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In the users guide for special functions 

- accuracy of current implementation of Beta.logBeta
  - standard deviation of error (in ulps) is now provided for all special
    functions that have already been evaluated.


git-svn-id: https://svn.apache.org/repos/asf/commons/proper/math/trunk@1407594 13f79535-47bb-0310-9956-ffa450edef68
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Sebastien Brisard 2012-11-09 19:03:28 +00:00
parent 6687706d44
commit a170601629
1 changed files with 57 additions and 38 deletions
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src/site/apt/userguide/special.apt
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@ -44,29 +44,29 @@
** Gamma
<<<Gamma.gamma(x)>>> computes the Gamma function, \u0393(x),
<<<Gamma.gamma(x)>>> computes the Gamma function, Γ(x)
(see {{{http://mathworld.wolfram.com/GammaFunction.html}MathWorld}},
{{{http://dlmf.nist.gov/5}DLMF}}). The accuracy of the Commons-Math
implementation is assessed by comparaison with high precision values computed
with the {{{http://maxima.sourceforge.net/}Maxima}} Computer Algebra System.
*---------------+----------------------------------------+-----------------+----------------+
|| Interval || Values tested || Average error || Maximum error |
*---------------+----------------------------------------+-----------------+----------------+
| (-5.0, -4.0) | <<<k / 1024, k = -5119, ..., -4097>>> | 0.49 ulps | 3.0 ulps |
*---------------+----------------------------------------+-----------------+----------------+
| (-4.0, -3.0) | <<<k / 1024, k = -4095, ..., -3073>>> | 0.36 ulps | 2.0 ulps |
*---------------+----------------------------------------+-----------------+----------------+
| (-3.0, -2.0) | <<<k / 1024, k = -3071, ..., -2049>>> | 0.41 ulps | 2.0 ulps |
*---------------+----------------------------------------+-----------------+----------------+
| (-2.0, -1.0) | <<<k / 1024, k = -2047, ..., -1025>>> | 0.37 ulps | 2.0 ulps |
*---------------+----------------------------------------+-----------------+----------------+
| (-1.0, 0.0) | <<<k / 1024, k = -1023, ..., -1>>> | 0.46 ulps | 2.0 ulps |
*---------------+----------------------------------------+-----------------+----------------+
| (0.0, 8.0] | <<<k / 1024, k = 1, ..., 8192>>> | 0.33 ulps | 2.0 ulps |
*---------------+----------------------------------------+-----------------+----------------+
| (8.0, 141.0] | <<<k / 64, k = 513, ..., 9024>>> | 1.32 ulps | 7.0 ulps |
*---------------+----------------------------------------+-----------------+----------------+
*----------------+-----------------------------------------------+-----------------+----------------------+----------------+
|| Interval || Values tested || Average error || Standard deviation || Maximum error |
*----------------+-----------------------------------------------+-----------------+----------------------+----------------+
| -5 \< x \< -4 | <<<x[i] = i / 1024, i = -5119, ..., -4097>>> | 0.49 ulps | 0.57 ulps | 3.0 ulps |
*----------------+-----------------------------------------------+-----------------+----------------------+----------------+
| -4 \< x \< -3 | <<<x[i] = i / 1024, i = -4095, ..., -3073>>> | 0.36 ulps | 0.51 ulps | 2.0 ulps |
*----------------+-----------------------------------------------+-----------------+----------------------+----------------+
| -3 \< x \< -2 | <<<x[i] = i / 1024, i = -3071, ..., -2049>>> | 0.41 ulps | 0.53 ulps | 2.0 ulps |
*----------------+-----------------------------------------------+-----------------+----------------------+----------------+
| -2 \< x \< -1 | <<<x[i] = i / 1024, i = -2047, ..., -1025>>> | 0.37 ulps | 0.50 ulps | 2.0 ulps |
*----------------+-----------------------------------------------+-----------------+----------------------+----------------+
| -1 \< x \< 0 | <<<x[i] = i / 1024, i = -1023, ..., -1>>> | 0.46 ulps | 0.54 ulps | 2.0 ulps |
*----------------+-----------------------------------------------+-----------------+----------------------+----------------+
| 0 \< x ≤ 8 | <<<x[i] = i / 1024, i = 1, ..., 8192>>> | 0.33 ulps | 0.48 ulps | 2.0 ulps |
*----------------+-----------------------------------------------+-----------------+----------------------+----------------+
| 8 \< x ≤ 141 | <<<x[i] = i / 64, i = 513, ..., 9024>>> | 1.32 ulps | 1.19 ulps | 7.0 ulps |
*----------------+-----------------------------------------------+-----------------+----------------------+----------------+
** Log Gamma
@ -77,33 +77,52 @@
implementation is assessed by comparaison with high precision values computed
with the {{{http://maxima.sourceforge.net/}Maxima}} Computer Algebra System.
