A practical example: computing the error bar of a Green’s function

In the following example, we construct a series of noisy imaginary-time Green’s functions and compute the associated averages and error bars.


So far, the interface of the statistical analysis tools is rudimentary. It may evolve to allow users to compute the error bar of a stack of Green’s functions directly.

#include <triqs/clef.hpp>
#include <triqs/gfs.hpp>
#include <triqs/mc_tools/random_generator.hpp>
#include <triqs/statistics.hpp>
using namespace triqs::statistics;
using triqs::clef::placeholder;
using namespace triqs::gfs;
int main() {

  //generate Green's functions with random noise
  triqs::mc_tools::random_generator RND;
  placeholder<0> w_;
  auto gw = gf<imfreq, scalar_valued>{{10, Fermion, 200}};
  gw(w_) << 1 / (w_ - 2.0);
  int n_tau = 401;
  auto gt   = gf<imtime, scalar_valued>{{10, Fermion, n_tau}};
  gt()      = fourier(gw);

  int n_measures = 20;
  std::vector<gf<imtime, scalar_valued>> G_measurements(n_measures, gt);
  for (auto &g : G_measurements)
    for (auto const &t : g.mesh()) g[t] += RND(0.1) - 0.05; //adding uniform noise of 0.05

  //put the generated Green's functions into a vector of observables
  std::vector<observable<dcomplex>> gt_as_observable(n_tau);
  int i_tau = 0;
  for (auto &o : gt_as_observable) {
    for (int i = 0; i < n_measures; i++) o << G_measurements[i][i_tau];
  //compute the average and error bar
  i_tau = 0;
  std::cout << "Exact value \t Average +/- Error" << std::endl;
  for (auto const &o : gt_as_observable) {
    std::cout << gt[i_tau] << "\t" << average_and_error(o) << std::endl;
  return 0;