Omega Meshes

The following plot shows how a subset of those behaves. Typically, all other meshes aim to be denser around 0 than a linear mesh.

import matplotlib.pyplot as plt
import numpy as np
from triqs_maxent.omega_meshes import *

for mesh in [LinearOmegaMesh, LorentzianOmegaMesh,
     LorentzianSmallerOmegaMesh, HyperbolicOmegaMesh]:

     m = mesh(omega_min=-10,omega_max=10,n_points=100)
     plt.plot(list(m),label=mesh.__name__)
plt.xlabel('Data point number $i$')
plt.ylabel('Value of $\\omega_i$')
plt.legend()

import matplotlib.pyplot as plt
import numpy as np
from triqs_maxent.omega_meshes import *

N = 100000
gauge = None
for mesh in [LinearOmegaMesh, LorentzianOmegaMesh,
     LorentzianSmallerOmegaMesh, HyperbolicOmegaMesh]:

     m = mesh(omega_min=-10,omega_max=10,n_points=N)
     h,b=np.histogram(list(m),bins=np.linspace(-10,10,100))
     if gauge is None:
         gauge = h[0]
     plt.semilogy((b[:-1]+b[1:])/2.0,np.array(h)/float(gauge),label=mesh.__name__)
plt.xlabel('$\\omega$')
plt.ylabel('density of points (arb. u.)')
plt.legend(loc='lower right')

class triqs_maxent.omega_meshes.BaseOmegaMesh(omega_min=-10, omega_max=10, n_points=100, *args, **kwargs)[source]

Bases: ndarray

Base class for omega meshes. All meshes inherit from this class.

class triqs_maxent.omega_meshes.LinearOmegaMesh(omega_min=-10, omega_max=10, n_points=100, *args, **kwargs)[source]

Bases: BaseOmegaMesh

Omega mesh with linear spacing

The \(i\)-th \(\omega\)-point is given by

\[\omega_i = \omega_{min} + i \frac{\omega_{max}-\omega_{min}}{n_{max}-1},\]

where \(i\) runs from \(0\) to \(n_{max}-1\).

Parameters:

omega_minfloat: the minimal omega
omega_maxfloat: the maximal omega
n_pointsint: the number of omega points

class triqs_maxent.omega_meshes.DataOmegaMesh(data)[source]

Bases: BaseOmegaMesh

Omega mesh from data array

The \(\omega\)-points are picked from a user-supplied array.

Parameters:

dataarray: an array giving the omega points

class triqs_maxent.omega_meshes.LorentzianOmegaMesh(omega_min=-10, omega_max=10, n_points=100, *args, **kwargs)[source]

Bases: BaseOmegaMesh

Omega mesh with Lorentzian spacing

This mesh is a lot denser than the linear mesh around \(\\omega=0\) and far less denser for high \(|\omega|\). The lowest value is at \(\omega_{min}\), the largest at \(\omega_{max}\).

Parameters:

omega_minfloat: the minimal omega
omega_maxfloat: the maximal omega
n_pointsint: the number of omega points
cutfloat: a parameter influencing the relative density between the middle and the edge of the interval

class triqs_maxent.omega_meshes.LorentzianSmallerOmegaMesh(omega_min=-10, omega_max=10, n_points=100, *args, **kwargs)[source]

Bases: BaseOmegaMesh

Omega mesh with Lorentzian spacing

This mesh is a lot denser than the linear mesh around \(\\omega=0\) and far less denser for high \(|\omega|\). The lowest value is not at \(\omega_{min}\), the largest at \(\omega_{max}\); this is the main difference related to LorentzianOmegaMesh.

Parameters:

omega_minfloat: the minimal omega
omega_maxfloat: the maximal omega
n_pointsint: the number of omega points
cutfloat: a parameter influencing the relative density between the middle and the edge of the interval

class triqs_maxent.omega_meshes.HyperbolicOmegaMesh(omega_min=-10, omega_max=10, n_points=100, *args, **kwargs)[source]

Bases: BaseOmegaMesh

Omega mesh with hyperbolic spacing

This mesh is denser than the linear mesh around \(\omega=0\) and behaves like a sparser variant of a linear mesh at \(|\omega|\to\infty\).

Parameters:

omega_minfloat: the minimal omega
omega_maxfloat: the maximal omega
n_pointsint: the number of omega points