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First commit : triqs libs version 1.0 alpha1 for earlier commits, see TRIQS0.x repository.
2013-07-17 19:24:07 +02:00
.. highlight:: c
Green functions
#################
Here are the concepts and techniques related to the Green functions.
All Green functions are implemented as a gf<Descriptor> and gf_view<Descriptor> where :
* Descriptor contains everything specific to the Green function
(number of variables, domain of definition, interpolation techniques, etc...).
* The gf/gf_view class implements the generic part (hdf5, view mechanism, interaction with clef library, etc...).
We first present the concept of the Descriptor, and the concepts of its elements.
In a second part, we present a more detailed description of gf/gf_view.
Pure functions on domains
==============================
This formalize a mathematical function :
* 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.
Domain concept
-------------------------------------------------
* **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** : CopyConstructible, DefaultContructible, EqualComparable, BoostSerializable, H5-serializable.
* **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. In particular, in |
| | Matsubara, it is a complex. |
+----------------------------------------------------------------------------+---------------------------------------------------------------------+
* **Examples** :
* Matsubara frequencies (boson/fermion)
* Matsubara time
* Real frequencies
* Real time
* Brillouin zone
PureFunctionOnDomain
-----------------------
* **Purpose** : A function from a domain to a target space.
* **Refines** :
* **Definition** :
+--------------------------------------+----------------------------------------------------------+
| Elements | Comment |
+======================================+==========================================================+
| domain_t | Type of the Domain represented, modelling Domain concept |
+--------------------------------------+----------------------------------------------------------+
| domain_t const & domain() const | Returns the domain |
+--------------------------------------+----------------------------------------------------------+
| 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..).
Functions stored on a mesh
================================
A special kind of function is those stored on a mesh.
The mesh can be regular or not, multi-dimensionnal.
We first define the concept of a mesh, and then the concept of a function discretized over that mesh.
Mesh concept
-------------------------------------------------
* **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** : CopyConstructible, DefaultContructible, EqualComparable, BoostSerializable, H5-serializable.
* **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 |
+--------------------------------------------------------------+-------------------------------------------------------------------------------+
| size_t 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 |
+--------------------------------------------------------------+-------------------------------------------------------------------------------+
| size_t index_to_linear(index_t const &) const | Flattening the index of the mesh into a contiguous linear index |
+--------------------------------------------------------------+-------------------------------------------------------------------------------+
| mesh_point_t | 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> iterator | A generator of all the mesh points. |
+--------------------------------------------------------------+-------------------------------------------------------------------------------+
| iterator begin()/end() const | Standard access to iterator on the mesh |
+--------------------------------------------------------------+-------------------------------------------------------------------------------+
| friend bool operator == (mesh_t const& M1, mesh_t const &M2) | Comparison between meshes |
+--------------------------------------------------------------+-------------------------------------------------------------------------------+
* **Examples** : Some domains and the corresponding possible meshes.
+-----------------------------------------------------+--------------------------------------------------------+
| Domain Type | What does the corresponding mesh type contain ? |
+=====================================================+========================================================+
| Matsubara frequencies | Nmax, the max number of Matsubara Freq. |
+-----------------------------------------------------+--------------------------------------------------------+
| Matsubara time | The time slicing is on a mesh, or the Legendre mesh is |
| | we store the function with Legendre representation. |
+-----------------------------------------------------+--------------------------------------------------------+
| Real frequencies | Parameters of the mesh in frequency |
+-----------------------------------------------------+--------------------------------------------------------+
| Real time | |
+-----------------------------------------------------+--------------------------------------------------------+
| Brillouin zone | parameters of the mesh over the BZ, symmetry ? |
+-----------------------------------------------------+--------------------------------------------------------+
MeshPoint concept
-------------------------------------------------
* **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 |
+------------------------------------------------+-----------------------------------------------------------------------------+
| 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 |
+------------------------------------------------+-----------------------------------------------------------------------------+
| operator mesh_t::domain_t::point_t() const | cast to the corresponding domain point |
+------------------------------------------------+-----------------------------------------------------------------------------+
| Implements the basic operations on the domain | Only for non composite mesh |
| by using the cast operation | |
+------------------------------------------------+-----------------------------------------------------------------------------+
The MeshPoint mechanism
---------------------------
A MeshPoint is just a storage of a reference to the mesh and the index of the point in a custom structure.
