Symmetries

Detailed Description

Tools to use and detect symmetries in nda::Array objects.

Here is a simple example of how symmetries can be used in nda:

#include <nda/nda.hpp>
#include <nda/sym_grp.hpp>
 
#include <array>
#include <complex>
#include <functional>
#include <initializer_list>
#include <iostream>
#include <tuple>
#include <vector>
 
int main() {
  constexpr int N = 3;
 
  // typedefs for the symmetry group
  using idx_t      = std::array<long, 2>;
  using sym_t      = std::tuple<idx_t, nda::operation>;
  using sym_func_t = std::function<sym_t(idx_t const &)>;
 
  // create an hermitian matrix
  auto A = nda::array<std::complex<double>, 2>::rand(N, N);
  for (int i = 0; i < N; ++i) {
    for (int j = i + 1; j < N; ++j) A(i, j) = std::conj(A(j, i));
  }
 
  // hermitian symmetry (satisfies nda::NdaSymmetry)
  auto h_symmetry = [](idx_t const &x) {
    idx_t xp = {x[1], x[0]};
    return sym_t{xp, nda::operation{false, true}}; // sign flip = false, complex conjugate = true
  };
 
  // construct the symmetry group
  auto grp = nda::sym_grp{A, std::vector<sym_func_t>{h_symmetry}};
 
  // create an initializer function (satisfies nda::NdaInitFunc)
  auto init_func = [&A](idx_t const &x) { return std::apply(A, x); };
 
  // initialize a second array using the symmetry group and array A
  nda::array<std::complex<double>, 2> B(N, N);
  grp.init(B, init_func);
 
  // output A and B
  std::cout << A << std::endl;
  std::cout << B << std::endl;
 
  // get representative data (should be a vector of size 6)
  auto vec = grp.get_representative_data(A);
  std::cout << "\n" << nda::array_view<std::complex<double>, 1>(vec) << std::endl;
}

Output:

A:
[[(0.79407,0.479187),(0.0316266,-0.885449),(0.620883,-0.589129)]
 [(0.0316266,0.885449),(0.431919,0.784867),(0.25658,-0.197428)]
 [(0.620883,0.589129),(0.25658,0.197428),(0.972195,0.354817)]]
 
B:
[[(0.79407,0.479187),(0.0316266,-0.885449),(0.620883,-0.589129)]
 [(0.0316266,0.885449),(0.431919,0.784867),(0.25658,-0.197428)]
 [(0.620883,0.589129),(0.25658,0.197428),(0.972195,0.354817)]]
 
Representative data:
[(0.79407,0.479187),(0.0316266,-0.885449),(0.620883,-0.589129),(0.431919,0.784867),(0.25658,-0.197428),(0.972195,0.354817)]

It first creates a hermitian matrix A and uses the symmetry h_symmetry, i.e. A(i, j) = std::conj(A(j, i)), to construct the corresponding symmetry group. The symmetry group can then be used to initialize other arrays, get the representative data of an array with the same symmetry and more.

Concepts
concept	nda::NdaInitFunc
	Concept defining an initializer function.
concept	nda::NdaSymmetry
	Concept defining a symmetry in nda.

Classes
struct	nda::operation
	A structure to capture combinations of complex conjugation and sign flip operations. More...
class	nda::sym_grp< F, A >
	Class representing a symmetry group. More...

Functions
template<Array A>
bool	nda::is_valid (A const &a, std::array< long, static_cast< std::size_t >(get_rank< A >)> const &idx)
	Check if a multi-dimensional index is valid, i.e. not out of bounds, w.r.t. to a given nda::Array object.

Function Documentation

is_valid()

template<Array A>

bool nda::is_valid	(	A const &	a,
		std::array< long, static_cast< std::size_t >(get_rank< A >)> const &	idx )

#include <nda/sym_grp.hpp>

Check if a multi-dimensional index is valid, i.e. not out of bounds, w.r.t. to a given nda::Array object.

Template Parameters

A	nda::Array type.

Parameters

a	nda::Array object.
idx	Multi-dimensional index.

Returns

True if the index is not out of bounds, false otherwise.

Definition at line 78 of file sym_grp.hpp.