| Current File : //usr/include/eigen3/unsupported/Eigen/CXX11/src/TensorSymmetry/util/TemplateGroupTheory.h |
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2013 Christian Seiler <christian@iwakd.de>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
#ifndef EIGEN_CXX11_TENSORSYMMETRY_TEMPLATEGROUPTHEORY_H
#define EIGEN_CXX11_TENSORSYMMETRY_TEMPLATEGROUPTHEORY_H
namespace Eigen {
namespace internal {
namespace group_theory {
/** \internal
* \file CXX11/src/TensorSymmetry/util/TemplateGroupTheory.h
* This file contains C++ templates that implement group theory algorithms.
*
* The algorithms allow for a compile-time analysis of finite groups.
*
* Currently only Dimino's algorithm is implemented, which returns a list
* of all elements in a group given a set of (possibly redundant) generators.
* (One could also do that with the so-called orbital algorithm, but that
* is much more expensive and usually has no advantages.)
*/
/**********************************************************************
* "Ok kid, here is where it gets complicated."
* - Amelia Pond in the "Doctor Who" episode
* "The Big Bang"
*
* Dimino's algorithm
* ==================
*
* The following is Dimino's algorithm in sequential form:
*
* Input: identity element, list of generators, equality check,
* multiplication operation
* Output: list of group elements
*
* 1. add identity element
* 2. remove identities from list of generators
* 3. add all powers of first generator that aren't the
* identity element
* 4. go through all remaining generators:
* a. if generator is already in the list of elements
* -> do nothing
* b. otherwise
* i. remember current # of elements
* (i.e. the size of the current subgroup)
* ii. add all current elements (which includes
* the identity) each multiplied from right
* with the current generator to the group
* iii. add all remaining cosets that are generated
* by products of the new generator with itself
* and all other generators seen so far
*
* In functional form, this is implemented as a long set of recursive
* templates that have a complicated relationship.
*
* The main interface for Dimino's algorithm is the template
* enumerate_group_elements. All lists are implemented as variadic
* type_list<typename...> and numeric_list<typename = int, int...>
* templates.
*
* 'Calling' templates is usually done via typedefs.
*
* This algorithm is an extended version of the basic version. The
* extension consists in the fact that each group element has a set
* of flags associated with it. Multiplication of two group elements
* with each other results in a group element whose flags are the
* XOR of the flags of the previous elements. Each time the algorithm
* notices that a group element it just calculated is already in the
* list of current elements, the flags of both will be compared and
* added to the so-called 'global flags' of the group.
*
* The rationale behind this extension is that this allows not only
* for the description of symmetries between tensor indices, but
* also allows for the description of hermiticity, antisymmetry and
* antihermiticity. Negation and conjugation each are specific bit
* in the flags value and if two different ways to reach a group
* element lead to two different flags, this poses a constraint on
* the allowed values of the resulting tensor. For example, if a
* group element is reach both with and without the conjugation
* flags, it is clear that the resulting tensor has to be real.
*
* Note that this flag mechanism is quite generic and may have other
* uses beyond tensor properties.
*
* IMPORTANT:
* This algorithm assumes the group to be finite. If you try to
* run it with a group that's infinite, the algorithm will only
* terminate once you hit a compiler limit (max template depth).
* Also note that trying to use this implementation to create a
* very large group will probably either make you hit the same
* limit, cause the compiler to segfault or at the very least
* take a *really* long time (hours, days, weeks - sic!) to
* compile. It is not recommended to plug in more than 4
* generators, unless they are independent of each other.
*/
/** \internal
*
* \class strip_identities
* \ingroup CXX11_TensorSymmetry_Module
*
* \brief Cleanse a list of group elements of the identity element
*
* This template is used to make a first pass through all initial
* generators of Dimino's algorithm and remove the identity
* elements.
*
* \sa enumerate_group_elements
*/
template<template<typename, typename> class Equality, typename id, typename L> struct strip_identities;
template<
template<typename, typename> class Equality,
typename id,
typename t,
typename... ts
>
struct strip_identities<Equality, id, type_list<t, ts...>>
{
typedef typename conditional<
Equality<id, t>::value,
typename strip_identities<Equality, id, type_list<ts...>>::type,
typename concat<type_list<t>, typename strip_identities<Equality, id, type_list<ts...>>::type>::type
>::type type;
constexpr static int global_flags = Equality<id, t>::global_flags | strip_identities<Equality, id, type_list<ts...>>::global_flags;
};
template<
template<typename, typename> class Equality,
typename id
EIGEN_TPL_PP_SPEC_HACK_DEFC(typename, ts)
>
struct strip_identities<Equality, id, type_list<EIGEN_TPL_PP_SPEC_HACK_USE(ts)>>
{
typedef type_list<> type;
constexpr static int global_flags = 0;
};
/** \internal
*
* \class dimino_first_step_elements_helper
* \ingroup CXX11_TensorSymmetry_Module
*
* \brief Recursive template that adds powers of the first generator to the list of group elements
*
* This template calls itself recursively to add powers of the first
* generator to the list of group elements. It stops if it reaches
* the identity element again.
