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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