1 | /** |
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2 | * Spherical Bessel function 3*j1(x)/x |
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3 | * |
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4 | * Used for low q to avoid cancellation error. |
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5 | * Note that the values differ from sasview ~ 5e-12 rather than 5e-14, but |
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6 | * in this case it is likely cancellation errors in the original expression |
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7 | * using double precision that are the source. |
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8 | */ |
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9 | |
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10 | // The choice of the number of terms in the series and the cutoff value for |
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11 | // switching between series and direct calculation depends on the numeric |
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12 | // precision. |
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13 | // |
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14 | // Point where direct calculation reaches machine precision: |
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15 | // |
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16 | // single machine precision eps 3e-8 at qr=1.1 ** |
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17 | // double machine precision eps 4e-16 at qr=1.1 |
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18 | // |
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19 | // Point where Taylor series reaches machine precision (eps), where taylor |
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20 | // series matches direct calculation (cross) and the error at that point: |
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21 | // |
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22 | // prec n eps cross error |
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23 | // single 3 0.28 0.4 6.2e-7 |
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24 | // single 4 0.68 0.7 2.3e-7 |
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25 | // single 5 1.18 1.2 7.5e-8 |
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26 | // double 3 0.01 0.03 2.3e-13 |
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27 | // double 4 0.06 0.1 3.1e-14 |
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28 | // double 5 0.16 0.2 5.0e-15 |
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29 | // |
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30 | // ** Note: relative error on single precision starts increase on the direct |
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31 | // method at qr=1.1, rising from 3e-8 to 5e-5 by qr=1e3. This should be |
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32 | // safe for the sans range, with objects of 100 nm supported to a q of 0.1 |
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33 | // while maintaining 5 digits of precision. For usans/sesans, the objects |
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34 | // are larger but the q is smaller, so again it should be fine. |
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35 | // |
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36 | // See explore/sph_j1c.py for code to explore these ranges. |
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37 | |
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38 | // Use 4th order series |
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39 | #if FLOAT_SIZE>4 |
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40 | #define SPH_J1C_CUTOFF 0.1 |
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41 | #else |
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42 | #define SPH_J1C_CUTOFF 0.7 |
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43 | #endif |
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44 | |
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45 | double sph_j1c(double q); |
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46 | double sph_j1c(double q) |
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47 | { |
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48 | if (q < SPH_J1C_CUTOFF) { |
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49 | const double q2 = q*q; |
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50 | return (1.0 + q2*(-3./30. + q2*(3./840. + q2*(-3./45360.))));// + q2*(3./3991680.))))); |
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51 | } else { |
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52 | double sin_q, cos_q; |
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53 | SINCOS(q, sin_q, cos_q); |
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54 | return 3.0*(sin_q/q - cos_q)/(q*q); |
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55 | } |
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56 | } |
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