*
* \sa enumerate_group_elements, dimino_first_step_elements
*/
template<
template<typename, typename> class Multiply,
template<typename, typename> class Equality,
typename id,
typename g,
typename current_element,
typename elements,
bool dont_add_current_element // = false
>
struct dimino_first_step_elements_helper
#ifndef EIGEN_PARSED_BY_DOXYGEN
: // recursive inheritance is too difficult for Doxygen
public dimino_first_step_elements_helper<
Multiply,
Equality,
id,
g,
typename Multiply<current_element, g>::type,
typename concat<elements, type_list<current_element>>::type,
Equality<typename Multiply<current_element, g>::type, id>::value
> {};
template<
template<typename, typename> class Multiply,
template<typename, typename> class Equality,
typename id,
typename g,
typename current_element,
typename elements
>
struct dimino_first_step_elements_helper<Multiply, Equality, id, g, current_element, elements, true>
#endif // EIGEN_PARSED_BY_DOXYGEN
{
typedef elements type;
constexpr static int global_flags = Equality<current_element, id>::global_flags;
};
/** \internal
*
* \class dimino_first_step_elements
* \ingroup CXX11_TensorSymmetry_Module
*
* \brief Add all powers of the first generator to the list of group elements
*
* This template takes the first non-identity generator and generates the initial
* list of elements which consists of all powers of that generator. For a group
* with just one generated, it would be enumerated after this.
*
* \sa enumerate_group_elements
*/
template<
template<typename, typename> class Multiply,
template<typename, typename> class Equality,
typename id,
typename generators
>
struct dimino_first_step_elements
{
typedef typename get<0, generators>::type first_generator;
typedef typename skip<1, generators>::type next_generators;
typedef type_list<first_generator> generators_done;
typedef dimino_first_step_elements_helper<
Multiply,
Equality,
id,
first_generator,
first_generator,
type_list<id>,
false
> helper;
typedef typename helper::type type;
constexpr static int global_flags = helper::global_flags;
};
/** \internal
*
* \class dimino_get_coset_elements
* \ingroup CXX11_TensorSymmetry_Module
*
* \brief Generate all elements of a specific coset
*
* This template generates all the elements of a specific coset by
* multiplying all elements in the given subgroup with the new
* coset representative. Note that the first element of the
* subgroup is always the identity element, so the first element of
* ther result of this template is going to be the coset
* representative itself.
*
* Note that this template accepts an additional boolean parameter
* that specifies whether to actually generate the coset (true) or
* just return an empty list (false).
*
* \sa enumerate_group_elements, dimino_add_cosets_for_rep
*/
template<
template<typename, typename> class Multiply,
typename sub_group_elements,
typename new_coset_rep,
bool generate_coset // = true
>
struct dimino_get_coset_elements
{
typedef typename apply_op_from_right<Multiply, new_coset_rep, sub_group_elements>::type type;
};
template<
template<typename, typename> class Multiply,
typename sub_group_elements,
typename new_coset_rep
>
struct dimino_get_coset_elements<Multiply, sub_group_elements, new_coset_rep, false>
{
typedef type_list<> type;
};
/** \internal
*
* \class dimino_add_cosets_for_rep
* \ingroup CXX11_TensorSymmetry_Module
*
* \brief Recursive template for adding coset spaces
*
* This template multiplies the coset representative with a generator
* from the list of previous generators. If the new element is not in
* the group already, it adds the corresponding coset. Finally it
* proceeds to call itself with the next generator from the list.
