Leg - 3 years ago 144
C++ Question

# How to write a matrix matrix product that can compete with Eigen?

Below is the C++ implementation comparing the time taken by Eigen and For Loop to perform matrix-matrix products. The For loop has been optimised to minimise cache misses. The for loop is faster than Eigen initially but then eventually becomes slower (upto a factor of 2 for 500 by 500 matrices). What else should I do to compete with Eigen? Is blocking the reason for the better Eigen performance? If so, how should I go about adding blocking to the for loop?

``````#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";
}
``````

There is no need to mystifying how a high performance implementation of the matrix-matrix product can be achieved. In fact we need more people knowing about it, in order to face future challenges in high-performance computing. In order to get into this topic reading BLIS: A Framework for Rapidly Instantiating BLAS Functionality is a good starting point.

So in order to demystify and to answer the question (How to write a matrix matrix product that can compete with Eigen) I extended the code posted by ggael to a total of 400 lines. I just tested it on an AVX machine (Intel(R) Core(TM) i5-3470 CPU @ 3.20GHz). Here some results:

``````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
``````

If you have FMA you can compile with

``````g++-5.3 -O3 -DNDEBUG -std=c++11 -mfma -m64 -I ../eigen.3.2.8/ -DHAVE_FMA gemm.cc -lrt
``````

If you also want multithreading with openMP also compile with `-fopenmp`

Here the complete code based on the ideas of the BLIS paper. It is self-contained except that it needs the complete Eigen source files as ggael already noted:

``````#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() << "\n\n";
}
``````
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