org.apache.commons.numbers.gamma.LanczosApproximation Maven / Gradle / Ivy
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package org.apache.commons.numbers.gamma;
/**
*
* Lanczos approximation to the Gamma function.
*
* It is related to the Gamma function by the following equation
* \[
* \Gamma(x) = \sqrt{2\pi} \, \frac{(g + x + \frac{1}{2})^{x + \frac{1}{2}} \, e^{-(g + x + \frac{1}{2})} \, \mathrm{lanczos}(x)}
* {x}
* \]
* where \( g \) is the Lanczos constant.
*
* See equations (1) through (5), and Paul Godfrey's
* Note on the computation
* of the convergent Lanczos complex Gamma approximation.
*/
public final class LanczosApproximation {
/** \( g = \frac{607}{128} \). */
private static final double LANCZOS_G = 607d / 128d;
/** Lanczos coefficients. */
private static final double[] LANCZOS = {
0.99999999999999709182,
57.156235665862923517,
-59.597960355475491248,
14.136097974741747174,
-0.49191381609762019978,
.33994649984811888699e-4,
.46523628927048575665e-4,
-.98374475304879564677e-4,
.15808870322491248884e-3,
-.21026444172410488319e-3,
.21743961811521264320e-3,
-.16431810653676389022e-3,
.84418223983852743293e-4,
-.26190838401581408670e-4,
.36899182659531622704e-5,
};
/** Private constructor. */
private LanczosApproximation() {
// intentional empty.
}
/**
* Computes the Lanczos approximation.
*
* @param x Argument.
* @return the Lanczos approximation.
*/
public static double value(final double x) {
double sum = 0;
for (int i = LANCZOS.length - 1; i > 0; i--) {
sum += LANCZOS[i] / (x + i);
}
return sum + LANCZOS[0];
}
/**
* @return the Lanczos constant \( g = \frac{607}{128} \).
*/
public static double g() {
return LANCZOS_G;
}
}