*
* \sa enumerate_group_elements, dimino_add_all_coset_spaces
*/
template<
template<typename, typename> class Multiply,
template<typename, typename> class Equality,
typename id,
typename sub_group_elements,
typename elements,
typename generators,
typename rep_element,
int sub_group_size
>
struct dimino_add_cosets_for_rep;
template<
template<typename, typename> class Multiply,
template<typename, typename> class Equality,
typename id,
typename sub_group_elements,
typename elements,
typename g,
typename... gs,
typename rep_element,
int sub_group_size
>
struct dimino_add_cosets_for_rep<Multiply, Equality, id, sub_group_elements, elements, type_list<g, gs...>, rep_element, sub_group_size>
{
typedef typename Multiply<rep_element, g>::type new_coset_rep;
typedef contained_in_list_gf<Equality, new_coset_rep, elements> _cil;
constexpr static bool add_coset = !_cil::value;
typedef typename dimino_get_coset_elements<
Multiply,
sub_group_elements,
new_coset_rep,
add_coset
>::type coset_elements;
typedef dimino_add_cosets_for_rep<
Multiply,
Equality,
id,
sub_group_elements,
typename concat<elements, coset_elements>::type,
type_list<gs...>,
rep_element,
sub_group_size
> _helper;
typedef typename _helper::type type;
constexpr static int global_flags = _cil::global_flags | _helper::global_flags;
/* Note that we don't have to update global flags here, since
* we will only add these elements if they are not part of
* the group already. But that only happens if the coset rep
* is not already in the group, so the check for the coset rep
* will catch this.
*/
};
template<
template<typename, typename> class Multiply,
template<typename, typename> class Equality,
typename id,
typename sub_group_elements,
typename elements
EIGEN_TPL_PP_SPEC_HACK_DEFC(typename, empty),
typename rep_element,
int sub_group_size
>
struct dimino_add_cosets_for_rep<Multiply, Equality, id, sub_group_elements, elements, type_list<EIGEN_TPL_PP_SPEC_HACK_USE(empty)>, rep_element, sub_group_size>
{
typedef elements type;
constexpr static int global_flags = 0;
};
/** \internal
*
* \class dimino_add_all_coset_spaces
* \ingroup CXX11_TensorSymmetry_Module
*
* \brief Recursive template for adding all coset spaces for a new generator
*
* This template tries to go through the list of generators (with
* the help of the dimino_add_cosets_for_rep template) as long as
* it still finds elements that are not part of the group and add
* the corresponding cosets.
*
* \sa enumerate_group_elements, dimino_add_cosets_for_rep
*/
template<
template<typename, typename> class Multiply,
template<typename, typename> class Equality,
typename id,
typename sub_group_elements,
typename elements,
typename generators,
int sub_group_size,
int rep_pos,
bool stop_condition // = false
>
struct dimino_add_all_coset_spaces
{
typedef typename get<rep_pos, elements>::type rep_element;
typedef dimino_add_cosets_for_rep<
Multiply,
Equality,
id,
sub_group_elements,
elements,
generators,
rep_element,
sub_group_elements::count
> _ac4r;
typedef typename _ac4r::type new_elements;
constexpr static int new_rep_pos = rep_pos + sub_group_elements::count;
constexpr static bool new_stop_condition = new_rep_pos >= new_elements::count;
typedef dimino_add_all_coset_spaces<
Multiply,
Equality,
id,
sub_group_elements,
new_elements,
generators,
sub_group_size,
new_rep_pos,
new_stop_condition
> _helper;
typedef typename _helper::type type;
constexpr static int global_flags = _helper::global_flags | _ac4r::global_flags;
};
template<
template<typename, typename> class Multiply,
template<typename, typename> class Equality,
typename id,
typename sub_group_elements,
typename elements,
typename generators,
int sub_group_size,
int rep_pos
>
struct dimino_add_all_coset_spaces<Multiply, Equality, id, sub_group_elements, elements, generators, sub_group_size, rep_pos, true>
{
typedef elements type;
constexpr static int global_flags = 0;
};
/** \internal
*
* \class dimino_add_generator
* \ingroup CXX11_TensorSymmetry_Module
*
* \brief Enlarge the group by adding a new generator.
*
* It accepts a boolean parameter that determines if the generator is redundant,
* i.e. was already seen in the group. In that case, it reduces to a no-op.
*
* \sa enumerate_group_elements, dimino_add_all_coset_spaces
*/
template<
template<typename, typename> class Multiply,
template<typename, typename> class Equality,
typename id,
typename elements,
typename generators_done,
typename current_generator,
bool redundant // = false
>
struct dimino_add_generator
{
/* this template is only called if the generator is not redundant
* => all elements of the group multiplied with the new generator
* are going to be new elements of the most trivial coset space
*/
typedef typename apply_op_from_right<Multiply, current_generator, elements>::type multiplied_elements;
typedef typename concat<elements, multiplied_elements>::type new_elements;
constexpr static int rep_pos = elements::count;
typedef dimino_add_all_coset_spaces<
Multiply,
Equality,
id,
elements, // elements of previous subgroup
new_elements,
typename concat<generators_done, type_list<current_generator>>::type,
elements::count, // size of previous subgroup
rep_pos,
false // don't stop (because rep_pos >= new_elements::count is always false at this point)
> _helper;
typedef typename _helper::type type;
constexpr static int global_flags = _helper::global_flags;
};
template<
template<typename, typename> class Multiply,
template<typename, typename> class Equality,
typename id,
typename elements,
typename generators_done,
typename current_generator
>
struct dimino_add_generator<Multiply, Equality, id, elements, generators_done, current_generator, true>
{
// redundant case
typedef elements type;
constexpr static int global_flags = 0;
};
/** \internal
*
* \class dimino_add_remaining_generators
* \ingroup CXX11_TensorSymmetry_Module
*
* \brief Recursive template that adds all remaining generators to a group
*
* Loop through the list of generators that remain and successively
* add them to the group.
