# The ImpedanceFitter is a package to fit impedance spectra to
# equivalent-circuit models using open-source software.
#
# Copyright (C) 2021, 2022 Julius Zimmermann,
# julius.zimmermann[AT]uni-rostock.de
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <https://www.gnu.org/licenses/>.
from cmath import sqrt
import numpy as np
[docs]
def get_cubic_roots(a, b, c):
"""Get the roots of a cubic equation.
Parameters
----------
a: complex
polynomial coefficient
b: complex
polynomial coefficient
c: complex
polynomial coefficient
Returns
-------
:class:`numpy.ndarray`, complex
Complex permittivity array
Notes
-----
Return the roots of an expression
.. math::
z^3 + a z^2 + b z + c = 0
More details can be found in [Press2007]_.
References
----------
.. [Press2007] Press, W.H., Teukolsky, S.A., Vetterling, W.T.,
and Flannery, B.P. (2007)
Numerical recipes : the art of scientific computing.
Cambridge Univ. Press, USA, 3rd edition
"""
Q = (a * a - 3.0 * b) / 9.0
R = (2.0 * a * a * a - 9.0 * a * b + 27.0 * c) / 54.0
A = -((R + sqrt(R * R - Q * Q * Q)) ** (1.0 / 3.0))
if np.isclose(abs(A), 0):
B = 0
else:
B = Q / A
# first root
x1 = (A + B) - a / 3.0
x2 = -0.5 * (A + B) - a / 3.0 + 1j * 0.5 * sqrt(3.0) * (A - B)
x3 = -0.5 * (A + B) - a / 3.0 - 1j * 0.5 * sqrt(3.0) * (A - B)
return x1, x2, x3