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spc.py
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spc.py
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# -*- coding: utf-8 -*-
"""
Special functions
Created on Fri May 25 01:23:04 2018
@author: luiz_
"""
import scipy.special as spc
import numpy as np
MIE_ARGS = ['diameter', 'wavelength', 'index', 'mu_sp', 'mu']
BSC_ARGS = ['v', 'rho', 'phi', 'z',
'coeffs', 'axicons', 'kzs', 'method']
def hankel(order, argument, kind=2):
sign = 1 if kind == 1 else -1
return spc.jv(order, argument) + sign * 1j * spc.yv(order, argument)
def spherical_jn(order, argument):
"""" Reimplementation of Spherical Bessel Function of first kind as
scipy crashes for complex arguments on Python 3.6.
"""
return np.sqrt(np.pi / (2 * argument)) * spc.jv(order + 1 / 2, argument)
def spherical_yn(order, argument):
"""" Reimplementation of Spherical Bessel Function of second kind as
scipy crashes for complex arguments on Python 3.6.
"""
return np.sqrt(np.pi / (2 * argument)) * spc.yv(order + 1 / 2, argument)
def spherical_hankel2(order, argument):
return (spherical_jn(order, argument) \
- 1j * spherical_yn(order, argument))
def riccati_bessel(order, argument, kind=1, overflow_protection=True):
if overflow_protection:
if kind == 1:
return np.sqrt(np.pi * argument / 2) * spc.jv(order + 1 / 2, argument)
elif kind == 2:
return (np.sqrt(np.pi * argument / 2) \
* (spc.jv(order + 1 / 2, argument) \
- 1j * spc.yv(order + 1 / 2, argument)))
elif kind == 3:
return np.sqrt(np.pi * argument / 2) * spc.yv(order + 1 / 2, argument)
else:
raise ValueError("Only kind=1 and kind=2 allowed.")
if kind == 1:
return argument * spherical_jn(order, argument)
elif kind == 2:
return argument * spherical_hankel2(order, argument)
elif kind == 3:
return argument * spherical_yn(order, argument)
else:
raise ValueError("Only kind=1 and kind=2 allowed.")
def d_riccati_bessel(order, argument, kind=1):
if kind == 1:
sp_function = spherical_jn
elif kind == 2:
sp_function = spherical_hankel2
elif kind == 3:
sp_function = spherical_yn
else:
raise ValueError("Only kind=1 and kind=2 allowed.")
return (argument * sp_function(order - 1, argument) \
- (order) * sp_function(order, argument))
def riccati_semi_wronskian1(n, x, M, f_bessel=[spc.jv, spc.jv]):
result = (x * f_bessel[0](n - 1 / 2, x) - n * f_bessel[0](n + 1 / 2, x))
return result * np.sqrt(M) * np.pi / 2 * f_bessel[1](n + 1 / 2, M * x)
def riccati_semi_wronskian2(n, x, M, f_bessel=[spc.jv, spc.jv]):
return riccati_semi_wronskian1(n, M * x, 1 / M,
f_bessel=[f_bessel[1], f_bessel[0]])
def mie_coeff_a(n, diameter=35E-6, wavelength=1064E-9, index=1.01,
mu_sp=1, mu=1, overflow_protection=True):
a = np.pi * diameter / wavelength
b = index * a
numerator = (mu_sp * riccati_bessel(n, a) * d_riccati_bessel(n, b) \
- mu * index * d_riccati_bessel(n, a) * riccati_bessel(n, b))
denominator = (mu_sp * riccati_bessel(n, a, kind=2) \
* d_riccati_bessel(n, b) \
- mu * index * d_riccati_bessel(n, a, kind=2) \
* riccati_bessel(n, b))
return numerator / denominator
def mie_coeff_a_alt(n, diameter=35E-6, wavelength=1064E-9, index=1.01,
mu_sp=1, mu=1, overflow_protection=True):
a = np.pi * diameter / wavelength
b = index * a
if overflow_protection:
fa = (mu_sp * riccati_semi_wronskian2(n, a, index) \
- index * mu * riccati_semi_wronskian1(n, a, index))
bess = [spc.yv, spc.jv]
fb = (mu_sp * riccati_semi_wronskian2(n, a, index, f_bessel=bess) \
- index * mu * riccati_semi_wronskian1(n, a, index,
f_bessel=bess))
if fb == 0:
return 1
return fa * (fa - 1j * fb) / (fa ** 2 + fb ** 2)
fa = (mu_sp * riccati_bessel(n, a) * d_riccati_bessel(n, b) \
