""" Utilities functions used in testing and demonstration."""
"""
Copyright 2021 Hugo U.R. Strand
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express
or implied. See the License for the specific language governing
permissions and limitations under the License."""
import numpy as np
from scipy.integrate import quad
from pydlr import kernel
[docs]def analytic_bethe_G_tau(tau, beta):
"""Get Bethe graph imaginary time Green's function.
Obtained by semi-analytic evaluation using adaptive integration of
.. math:: G(\\tau) = -\\frac{2}{\\pi} \\int_{-1}^{1} K\\left(\\frac{\\tau}{\\beta}, \\beta \\omega\\right) \\sqrt{1 - \\omega^2} \, d\\omega
Parameters
----------
tau : (n), ndarray
Imaginary time :math:`\\tau` points to evaulate the Bethe Green's function on.
beta : float
Inverse temperature :math:`\\beta`
Returns
-------
G_iaa : (n, 1, 1), ndarray
Imaginary time Green's function :math:`G(\\tau)` evaluated on the given imaginary times.
"""
I = lambda omega, tau: -2/np.pi * kernel(np.array([tau/beta]), np.array([beta*omega]))[0,0]
integral = lambda tau: quad(I, -1, 1, weight='alg', wvar=(0.5, 0.5), args=tau)[0]
return np.vectorize(integral)(tau).reshape((len(tau), 1, 1))