Expressions in depth

Creation of the expression

Because of the expression template mechanism, every glas-expression is represented by an object that refers to all necessary data to evaluate (part of) the expression. For being generic, these objects are instances of a template class. To avoid having to list all template arguments of the template class when constructing the object, free functions are available that rely on template argument deduction. Thus instead of writing following expression (which most probably will not fit on your screen but that is not necessarily a bad thing because it exactly proves the point we want to make here ;-) 
  scalar_vector_mult_expression< vector_norm_2_expression< dense_vector< double > >, dense_vector< double > >( vector_norm_2_expression< dense_vector< double > >(v), dense_vector< double >(w) )
we can write 
  scalar_vector_mult( vector_norm_2(v), w )
These free functions are only an aid to construct the expression-object using template argument deduction. Their implementation is therefore straightforward: 
scalar_vector_mult_expression<S,V> scalar_vector_mult(S const& s, V const& v ) 
{ return scalar_vector_mult_expression<S,V>( s, v ) ; }
Similarly, vector_norm_2(v) creates the UnaryExpression 
vector_norm_2_expression<V> vector_norm_2( V const& v ) 
{ return vector_norm_2_expression<V>( v ) ; }

The expression-object

The creation of the scalar_vector_mult_expression does not compute the result of this expression, but only stores the data that are needed to compute it. For instance scalar_vector_mult_expression might be implemented like 
  template <class S, class V>
  struct scalar_vector_mult_expression
  {
    scalar_vector_mult_expression(S const& s, V const& v )
      : s_( s )
      , v_( v )
    {}
  
    S const& s_ ;
    V const& v_ ;
  };
while vector_norm_2_expression might be implemented like 
template <class V>
struct vector_norm_2_expression
{
  norm_1_expression( V const& v )
    : v_( v )
  {}

  V cons& argument() const 
  { return v_ ; }

  V const& v_ ;
};

Further reading