- mu * index * d_riccati_bessel(n, a) * riccati_bessel(n, b))
fb = (mu_sp * riccati_bessel(n, a, kind=3) \
* d_riccati_bessel(n, b) \
- mu * index * d_riccati_bessel(n, a, kind=3) \
* riccati_bessel(n, b))
if fb == 0:
return 1
return fa * (fa + 1j * fb) / (fa ** 2 + fb ** 2)
def mie_coeff_b(n, diameter=35E-6, wavelength=1064E-9, index=1.01,
mu_sp=1, mu=1, overflow_protection=True):
a = np.pi * diameter / wavelength
b = index * a
if overflow_protection:
return mie_coeff_b_alt(n, diameter=diameter,
wavelength=wavelength, index=index,
mu_sp=mu_sp, mu=mu, overflow_protection=True)
numerator = (mu * index * riccati_bessel(n, a) * d_riccati_bessel(n, b) \
- mu_sp * d_riccati_bessel(n, a) * riccati_bessel(n, b))
denominator = (mu * index * riccati_bessel(n, a, kind=2) \
* d_riccati_bessel(n, b) \
- mu_sp * index * d_riccati_bessel(n, a, kind=2) \
* riccati_bessel(n, b))
return numerator / denominator
def mie_coeff_b_alt(n, diameter=35E-6, wavelength=1064E-9, index=1.01,
mu_sp=1, mu=1, overflow_protection=True):
a = np.pi * diameter / wavelength
b = index * a
if overflow_protection:
fa = (index * mu * riccati_semi_wronskian2(n, a, index) \
- mu_sp * riccati_semi_wronskian1(n, a, index))
bess = [spc.yv, spc.jv]
fb = (index * mu * riccati_semi_wronskian2(n, a, index, f_bessel=bess) \
- mu_sp * riccati_semi_wronskian1(n, a, index,
f_bessel=bess))
if fb == 0:
return 1
return fa * (fa + 1j * fb) / (fa ** 2 + fb ** 2)
fa = (mu * index * riccati_bessel(n, a) * d_riccati_bessel(n, b) \
- mu_sp * d_riccati_bessel(n, a) * riccati_bessel(n, b))
fb = (mu * index * riccati_bessel(n, a, kind=3) \
* d_riccati_bessel(n, b) \
- mu_sp * index * d_riccati_bessel(n, a, kind=3) \
* riccati_bessel(n, b))
return fa * (fa + 1j * fb) / (fa ** 2 + fb ** 2)
def scatt_eff(x, wavelength=1064E-9, index=1.01,
mu_sp=1, mu=1):
result = 0
kwargs = {'diameter': x, 'wavelength': wavelength, 'index': index,
'mu_sp': mu_sp, 'mu': mu}
for n in range(1, get_max_it(x, wave_number_k=1) + 160):
increment = (2 * n + 1) * (np.abs(mie_coeff_a_alt(n, **kwargs) ** 2) \
* np.abs(mie_coeff_b_alt(n, **kwargs)) ** 2)
result += increment
return (2 / x ** 2) * result
def fac_plus_minus(n, m):
""" Calculates the expression below avoiding overflows.
.. math::
\\frac{(n + m)!}{(n - m)!}
"""
product = 1
if n > 0:
if m > 0:
if n - m >= 0:
for factor in range(n - m + 1, n + m + 1):
product *= factor
return product
else:
for factor in range(m - n + 1, m + n + 1):
product *= factor
return pow(-1, n - m) * product
if m == 0:
return 1
def get_max_it(x_max, wave_number_k):
""" Calculates stop iteration number """
max_it = np.ceil(wave_number_k * x_max + np.longdouble(4.05) \
* pow(wave_number_k * x_max, 1/3)) + 2
if np.isnan(max_it):
return 2
return int(max_it)
def gamma_quotient(num_arg, den_arg):
return np.exp(spc.loggamma(num_arg) - spc.loggamma(den_arg))
def gamma_quotients(num_args=[], den_args=[]):
import warnings
warnings.filterwarnings("error")
log = 0
sgn = 1
for arg in (num_args + den_args):
sgn *= spc.gammasgn(arg)
for num in num_args:
log += spc.loggamma(num)
for den in den_args:
log -= spc.loggamma(den)
try:
res = sgn * np.exp(log)
warnings.filterwarnings("default")
return res
except RuntimeWarning:
with open('./runtimewarnings.txt', 'a') as f:
f.writelines(str([num_args, den_args]) + '\n')
warnings.filterwarnings("default")
return sgn * np.exp(log)
def exponentials(kwargs):
log = 0
for base in kwargs:
log += kwargs[base] * np.log(base)
return np.exp(log)
def partial_factorial(n, m):
""" Computes n! / (n - m)! """
factorial = 1
for p in range(n - m + 1, n):
factorial *= p
return factorial
def gouesbet_epsilon(n, u):
return 1 if n > u else 0