# A quick trick for faster naïve matrix multiplication

If you need to multiply some matrices together very quickly, usually it's best to use a highly optimized library like ATLAS. But sometimes adding such a dependency isn't worth it, if you're worried about portability, code size, etc. If you just need good performance, rather than the best possible performance, it can make sense to hand-roll your own matrix multiplication function.

Unfortunately, the way that matrix multiplication is usually taught:

$$C_{i,j} = \sum_k A_{i,k} \, B_{k,j}$$

$$\Bigg($$$$\Bigg) = \Bigg($$$$\Bigg)\,\Bigg($$$$\Bigg)$$

void matmul(double *dest, const double *lhs, const double *rhs,
size_t rows, size_t mid, size_t cols) {
for (size_t i = 0; i < rows; ++i) {
for (size_t j = 0; j < cols; ++j) {
const double *rhs_row = rhs;
double sum = 0.0;
for (size_t k = 0; k < mid; ++k) {
sum += lhs[k] * rhs_row[j];
rhs_row += cols;
}
*dest++ = sum;
}
lhs += mid;
}
}

This function multiplies a rows×mid matrix with a mid×cols matrix using the "linear algebra 101" algorithm. Unfortunately, it has a bad memory access pattern: we loop over dest and lhs pretty much in order, but jump all over the place in rhs, since it's stored row-major but we need its columns.

Luckily there's a simple fix that's dramatically faster: instead of computing each cell of the destination separately, we can update whole rows of it at a time. Effectively, we do this:

$$C_{i} = \sum_j A_{i,j} \, B_j$$

$$\Bigg($$$$\Bigg) = \Bigg($$$$\Bigg)\,\Bigg($$$$\Bigg)$$

In code, it looks like this:

void matmul(double *dest, const double *lhs, const double *rhs,
size_t rows, size_t mid, size_t cols) {
memset(dest, 0, rows * cols * sizeof(double));

for (size_t i = 0; i < rows; ++i) {
const double *rhs_row = rhs;
for (size_t j = 0; j < mid; ++j) {
for (size_t k = 0; k < cols; ++k) {
dest[k] += lhs[j] * rhs_row[k];
}
rhs_row += cols;
}
dest += cols;
lhs += mid;
}
}

On my computer, that drops the time to multiply two 256×256 matrices from 37ms to 13ms (with gcc -O3). ATLAS does it in 5ms, though, so always use something like it if it's available.