doc : gf, fourier,meshes

doc/reference/c++/gf/fourier.rst

@@ -37,10 +37,10 @@ It will be used to calculate the (inverse) Fourier transform, in real/imaginary
 
 The DFT transforms of a sequence of :math:`N` complex numbers :math:`f_0...,f_{N-1}` into a sequence of :math:`N` complex numbers :math:`\tilde f_0...,\tilde f_{N-1}` according to the formula:
        :label: _DFT
-    .. math:: \tilde f_m = \sum_{k=0}^{N-1} f_k e^{-i 2 \pi m k / N}.
+    .. math:: \tilde f_k = \sum_{n=0}^{N-1} f_n e^{-i 2 \pi k n / N}.
 The inverse DFT formula is
        :label: _inv_DFT
-    .. math:: f_k = \frac{1}{N} \sum_{m=0}^{N-1} \tilde f_m e^{i 2 \pi m k / N}.
+    .. math:: f_n = \frac{1}{N} \sum_{k=0}^{N-1} \tilde f_k e^{i 2 \pi k n / N}.
 
 
 
@@ -54,17 +54,15 @@ The times are :math:`t_k=t_{min}+k\delta t` and the frequencies :math:`\omega_m=
 
 By approximating Eq. :ref:`TF_R` by 
     .. math:: \tilde G(\omega_m) = \delta t \sum_{k=0}^{N_t} G(t_k) e^{i\omega_m t_k}, 
-we recognize a DFT (Eq. :ref:`DFT`). To calculate it using FFTW, we first need to prepare the input:
+we recognize an inverse DFT (Eq. :ref:`inv_DFT`). To calculate it using FFTW, we first need to prepare the input :math:`\tilde f_k`, then to do the DFT and finally to modify the output to obtain :math:`\tilde G(\omega_m)` using the two formulas:
-    .. math:: f_k = G(t_k) e^{i \omega_{min}t_k},
+    .. math:: \tilde f_k = G(t_k) e^{i \omega_{min}t_k},
-then to do the DFT and finally to modify the output to obtain :math:`\tilde G(\omega_m)` as
+    .. math:: \tilde G(\omega_m) = \delta t f_m e^{i t_{min}(\omega_m-\omega_{min})}.
-    .. math:: \tilde G(\omega_m) = \delta t \tilde f_m e^{i t_{min}(\omega_m-\omega_{min})}.
 
 Similarly, the inverse transformation is obtained by approximating Eq. :ref:`eq_inv_TF_R` by 
     .. math:: G(t_k)=\frac{\delta\omega}{2\pi}\sum_{m=0}^{N_\omega} \tilde G(\omega_m)e^{-i\omega_m t_k},
-we recognize an inverse DFT (Eq. :ref:`inv_DFT`). To calculate it using FFTW, we first need to prepare the input: 
+we recognize a DFT (Eq. :ref:`DFT`). To calculate it using FFTW, we first need to prepare the input :math:`f_m`, then to do the inverse DFT and finally to modify the output to obtain :math:`G(t_k)`:
-    .. math:: \tilde f_m = \tilde G(\omega_m) e^{-i t_{min}\omega_m},
+    .. math:: f_m = \tilde G(\omega_m) e^{-i t_{min}\omega_m},
-then to do the inverse DFT and finally to modify the output to obtain :math:`G(t_k)` as
+    .. math:: G(t_k) = \frac{1}{N_t \delta t}\tilde f_k e^{-i \omega_{min}(t_k-t_{min})}.
-    .. math:: G(t_k) = \frac{1}{N_t \delta t}f_k e^{-i \omega_{min}(t_k-t_{min})},
 
 
 
@@ -74,24 +72,55 @@ Implementation in imaginary time/frequency using FFTW
 
 The imaginary time mesh parameters are :math:`\beta` and :math:`N_\tau`, plus a tag ``half_bins``, ``full_bins`` or ``without_last``. 
 In the ``full_bins`` case, one point of the time GF has to be removed for the fourier transform. 
-From these parameters, we deduce :math:`\delta\tau=\beta/N_\tau`
+From these parameters, we deduce :math:`\delta\tau=\beta/N_\tau`. 
 
