.. highlight:: c

Linear algebra 
===============================================

Several standard linear algebra operations are provided for the matrix and vector.
Note however that matrix and vector are not the main purpose of this library, 
hence the linear algebra support is less extended than other purely matrix library, 
like e.g. Eigen.

The computation are done when possible by calling lapack and blas.
In some important cases (like matrix product), a slow but generic version is also
provided for type that lapack do not treat (e.g. matrix of int, matrix of custom objects)
but be aware that these are not optimized for performance and should be used only in 
non critical part of the codes.

matrix product
--------------------

The * operator map the matrix x matrix and matrix x vector product.

Example : matrix * matrix and * vector ...

.. compileblock::

   #include <triqs/arrays.hpp>
   using triqs::arrays::matrix; using triqs::clef::placeholder;
   int main() {  
    // declare an init 2 matrices 
    placeholder<0> i_; placeholder<1> j_;
    matrix<double> A (2,2), B(2,2), C; 
    A(i_,j_) <<  i_ + j_ ; B(i_,j_) <<  2*i_ + j_ ;

    // Making the product
    C= A*B;
    // Note that the * returns a lazy object 
    // that has ImmutableArray concept, and defines a specialized version assignment.
    // compiler rewrites this as something like matmul_with_lapack (A,B,C);
    // There are no temporary here.

   std::cout<< " C = " << C<< std::endl;   
   std::cout<< " A*B = " << A*B << std::endl; // NB A*B does not return a matrix, but 
   // a lazy expression which is not evaluated until needed. 
   }
 
For types that lapack do not use, a generic version of the matrix product is provided.
(same syntax, the dispatch is made at compile time depending of the type of the matrices).

NB : 

* The **generic** version of the product will be used if
both matrices do not have the same type.
**Is this reasonnable ??**.

* Is the matrix's slowest index is not contiguous, blas/lapack can not be called directly.
  So a temporary copy will be made to reorganize the matrix before calling blas/lapack.
  Code is therefore (normally !) always correct, but this can produce a serious performance hit.


Matrix inversion
----------------------

The inverse function return a lazy inverse of any object which has ImmutableMatrix concept
and can therefore be mixed with any other matrix expression.

Example : TO BE WRITTEN


LU decomposition
----------------------

Done. doc to be written

Diagonalization 
-------------------
Done. doc to be written

SVD decomposition
-------------------

To be done 

Interface with Eigen
------------------------

To be implemented. Only possible for when order is known at compile time.