Author: celestin
Date: Fri Nov 9 19:03:28 2012
New Revision: 1407594
URL: http://svn.apache.org/viewvc?rev=1407594&view=rev
Log:
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.
Modified:
commons/proper/math/trunk/src/site/apt/userguide/special.apt
Modified: commons/proper/math/trunk/src/site/apt/userguide/special.apt
URL:
http://svn.apache.org/viewvc/commons/proper/math/trunk/src/site/apt/userguide/special.apt?rev=1407594&r1=1407593&r2=1407594&view=diff
==============================================================================
--- commons/proper/math/trunk/src/site/apt/userguide/special.apt (original)
+++ commons/proper/math/trunk/src/site/apt/userguide/special.apt Fri Nov 9
19:03:28 2012
@@ -44,29 +44,29 @@
** Gamma
- <<<Gamma.gamma(x)>>> computes the Gamma function, \u0393(x),
+ <<<Gamma.gamma(x)>>> computes the Gamma function, Î(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}})
+
