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