Does MKL optimize cblas for *major order?

Question

I am using mkl cblas_dgemm and currently have it with CblasRowMajor, CblasNoTrans, CblasNotrans, for my matrices.

I know that c is a row major language, whereas dgemm is a column major algorithm. I am interested to know if switching the ordering of the matrices will have any affect on the cblas_dgemm algorithm if I am linking against mkl. Is mkl smart enough to do things behind the scenes that I would try to do to optimized the matrix multiplcations? If not, what is the best way to perform matrix multiplications with mkl?

Answer 1

TL;DR: In short it does not matter whether you perform matrix-matrix multiplications using row-major or column-major ordering with MKL (and other BLAS implementations).

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I know that c is a row major language, whereas dgemm is a column major algorithm.

DGEMM is not a column-major algorithm, it is the BLAS interface for computing a matrix-matrix product with general matrices. The common reference implementation for DGEMM (and most of BLAS) is Netlib's which is written in Fortran. The only reason it assumes column-major ordering is because Fortran is a column-major order language. DGEMM (and the corresponding BLAS Level 3 functions) is not specifically for column-major data.

What Does DGEMM Compute?

DGEMM in basic maths performs a 2D matrix-matrix multiplication. The standard algorithm for multiplying 2D matrices requires you to traverse one matrix along its rows and the other along its columns. To perform the matrix-matrix multiplication, AB = C, we would multiply the rows of A by the columns of B to produce C. Therefore, the ordering of the input matrices does not matter as one matrix must be traversed along its rows and the other along its columns.

Investigating Row-Major and Column-Major DGEMM Computation with MKL

Intel MKL is smart enough to utilise this under the hood and provides the exact same performance for row-major and column-major data.

cblas_dgemm(CblasRowMajor, CblasNoTrans, CblasNoTrans, ...);

and

cblas_dgemm(CblasColMajor, CblasNoTrans, CblasNoTrans, ...);

will execute with similar performance. We can test this with a relatively simple program

#include <float.h>
#include <mkl.h>
#include <omp.h>
#include <stdio.h>

void init_matrix(double *A, int n, int m, double d);
void test_dgemm(CBLAS_LAYOUT Layout, double *A, double *B, double *C, const MKL_INT m, const MKL_INT n, const MKL_INT k, int nSamples, double *timing);
void print_summary(const MKL_INT m, const MKL_INT n, const MKL_INT k, const int nSamples, const double *timing);

int main(int argc, char **argv) {
    MKL_INT n, k, m;
    double *a, *b, *c;
    double *timing;
    int nSamples = 1;

    if (argc != 5){
        fprintf(stderr, "Error: Wrong number of arguments!\n");
        fprintf(stderr, "usage: %s mMatrix nMatrix kMatrix NSamples\n", argv[0]);
        return -1;
    }

    m = atoi(argv[1]);
    n = atoi(argv[2]);
    k = atoi(argv[3]);

    nSamples = atoi(argv[4]);

    timing = malloc(nSamples * sizeof *timing);

    a = mkl_malloc(m*k * sizeof *a, 64);
    b = mkl_malloc(k*n * sizeof *a, 64);
    c = mkl_calloc(m*n, sizeof *a, 64);

    /** ROW-MAJOR ORDERING **/
    test_dgemm(CblasRowMajor, a, b, c, m, n, k, nSamples, timing);

    /** COLUMN-MAJOR ORDERING **/
    test_dgemm(CblasColMajor, a, b, c, m, n, k, nSamples, timing);

    mkl_free(a);
    mkl_free(b);
    mkl_free(c);
    free(timing);
}

void init_matrix(double *A, int n, int m, double d) {
    int i, j;
    #pragma omp for schedule (static) private(i,j)
    for (i = 0; i < n; ++i) {
        for (j = 0; j < m; ++j) {
            A[j + i*n] = d * (double) ((i - j) / n);
        }
    }
}

void test_dgemm(CBLAS_LAYOUT Layout, double *A, double *B, double *C, const MKL_INT m, const MKL_INT n, const MKL_INT k, int nSamples, double *timing) {
    int i;
    MKL_INT lda = m, ldb = k, ldc = m;
    double alpha = 1.0, beta = 0.0;

    if (CblasRowMajor == Layout) {
        printf("\n*****ROW-MAJOR ORDERING*****\n\n");
    } else if (CblasColMajor == Layout) {
        printf("\n*****COLUMN-MAJOR ORDERING*****\n\n");
    }

    init_matrix(A, m, k, 0.5);
    init_matrix(B, k, n, 0.75);
    init_matrix(C, m, n, 0);

