1 /*
2 * Licensed to the Apache Software Foundation (ASF) under one or more
3 * contributor license agreements. See the NOTICE file distributed with
4 * this work for additional information regarding copyright ownership.
5 * The ASF licenses this file to You under the Apache License, Version 2.0
6 * (the "License"); you may not use this file except in compliance with
7 * the License. You may obtain a copy of the License at
8 *
9 * http://www.apache.org/licenses/LICENSE-2.0
10 *
11 * Unless required by applicable law or agreed to in writing, software
12 * distributed under the License is distributed on an "AS IS" BASIS,
13 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14 * See the License for the specific language governing permissions and
15 * limitations under the License.
16 */
17 package org.apache.commons.numbers.gamma;
18
19 /**
20 * <a href="http://mathworld.wolfram.com/LanczosApproximation.html">
21 * Lanczos approximation</a> to the Gamma function.
22 *
23 * It is related to the Gamma function by the following equation
24 * \[
25 * \Gamma(x) = \sqrt{2\pi} \, \frac{(g + x + \frac{1}{2})^{x + \frac{1}{2}} \, e^{-(g + x + \frac{1}{2})} \, \mathrm{lanczos}(x)}
26 * {x}
27 * \]
28 * where \( g \) is the Lanczos constant.
29 *
30 * See equations (1) through (5), and Paul Godfrey's
31 * <a href="http://my.fit.edu/~gabdo/gamma.txt">Note on the computation
32 * of the convergent Lanczos complex Gamma approximation</a>.
33 */
34 public final class LanczosApproximation {
35 /** \( g = \frac{607}{128} \). */
36 private static final double LANCZOS_G = 607d / 128d;
37 /** Lanczos coefficients. */
38 private static final double[] LANCZOS = {
39 0.99999999999999709182,
40 57.156235665862923517,
41 -59.597960355475491248,
42 14.136097974741747174,
43 -0.49191381609762019978,
44 .33994649984811888699e-4,
45 .46523628927048575665e-4,
46 -.98374475304879564677e-4,
47 .15808870322491248884e-3,
48 -.21026444172410488319e-3,
49 .21743961811521264320e-3,
50 -.16431810653676389022e-3,
51 .84418223983852743293e-4,
52 -.26190838401581408670e-4,
53 .36899182659531622704e-5,
54 };
55
56 /** Private constructor. */
57 private LanczosApproximation() {
58 // intentionally empty.
59 }
60
61 /**
62 * Computes the Lanczos approximation.
63 *
64 * @param x Argument.
65 * @return the Lanczos approximation.
66 */
67 public static double value(final double x) {
68 double sum = 0;
69 for (int i = LANCZOS.length - 1; i > 0; i--) {
70 sum += LANCZOS[i] / (x + i);
71 }
72 return sum + LANCZOS[0];
73 }
74
75 /**
76 * @return the Lanczos constant \( g = \frac{607}{128} \).
77 */
78 public static double g() {
79 return LANCZOS_G;
80 }
81 }