| 1 | r""" |
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| 2 | Show numerical precision of $\ln \Gamma(x)$. |
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| 3 | """ |
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| 4 | |
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| 5 | import numpy as np |
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| 6 | import scipy.special |
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| 7 | from sympy.mpmath import mp |
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| 8 | #import matplotlib; matplotlib.use('TkAgg') |
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| 9 | import pylab |
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| 10 | |
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| 11 | mp.dps = 150 # number of digits to use in estimating true value |
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| 12 | |
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| 13 | SHOW_DIFF = True # True if show diff rather than function value |
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| 14 | LINEAR_X = False # True if q is linearly spaced instead of log spaced |
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| 15 | |
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| 16 | def mp_gamma(vec): |
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| 17 | """ |
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| 18 | Direct calculation using sympy multiprecision library. |
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| 19 | """ |
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| 20 | return [_mp_fn(mp.mpf(x)) for x in vec] |
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| 21 | |
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| 22 | def _mp_fn(x): |
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| 23 | """ |
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| 24 | Helper funciton for mp_j1c |
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| 25 | """ |
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| 26 | #return mp.gamma(x) |
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| 27 | return mp.loggamma(x) |
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| 28 | |
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| 29 | def np_gamma(x, dtype): |
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| 30 | """ |
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| 31 | Direct calculation using scipy. |
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| 32 | """ |
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| 33 | x = np.asarray(x, dtype) |
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| 34 | return scipy.special.gammaln(x) |
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| 35 | #return scipy.special.gamma(x) |
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| 36 | |
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| 37 | def lanczos_gamma(x, dtype): |
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| 38 | coeff = np.asarray(( |
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| 39 | 76.18009172947146, -86.50532032941677, |
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| 40 | 24.01409824083091, -1.231739572450155, |
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| 41 | 0.1208650973866179e-2,-0.5395239384953e-5), dtype) |
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| 42 | |
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| 43 | x = np.asarray(x, dtype) |
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| 44 | tmp = x + np.asarray(5.5, dtype) |
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| 45 | tmp -= (x + np.asarray(0.5, dtype))*np.log(tmp) |
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| 46 | ser = np.ones_like(x)*np.asarray(1.000000000190015, dtype) |
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| 47 | for k,c in enumerate(coeff): |
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| 48 | ser += c/(x + np.asarray(k+1, dtype)) |
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| 49 | return -tmp+np.log(np.asarray(2.5066282746310005, dtype)*ser/x); |
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| 50 | |
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| 51 | def plotdiff(x, target, actual, label): |
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| 52 | """ |
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| 53 | Plot the computed value. |
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| 54 | |
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| 55 | Use relative error if SHOW_DIFF, otherwise just plot the value directly. |
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| 56 | """ |
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| 57 | if SHOW_DIFF: |
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| 58 | err = np.clip(abs((target-actual)/target), 0, 1) |
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| 59 | pylab.loglog(x, err, '-', label=label) |
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| 60 | else: |
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| 61 | limits = np.min(target), np.max(target) |
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| 62 | pylab.loglog(x, np.clip(actual,*limits), '-', label=label) |
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| 63 | |
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| 64 | def compare(x, precision): |
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| 65 | r""" |
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| 66 | Compare the different computation methods using the given precision. |
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| 67 | """ |
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| 68 | target = np.asarray(mp_gamma(x), 'double') |
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| 69 | direct = np_gamma(x, precision) |
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| 70 | approx = lanczos_gamma(x, precision) |
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| 71 | plotdiff(x, target, direct, 'scipy '+precision) |
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| 72 | plotdiff(x, target, approx, 'sasmodels '+precision) |
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| 73 | pylab.xlabel("x (arbitrary)") |
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| 74 | if SHOW_DIFF: |
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| 75 | pylab.ylabel("relative error") |
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| 76 | else: |
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| 77 | pylab.ylabel("ln(gamma(x))") |
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| 78 | pylab.loglog(x, target, '-', label="true value") |
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| 79 | if LINEAR_X: |
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| 80 | pylab.xscale('linear') |
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| 81 | |
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| 82 | def main(): |
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| 83 | r""" |
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| 84 | Compare accuracy of different methods for computing $3 j_1(x)/x$. |
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| 85 | :return: |
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| 86 | """ |
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| 87 | if LINEAR_X: |
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| 88 | qr = np.linspace(1,1000,2000) |
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| 89 | else: |
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| 90 | qr = np.logspace(-3,5,400) |
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| 91 | pylab.subplot(121) |
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| 92 | compare(qr, 'single') |
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| 93 | pylab.legend(loc='best') |
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| 94 | pylab.subplot(122) |
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| 95 | compare(qr, 'double') |
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| 96 | pylab.legend(loc='best') |
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| 97 | pylab.suptitle('ln gamma') |
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| 98 | |
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| 99 | if __name__ == "__main__": |
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| 100 | main() |
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| 101 | pylab.show() |
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