Eigen3 Developer Documentation

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Dense classes

This section can now be found in the doxygen-generated documentation; see

Nested Expression Templates

This section is completely outdated. It needs to be rewritten to match with Eigen 3.3 internals.


   MatrixXf A, B, C; 
C = A.transpose() + B;

A.transpose() returns an object of type Transpose<MatrixXf> where the MatrixXf A is nested into the Transpose expression. The nested type tells how to store the nested object. Here MatrixXf::Nested boils down to a MatrixXf&, and thus "A.transpose()" stores a reference to A.

There are two main reasons we introduced such a nesting type mechanism and not always use a reference:

[I] Expressions other than Matrix or Array are lightweight and better nested by value. In the previous example, A.transpose() + B returns an object of type CwiseBinaryOp<ei_scalar_sum_op<float>, Transpose<MatrixXf>, MatrixXf> storing both sides of the addition as follows:

   Transpose<MatrixXf>::Nested lhs; // left hand side 
MatrixXf::Nested rhs; // right hand side

which boils down to:

   const Transpose<MatrixXf> lhs; // nesting by value 
const MatrixXf& rhs; // nesting by reference

Nesting by value small object avoids temporary headache when a function has to return complex expressions, e.g.:

   template<typename A, typename B> 
CwiseBinaryOp<ei_scalar_sum_op<float>, Transpose<A>, B>
adjoint(const A& a, const B& b)
return a.transpose() + b;

If the temporary "a.transpose()" was stored by reference by the CwiseBinaryOp expression, then you would end up with a segfault because the "a.transpose()" temporary is destroyed just before the function returns, and so the CwiseBinaryOp expression would store a reference to dead object.

[II] Some expressions must be evaluated into temporaries before being used. For instance, in the following example:d = a * b + c; for performance reason, the matrix product a * b has to be evaluated into a temporary before evaluating the addition. This is achieved as follows. Here we build the expression of type:

   CwiseBinaryOp<ei_scalar_sum_op<float>, Product<MatrixXf,MatrixXf>, MatrixXf>

which stores a Product<MatrixXf,MatrixXf>::Nested for the lhs, and Product<MatrixXf,MatrixXf>::Nested is ... MatrixXf !!

Something more complicated:

   (a + b) * c 

Here, if c is not too small, it is better to evaluate (a+b) into a temporary before doing the matrix product, otherwise, a+b would be computed c.cols() times. To this end we have a ei_nested<> helper class to determine the ideal nesting type. In Product, we have something like:

   ei_nested<CwiseBinaryOp<ei_scalar_sum_op<float>, type_of_a, type_of_b>, type_of_c::ColsAtCompileTime>::type

giving us the nesting type of the left hand side of the product (here a MatrixXf if a, b, and c are MatrixXf). For the right hand side here we have:

   ei_nested<CwiseBinaryOp<ei_scalar_sum_op<float>, type_of_c, type_of_a_plus_b::RowsAtCompileTime>::type 

which gives us a MatrixXf&.

ei_nested<> determines whether the nested expression has to be evaluated or not in function of an estimation of the evaluation cost of one coefficient. This cost is automatically computed by the expressions in the ei_traits<> specializations.

Very important: When you write a generic function taking, e.g., a MatrixBase<Derived> object you should really honor the nesting type of the Derived class:

   template<typename Derived> 
void foo(const MatrixBase<Derived>& _x)
typename Derived::Nested x(_x.derived());

// use x safely

Actually, if you use the argument more than once, you should even use the ei_nested<> helper:

   template<typename Derived> </br>
   typename Derived::Scalar foo(const MatrixBase<Derived>& _x) 
typename ei_nested<Derived,2>::type x(_x.derived());

return (x + x.adjoint()).maxCoeff();

If you don't do so, and call foo(a*b); then the expensive product a*b will be computed twice !!