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Gilles 2016-05-16 19:05:24 +02:00
parent 2f8ddc3985
commit d731d1645a
1 changed files with 39 additions and 35 deletions
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src/main/java/org/apache/commons/math4/special/Gamma.java
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@ -53,13 +53,14 @@ import org.apache.commons.math4.util.FastMath;
public class Gamma {
/**
* <a href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant">Euler-Mascheroni constant</a>
*
* @since 2.0
*/
public static final double GAMMA = 0.577215664901532860606512090082;
/**
* The value of the {@code g} constant in the Lanczos approximation, see
* {@link #lanczos(double)}.
* Constant \( g = \frac{607}{128} \) in the {@link #lanczos(double) Lanczos approximation}.
*
* @since 3.1
*/
public static final double LANCZOS_G = 607.0 / 128.0;
@ -218,13 +219,12 @@ public class Gamma {
private Gamma() {}
/**
* Returns the value of \( \log \Gamma(x) \) for \( x > 0 \).
*
* <p>
* Returns the value of log&nbsp;&Gamma;(x) for x&nbsp;&gt;&nbsp;0.
* </p>
* <p>
* For x &le; 8, the implementation is based on the double precision
* For \( x \leq 8 \), the implementation is based on the double precision
* implementation in the <em>NSWC Library of Mathematics Subroutines</em>,
* {@code DGAMLN}. For x &gt; 8, the implementation is based on
* {@code DGAMLN}. For \( x \geq 8 \), the implementation is based on
* </p>
* <ul>
* <li><a href="http://mathworld.wolfram.com/GammaFunction.html">Gamma
@ -237,8 +237,7 @@ public class Gamma {
* </ul>
*
* @param x Argument.
* @return the value of {@code log(Gamma(x))}, {@code Double.NaN} if
* {@code x <= 0.0}.
* @return the value of {@code log(Gamma(x))} or {@code NaN} if {@code x <= 0}.
*/
public static double logGamma(double x) {
double ret;
@ -267,7 +266,7 @@ public class Gamma {
}
/**
* Returns the regularized gamma function P(a, x).
* Returns the regularized gamma function \( P(a, x) \).
*
* @param a Parameter.
* @param x Value.
@ -279,7 +278,7 @@ public class Gamma {
}
/**
* Returns the regularized gamma function P(a, x).
* Returns the regularized gamma function \( P(a, x) \).
*
* The implementation of this method is based on:
* <ul>
@ -360,7 +359,7 @@ public class Gamma {
}
/**
* Returns the regularized gamma function Q(a, x) = 1 - P(a, x).
* Returns the regularized gamma function \( Q(a, x) = 1 - P(a, x) \).
*
* The implementation of this method is based on:
* <ul>
@ -424,21 +423,24 @@ public class Gamma {
/**
* <p>Computes the digamma function of x.</p>
* Computes the digamma function.
*
* <p>This is an independently written implementation of the algorithm described in
* Jose Bernardo, Algorithm AS 103: Psi (Digamma) Function, Applied Statistics, 1976.
* A reflection formula (https://en.wikipedia.org/wiki/Digamma_function#Reflection_formula)
* is incorporated to improve performance on negative values.</p>
* A <a href="https://en.wikipedia.org/wiki/Digamma_function#Reflection_formula">
* reflection formula</a> is incorporated to improve performance on negative values.</p>
*
* <p>Some of the constants have been changed to increase accuracy at the moderate expense
* of run-time. The result should be accurate to within 10^-8 relative tolerance for
* 0 < x < 10^-5 and within 10^-8 absolute tolerance otherwise.</p>
* <p>Some of the constants have been changed to increase accuracy at the moderate
* expense of run-time. The result should be accurate to within \( 10^{-8} \)
* relative tolerance for \( 0 < x < 10^{-5} \) and within \( 10^{-8} \) absolute
* tolerance otherwise.</p>
*
* @param x Argument.
* @return digamma(x) to within 10^-8 relative or absolute error whichever is larger.
* @return digamma(x) to within \( 10^{-8} \) relative or absolute error whichever is larger.
*
* @see <a href="http://en.wikipedia.org/wiki/Digamma_function">Digamma</a>
* @see <a href="http://www.uv.es/~bernardo/1976AppStatist.pdf">Bernardo&apos;s original article </a>
* @see <a href="http://www.uv.es/~bernardo/1976AppStatist.pdf">Bernardo's original article</a>
*
* @since 2.0
*/
public static double digamma(double x) {
@ -475,14 +477,17 @@ public class Gamma {
}
/**
* Computes the trigamma function of x.
* Computes the trigamma function.
* This function is derived by taking the derivative of the implementation
* of digamma.
*
* @param x Argument.
* @return trigamma(x) to within 10^-8 relative or absolute error whichever is smaller
* @return {@code trigamma(x)} to within \( 10^{-8} \) relative or absolute
* error whichever is smaller
*
* @see <a href="http://en.wikipedia.org/wiki/Trigamma_function">Trigamma</a>
* @see Gamma#digamma(double)
*
* @since 2.0
*/
public static double trigamma(double x) {
@ -511,11 +516,11 @@ public class Gamma {
* Returns the Lanczos approximation used to compute the gamma function.
* The Lanczos approximation is related to the Gamma function by the
* following equation
* <center>
* {@code gamma(x) = sqrt(2 * pi) / x * (x + g + 0.5) ^ (x + 0.5)
* * exp(-x - g - 0.5) * lanczos(x)},
* </center>
* where {@code g} is the Lanczos constant.
* \[
* \Gamma(x) = \sqrt{2\pi} \, \frac{(x + g + 1/2)^{x + \frac{1}{2}} \, e^{-x - g - \frac{1}{2}} \, \mathrm{lanczos}(x)}
* {x}
* \]
* where \(g\) is the Lanczos constant.
* </p>
*
* @param x Argument.
@ -535,10 +540,9 @@ public class Gamma {
}
/**
* Returns the value of 1 / &Gamma;(1 + x) - 1 for -0&#46;5 &le; x &le;
* 1&#46;5. This implementation is based on the double precision
* implementation in the <em>NSWC Library of Mathematics Subroutines</em>,
* {@code DGAM1}.
* Returns the value of \( 1 / \Gamma(1 + x) - 1 \) for \( -0.5 \leq x \leq 1.5 \).
* This implementation is based on the double precision implementation in
* the <em>NSWC Library of Mathematics Subroutines</em>, {@code DGAM1}.
*
* @param x Argument.
* @return The value of {@code 1.0 / Gamma(1.0 + x) - 1.0}.
@ -629,7 +633,7 @@ public class Gamma {
}
/**
* Returns the value of log &Gamma;(1 + x) for -0&#46;5 &le; x &le; 1&#46;5.
* Returns the value of \( \log \Gamma(1 + x) \) for \( -0.5 \leq x \leq 1.5 \).
* This implementation is based on the double precision implementation in
* the <em>NSWC Library of Mathematics Subroutines</em>, {@code DGMLN1}.
*
@ -654,9 +658,9 @@ public class Gamma {
/**
* Returns the value of Γ(x). Based on the <em>NSWC Library of
* Mathematics Subroutines</em> double precision implementation,
* {@code DGAMMA}.
* Returns the value of \( \Gamma(x) \).
* Based on the <em>NSWC Library of Mathematics Subroutines</em> double
* precision implementation, {@code DGAMMA}.
*
* @param x Argument.
* @return the value of {@code Gamma(x)}.