hi all

I am starting to find that the technique of ET is very interesting thus and am looking into how it is used in practical library. I hope someone can help me here. Most of the articles online are about optimising the following addition expression

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a = b + c + d + e;


which ET can transform it into something like

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for(int i = 0; i < n; ++i){
a[i] = b[i] + c[i] + d[i] + e[i];
}


by walking the expression tree

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Assign<a, sum<sum<sum<d,e>, c>, b>>


This optimisation is rather for element-wise operation. Now suppose that I have the follow matrices computation (not exactly Eigen notation)

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int m, n, p, q, r;
MatrixXd a(m, n), b(n, p), c(p,q), d(q,r), result(m, r);
//Fill them up with doubles
result = a*b*c*d; //where * is matrix product


What kind of optimisations are performed here at compiled time?

Eigen evaluates matrix-matrix product immediately. So in the case of an expression involving only matrix-matrix products, such as in your last example, Eigen's ETs won't make a big difference: they will only help ensure that no redundant copy of matrices is performed (as would be the case if operator* returned a new matrix by value).

If you are sure that you want matrix products to be lazy i.e. not evaluate immediately, you can do:

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result = a.lazyProduct(b).lazyProduct(c).lazyProduct(d);


but be very careful: this can be a very bad idea as the complexity of evaluating such a chained lazy product can be far worse than the complexity of the default i.e. evaluating each product immediately!!