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dft_tools/doc/reference/c++/gf/concepts.rst
Olivier Parcollet 1ecec0b933 gfs: Fix bug in imfreq with <0 freq.
- there was a confusion in gf imfreq, in the new case
where freq can be <0 (non real gf, or for product gf).
- index: is the matsubara n, as in the struct matsubara_freq
  index can be >0 or <0
- linear_index : is the shift from the 0. It is always >0.
  Fixed function to compute it.

- Also changed the construction of mesh_point in the generic iterator.
  Before, was constructed with a mesh point of index 0
  Now, added a new constructor on mesh_point_t, just taking the mesh
  which construct the *first* mesh_point.
  Fixed linear, discrete, product accordingly.
  Added to the documentation of the concepts of gf.
2014-01-05 20:10:33 +01:00

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.. highlight:: c
Concepts
#################
A Green function is simply a function, which has :
* a `domain` for its variable(s) (e.g. Matsubara/real time/frequencies, Legendre coefficients).
* a `target` space, i.e. the value of the Green function which can be :
* a scalar (double, complex)
* a matrix,
* another Green function (See below, currying Green functions ... REF ... ).
In this section, we define the general concepts for these objects.
First, we need to distinguish the `domain` on which the function is defined
from its representation in a computer, which we call a `mesh`.
.. note::
"mesh" should be understood here in a general and abstract way,
as the representation of the domain in the computer.
In most cases, it is indeed a real mesh on a domain (e.g. a Brillouin zone),
but the set of Legendre coefficients is also a mesh in our sense.
We will therefore now formally define the concept for `domain`, for `mesh`,
the notion of `pure function on a domain` (i.e. a mathematical Green function)
and the notion of `function on a grid`.
.. _Concept_Domain:
Domain
-------------------------------------------------
* **Purpose** : The domain of definition of a function. It is a mathematical definition of the domain,
and does not contain any mesh, or details on its representation in a computer.
* **Refines** : RegularType.
* **Definition** :
+----------+--------------------------------------------------------------------+
| Elements | Comment |
+==========+====================================================================+
| point_t | Type of element in the domain (int, int, double, k_vector, ...) as |
| | in the call of a function over this domain. |
+----------+--------------------------------------------------------------------+
* **Examples** :
* Matsubara time
* Matsubara frequencies (boson/fermion) : in this case, point_t is `matsubara_freq`, a simple type containing (n, beta, statistics).
* Real frequencies
* Real time
* Brillouin zone
* Cartesian product of previous domains to build multi-variable functions.
.. _Concept_PureFunctionOnDomain:
PureFunctionOnDomain
-----------------------
* **Purpose** :
A mathematical (pure) function from a domain to a target space.
* it has a domain of definition
* it can be called on any point of the domain, as a *pure* function, i.e. without any side effect.
* **Refines** :
* **Definition** :
+--------------------------------------+----------------------------------------------------------+
| Elements | Comment |
+======================================+==========================================================+
| domain_t const & domain() const | Returns the domain (deduced as domain_t) |
+--------------------------------------+----------------------------------------------------------+
| operator (domain_t::point_t) const | Calling for all elements of the Domain (including infty |
| | if it is in the domain... |
+--------------------------------------+----------------------------------------------------------+
* NB : Note that the return type of the function is *NOT* part of the concept,
it has to be deduced by the compiler (using C++11 decltype, std::result_of, eg..).
.. note::
Probably domain_t should also be deduced from the return type of domain ... TO BE CORRECTED
.. _Concept_Mesh:
Mesh
-------------------------------------------------
* **Purpose** : A mesh over a domain, and more generally the practical representation of the domain in a computer.
It does not really need to be a mesh : e.g. if the function is represented on a polynomial basis,
it is the parameters of this representation (max number of coordinates, e.g.)
* **Refines** : RegularType, H5-serializable, Printable.
