.. highlight:: c

array and matrix/vector algebra 
=======================================================

Arrays and matrices can be combined in formal algebraic expressions, which models the :ref:`HasImmutableArrayInterface` concept.
This algebraic expressions can therefore be used as RHS of assignment (SEE) or in array/matrix contructors.

.. warning::
   To use this technique, you have to include the <arrays/expressions/array_algebra.hpp> or <arrays/expressions/matrix_algebra.hpp> 
   files.

For example ::
  
   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

The technique is called `expression templates`. It allows the elimination of temporaries, so that the clear 
and readable code::

   Z= A + 2*B + C/2;

is in fact rewritten by the compiler into ::
 
   for (i,j,...) indices of A, B  : 
     C(i,j) = A(i,j) + 2* B(i,j) + C(i,j)/2

instead of making a chain of temporaries (C/2, 2*B, 2*B + C/2...) that "ordinary" object-oriented programming would produce.
As a result, the produced code is as fast as if you were writing the loop yourself,
but with several advantages : 

* It is more **compact** and **readable** : you don't have to write the loop, and he indices range are computed automatically.
* It is much better for **optimization** : 
  
  * What you want is to tell the compiler/library to compute this expression, not *how* to do it optimally on a given machine.
  * For example, since the memory layout is decided at compile time, the library can traverse the data
    in an optimal way, allowing machine-dependent optimization.
  * The library can perform easy optimisations, e.g. for vector it will use blas if possible.
  
Arrays vs matrices
----------------------

Because their multiplication is not the same, arrays and matrices don't form the same algebra.
Mixing them in expression  would therefore be meaningless and it is therefore not allowed ::

   array<long,2> A;
   matrix<long,2> M;
   M + A; // --> ERROR

However, you can always make a matrix_view from a array of rank 2 ::
  
   A + make_matrix_view(M); //--> OK.

.. note::

   Making view is very cheap, it only copies the index systems. 

Expressions are lazy
---------------------------

This means that constructing an expression is separated from evaluating it ::

   auto e =  A + 2*B;             // expression, purely formal, no computation is done
   cout<< e <<endl ;              // prints the expression
   cout<< e(1,2) <<endl ;         // evaluates just at a point
   cout<< e.domain() <<endl ;     // just computes its domain
   array<long,2> D(e);            // now really makes the computation and store the result in D.

The expression type is complicated (the expression in stored in the C++ type), so we used here 
the C++0x `auto` to make simple things simple...
  
FAQ
----------

Where can expressions be used ?
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Expressions models :ref:`HasImmutableArrayInterface` concept, so they can used *anywhere* 
an object modeling this concept is accepted, e.g. : 

* array, matrix contruction
* operator =, +=, -=, ...

They behave like an immutable array : they have a domain, they can be evaluated.
When `C` is assigned to the expression in the previous example, 
the compiler just needs to compute the new domain for `C`, resize it and fill it by evaluation of the expression.

What is the cost of this technique ?
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Thanks to the Boost Proto library, this can be done :

* with an acceptable increase in compilation time (try it !).
* in about *2 pages of readable code* !