C++ for Scientists - Technische Universität Dresden

162 CHAPTER 5. META-PROGRAMMING
number of columns in the matrix is taken from an internal definition in the matrix type for the
sake of simplicity. Passing this as extra template argument or taking a type traits would have
been more general because we are now limited to types where my cols is defined in the class.
We still need a (full) specialization to terminate the recursion:
template
struct fsize mat vec mult
{
template
void operator()(const Matrix& A, const VecIn& v in, VecOut& v out)
{
v out[0]= A[0][0] ∗ v in[0];
}
};
With the inlining, our program will execute the operation w= A ∗ v for vectors of size 4 as:
w[0]= A[0][0] ∗ v[0];
w[0]+= A[0][1] ∗ v[1];
w[0]+= A[0][2] ∗ v[2];
w[0]+= A[0][3] ∗ v[3];
w[1]= A[1][0] ∗ v[0];
w[1]+= A[1][1] ∗ v[1];
w[1]+= A[1][2] ∗ v[2];
w[1]+= A[1][3] ∗ v[3];
w[2]= A[2][0] ∗ v[0];
w[2]+= A[2][1] ∗ v[1];
w[2]+= A[2][2] ∗ v[2];
w[2]+= A[2][3] ∗ v[3];
w[3]= A[3][0] ∗ v[0];
w[3]+= A[3][1] ∗ v[1];
w[3]+= A[3][2] ∗ v[2];
w[3]+= A[3][3] ∗ v[3];
Our tests have shown that such an implementation is really faster than the compiler optimization
on loops. 18
Increasing Concurrency
A disadvantage of the preceeding implementation is that all operations on an entry of the target
vector are performed in one sweep. Therefore, the second operation must wait for the first the
third for the second on so on. The fifth operation can be done in parallel with the forth,
the ninth with the eighth but this is not satisfying. We like to have more concurrency in our
program that enables parallel pipelines in superscalar processors. Again, we can twiddle our
thumbs and hope that the compiler will reorder the statements or take it in our hands. More
concurrency is provided by the following operation sequence:
w[0]= A[0][0] ∗ v[0];
w[1]= A[1][0] ∗ v[0];
w[2]= A[2][0] ∗ v[0];
w[3]= A[3][0] ∗ v[0];
w[0]+= A[0][1] ∗ v[1];
18 TODO: Give numbers