    // First call performs any buffer/thread initialisation
    cblas_dgemm(Layout, CblasNoTrans, CblasNoTrans, m, n, k, alpha, A, lda, B, ldb, beta, C, ldc);

    double tmin = DBL_MAX, tmax = 0.0;
    for (i = 0; i < nSamples; ++i) {
        init_matrix(A, m, k, 0.5);
        init_matrix(B, k, n, 0.75);
        init_matrix(C, m, n, 0);

        timing[i] = dsecnd();
        cblas_dgemm(Layout, CblasNoTrans, CblasNoTrans, m, n, k, alpha, A, lda, B, ldb, beta, C, ldc);
        timing[i] = dsecnd() - timing[i];

        if (tmin > timing[i]) tmin = timing[i];
        else if (tmax < timing[i]) tmax = timing[i];
    }

    print_summary(m, n, k, nSamples, timing);
}

void print_summary(const MKL_INT m, const MKL_INT n, const MKL_INT k, const int nSamples, const double *timing) {
    int i;

    double tavg = 0.0;
    for(i = 0; i < nSamples; i++) {
        tavg += timing[i];
    }
    tavg /= nSamples;

    printf("#Loop | Sizes  m   n   k  | Time (s)\n");
    for(i = 0; i < nSamples; i++) {
        printf("%4d %12d %3d %3d  %6.4f\n", i + 1 , m, n, k, timing[i]);
    }

    printf("Summary:\n");
    printf("Sizes  m   n   k  | Avg. Time (s)\n");
    printf(" %8d %3d %3d %12.8f\n", m, n, k, tavg);
}

on my system this produces

$ ./benchmark_dgemm 1000 1000 1000 5
*****ROW-MAJOR ORDERING*****

#Loop | Sizes  m   n   k  | Time (s)
   1         1000 1000 1000  0.0589
   2         1000 1000 1000  0.0596
   3         1000 1000 1000  0.0603
   4         1000 1000 1000  0.0626
   5         1000 1000 1000  0.0584
Summary:
Sizes  m   n   k  | Avg. Time (s)
     1000 1000 1000   0.05995692

*****COLUMN-MAJOR ORDERING*****

#Loop | Sizes  m   n   k  | Time (s)
   1         1000 1000 1000  0.0597
   2         1000 1000 1000  0.0610
   3         1000 1000 1000  0.0581
   4         1000 1000 1000  0.0594
   5         1000 1000 1000  0.0596
Summary:
Sizes  m   n   k  | Avg. Time (s)
     1000 1000 1000   0.05955171

where we can see that there is very little difference between the column-major ordering time and the row-major ordering time. 0.0595 seconds for column-major versus 0.0599 seconds for row-major. Executing this again might produce the following, where the row-major calculation is faster by 0.00003 seconds.

$ ./benchmark_dgemm 1000 1000 1000 5
*****ROW-MAJOR ORDERING*****

#Loop | Sizes  m   n   k  | Time (s)
   1         1000 1000 1000  0.0674
   2         1000 1000 1000  0.0598
   3         1000 1000 1000  0.0595
   4         1000 1000 1000  0.0587
   5         1000 1000 1000  0.0584
Summary:
Sizes  m   n   k  | Avg. Time (s)
     1000 1000 1000   0.06075310

*****COLUMN-MAJOR ORDERING*****

#Loop | Sizes  m   n   k  | Time (s)
   1         1000 1000 1000  0.0634
   2         1000 1000 1000  0.0596
   3         1000 1000 1000  0.0582
   4         1000 1000 1000  0.0582
   5         1000 1000 1000  0.0645
Summary:
Sizes  m   n   k  | Avg. Time (s)
     1000 1000 1000   0.06078266