-CHAPTER NOT FINISHED !!!! It seems that only real GF's in time are considered (w_n is always >0)... 
+For the imaginary frequency mesh, the mesh parameters are :math:`\beta`, :math:`n_{min}` and :math:`N_{\omega_n}`. 
-
-For the imaginary frequency mesh, they are :math:`n_{min}`, :math:`\beta` and :math:`N_\omega`. 
 From them, we deduce :math:`\delta\omega=\frac{2\pi}{\beta}`. 
+
 The Fourier transform requires :math:`N_\omega=N_\tau`. 
 The times are :math:`\tau_k=\tau_{min}+k\delta\tau` and the frequencies :math:`\omega_n=\omega_{min}+n\delta \omega`. 
 :math:`\tau_{min}` is either 0 or :math:`\delta\tau/2` depending on the mesh kind.  
-:math:`\omega_{min}` is either :math:`\frac{2\pi(n_{min}+1)}{\beta}` or :math:`\frac{2\pi n_{min}}{\beta}` depending on the statistic.
+:math:`\omega_{min}` is either :math:`\frac{2\pi(n_{min}+1)}{\beta}` or :math:`\frac{2\pi n_{min}}{\beta}` depending on the statistics.
 
 We approximate the TF and its inverse by
     .. math:: \tilde G(i\omega_n) = \delta\tau \sum_{k=0}^{N_\tau} G(\tau_k)e^{i\omega_n \tau_k}
-    .. math:: G(\tau_k) = \sum_{n=0}^{N_\tau} \frac{1}{\beta} \tilde G(i\omega_n)e^{-i\omega_n \tau_k}
+    .. math:: G(\tau_k) = \sum_{n=n_{min}}^{N_\tau} \frac{1}{\beta} \tilde G(i\omega_n)e^{-i\omega_n \tau_k}
 
 We use for the TF:
-    .. math:: f_k = G(\tau_k) e^{i \omega_{min}\tau_k},
+    .. math:: \tilde f_k = G(\tau_k) e^{i \omega_{min}\tau_k},
-    .. math:: \tilde G(i\omega_m) = \frac{\beta}{N_\tau} \tilde f_m e^{i \tau_{min}(\omega_m-\omega_{min})}.
+    .. math:: \tilde G(i\omega_n) = \frac{\beta}{N_\tau} f_n e^{i \tau_{min}(\omega_n-\omega_{min})}.
+and for the inverse TF:
+    .. math:: f_m = \frac{1}{\beta}\tilde G(i\omega_n) e^{-i t_{min}\omega_n},
+    .. math:: G(t_k) = \tilde f_k e^{-i \omega_{min}(\tau_k-\tau_{min})},
+
+
+Special case of real functions in time for fermions
+----------------------------------------------------
+
+In this case, :math:`G(i\omega_n)=conj(G(i\omega_n))` and we only store the values of :math:`G(i\omega_n)` for :math:`\omega_n > 0`. 
+The Eq. :ref:`inv_DFT_I` becomes: 
+
+    :label: _inv_TF_I_real_fermion
+    .. math:: G(\tau)=\sum_{n=0}^\infty \frac{2}{\beta} \tilde G(i\omega_n)\cos(\omega_n \tau)
+
+The inverse TF formulas are in this case 
+    .. math:: f_m = \frac{2}{\beta}\tilde G(i\omega_n) e^{-i t_{min}\omega_n},
+    .. math:: G(t_k) = \tilde f_k e^{-i \omega_{min}(\tau_k-\tau_{min})},
+
+Special case of real functions in time for bosons
+--------------------------------------------------
+
+In this case, :math:`G(i\omega_n)=conj(G(i\omega_n))` and we only store the values of :math:`G(i\omega_n)` for :math:`\omega_n \ge 0`. 
+The Eq. :ref:`inv_DFT_I` becomes: 
+
+    :label: _inv_TF_I_real_bosons
+    .. math:: G(\tau)=\frac{1}{\beta} \tilde G(0)+\sum_{n=1}^\infty \frac{2}{\beta} \tilde G(i\omega_n)\cos(\omega_n \tau)
+
+The inverse TF formulas are in this case 
+    .. math:: f_0 = \frac{1}{\beta}\tilde G(0),
+    .. math:: f_m = \frac{2}{\beta}\tilde G(i\omega_n) \cos(t_{min}\omega_n),
+    .. math:: G(t_k) = \tilde f_k e^{-i \omega_{min}(\tau_k-\tau_{min})},
+
 
 Effect of a TF on the tail
 ===========================

doc/reference/c++/gf/meshes.rst

@@ -25,9 +25,63 @@ Briefly, if we want to describe a function on an interval:
 
 We then have to be careful for example when we fourier transform the function (to not take twice the same point).
 
