Expression templates and matrix multiplication

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C++

templates

mathematics

Matrix

Hello,

I'm trying to understand how mathematical libraries that use Expression Templates techniques ( like Eigen for exmaple ) handle Matrix Multiplication.

What I have tried:

In many examples I found there is always just the element-wise operation implemented:

A = B*C + D

is compiled as a single loop

for ( i = 0 ; i < size; i++ )
{
A[i] = B[i]*C[i] + D[i];
}

but how to manage matrix multiplication ( that needs nested loops )?

Thank you