.. highlight:: c

.. _arith_expression: 

Arithmetic Expressions
-------------------------------------------------

* **Definition** :
  
  By `expression`, we mean here an object, typically representing a mathematical expression, 
  which models the :ref:`HasImmutableArrayInterface` concept.

* **Use** :
  
  Expression can be used : 
   - as RHS (Right Hand Side) of various operators (=, +=, -=, ...)
   - at any place where an `expression` is expected.

* **How to make expression ?**

  Expressions can be build easily in various ways : 

   - Using arithmetic operators, e.g. A + 2*B
     Examples ::

      array<long,2> A (2,2), B(2,2),C;
      C= A + 2*B;
      array<long,2> D( A+ 2*B);
      array<double,2> F( 0.5 * A);  // Type promotion is automatic
     
      // or even in C++0x : 
      auto e =  A + 2*B;           // expression, purely formal
      array<long,2> D(e);          // really makes the computation
      cout<< e <<endl ;            // prints the expression
      cout<< e(1,2) <<endl ;       // evaluates just at a point
      cout<< e.domain() <<endl ;   // just computes the domain

   - Using predefined expressions : 
      
       - matmul
 
         The code::
          
          matrix<double> M1(2,2), M2(2,2), M3;
          M3 = matmul(M1,M2);
         
         will do the following : 
           -  matmul returns a lazy object modelling the :ref:`Expression` concept.
           -  a result this is compiled as some `matmul_with_lapack(M1,M2,M3)` : **there is NO intermediate copy**.

        - mat_vec_mul

   - Building custom expressions. The transpose example...
      
      Describe the concept, what to implement, etc....
        
        - Transpose:: 

           array<long,2> A (2,2), B(2,2),C(2,2);
           C= A + 2*B;
           C = A + Transpose(B);         // Transpose(X) returns a lazy object that models HasImmutableArrayInterface. 
           C = A + Transpose(B + B);     // X can also be an expression...
           C = Transpose(B);             //
          
           // non square
           array<long,2> R(2,3),Rt(3,2);
           cout<<" R = "<< array<long,2>(Transpose(R)) <<endl;
          
           // mapping any function 
           C =  map_expr(&sqr,A);
           cout<<" C = "<< map_expr(&sqr,A,"SQR")<<" = "<<C<<endl;
          
           // matmul as expression Oi are 'C' or 'F'
           matrix<double,O1> M1(2,2); matrix<double,O2> M2(2,2); matrix<double,O3> M3;
           // The central instruction : note that matmul returns a lazy object 
           // that has ImmutableArray interface, and defines a specialized version assignment
           // As a result this is equivalent to some matmul_with_lapack(M1,M2,M3) : there is NO intermediate copy.
           M3 = matmul(M1,M2);
    

     See expression.hpp.
     At the moment, only +,-, * and / by scalar are implemented.
     
     An expression models HasImmutableArrayInterface, i.e. :
      
     * It has a domain (computed from the expression)
     * It can be evaluated.
     
     It is then easy to mix them with other objects,
     that model the same concept. See e.g. expr2.cpp (map_expr) for examples.
     
     
     * *Multiplication* : 
     not decided, since it is not the same for array or matrices.
     Two choices : 

      * Do not add \* for array, matrices (use matmul e.g.) and allow mixing array, matrix
        e.g. add an array<int,2> and a matrix <int>

      * Add the \*, but then do different expression for array and matrix/vector,
        then one can not mix them.
        In that case, it is however trivial to say e.g. M + matrix_view<int>(A) if A is an array.