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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.math.special;
18  
19  import java.io.Serializable;
20  
21  import org.apache.commons.math.MathException;
22  import org.apache.commons.math.util.ContinuedFraction;
23  
24  /**
25   * This is a utility class that provides computation methods related to the
26   * Beta family of functions.
27   *
28   * @version $Revision: 549278 $ $Date: 2007-06-20 15:24:04 -0700 (Wed, 20 Jun 2007) $
29   */
30  public class Beta implements Serializable {
31  
32      /** Serializable version identifier */
33      private static final long serialVersionUID = -3833485397404128220L;
34  
35      /** Maximum allowed numerical error. */
36      private static final double DEFAULT_EPSILON = 10e-15;
37  
38      /**
39       * Default constructor.  Prohibit instantiation.
40       */
41      private Beta() {
42          super();
43      }
44  
45      /**
46       * Returns the
47       * <a href="http://mathworld.wolfram.com/RegularizedBetaFunction.html">
48       * regularized beta function</a> I(x, a, b).
49       * 
50       * @param x the value.
51       * @param a the a parameter.
52       * @param b the b parameter.
53       * @return the regularized beta function I(x, a, b)
54       * @throws MathException if the algorithm fails to converge.
55       */
56      public static double regularizedBeta(double x, double a, double b)
57          throws MathException
58      {
59          return regularizedBeta(x, a, b, DEFAULT_EPSILON, Integer.MAX_VALUE);
60      }
61  
62      /**
63       * Returns the
64       * <a href="http://mathworld.wolfram.com/RegularizedBetaFunction.html">
65       * regularized beta function</a> I(x, a, b).
66       * 
67       * @param x the value.
68       * @param a the a parameter.
69       * @param b the b parameter.
70       * @param epsilon When the absolute value of the nth item in the
71       *                series is less than epsilon the approximation ceases
72       *                to calculate further elements in the series.
73       * @return the regularized beta function I(x, a, b)
74       * @throws MathException if the algorithm fails to converge.
75       */
76      public static double regularizedBeta(double x, double a, double b,
77          double epsilon) throws MathException
78      {
79          return regularizedBeta(x, a, b, epsilon, Integer.MAX_VALUE);
80      }
81  
82      /**
83       * Returns the regularized beta function I(x, a, b).
84       * 
85       * @param x the value.
86       * @param a the a parameter.
87       * @param b the b parameter.
88       * @param maxIterations Maximum number of "iterations" to complete. 
89       * @return the regularized beta function I(x, a, b)
90       * @throws MathException if the algorithm fails to converge.
91       */
92      public static double regularizedBeta(double x, double a, double b,
93          int maxIterations) throws MathException
94      {
95          return regularizedBeta(x, a, b, DEFAULT_EPSILON, maxIterations);
96      }
97      
98      /**
99       * Returns the regularized beta function I(x, a, b).
100      * 
101      * The implementation of this method is based on:
102      * <ul>
103      * <li>
104      * <a href="http://mathworld.wolfram.com/RegularizedBetaFunction.html">
105      * Regularized Beta Function</a>.</li>
106      * <li>
107      * <a href="http://functions.wolfram.com/06.21.10.0001.01">
108      * Regularized Beta Function</a>.</li>
109      * </ul>
110      * 
111      * @param x the value.
112      * @param a the a parameter.
113      * @param b the b parameter.
114      * @param epsilon When the absolute value of the nth item in the
115      *                series is less than epsilon the approximation ceases
116      *                to calculate further elements in the series.
117      * @param maxIterations Maximum number of "iterations" to complete. 
118      * @return the regularized beta function I(x, a, b)
119      * @throws MathException if the algorithm fails to converge.
120      */
121     public static double regularizedBeta(double x, final double a,
122         final double b, double epsilon, int maxIterations) throws MathException
123     {
124         double ret;
125 
126         if (Double.isNaN(x) || Double.isNaN(a) || Double.isNaN(b) || (x < 0) ||
127             (x > 1) || (a <= 0.0) || (b <= 0.0))
128         {
129             ret = Double.NaN;
130         } else if (x > (a + 1.0) / (a + b + 2.0)) {
131             ret = 1.0 - regularizedBeta(1.0 - x, b, a, epsilon, maxIterations);
132         } else {
133             ContinuedFraction fraction = new ContinuedFraction() {
134 
135                 private static final long serialVersionUID = -7658917278956100597L;
136 
137                 protected double getB(int n, double x) {
138                     double ret;
139                     double m;
140                     if (n % 2 == 0) { // even
141                         m = n / 2.0;
142                         ret = (m * (b - m) * x) /
143                             ((a + (2 * m) - 1) * (a + (2 * m)));
144                     } else {
145                         m = (n - 1.0) / 2.0;
146                         ret = -((a + m) * (a + b + m) * x) /
147                                 ((a + (2 * m)) * (a + (2 * m) + 1.0));
148                     }
149                     return ret;
150                 }
151 
152                 protected double getA(int n, double x) {
153                     return 1.0;
154                 }
155             };
156             ret = Math.exp((a * Math.log(x)) + (b * Math.log(1.0 - x)) -
157                 Math.log(a) - logBeta(a, b, epsilon, maxIterations)) *
158                 1.0 / fraction.evaluate(x, epsilon, maxIterations);
159         }
160 
161         return ret;
162     }
163 
164     /**
165      * Returns the natural logarithm of the beta function B(a, b).
166      * 
167      * @param a the a parameter.
168      * @param b the b parameter.
169      * @return log(B(a, b))
170      */
171     public static double logBeta(double a, double b) {
172         return logBeta(a, b, DEFAULT_EPSILON, Integer.MAX_VALUE);
173     }
174     
175     /**
176      * Returns the natural logarithm of the beta function B(a, b).
177      *
178      * The implementation of this method is based on:
179      * <ul>
180      * <li><a href="http://mathworld.wolfram.com/BetaFunction.html">
181      * Beta Function</a>, equation (1).</li>
182      * </ul>
183      * 
184      * @param a the a parameter.
185      * @param b the b parameter.
186      * @param epsilon When the absolute value of the nth item in the
187      *                series is less than epsilon the approximation ceases
188      *                to calculate further elements in the series.
189      * @param maxIterations Maximum number of "iterations" to complete. 
190      * @return log(B(a, b))
191      */
192     public static double logBeta(double a, double b, double epsilon,
193         int maxIterations) {
194             
195         double ret;
196 
197         if (Double.isNaN(a) || Double.isNaN(b) || (a <= 0.0) || (b <= 0.0)) {
198             ret = Double.NaN;
199         } else {
200             ret = Gamma.logGamma(a) + Gamma.logGamma(b) -
201                 Gamma.logGamma(a + b);
202         }
203 
204         return ret;
205     }
206 }