*
* \sa enumerate_group_elements, dimino_add_generator
*/
template<
template<typename, typename> class Multiply,
template<typename, typename> class Equality,
typename id,
typename generators_done,
typename remaining_generators,
typename elements
>
struct dimino_add_remaining_generators
{
typedef typename get<0, remaining_generators>::type first_generator;
typedef typename skip<1, remaining_generators>::type next_generators;
typedef contained_in_list_gf<Equality, first_generator, elements> _cil;
typedef dimino_add_generator<
Multiply,
Equality,
id,
elements,
generators_done,
first_generator,
_cil::value
> _helper;
typedef typename _helper::type new_elements;
typedef dimino_add_remaining_generators<
Multiply,
Equality,
id,
typename concat<generators_done, type_list<first_generator>>::type,
next_generators,
new_elements
> _next_iter;
typedef typename _next_iter::type type;
constexpr static int global_flags =
_cil::global_flags |
_helper::global_flags |
_next_iter::global_flags;
};
template<
template<typename, typename> class Multiply,
template<typename, typename> class Equality,
typename id,
typename generators_done,
typename elements
>
struct dimino_add_remaining_generators<Multiply, Equality, id, generators_done, type_list<>, elements>
{
typedef elements type;
constexpr static int global_flags = 0;
};
/** \internal
*
* \class enumerate_group_elements_noid
* \ingroup CXX11_TensorSymmetry_Module
*
* \brief Helper template that implements group element enumeration
*
* This is a helper template that implements the actual enumeration
* of group elements. This has been split so that the list of
* generators can be cleansed of the identity element before
* performing the actual operation.
*
* \sa enumerate_group_elements
*/
template<
template<typename, typename> class Multiply,
template<typename, typename> class Equality,
typename id,
typename generators,
int initial_global_flags = 0
>
struct enumerate_group_elements_noid
{
typedef dimino_first_step_elements<Multiply, Equality, id, generators> first_step;
typedef typename first_step::type first_step_elements;
typedef dimino_add_remaining_generators<
Multiply,
Equality,
id,
typename first_step::generators_done,
typename first_step::next_generators, // remaining_generators
typename first_step::type // first_step elements
> _helper;
typedef typename _helper::type type;
constexpr static int global_flags =
initial_global_flags |
first_step::global_flags |
_helper::global_flags;
};
// in case when no generators are specified
template<
template<typename, typename> class Multiply,
template<typename, typename> class Equality,
typename id,
int initial_global_flags
>
struct enumerate_group_elements_noid<Multiply, Equality, id, type_list<>, initial_global_flags>
{
typedef type_list<id> type;
constexpr static int global_flags = initial_global_flags;
};
/** \internal
*
* \class enumerate_group_elements
* \ingroup CXX11_TensorSymmetry_Module
*
* \brief Enumerate all elements in a finite group
*
* This template enumerates all elements in a finite group. It accepts
* the following template parameters:
*
* \tparam Multiply The multiplication operation that multiplies two group elements
* with each other.
* \tparam Equality The equality check operation that checks if two group elements
* are equal to another.
* \tparam id The identity element
* \tparam _generators A list of (possibly redundant) generators of the group
*/
template<
template<typename, typename> class Multiply,
template<typename, typename> class Equality,
typename id,
typename _generators
>
struct enumerate_group_elements
: public enumerate_group_elements_noid<
Multiply,
Equality,
id,
typename strip_identities<Equality, id, _generators>::type,
strip_identities<Equality, id, _generators>::global_flags
>
{
};
} // end namespace group_theory
} // end namespace internal
} // end namespace Eigen
#endif // EIGEN_CXX11_TENSORSYMMETRY_TEMPLATEGROUPTHEORY_H
/*
* kate: space-indent on; indent-width 2; mixedindent off; indent-mode cstyle;
*/