版本（您需要完整的 Eigen 源代码才能获得bench/BenchTimer.h）：
#include<iostream>
#include<Eigen/Dense>
#include<bench/BenchTimer.h>
void myprod(double *c, const double* a, const double* b, int N) {
int count = 0;
int count1, count2;
for (int j=0; j<N; ++j) {
count1 = j*N;
for (int k=0; k<N; ++k) {
c[count] = a[count1]*b[k];
++count;
}
}
for (int j=0; j<N; ++j) {
count2 = N;
for (int l=1; l<N; ++l) {
count = j*N;
count1 = count+l;
for (int k=0; k<N; ++k) {
c[count]+=a[count1]*b[count2];
++count;
++count2;
}
}
}
}
int main(int argc, char* argv[]) {
int N = atoi(argv[1]);
int tries = 4;
int rep = std::max<int>(1,10000000/N/N/N);
Eigen::MatrixXd a_E = Eigen::MatrixXd::Random(N,N);
Eigen::MatrixXd b_E = Eigen::MatrixXd::Random(N,N);
Eigen::MatrixXd c_E(N,N);
Eigen::BenchTimer t1, t2;
BENCH(t1, tries, rep, c_E.noalias() = a_E*b_E );
BENCH(t2, tries, rep, myprod(c_E.data(), a_E.data(), b_E.data(), N));
std::cout << "nTime taken by Eigen is: " << t1.best() << "n";
std::cout << "nTime taken by for-loop is: " << t2.best() << "n";
}
在启用3.3-beta1和FMA的情况下进行编译（-mfma），那么差距就会变大，对于N=2000来说几乎是一个数量级。
我可以建议两个简单的优化。
1）矢量化它。如果使用内联汇编或编写汇编过程对其进行矢量化会更好，但您也可以使用编译器内部函数。您甚至可以让编译器对循环进行矢量化，但有时很难编写适当的循环以供编译器矢量化。
2）使其平行。尝试使用 OpenMP。
相关文章：
- 函数何时会在c++中包含stack_Unwind_Resume调用
- 正在为Xtensa simcall函数编写回调函数
- Python中的for循环与C++有何不同
- 在程序中编写脚本来编写和编译代码
- 编写时C++中的输入重定向问题
- 在C++中，如何在类和函数(可能是模板化的）的头中编写完整的实现
- c++多进程编写一个唯一的文件
- 如何用C++编写BFS函数
- 编写代码时C++出现错误：错误 1 错误 C2601：'circle'：本地函数定义是非法的
- 如何编写一个使用n倍三元条件语句的C++布尔函数
- 为C++03编译器编写部分unique_ptr，该编译器与较新的编译器在公共代码库上运行
- 如何为 C 型字符串数组编写 getter 和 setter?
- 如何在没有函数的情况下编写此代码并使C++更简单?
- 编写一个函数以使用 n 百分比的 CPU 使用率
- 如何使用OpenCV-C++编写*.mp4视频?
- 3-3. 编写一个程序来计算每个不同单词在其输入中出现的次数
- 我的目标是编写一个程序来计算和存储字符串在字符数组中出现的位置
- 字符 * 和字符串在C++中有何不同?(在描述中编写代码）
- 如何编写可以与本征竞争的矩阵矩阵积
- 在C#中为windows API编写COM服务器，从何开始
