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#+TITLE: Hermitian Grid
* Description
The =GridHermitian= class is an advanced container in /Tamaas/. It aims at optimizing storage and computation times on complex data that exhibits the Hermitian symetry, i.e.:
$$f(-q) = \bar{f}(q)\text{ or in 2D }f(-q_1, -q_2) = \bar{f}(q_1, q_2)$$
Where the $\bar{\cdot}$ symbol denotes the complex conjugate. The template parameters of
=GridHermitian= are:
- =T= the underlying type stored[fn:1]
- =dim= the dimension of the grid
[fn:1] the actual type stored is =std::complex<T>=.
* Surface size
Let =g= be a grid of size $n_1\times n_2\times\dotsb\times n_d$ containing real numbers. Its Fourier transform has an Hermitian symetry. Due to the nature of this symetry, the actual size of the result of the Fourier transform can be reduced to:
#+LABEL: eqn:hermitian
$$n_1\times n_2\times\dotsb\times \lfloor\frac{n_d}{2}+1\rfloor$$
Which is about half of the original grid.
* Creation
Like =Grid=, there are two ways to create a =GridHermitian<T, dim>= instance:
- =GridHermitian()= initializes an empty grid
- =GridHermitian(n[], L[], nb_components)= initializes a grid with specified number of elements, physical dimensions and number of components per-point of grid.
Note that the second constructor needs the true size of the Hermitian grid (equation above) not the size of the normal grid.
* Accessing elements
The =()= operator is used in the same way as that of =Grid=, however the range of values accessible is larger than the acutal size of the grid. If an element outside the grid is accessed, two things can happen:
- If the =()= operator was called with only one parameter:
- an exception is raised (in debug mode)
- 0 is returned
- If the =()= operator was called with more than one parameter:
- the conjugate of the corresponding value is computed and returned
This system allows backward compatibility with previous loops that were written for full surfaces.
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