File size: 3,755 Bytes
dffca60 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 |
"""
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
"""
import numpy as np
import matplotlib.pyplot as plt
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:
def sh(v, z):
return float(mpmath.struveh(mpmath.mpf(v), mpmath.mpf(z)))
else:
def sh(v, z):
return 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__":
main()
|