[e6f1410] | 1 | r""" |
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| 2 | Show numerical precision of $2 J_1(x)/x$. |
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| 3 | """ |
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| 4 | |
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| 5 | import numpy as np |
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| 6 | from sympy.mpmath import mp |
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| 7 | #import matplotlib; matplotlib.use('TkAgg') |
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| 8 | import pylab |
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| 9 | |
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| 10 | mp.dps = 150 # number of digits to use in estimating true value |
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| 11 | |
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| 12 | SHOW_DIFF = True # True if show diff rather than function value |
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| 13 | LINEAR_X = False # True if q is linearly spaced instead of log spaced |
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| 14 | |
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| 15 | def mp_J1c(vec): |
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| 16 | """ |
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| 17 | Direct calculation using sympy multiprecision library. |
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| 18 | """ |
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| 19 | return [_mp_J1c(mp.mpf(x)) for x in vec] |
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| 20 | |
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| 21 | def _mp_J1c(x): |
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| 22 | """ |
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| 23 | Helper funciton for mp_j1c |
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| 24 | """ |
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| 25 | return mp.mpf(2)*mp.j1(x)/x |
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| 26 | |
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| 27 | def np_j1c(x, dtype): |
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| 28 | """ |
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| 29 | Direct calculation using scipy. |
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| 30 | """ |
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| 31 | from scipy.special import j1 |
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| 32 | x = np.asarray(x, dtype) |
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| 33 | return np.asarray(2, dtype)*j1(x)/x |
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| 34 | |
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| 35 | def cephes_j1c(x, dtype, n): |
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| 36 | """ |
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| 37 | Calculation using pade approximant. |
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| 38 | """ |
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| 39 | x = np.asarray(x, dtype) |
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| 40 | ans = np.empty_like(x) |
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| 41 | ax = abs(x) |
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| 42 | |
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| 43 | # Branch a |
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| 44 | a_idx = ax < 8.0 |
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| 45 | a_xsq = x[a_idx]**2 |
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| 46 | a_coeff1 = list(reversed((72362614232.0, -7895059235.0, 242396853.1, -2972611.439, 15704.48260, -30.16036606))) |
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| 47 | a_coeff2 = list(reversed((144725228442.0, 2300535178.0, 18583304.74, 99447.43394, 376.9991397, 1.0))) |
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| 48 | a_ans1 = np.polyval(a_coeff1[n:], a_xsq) |
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| 49 | a_ans2 = np.polyval(a_coeff2[n:], a_xsq) |
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| 50 | ans[a_idx] = 2*a_ans1/a_ans2 |
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| 51 | |
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| 52 | # Branch b |
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| 53 | b_idx = ~a_idx |
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| 54 | b_ax = ax[b_idx] |
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| 55 | b_x = x[b_idx] |
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| 56 | |
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| 57 | b_y = 64.0/(b_ax**2) |
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| 58 | b_xx = b_ax - 2.356194491 |
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| 59 | b_coeff1 = list(reversed((1.0, 0.183105e-2, -0.3516396496e-4, 0.2457520174e-5, -0.240337019e-6))) |
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| 60 | b_coeff2 = list(reversed((0.04687499995, -0.2002690873e-3, 0.8449199096e-5, -0.88228987e-6, 0.105787412e-6))) |
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| 61 | b_ans1 = np.polyval(b_coeff1[n:], b_y) |
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| 62 | b_ans2 = np.polyval(b_coeff2[n:], b_y) |
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| 63 | b_sn, b_cn = np.sin(b_xx), np.cos(b_xx) |
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| 64 | ans[b_idx] = np.sign(b_x)*np.sqrt(0.636619772/b_ax) * (b_cn*b_ans1 - (8.0/b_ax)*b_sn*b_ans2)*2.0/b_x |
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| 65 | |
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| 66 | return ans |
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| 67 | |
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| 68 | def plotdiff(x, target, actual, label): |
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| 69 | """ |
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| 70 | Plot the computed value. |
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| 71 | |
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| 72 | Use relative error if SHOW_DIFF, otherwise just plot the value directly. |
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| 73 | """ |
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| 74 | if SHOW_DIFF: |
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| 75 | err = np.clip(abs((target-actual)/target), 0, 1) |
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| 76 | pylab.loglog(x, err, '-', label=label) |
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| 77 | else: |
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| 78 | limits = np.min(target), np.max(target) |
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| 79 | pylab.semilogx(x, np.clip(actual,*limits), '-', label=label) |
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| 80 | |
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| 81 | def compare(x, precision): |
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| 82 | r""" |
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| 83 | Compare the different computation methods using the given precision. |
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| 84 | """ |
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| 85 | target = np.asarray(mp_J1c(x), 'double') |
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| 86 | direct = np_j1c(x, precision) |
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| 87 | approx0 = cephes_j1c(x, precision, 0) |
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| 88 | approx1 = cephes_j1c(x, precision, 1) |
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| 89 | plotdiff(x, target, direct, 'direct '+precision) |
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| 90 | plotdiff(x, target, approx0, 'cephes '+precision) |
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| 91 | #plotdiff(x, target, approx1, 'reduced '+precision) |
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| 92 | pylab.xlabel("qr (1/Ang)") |
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| 93 | if SHOW_DIFF: |
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| 94 | pylab.ylabel("relative error") |
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| 95 | else: |
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| 96 | pylab.ylabel("2 J1(x)/x") |
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| 97 | pylab.semilogx(x, target, '-', label="true value") |
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| 98 | if LINEAR_X: |
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| 99 | pylab.xscale('linear') |
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| 100 | |
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| 101 | def main(): |
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| 102 | r""" |
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| 103 | Compare accuracy of different methods for computing $3 j_1(x)/x$. |
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| 104 | :return: |
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| 105 | """ |
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| 106 | if LINEAR_X: |
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| 107 | qr = np.linspace(1,1000,2000) |
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| 108 | else: |
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| 109 | qr = np.logspace(-3,5,400) |
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| 110 | pylab.subplot(121) |
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| 111 | compare(qr, 'single') |
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| 112 | pylab.legend(loc='best') |
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| 113 | pylab.subplot(122) |
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| 114 | compare(qr, 'double') |
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| 115 | pylab.legend(loc='best') |
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| 116 | pylab.suptitle('2 J1(x)/x') |
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| 117 | |
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| 118 | if __name__ == "__main__": |
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| 119 | print "\n".join(str(x) for x in mp_J1c([1e-6,1e-5,1e-4,1e-3])) |
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| 120 | main() |
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| 121 | pylab.show() |
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