*---------------------------------------------+---------------------------------------+-----------------+----------------+
|| Interval || Values tested || Average error || Maximum error |
*---------------------------------------------+---------------------------------------+-----------------+----------------+
| (0.0, 8.0] | <<<k / 1024, k = 1, ..., 8192>>> | 0.32 ulps | 4.0 ulps |
*---------------------------------------------+---------------------------------------+-----------------+----------------+
| (8.0, 1024.0] | <<<k / 8, k = 65, ..., 8192>>> | 0.43 ulps | 3.0 ulps |
*---------------------------------------------+---------------------------------------+-----------------+----------------+
| (1024.0, 8192.0] | <<<k, k = 1025, 8192>>> | 0.53 ulps | 3.0 ulps |
*---------------------------------------------+---------------------------------------+-----------------+----------------+
| [8933.439345993791, 1.75555970201398e+305] | <<<2**(k / 8), k = 105, ..., 8112>>> | 0.35 ulps | 2.0 ulps |
*---------------------------------------------+---------------------------------------+-----------------+----------------+
*------------------------------------------------+-----------------------------------------------+-----------------+----------------------+----------------+
|| Interval || Values tested || Average error || Standard deviation || Maximum error |
*------------------------------------------------+-----------------------------------------------+-----------------+----------------------+----------------+
| 0 \< x ≤ 8 | <<<x[i] = i / 1024, i = 1, ..., 8192>>> | 0.32 ulps | 0.50 ulps | 4.0 ulps |
*------------------------------------------------+-----------------------------------------------+-----------------+----------------------+----------------+
| 8 \< x ≤ 1024 | <<<x[i] = i / 8, i = 65, ..., 8192>>> | 0.43 ulps | 0.53 ulps | 3.0 ulps |
*------------------------------------------------+-----------------------------------------------+-----------------+----------------------+----------------+
| 1024 \< x ≤ 8192 | <<<x[i], i = 1025, ..., 8192>>> | 0.53 ulps | 0.56 ulps | 3.0 ulps |
*------------------------------------------------+-----------------------------------------------+-----------------+----------------------+----------------+
| 8933.439345993791 ≤ x ≤ 1.75555970201398e+305 | <<<x[i] = 2**(i / 8), i = 105, ..., 8112>>> | 0.35 ulps | 0.49 ulps | 2.0 ulps |
*------------------------------------------------+-----------------------------------------------+-----------------+----------------------+----------------+
** Regularized Gamma
<<<Gamma.regularizedGammaP(a, x)>>> computes the value of the regularized
Gamma function, P(a, x)
(see {{{http://mathworld.wolfram.com/RegularizedGammaFunction.html}MathWorld}}).
* 5.4 Beta funtions
* 5.4 Beta functions
{{{../apidocs/org/apache/commons/math3/special/Beta.html}Beta}} contains
several useful functions involving the Beta Function.
*------------------+-----------------+-------------------------------------------------------------------------------------------------------------+
|| Function || Method || Reference |
*------------------+-----------------+-------------------------------------------------------------------------------------------------------------+
| Log Beta | logBeta | See {{{http://mathworld.wolfram.com/BetaFunction.html}Beta Function}} from MathWorld |
*------------------+-----------------+-------------------------------------------------------------------------------------------------------------+
| Regularized Beta | regularizedBeta | See {{{http://mathworld.wolfram.com/RegularizedBetaFunction.html}Regularized Beta Function}} from MathWorld |
*------------------+-----------------+-------------------------------------------------------------------------------------------------------------+
** Log Beta
<<<Beta.logBeta(a, b)>>> computes the value of the natural logarithm of the
Beta function, log B(a, b).
(see {{{http://mathworld.wolfram.com/BetaFunction.html}MathWorld}},
{{{http://dlmf.nist.gov/5.12}DLMF}}). The accuracy of the Commons-Math
implementation is assessed by comparaison with high precision values computed
with the {{{http://maxima.sourceforge.net/}Maxima}} Computer Algebra System.
*----------------+----------------------------------------+-----------------+----------------------+----------------+
|| Interval || Values tested || Average error || Standard deviation || Maximum error |
*----------------+----------------------------------------+-----------------+----------------------+-----------------+
| 0 \< x ≤ 8\ | <<<x[i] = i / 32, i = 1, ..., 256>>>\ | 5.04 ulps | 270.99 ulps | 46696.0 ulps |
| 0 \< y ≤ 8 | <<<y[j] = j / 32, j = 1, ..., 256>>> | | | |
*----------------+----------------------------------------+-----------------+----------------------+-----------------+
| 0 \< x ≤ 8\ | <<<x[i] = i / 32, i = 1, ..., 256>>>\ | 9.75 ulps | 149.42 ulps | 19126.0 ulps |
| 8 \< y ≤ 16 | <<<y[j] = j / 32, j = 257, ..., 512>>> | | | |
*----------------+----------------------------------------+-----------------+----------------------+-----------------+
| 0 \< x ≤ 8\ | <<<x[i] = i / 32, i = 1, ..., 256>>>\ | 357.82 ulps | 39297.58 ulps | 8635522.0 ulps |
| 17 \< y ≤ 256 | <<<y[j] = j, j = 17, ..., 256>>> | | | |
*----------------+----------------------------------------+-----------------+----------------------+-----------------+
** Regularized Beta
(see {{{http://mathworld.wolfram.com/RegularizedBetaFunction.html}MathWorld}})