The interest of having such a structure is that :
* The gf function has a operator()(mesh_t::mesh_point_t) (see below) which is a direct access to the data on the grid.
Hence if MP is a such a MeshPoint, g(MP) is equivalent to something like g.data_on_grid_at_index( MP.index)
* MP however can be casted to a point in the domain and therefore it *is* a domain_t::point_t as well.
As a result, g(MP) = 1/(MP + 2) makes senses iif it makes senses in the domain.
* Moreover, because of the iterator on the mesh, one can write ::
// 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
FunctionOnMesh concept
-----------------------------
* **Purpose** : A function from a domain to a target space, represented in a mesh.
This function can only be evaluated on the mesh points.
* **Refines** : BoostSerializable, H5-serializable.
* **Definition** :
+--------------------------------------------------+------------------------------------------------------------------------+
| Elements | Comment |
+==================================================+========================================================================+
| mesh_t | Type of the mesh representing the domain. |
+--------------------------------------------------+------------------------------------------------------------------------+
| mesh_t const & mesh() const | Returns the mesh. |
+--------------------------------------------------+------------------------------------------------------------------------+
| auto operator ( grid_pt<mesh_t> const &) const | Calling on a grid_pt gives direct access to the value on a grid point. |
| auto & operator ( grid_pt<mesh_t> const &) | Const and non const version. |
+--------------------------------------------------+------------------------------------------------------------------------+
* **NB** : the result type of the () operator is deduced. Not needed in the concept.
descriptors
======================================
A descriptor is a structure that contains everything specific to the Green function
(number of variables, domain of definition, interpolation techniques, etc...).
Descriptor concept
---------------------------
* **Purpose** :
* **Refines** :
* **NB** : Pure static, does NOT contains any data.
* **Definition** :
+------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+
| Elements | Comment |
+====================================================================================+===============================================================================+
| struct tag {}; | A tag for the gf |
+------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+
| mesh_t | Mesh for the gf, modeling Mesh concept |
+------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+
| storage_t | The type of the storage of the data (array, vector, etc....) |
+------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+
| singularity_t | Type of object storing the singularities of the gf. It is used e.g. in the |
| | Fourier transformation, density computation, etc... For a simple g(omega), |
| | g(t), it is typically a high frequency tail. For a more complex function |
| | g(nu,nu'), it can be different. |
+------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+
| symmetry_t | Type of the object storing the symmetry property of the Green function. It is |
| | *nothing* by default. This type must be a value (DefaultConstructible, |
| | CopyConstructible, Assignable) |
+------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+
| target_indices_t | Type of the indices of the gf, typically array<std::string,arity> |
+------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+
| static const int arity | Number of variable authorized in calling the gf (just for compile time check |
| | and nice error message, it is not really necessary) |
+------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+
| struct evaluator { auto operator()( mesh_t const &, DATA_t const &, S_t const &, | All the permitted const call of the gf ! (DATA_t defined below) with the |
| Args&&... args) .... as many overload as necessary } | parenthesis operator The gf<...> function create such a struct, so it can |
| | hold some data ... |
+------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+
| static std::string h5_name() | Name for hdf5 naming (attribute of the tree in which the gf is stored). |
+------------------------------------------------------------------------------------+-------------------------------------------------------------------------------+
* **Values vs Views**
target_t, singularity_t, indices_t are expected to be *values*.
The corresponding views, i.e., target_view_t, singularity_view_t, indices_view_t will be deduced from the value type, and
replaced by the value_type if no view is available.
* S_t is singularity_t or its corresponding view type (if it exists).
The gf/gf_view class
====================
The gf/gf_view classes are generic Green function, templated on Descriptor.
They handle :
* view/non view aspect, copy, etc...
* hdf5 interface
* clef interface
* the MeshPoint mechanism as explained above.
* perfect forwarding of all other const call of operator() to Descriptor.
Constructors are limited to a minimal number :
* empty one for gf (value semantics).
* copy construction, from gf and gf_view of the same descriptor.
* construction from the data of the GF.
The other custom construction are delegated to make_gf functions::
gf<Descriptor> make_gf (Descriptor, my_favorite arguments).
We use here the simplest dispatch using the fact that Descriptor is an empty struct,
so we can dispath the make_gf. Example of use ::
auto G = make_gf (matsubara_freq(), beta, Fermion, make_shape(2,2));
Doxygen documentation
======================
The :doxy:`full C++ documentation<triqs::gf::gf>` is available here.
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