Beta


1   /*
2    * Copyright 2003-2004 The Apache Software Foundation.
3    *
4    * Licensed under the Apache License, Version 2.0 (the "License");
5    * you may not use this file except in compliance with the License.
6    * You may obtain a copy of the License at
7    *
8    *      http://www.apache.org/licenses/LICENSE-2.0
9    *
10   * Unless required by applicable law or agreed to in writing, software
11   * distributed under the License is distributed on an "AS IS" BASIS,
12   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
13   * See the License for the specific language governing permissions and
14   * limitations under the License.
15   */
16  package org.apache.commons.math.special;
17  
18  import java.io.Serializable  ;
19  
20  import org.apache.commons.math.MathException;
21  import org.apache.commons.math.util.ContinuedFraction;
22  
23  /**
24   * This is a utility class that provides computation methods related to the
25   * Beta family of functions.
26   *
27   * @version $Revision$ $Date: 2005-02-26 05:11:52 -0800 (Sat, 26 Feb 2005) $
28   */
29  public class Beta implements Serializable   {
30      /** Maximum allowed numerical error. */
31      private static final double DEFAULT_EPSILON = 10e-9;
32  
33      /**
34       * Default constructor.  Prohibit instantiation.
35       */
36      private Beta() {
37          super();
38      }
39  
40      /**
41       * Returns the
42       * <a HREF="http://mathworld.wolfram.com/RegularizedBetaFunction.html">
43       * regularized beta function</a> I(x, a, b).
44       * 
45       * @param x the value.
46       * @param a the a parameter.
47       * @param b the b parameter.
48       * @return the regularized beta function I(x, a, b)
49       * @throws MathException if the algorithm fails to converge.
50       */
51      public static double regularizedBeta(double x, double a, double b)
52          throws MathException
53      {
54          return regularizedBeta(x, a, b, DEFAULT_EPSILON, Integer.MAX_VALUE);
55      }
56  
57      /**
58       * Returns the
59       * <a HREF="http://mathworld.wolfram.com/RegularizedBetaFunction.html">
60       * regularized beta function</a> I(x, a, b).
61       * 
62       * @param x the value.
63       * @param a the a parameter.
64       * @param b the b parameter.
65       * @param epsilon When the absolute value of the nth item in the
66       *                series is less than epsilon the approximation ceases
67       *                to calculate further elements in the series.
68       * @return the regularized beta function I(x, a, b)
69       * @throws MathException if the algorithm fails to converge.
70       */
71      public static double regularizedBeta(double x, double a, double b,
72          double epsilon) throws MathException
73      {
74          return regularizedBeta(x, a, b, epsilon, Integer.MAX_VALUE);
75      }
76  
77      /**
78       * Returns the regularized beta function I(x, a, b).
79       * 
80       * @param x the value.
81       * @param a the a parameter.
82       * @param b the b parameter.
83       * @param maxIterations Maximum number of "iterations" to complete. 
84       * @return the regularized beta function I(x, a, b)
85       * @throws MathException if the algorithm fails to converge.
86       */
87      public static double regularizedBeta(double x, double a, double b,
88          int maxIterations) throws MathException
89      {
90          return regularizedBeta(x, a, b, DEFAULT_EPSILON, maxIterations);
91      }
92      
93      /**
94       * Returns the regularized beta function I(x, a, b).
95       * 
96       * The implementation of this method is based on:
97       * <ul>
98       * <li>
99       * <a HREF="http://mathworld.wolfram.com/RegularizedBetaFunction.html">
100      * Regularized Beta Function</a>.</li>
101      * <li>
102      * <a HREF="http://functions.wolfram.com/06.21.10.0001.01">
103      * Regularized Beta Function</a>.</li>
104      * </ul>
105      * 
106      * @param x the value.
107      * @param a the a parameter.
108      * @param b the b parameter.
109      * @param epsilon When the absolute value of the nth item in the
110      *                series is less than epsilon the approximation ceases
111      *                to calculate further elements in the series.
112      * @param maxIterations Maximum number of "iterations" to complete. 
113      * @return the regularized beta function I(x, a, b)
114      * @throws MathException if the algorithm fails to converge.
115      */
116     public static double regularizedBeta(double x, final double a,
117         final double b, double epsilon, int maxIterations) throws MathException
118     {
119         double ret;
120 
121         if (Double.isNaN(x) || Double.isNaN(a) || Double.isNaN(b) || (x < 0) ||
122             (x > 1) || (a <= 0.0) || (b <= 0.0))
123         {
124             ret = Double.NaN;
125         } else if (x > (a + 1.0) / (a + b + 2.0)) {
126             ret = 1.0 - regularizedBeta(1.0 - x, b, a, epsilon, maxIterations);
127         } else {
128             ContinuedFraction fraction = new ContinuedFraction() {
129                 protected double getB(int n, double x) {
130                     double ret;
131                     double m;
132                     if (n % 2 == 0) { // even
133                         m = n / 2.0;
134                         ret = (m * (b - m) * x) /
135                             ((a + (2 * m) - 1) * (a + (2 * m)));
136                     } else {
137                         m = (n - 1.0) / 2.0;
138                         ret = -((a + m) * (a + b + m) * x) /
139                                 ((a + (2 * m)) * (a + (2 * m) + 1.0));
140                     }
141                     return ret;
142                 }
143 
144                 protected double getA(int n, double x) {
145                     return 1.0;
146                 }
147             };
148             ret = Math.exp((a * Math.log(x)) + (b * Math.log(1.0 - x)) -
149                 Math.log(a) - logBeta(a, b, epsilon, maxIterations)) *
150                 1.0 / fraction.evaluate(x, epsilon, maxIterations);
151         }
152 
153         return ret;
154     }
155 
156     /**
157      * Returns the natural logarithm of the beta function B(a, b).
158      * 
159      * @param a the a parameter.
160      * @param b the b parameter.
161      * @return log(B(a, b))
162      */
163     public static double logBeta(double a, double b) {
164         return logBeta(a, b, DEFAULT_EPSILON, Integer.MAX_VALUE);
165     }
166     
167     /**
168      * Returns the natural logarithm of the beta function B(a, b).
169      *
170      * The implementation of this method is based on:
171      * <ul>
172      * <li><a HREF="http://mathworld.wolfram.com/BetaFunction.html">
173      * Beta Function</a>, equation (1).</li>
174      * </ul>
175      * 
176      * @param a the a parameter.
177      * @param b the b parameter.
178      * @param epsilon When the absolute value of the nth item in the
179      *                series is less than epsilon the approximation ceases
180      *                to calculate further elements in the series.
181      * @param maxIterations Maximum number of "iterations" to complete. 
182      * @return log(B(a, b))
183      */
184     public static double logBeta(double a, double b, double epsilon,
185         int maxIterations) {
186             
187         double ret;
188 
189         if (Double.isNaN(a) || Double.isNaN(b) || (a <= 0.0) || (b <= 0.0)) {
190             ret = Double.NaN;
191         } else {
192             ret = Gamma.logGamma(a) + Gamma.logGamma(b) -
193                 Gamma.logGamma(a + b);
194         }
195 
196         return ret;
197     }
198 }
199
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