* **Definition** :
+-------------------------------------------------------+-------------------------------------------------------------------------------+
| Elements | Comment |
+=======================================================+===============================================================================+
| domain_t | Type of the Domain represented, modeling the Domain concept |
+-------------------------------------------------------+-------------------------------------------------------------------------------+
| domain_t const & domain() const | Access to the domain |
+-------------------------------------------------------+-------------------------------------------------------------------------------+
| index_t | Type of indices of a point on the grid. Typically a tuple of long or a long |
+-------------------------------------------------------+-------------------------------------------------------------------------------+
| long size() const | The number of points in the mesh. |
+-------------------------------------------------------+-------------------------------------------------------------------------------+
| domain_t::point_t index_to_point(index_t) const | From the index of a mesh point, compute the corresponding point in the domain |
+-------------------------------------------------------+-------------------------------------------------------------------------------+
| long index_to_linear(index_t const &) const | Flattening the index of the mesh into a contiguous linear index |
+-------------------------------------------------------+-------------------------------------------------------------------------------+
| mesh_point_t | A type modeling MeshPoint concept (see below). |
+-------------------------------------------------------+-------------------------------------------------------------------------------+
| mesh_point_t operator[](index_t const & index ) const | From an index, return a mesh_point_t containing this a ref to this mesh and |
| | the index. |
+-------------------------------------------------------+-------------------------------------------------------------------------------+
| mesh_pt_generator<mesh_t> const_iterator | A generator of all the mesh points. |
+-------------------------------------------------------+-------------------------------------------------------------------------------+
| const_iterator begin()/end() const | Standard access to iterator on the mesh Standard access to iterator on the |
| cbegin()/cend() const | mesh |
+-------------------------------------------------------+-------------------------------------------------------------------------------+
+---------------------------+-----------------------------------------------------------+
| Free functions | Comment |
+===========================+===========================================================+
| void foreach (mesh_t, F) | If F is a function of synopsis |
| | auto F( mesh_t::mesh_point_t) |
| | it calls F for each point on the mesh, in arbitrary order |
+---------------------------+-----------------------------------------------------------+
.. _Concept_MeshPoint:
MeshPoint
-------------------------------------------------
* **Purpose** : Abstraction of a point on a mesh. A little more than a ref to the mesh and a index.
* **Refines** : CopyConstructible.
* **Definition** :
+------------------------------------------------+--------------------------------------------------------------------+
| Elements | Comment |
+================================================+====================================================================+
| mesh_t | Type of the mesh |
+------------------------------------------------+--------------------------------------------------------------------+
| mesh_t const * m | A pointer to the mesh to which the point belongs. |
+------------------------------------------------+--------------------------------------------------------------------+
| mesh_t::index_t index | The index of the point |
+------------------------------------------------+--------------------------------------------------------------------+
| mesh_point_t( mesh_t const &, index_t const &) | Constructor : a mesh point at the given index |
+------------------------------------------------+--------------------------------------------------------------------+
| mesh_point_t( mesh_t const &) | Constructor : the first mesh point |
+------------------------------------------------+--------------------------------------------------------------------+
| mesh_t::index_t [const &,] index() const | The index corresponding to the point |
+------------------------------------------------+--------------------------------------------------------------------+
| size_t linear_index() const | The linear index of the point (same as m->index_to_linear(index()) |
+------------------------------------------------+--------------------------------------------------------------------+
| void advance() | Advance to the next point on the mesh (used by iterators). |
+------------------------------------------------+--------------------------------------------------------------------+
| void at_end() | Is the point at the end of the grid |
+------------------------------------------------+--------------------------------------------------------------------+
| void reset() | Reset the mesh point to the first point |
+------------------------------------------------+--------------------------------------------------------------------+
| cast_t operator cast_t() const | == mesh_t::domain_t::point_t *implicit* cast to the corresponding |
| | domain point |
+------------------------------------------------+--------------------------------------------------------------------+
For one dimensional mesh, we also require that the MeshPoint implement the basic arithmetic operations
using the cast.
* **Discussion** :
A MeshPoint is just an index of a point on the mesh, and containers like gf can easily be overloaded for this type
to have a direct access to the grid (Cf [] operator of gf).
However, since the MeshPoint can be implicitely casted into the domain point, simple
expression like ::
g[p] = 1/ (p +2)
make sense and fill the corresponding point wiht the evaluation of 1/ (p+2) in the domain.
As a result, because iterating on a mesh result in a series of object modelling MeshPoint,
one can write naturally ::
// example of g, a Green function in Matsubara frequencies w
for (auto w : g.mesh())
g[w] = 1/(w + 2)
// This runs overs the mesh, and fills the function with 1/(w+2)
// In this expression, w is casted to the domain_t::point_t, here a complex<double>
// which allows to evaluate the function