# Implementation of the GEMM error estimator

## Exercise: Implement and test the GEMM error estimator

Use the tools from above and implement the following functions:

• dgemm_err_est receives matrices $$A$$, $$B$$, $$C_0$$, $$\hat{C}$$ and $$C$$ as well as scalars $$\alpha$$ and $$\beta$$. It computes the error estimator

$\frac{\|C - \hat{C}\|_\infty}{ \text{eps}\cdot\left( \max\{m,n,k\} \cdot |\alpha| \cdot \|A\|_\infty \|B\|_\infty + |\beta|\cdot\|C_0\|_\infty \right)}$

Hereby matrix $$C$$ gets overwritten with intermediate results.

Note:

• For type double the constant $$\text{eps}$$ is defined in the standard header float.h through the macro DBL_EPSILON.

• The skeleton contains a macro function for computing the maximum of to values.

• dgemm_ijl performs the GEMM operation with three nested loops (from most outer loop to most inner loop this variant iterates over i, j, l).

#include <float.h>
#include <math.h>
#include <stdbool.h>
#include <stddef.h>
#include <stdio.h>
#include <stdlib.h>
#include <sys/times.h>
#include <unistd.h>

//------------------------------------------------------------------------------

#define MY_ABS(x)   ((x)<0 ? -(x) : (x))

//------------------------------------------------------------------------------

void
initMatrix(size_t m, size_t n, double *A, ptrdiff_t incRowA, ptrdiff_t incColA,
bool withNan)
{
// if A is row major initialize A^T
if (MY_ABS(incRowA) > MY_ABS(incColA)) {
initMatrix(n, m, A, incColA, incRowA, withNan);
return;
}
// if A is col major
if (withNan) {
for (size_t j=0; j<n; ++j) {
for (size_t i=0; i<m; ++i) {
A[i*incRowA+j*incColA] = nan("");
}
}
} else {
for (size_t j=0; j<n; ++j) {
for (size_t i=0; i<m; ++i) {
double rValue = ((double)rand() - RAND_MAX/2)*2/RAND_MAX;
A[i*incRowA+j*incColA] = rValue;
}
}
}
}

void
printMatrix(size_t m, size_t n,
const double *A, ptrdiff_t incRowA, ptrdiff_t incColA)
{
for (size_t i=0; i<m; ++i) {
for (size_t j=0; j<n; ++j) {
printf("%10.3lf ", A[i*incRowA+j*incColA]);
}
printf("\n");
}
printf("\n");
}

//------------------------------------------------------------------------------

void
dgeaxpy(size_t m, size_t n, double alpha,
const double *A, ptrdiff_t incRowA, ptrdiff_t incColA,
double *B, ptrdiff_t incRowB, ptrdiff_t incColB)
{
if (m==0 || n==0) {
return;
}
// if B is row major:   B^T <- alpha*A^T + B^T
if (MY_ABS(incRowB) > MY_ABS(incColB)) {
dgeaxpy(n, m, alpha, A, incColA, incRowA, B, incColB, incRowB);
return;
}
// B is col major:
for (size_t j=0; j<n; ++j) {
for (size_t i=0; i<m; ++i) {
B[i*incRowB+j*incColB] += alpha*A[i*incRowA+j*incColA];
}
}
}

void
dgecopy(size_t m, size_t n,
const double *A, ptrdiff_t incRowA, ptrdiff_t incColA,
double *B, ptrdiff_t incRowB, ptrdiff_t incColB)
{
if (m==0 || n==0) {
return;
}
// if B is row major:   B^T <- A^T
if (MY_ABS(incRowB) > MY_ABS(incColB)) {
dgecopy(n, m, A, incColA, incRowA, B, incColB, incRowB);
return;
}
// B is col major:
for (size_t j=0; j<n; ++j) {
for (size_t i=0; i<m; ++i) {
B[i*incRowB+j*incColB] = A[i*incRowA+j*incColA];
}
}
}

