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"""
Convergence regions of the expansions used in ``struve.c``
Note that for v >> z both functions tend rapidly to 0,
and for v << -z, they tend to infinity.
The floating-point functions over/underflow in the lower left and right
corners of the figure.
Figure legend
=============
Red region
Power series is close (1e-12) to the mpmath result
Blue region
Asymptotic series is close to the mpmath result
Green region
Bessel series is close to the mpmath result
Dotted colored lines
Boundaries of the regions
Solid colored lines
Boundaries estimated by the routine itself. These will be used
for determining which of the results to use.
Black dashed line
The line z = 0.7*|v| + 12
"""
from __future__ import absolute_import, division, print_function
import numpy as np
import matplotlib.pyplot as plt
try:
import mpmath
except:
from sympy import mpmath
def err_metric(a, b, atol=1e-290):
m = abs(a - b) / (atol + abs(b))
m[np.isinf(b) & (a == b)] = 0
return m
def do_plot(is_h=True):
from scipy.special._ufuncs import \
_struve_power_series, _struve_asymp_large_z, _struve_bessel_series
vs = np.linspace(-1000, 1000, 91)
zs = np.sort(np.r_[1e-5, 1.0, np.linspace(0, 700, 91)[1:]])
rp = _struve_power_series(vs[:,None], zs[None,:], is_h)
ra = _struve_asymp_large_z(vs[:,None], zs[None,:], is_h)
rb = _struve_bessel_series(vs[:,None], zs[None,:], is_h)
mpmath.mp.dps = 50
if is_h:
sh = lambda v, z: float(mpmath.struveh(mpmath.mpf(v), mpmath.mpf(z)))
else:
sh = lambda v, z: float(mpmath.struvel(mpmath.mpf(v), mpmath.mpf(z)))
ex = np.vectorize(sh, otypes='d')(vs[:,None], zs[None,:])
err_a = err_metric(ra[0], ex) + 1e-300
err_p = err_metric(rp[0], ex) + 1e-300
err_b = err_metric(rb[0], ex) + 1e-300
err_est_a = abs(ra[1]/ra[0])
err_est_p = abs(rp[1]/rp[0])
err_est_b = abs(rb[1]/rb[0])
z_cutoff = 0.7*abs(vs) + 12
levels = [-1000, -12]
plt.cla()
plt.hold(1)
plt.contourf(vs, zs, np.log10(err_p).T, levels=levels, colors=['r', 'r'], alpha=0.1)
plt.contourf(vs, zs, np.log10(err_a).T, levels=levels, colors=['b', 'b'], alpha=0.1)
plt.contourf(vs, zs, np.log10(err_b).T, levels=levels, colors=['g', 'g'], alpha=0.1)
plt.contour(vs, zs, np.log10(err_p).T, levels=levels, colors=['r', 'r'], linestyles=[':', ':'])
plt.contour(vs, zs, np.log10(err_a).T, levels=levels, colors=['b', 'b'], linestyles=[':', ':'])
plt.contour(vs, zs, np.log10(err_b).T, levels=levels, colors=['g', 'g'], linestyles=[':', ':'])
lp = plt.contour(vs, zs, np.log10(err_est_p).T, levels=levels, colors=['r', 'r'], linestyles=['-', '-'])
la = plt.contour(vs, zs, np.log10(err_est_a).T, levels=levels, colors=['b', 'b'], linestyles=['-', '-'])
lb = plt.contour(vs, zs, np.log10(err_est_b).T, levels=levels, colors=['g', 'g'], linestyles=['-', '-'])
plt.clabel(lp, fmt={-1000: 'P', -12: 'P'})
plt.clabel(la, fmt={-1000: 'A', -12: 'A'})
plt.clabel(lb, fmt={-1000: 'B', -12: 'B'})
plt.plot(vs, z_cutoff, 'k--')
plt.xlim(vs.min(), vs.max())
plt.ylim(zs.min(), zs.max())
plt.xlabel('v')
plt.ylabel('z')
def main():
plt.clf()
plt.subplot(121)
do_plot(True)
plt.title('Struve H')
plt.subplot(122)
do_plot(False)
plt.title('Struve L')
plt.savefig('struve_convergence.png')
plt.show()
if __name__ == "__main__":
import os
import sys
if '--main' in sys.argv:
main()
else:
import subprocess
subprocess.call([sys.executable, os.path.join('..', '..', '..', 'runtests.py'),
'-g', '--python', __file__, '--main'])
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