+How to access to a mesh point with its index
+---------------------------------------------
 
-The four basic meshes
+.. compileblock::
-=====================
+
+     #include <triqs/gf/refreq.hpp> 
+     using namespace triqs::gf;
+
+     int main() {
+      
+      //we construct a GF
+      double wmin = 0.0;
+      double wmax = 1.0;
+      int nw = 101;
+      auto Gw = make_gf<refreq, scalar_valued>(wmin, wmax, nw);
+
+      //we print the mesh parameters and print te value of the 10th point
+      std::cout << "The kind of the mesh is " << Gw.mesh().kind() << std::endl;
+      std::cout << "The smallest mesh point value is w_min=" << Gw.mesh().x_min() << std::endl;
+      std::cout << "The largest mesh point value is w_max=" << Gw.mesh().x_max() << std::endl;
+      std::cout << "The number of mesh points is n=" << Gw.mesh().size() << std::endl;
+      std::cout << "Between two consecutive mesh points: delta=" << Gw.mesh().delta() << std::endl;
+      std::cout << "The 10th mesh point is w=" << Gw.mesh()[10] << std::endl;
+
+     }
+
+How to access to a mesh point with a value
+-------------------------------------------
+
+In this case, we look for the closest mesh point, but can need the distance of the value to the mesh point. 
+``windowing`` gives all these informations: 
+
+.. compileblock::
+
+     #include <triqs/gf/refreq.hpp> 
+     using namespace triqs::gf;
+
+     int main() {
+      double wmin = 0.0;
+      double wmax = 1.0;
+      int nw=101;
+      auto Gw= make_gf<refreq, scalar_valued>(wmin, wmax, nw);
+      double w=0.25156;
+      size_t index; double wd; bool in;
+      std::tie(in, index, wd) = windowing ( Gw.mesh(), w);
+      std::cout << "Is the point w="<< w <<" in the mesh range ? " << in << std::endl;
+      if(in){
+       std::cout << "The point before is the " << index << "th" << std::endl;
+       std::cout << "The position in the intervall is " << wd << std::endl;
+      }
+     }
+
+
+
+
+The four basic linear meshes
+============================
 
 
 Real time
@@ -60,7 +114,57 @@ Depending on what one needs, we can choose ``full_bins``,  ``half_bins``  or ``w
 Matsubara frequency
 --------------------
 
-The domain is discrete. The Matsubara frequencies are :math:`\omega_n=\frac{(2n+1)\pi}{beta}` for fermions and :math:`\omega_n=\frac{2n\pi}{beta}` for bosons. 
+The domain is discrete. The Matsubara frequencies are :math:`\omega_n=\frac{(2n+1)\pi}{\beta}` for fermions and :math:`\omega_n=\frac{2n\pi}{\beta}` for bosons. 
+
+
+
+Products of meshes
+===================
+
+We detail the case of a two mesh product, but what follows is true for any number of meshes. 
+
+A mesh point can be labelled by a linear index, or by a tuple of indices. Each mesh point correspond to a point of the domain, which is a tuple of points of the subdomains. 
+We can navigate between these representations, through ``closest_mesh_pt``, ``get_closest_pt``, ``index_to_linear``,...
+
+
+How to access to the closest mesh point
+---------------------------------------
+
+.. compileblock::
+
+     #include <triqs/gf/two_real_times.hpp> 
+     using namespace triqs::gf;
+
+     int main() {
+      double tmax = 1.0;
+      int nt = 101;
+      auto Gtt = make_gf<two_real_times>(tmax, nt, triqs::arrays::make_shape(1,1));
+
+      //does not work for instance
+      //double t1 = 0.256, t2 = 0.758;
+      //Gtt(closest_mesh_pt(i1,i2)) = 1.5;
+      
+     }
+
+
+How to access to a mesh point with its index
+---------------------------------------------
+
+.. compileblock::
+
+     #include <triqs/gf/two_real_times.hpp> 
+     using namespace triqs::gf;
+
+     int main() {
+      double tmax = 1.0;
+      int nt = 101;
+      auto Gtt = make_gf<two_real_times>(tmax, nt, triqs::arrays::make_shape(1,1));
+
+      int i1 = 14, i2 = 86;
+      Gtt.on_mesh(i1, i2) = 1.8;
+      std::cout << Gtt.on_mesh(i1, i2)(0,0) << std::endl;
+      
+     }