void
dgescal(size_t m, size_t n, double alpha,
double *A, ptrdiff_t incRowA, ptrdiff_t incColA)
{
if (alpha==1 || m==0 || n==0) {
return;
}
// if A is row major: A^T <- alpha*A^T
if (MY_ABS(incRowA) > MY_ABS(incColA)) {
dgescal(n, m, alpha, A, incColA, incRowA);
return;
}
// A is col major:
if (alpha!=0) {
for (size_t j=0; j<n; ++j) {
for (size_t i=0; i<m; ++i) {
A[i*incRowA+j*incColA] *= alpha;
}
}
} else {
for (size_t j=0; j<n; ++j) {
for (size_t i=0; i<m; ++i) {
A[i*incRowA+j*incColA] = 0;
}
}
}
}

// This operation is not cache friendly!
double
dgenorm_inf(size_t m, size_t n,
const double *A, ptrdiff_t incRowA, ptrdiff_t incColA)
{
double res = 0;
for (size_t i=0; i<m; ++i) {
double asum = 0;
for (size_t j=0; j<n; ++j) {
asum += fabs(A[i*incRowA+j*incColA]);
}
if (asum>res) {
res = asum;
}
}
return res;
}

//------------------------------------------------------------------------------

void
dgemm_ijl(size_t m, size_t n, size_t k,
double alpha,
const double *A, ptrdiff_t incRowA, ptrdiff_t incColA,
const double *B, ptrdiff_t incRowB, ptrdiff_t incColB,
double beta,
double *C, ptrdiff_t incRowC, ptrdiff_t incColC)
{
dgescal(m, n, beta, C, incRowC, incColC);
if (m==0 || n==0 || k==0 || alpha==0) {
return;
}
}

//------------------------------------------------------------------------------

#define MAX(x,y)    ((x)>(y)) ? (x) : (y)

double
dgemm_err_est(size_t m, size_t n, size_t k,
double alpha,
const double *A, ptrdiff_t incRowA, ptrdiff_t incColA,
const double *B, ptrdiff_t incRowB, ptrdiff_t incColB,
const double *C0, ptrdiff_t incRowC0, ptrdiff_t incColC0,
double beta,
const double *C_, ptrdiff_t incRowC_, ptrdiff_t incColC_,
double *C, ptrdiff_t incRowC, ptrdiff_t incColC)
{
dgeaxpy(m, n, -1, C_, incRowC_, incColC_, C, incRowC, incColC);

double normD  = dgenorm_inf(m, n, C, incRowC, incColC);
double normC0 = dgenorm_inf(m, n, C0, incRowC0, incColC0);
double normA  = dgenorm_inf(m, k, A, incRowA, incColA);
double normB  = dgenorm_inf(k, n, B, incRowB, incColB);
size_t N      = MAX(m, MAX(n, k));

return normD; // TODO/FIXME
}

void
dgemm_ref(size_t m, size_t n, size_t k,
double alpha,
const double *A, ptrdiff_t incRowA, ptrdiff_t incColA,
const double *B, ptrdiff_t incRowB, ptrdiff_t incColB,
double beta,
double *C, ptrdiff_t incRowC, ptrdiff_t incColC)
{
if (beta!=1) {
if (beta!=0) {
for (size_t j=0; j<n; ++j) {
for (size_t i=0; i<m; ++i) {
C[i*incRowC+j*incColC] *= beta;
}
}
} else {
for (size_t j=0; j<n; ++j) {
for (size_t i=0; i<m; ++i) {
C[i*incRowC+j*incColC] = 0;
}
}
}
}
if (k==0 || alpha==0) {
return;
}
for (size_t j=0; j<n; ++j) {
for (size_t l=0; l<k; ++l) {
for (size_t i=0; i<m; ++i) {
C[i*incRowC+j*incColC] += alpha*A[i*incRowA+l*incColA]
*B[l*incRowB+j*incColB];
}
}
}
}

//------------------------------------------------------------------------------

double
wallTime()
{
static clock_t ticks_per_second = 0;
if (!ticks_per_second) {
ticks_per_second = sysconf(_SC_CLK_TCK);
}
struct tms timebuf;
/* times returns the number of real time ticks passed since start */
return (double) times(&timebuf) / ticks_per_second;
}

//------------------------------------------------------------------------------

#ifndef DIM_M
#define DIM_M 500
#endif

#ifndef DIM_N
#define DIM_N 600
#endif

#ifndef DIM_K
#define DIM_K 700
#endif

#ifndef ALPHA
#define ALPHA 1.5
#endif

#ifndef BETA
#define BETA 2.0
#endif

double A[DIM_M*DIM_K];
double B[DIM_K*DIM_N];
double C0[DIM_M*DIM_N];
double C_[DIM_M*DIM_N];
double C[DIM_M*DIM_N];

const bool rowMajorA[] = {0, 1, 0, 1, 0, 1, 0, 1};
const bool rowMajorB[] = {0, 0, 1, 1, 0, 0, 1, 1};
const bool rowMajorC[] = {0, 0, 0, 0, 1, 1, 1, 1};
const size_t numTests = sizeof(rowMajorA)/sizeof(bool);

int
main()
{
printf("A is %dx%d\n", DIM_M, DIM_K);
printf("B is %dx%d\n", DIM_K, DIM_N);
printf("C is %dx%d\n", DIM_M, DIM_N);

printf("%4s %4s %4s ", "A", "B", "C");
printf("%14s %14s ", "gemm_ijl", "time");
//printf("%14s %14s ", "gemm_ilj", "time");
//printf("%14s %14s ", "gemm_lij", "time");
//printf("%14s %14s ", "gemm_jil", "time");
//printf("%14s %14s ", "gemm_jli", "time");
//printf("%14s %14s ", "gemm_lji", "time");
//printf("%14s %14s ", "gemm_blk", "time");
printf("\n");

for (size_t test=0; test<numTests; ++test) {

ptrdiff_t incRowA = rowMajorA[test] ? DIM_K : 1;
ptrdiff_t incColA = rowMajorA[test] ? 1 : DIM_M;

ptrdiff_t incRowB = rowMajorB[test] ? DIM_N : 1;
ptrdiff_t incColB = rowMajorB[test] ? 1 : DIM_K;

ptrdiff_t incRowC = rowMajorC[test] ? DIM_N : 1;
ptrdiff_t incColC = rowMajorC[test] ? 1 : DIM_M;

printf("%4s ", incRowA>incColA ? "RM" : "CM");
printf("%4s ", incRowB>incColB ? "RM" : "CM");
printf("%4s ", incRowC>incColC ? "RM" : "CM");

srand(0);
initMatrix(DIM_M, DIM_K, A, incRowA, incColA, ALPHA==0.0);
initMatrix(DIM_K, DIM_N, B, incRowB, incColB, ALPHA==0.0);
initMatrix(DIM_M, DIM_N, C0, incRowC, incColC, BETA==0.0);

// compute reference solution
dgecopy(DIM_M, DIM_N, C0, incRowC, incColC, C_, incRowC, incColC);
dgemm_ref(DIM_M, DIM_N, DIM_K,
ALPHA,
A, incRowA, incColA,
B, incRowB, incColB,
BETA,
C_, incRowC, incColC);

// test other gemm implementations

double est, t;

// start testing: dgemm_ijl
dgecopy(DIM_M, DIM_N, C0, incRowC, incColC, C, incRowC, incColC);

t = wallTime();
dgemm_ijl(DIM_M, DIM_N, DIM_K,
ALPHA,
A, incRowA, incColA,
B, incRowB, incColB,
BETA,
C, incRowC, incColC);
t = wallTime() - t;

est = dgemm_err_est(DIM_M, DIM_N, DIM_K,
ALPHA,
A, incRowA, incColA,
B, incRowB, incColB,
C0, incRowC, incColC,
BETA,
C_, incRowC, incColC,
C, incRowC, incColC);

printf("%14.2e %14.2lf ", est, t);
// done testing: dgemm_ijl

printf("\n");
}
}


## Exercise: Implement other GEMM variants

Implement the remaining GEMM variants (e.g. dgemm_ilj, dgemm_lij, etc.).